Scaling Methods, Health Preferences and Health Effects

167
Scaling Methods, Health Preferences and Health Effects Patricia Cubí Mollá

Transcript of Scaling Methods, Health Preferences and Health Effects

Page 1: Scaling Methods, Health Preferences and Health Effects

Scaling Methods, Health Preferences and Health Effects

Patricia Cubí Mollá

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Scaling Methods, Health Preferences and HealthE¤ects

Patricia Cubí-Mollá

Supervisor: Carmen Herrero

Quantitative Economics DoctorateDepartamento de Fundamentos del Análisis Económico

Universidad de Alicante

March 2009

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A Alejandro, por motivarme a empezar.A Aitor, por ayudarme a seguir.A Juan, por animarme a acabar.

Especialmente a "Pato", por estar siempre ahí.

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Agradecimientos

Sin duda debo y quiero empezar por agradecer a Carmen Herrero el apoyo queme ha ofrecido durante estos años. Especialmente, porque han sido años difícilespara ella, y aún así se las ha arreglado para reunir energía, tiempo y paciencia quevolcar en mí.

Existen tres personas excepcionales que me han ayudado enormemente en estosinicios de la investigación, a las cuales les quiero expresar mi más sincera gratitud.Ildefonso Méndez, Elena Martínez y Climent Quintana.

Deseo agradecer también el apoyo inestimable, a nivel académico y personal, queme han ofrecido: José María Abellán, Eduardo Sánchez, Juan Oliva y María Seguí.

Estos años de doctorado hubieran resultado muy difíciles de llevar sin la compañíade Alicia, Constanza, Lari, Paco, Ricardo, y muy, pero muy especialmente, Aida yNatalia.

Agradezco a mis padres, con todo mi cariño, el haberme dado una educación queme ha permitido llegar a defender una tesis. A mi abuela Juana, que me acompañabaorgullosa a todos los actos universitarios, y que también me acompañará durantela defensa. A mis hermanas a las que tanto quiero, y a las que sin duda debo mivocación como docente. A María, Natasha, Alejandra, Laura y Ana, por ayudarmecon su amistad.

Por último agradecer el apoyo �nanciero que he recibido de la Generalitat Va-lenciana, de la Fundación BBVA y del IVIE.

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Contents

Agradecimientos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Introducción y Resumen General . . . . . . . . . . . . . . . . . . . . . . . 7Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

I Scaling Methods 27

1 Scaling Methods for Categorical Self-Assessed Health Measures 281.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.2.1 A standard normal latent health variable . . . . . . . . . . . . 301.2.2 A standard lognormal latent health variable . . . . . . . . . . 33

1.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

II Health E¤ects 52

2 Quality of Life Lost Due to Road Crashes 532.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.2.1 Measurement of health . . . . . . . . . . . . . . . . . . . . . . 562.2.2 Cardinalization of SAH . . . . . . . . . . . . . . . . . . . . . 582.2.3 Evaluation of health losses . . . . . . . . . . . . . . . . . . . . 61

2.3 Data and variable de�nitions . . . . . . . . . . . . . . . . . . . . . . . 642.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3 QALYs Lost Due to Road Crashes in Catalonia, 2002-2006 893.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

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3.3 Data and variable de�nitions . . . . . . . . . . . . . . . . . . . . . . . 963.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4 Estimating Health E¤ects 1054.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.2 Appropriate metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.3 Evaluating the direct health e¤ect . . . . . . . . . . . . . . . . . . . 1114.4 Combining values for health with life extension . . . . . . . . . . . . 1134.5 Aggregating health e¤ects among individuals . . . . . . . . . . . . . . 1154.6 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

III Health Preferences 122

5 Generalizing Quality-Adjusted Life Years in a Context of Certainty1235.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235.2 A general theorem for time preferences over chronic health states . . 126

5.2.1 Main De�nition . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.2.2 Axioms for Time Preferences . . . . . . . . . . . . . . . . . . . 1275.2.3 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.2.4 Transitive time preferences over health outcomes . . . . . . . . 133

5.3 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355.3.1 De�nition of preferences over H �T . . . . . . . . . . . . . . . 1355.3.2 Generalization of the axioms over H �T . . . . . . . . . . . . 1425.3.3 Proof of Theorem 2.1. . . . . . . . . . . . . . . . . . . . . . . 1475.3.4 Proof of corollary 2.1. . . . . . . . . . . . . . . . . . . . . . . 163

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

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Introducción y Resumen General

El análisis de los estados de salud experimentó un vuelco muy signi�cativo durante lasegunda mitad del siglo XX. Básicamente, podemos destacar tres directrices princi-pales: el establecimiento de un nuevo concepto de salud que incorpora elementos decalidad de vida, más allá de la simple ausencia de enfermedades; el método de clasi-�cación de estados de salud según la valoración individual en términos de utilidad ;y el creciente interés en el estudio de problemas de salud a nivel poblacional.

Una muestra de la creciente importancia del estudio de problemas de salud engrandes poblaciones es la creación de la Organización Mundial de la Salud (OMS), afecha 7 de abril de 1948, como el organismo de las Naciones Unidas especializado ensalud. La OMS de�ne la salud como un estado de completo bienestar físico, mentaly social, y no sólo como ausencia de afecciones o enfermedades (véase OMS, 1946- 1948). La de�nición de salud adoptada en la OMS desde su fundación intentabacapturar elementos esenciales de la calidad de vida, ausentes en las métricas en quesólo se toma en cuenta la longitud de la vida, como sucede con medidas de esperanzade vida, o tasas de mortalidad. Se introducen, por tanto, medidas combinadas decalidad y cantidad de vida, como la forma adecuada de estimar la salud de unapoblación.

La evolución de los estudios de salud a nivel individual experimenta un girosimilar al comentado anteriormente para grupos poblacionales. Desde los años 80una medida que combinaba calidad y cantidad de vida venía siendo utilizada am-pliamente en el análisis coste-utilidad de medicamentos y programas de salud, enespecial en los países del norte de Europa. Esta medida, denominada Quality Ad-justed Life Years (QALYs), o Años de Vida Ajustados por Calidad (AVACs) seinterpretó desde el primer momento en términos de utilidad derivada del disfrutede determinados estados de salud, en relación a disfrutar de salud perfecta. LosAVACs se computan ponderando cada año de vida por la utilidad del estado desalud disfrutado en dicho año.

El concepto más importante, y asimismo el más controvertido del cálculo deAVACs, es la estimación de la utilidad individual que reporta un estado de salud.Dicho valor debe ser determinado en una escala que va desde 0 (correspondiente ala utilidad asociada al estado �muerte�) hasta 1 (correspondiente a la utilidad quereporta la �salud perfecta�). La estimación de utilidades se basa en las preferenciasque el individuo establezca entre diferentes estados crónicos de salud. Los instrumen-tos más comúnmente empleados para obtener esos valores son: la lotería estándar(standard gamble, SG), el intercambio temporal (time tradeo¤, TTO), escalas depuntuación como la analógica visual (visual analogue scale, VAS) y los métodos devaluación contingente como la voluntad de pago (willingess to pay, WTP). Depen-diendo de la forma de derivar las valoraciones de los estados de salud en términosde utilidad, se han venido desarrollando diferentes métricas. Por una parte, dichasmétricas proporcionan nuevas herramientas en el análisis de los estados de salud;además, aunque están basadas en valoraciones de preferencias a nivel individual,pueden extenderse a tarifas de salud a nivel poblacional (véase, por ejemplo, la

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de�nición de EQ-5D VAS). Por otra parte, el uso de una métrica u otra frecuente-mente condiciona el resultado del análisis, lo que genera una discusión acerca de quémétricas son las más adecuadas (Dolan et al., 1996).

Por tanto, como decíamos al principio, durante las últimas décadas hemos asis-tido a un cambio trascendental en la evaluación de estados de salud: tanto a nivelindividual como poblacional, la salud se interpreta en términos no solo de cantidadsino también de calidad de vida; y se han desarrollado numerosos procedimientospara medir la calidad de vida a nivel poblacional basados en preferencias individualessobre estados de salud.

Gracias al nuevo enfoque y a las nuevas métricas, se han podido desarrollarnumerosos estudios que permiten re�ejar los problemas reales de salud en las pobla-ciones actuales. Cabe destacar un estudio realizado en 1992 por la OMS, a requer-imiento del Banco Mundial, conocido como Global Burden of Disease Study (GBD;ver Murray y Lopez, 1996). Este trabajo representa el primer intento de análisis delos principales problemas de salud en el mundo, además de intentar evaluar sus con-secuencias sobre la población mundial. Uno de los resultados más signi�cativos delestudio fue la elaboración de un ranking que re�ejaba las causas más importantesde pérdida de salud a nivel mundial y regional, en 1990, además de proyeccionesde tal ranking para el año 2020. En estudios posteriores (véase Mathers and Lon-car, 2006) se actualizaron las estimaciones, realizando proyecciones de las causasmás importantes de muerte en 2030. Destacamos a continuación los resultados másimportantes de dichos estudios.

Es muy signi�cativo que entre las seis principales causas de mortalidad a nivelmundial, solo precedida por cáncer, cardiopatías isquémicas, infartos, sida, e in-fecciones, �guren los accidentes de trá�co. Debemos pararnos a re�exionar sobreel hecho de que una de las principales causas de mortalidad no esté originada porproblemas de salud, sino por causas externas.

Si se analizan con más detalle los datos de accidentes de trá�co, se observa queen 2004, algo más del 50% de las muertes causadas por accidentes de trá�co estabanasociadas a jóvenes con edades comprendidas entre 15 y 44 años. Las heridas porcolisión vial resultaban la segunda causa principal de fallecimiento en el mundo paralos jóvenes de entre 5 y 29 años (datos obtenidos de la OMS, 2004). Además, en elestudio Mathers and Loncar (2006) se estimaba que los accidentes de trá�co seríanla principal causa del crecimiento del 40% en fallecimientos por causas externas, enproyecciones para el año 2030.

El problema de las colisiones de trá�co como causa de pérdida global de saludde la población sólo ha sido empezado a tomar en consideración recientemente. Sóloen los últimos años, y únicamente en los países desarrollados se ha empezado aconcienciar a la población sobre las necesidades de medidas preventivas para atajarlo que resulta ser un problema sanitario de primera magnitud. A partir del año2003, en la OMS han dedicado monografías especí�cas al análisis de este problema,y al estudio de la efectividad de algunas medidas preventivas (véase, OMS 2003,2004).

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Si centramos nuestra atención en España, la situación es preocupante por variasrazones. Por una parte, si bien en 2006 el número anual de víctimas (heridos yfallecidos) en accidentes de trá�co por cada 100.000 habitantes se sitúa en torno a389,1 (signi�cativamente menor que en otros países europeos, como Bélgica, Italia,Portugal, Holanda, Austria, Alemania o el Reino Unido), el número anual de falleci-dos por cada 100.000 habitantes está en torno a 13,8, muy por encima de los valoresde Italia, Holanda, Austria, y más que duplicando los de Alemania y el Reino Unido.Asimismo, el número de fallecidos sobre accidentados presenta un alto porcentaje,si lo comparamos con los países de nuestro entorno. En particular, estamos muy porencima de la media europea, y de las tasas de países como Portugal, Italia, Alemaniao Reino Unido.1

Las causas de las colisiones de trá�co son muy variadas: defectos en el diseño delas carreteras, falta de infraestructuras adecuadas, parques móviles obsoletos o fuerade control, falta de control policial, comportamiento de los conductores (exceso develocidad, consumo de alcohol u otras drogas, fatiga, falta de atención por el uso deteléfonos móviles y otros aparatos, falta de uso de instrumentos de prevención, comocinturones o cascos), de�ciencias en la cadena sanitaria (tardanza en la atención encarretera, problemas en los servicios de rehabilitación), etc. Y el problema es que lalista de razones sigue creciendo. Sólo si la población en su conjunto, los gobiernos, lapolicía, los servicios sanitarios y los medios de comunicación colaboran, será posibleatajar las funestas previsiones de la OMS para 2020 y 2030.

La implantación de medidas como las mencionadas, tendentes a disminuir lasiniestralidad de las colisiones de trá�co lleva a una pregunta crucial: ¿Merece lapena invertir en mejorar la coordinación de los servicios de emergencia? ¿Merecela pena invertir en sistemas informáticos más so�sticados y aumentar el controlpolicial para gestionar adecuadamente el permiso por puntos? Para responder aestas preguntas es esencial disponer de un instrumento adecuado para evaluar loscostes y bene�cios de tales medidas.

Como primer paso, pues, nos planteamos la siguiente pregunta: ¿Cómo podemoscuanti�car los bene�cios en salud derivados de reducir la accidentalidad en las car-reteras? Hay dos aproximaciones posibles: (1) Analizar el impacto en salud de lasdiferentes intervenciones para reducir la accidentalidad, y (2) analizar el impactoen salud de los accidentes para poder medir posteriormente los bene�cios de sureducción. La selección de uno u otro camino no es trivial, ya que condiciona lametodología y las métricas a utilizar. La primera aproximación, más generalizadaen los estudios coste-efectividad, tiene como ventaja importante la aplicabilidadinmediata de los resultados para la toma de decisiones de los responsables de lasdecisiones colectivas en la prevención de accidentes. La segunda alternativa, concen-trarse en el tamaño del problema, aunque criticada por algunos autores (Williams,1999), es de la mayor relevancia, habida cuenta que, a diferencia de lo que ocurrecon la mayor parte de las enfermedades, en el caso de los accidentes de trá�co no sedispone de información �able sobre los estados de salud previo y posterior al acci-dente, que permitan evaluar correctamente los bene�cios potenciales de las posibles

1Fuente: CARE (base de datos europea de accidentes de trá�co) y publicaciones nacionales.

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intervenciones para reducir la accidentalidad. Es por ello que en algunos de estostrabajos nos decantamos por la segunda opción. Es decir, queremos estimar cuál esel coste en salud global causado por las colisiones de trá�co.

Hay algunos estudios previos que han proporcionado descripciones del impacto delas colisiones viales sobre la salud de la población española. Entre ellos destacamos:EMAT-30 (2004); Pérez et al. (2006); Peiró-Pérez et al. (2006); Redondo et al.(2000); DGT (Dirección General de Trá�co) Estudio Piloto (1993); DGT Análisisde Morbilidad en Madrid y Barcelona (1994); DGT Estudio nacional multicéntrico(2005).

En el Estudio Piloto realizado por la DGT en Valladolid, en 1993, se realizabanlos primeros pasos hacia la medición de las características de las lesiones no fatales.Se obtuvieron datos de lesionados por carretera durante 1990 a través de la DGT,centros hospitalarios de la provincia, así como una encuesta de seguimiento de mor-bilidad, donde encontramos algunas preguntas relacionadas con la calidad de vida(estado actual, nivel de satisfacción) y la discapacidad funcional (incapacidad parasu trabajo habitual, tiempo medio de incapacidad temporal). El estudio es en sutotalidad descriptivo.

Un siguiente paso se llevaría a cabo mediante el Análisis de Morbilidad en Madridy Barcelona (DGT, 1994). En esta ocasión el estudio se centra en la morbilidad enBarcelona y Madrid, 1993-1994, con datos obtenidos a través de los diagnósticos deentrada y de alta. Las medidas empleadas para el análisis son, principalmente, larelación entre Tiempo Medio de Curación (TMC), incapacidad temporal e incapaci-dad permanente. Se realiza también un cálculo económico de los costes asociados adichos estados.

En el año 2000 se publica un análisis general a nivel estatal de las consecuencias delos accidentes de trá�co por Redondo et al., 2000. El estudio, de carácter descriptivo,toma como horizonte el periodo 1985-1994 y se basa en los datos de la DGT. Lasmedidas de mortalidad y morbilidad empleadas son: índice de mortalidad, tasa delesividad, tasa de letalidad. Como indicadores de la gravedad toma: densidad depoblación e índice de motorización.

La DGT analiza en el llamado Estudio Nacional Multicéntrico los accidentados enel periodo 2000-2004. En concreto, elige una muestra de 2.180 lesionados atendidosen centros hospitalarios en el territorio nacional y realiza un posterior seguimientodurante los 4 años posteriores al siniestro. Describe la morbilidad comparándolacon los datos de la DGT relativos a 2001 y, cuanti�ca el grado de lesividad de loslesionados mediante el Injury Severity Score ISS y el IDS (Índice de Dolor Social).

En los trabajos de EMAT-30 (Estudio de la Mortalidad a 30 días por Accidentesde Trá�co) y Pérez et al. (2006), encontramos la primera explotación de lesiona-dos por colisiones de trá�co analizado a partir del CMBDAH (Conjunto Mínimo yBásico de datos de las Altas Hospitalarias) en el conjunto de España. Los datosprovenientes de dicha fuente se comparan con las estadísticas de accidentes y vícti-mas de la DGT para 2001. Tales trabajos realizan una descripción de la mortalidad

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por variables socioeconómicas, así como una descripción de la morbilidad del acci-dentado, utilizando como medida de la gravedad el Injury Severity Score ISS.

En Peiró-Pérez et al. (2006), encontramos una breve pero exhaustiva descrip-ción epidemiológica de los accidentes de trá�co de 2002, a partir de los resultadosobtenidos en EMAT-30, utilizando por tanto diversas fuentes. En mortalidad citadatos provenientes de INE, CMBDAH y DGT, mientras que en morbilidad hacereferencia a la DGT, CMBDAH, RCSCV2 y ENS03.3 A partir de los datos delCMBDAH se calcula el ISS para evaluar el grado de lesividad.

Los trabajos comentados arriba tienen carácter principalmente descriptivo, y lasmedidas utilizadas por los más actuales atienden a la discapacidad funcional queel accidente causa en el accidentado. Sin embargo, varios estudios han probado laexistencia de importantes consecuencias psicológicas derivadas de un proceso mór-bido, como pueden ser estrés o depresión (O�Donnell et al., 2005), y en especial trasun accidente de trá�co (ver Mayou et al., 1993), por lo que las discapacidades en elespacio de la psicología no deberían ser obviadas.

Las di�cultades anteriormente señaladas nos llevan a proponer en este trabajouna nueva perspectiva, dentro del nuevo marco de salud de�nido por la OMS: con-templar los efectos de los accidentes en la calidad de vida del individuo; es decir,no sólo atendiendo al daño físico que el accidente ha provocado, sino centrándonostambién en el posible daño psicológico así como en el posible impacto del accidenteen el bienestar del accidentado. Estas tres perspectivas (física, emocional, y bien-estar del accidentado) de�nen el concepto general al que nos referimos como estadode salud del individuo.

Una vez establecido el marco metodológico del análisis, debemos recordar quela bondad de los estudios empíricos siempre está condicionada a la calidad de losdatos que los soporten. El interés creciente sobre los estados de salud poblacionales,además de la progresiva informatización de la sociedad, han impulsado la creación dediversas bases de datos, como altas hospitalarias, accidentes de trá�co, encuestas desalud realizadas por entidades públicas y privadas, etc., además de la informatizacióny accesibilidad a las bases de datos ya existentes.

Sin embargo, es más complicado incluir ciertas medidas especí�cas de salud enlos cuestionarios orientados a la población general. Por ejemplo, la valoración delestado de salud del individuo se obtiene normalmente de una pregunta básica so-bre la salud auto-percibida (SAP) del individuo, del tipo: �En su opinión, ¿cómovalora su estado de salud, en general?: Muy bueno/ Bueno / Regular/ Malo/ Muymalo�. Estas medidas de salud, que por una parte resultan ventajosas, ya que sonperfectamente entendibles por los encuestados, sin embargo no son apropiadas paraestablecer resultados en contextos de medidas basadas en la media (como puedenser los análisis de desigualdades en salud, medidas de efectos de tratamientos, etc.).El primer motivo, es el carácter ordinal de las medidas de SAP, que no permitenconsiderar el estado de salud como un continuo. Además, los criterios para valorar

2Red Centinela Sanitaria de la Comunidad Valenciana.3Encuesta Nacional de Salud de 2003.

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un estado de salud probablemente varían entre distintos individuos, por lo que larespuesta puede tener un signi�cado a nivel individual, pero no es correcto agregarlas observaciones a nivel poblacional, ignorando la heterogeneidad de las preferen-cias. Así, durante las últimas décadas se han ido desarrollando una amplia variedadde métodos dirigidos a cardinalizar las medidas categóricas de SAP, que son las másfrecuentemente utilizadas en las encuestas (véase van Doorslaer and Jones, 2003).

El objetivo principal de esta tesis consiste en analizar la valoración de los estadosde salud desde diferentes perspectivas. Tanto la primera como la segunda parte dela tesis presentan resultados más empíricos. Se han aplicado métodos econométricospara analizar las pérdidas de salud crónicas, en términos de calidad de vida perdida,reportadas por aquellos que han sufrido un accidente de trá�co de carácter grave,esto es, que les haya producido limitaciones en las actividades cotidianas, durantealgún periodo de tiempo. Para la obtención de resultados se han empleado los datosprovenientes de encuestas poblacionales de salud, en la mayoría de las cuales elestado de salud se describía mediante preguntas de SAP. Por tanto, ha sido necesarioanalizar varios métodos de cardinalización de dicha medida, y establecer el óptimo.Este análisis previo nos ha permitido obtener una visión general del estado de lasalud de la población, en términos de calidad de vida. En una tercera parte de latesis, enfocamos el tema de valoración de estados de salud desde una perspectivaaxiomática, re�exionando sobre la base teórica de las medidas que hemos empleadoen los análisis empíricos.

Más detalladamente:

En la primera parte de la tesis ("Scaling methods"), analizo los métodos másgeneralizados para escalar las ya mencionadas medidas ordinales de SAP: el modelode regresión probit ordenado y el modelo de regresión por intervalos. El objetivode estos procedimientos es el de estimar el peso de calidad asociado al estado desalud de cada uno de los individuos, condicionando por la salud auto-percibida quedeclaran y por diversas características socioeconómicas de los encuestados. Tantoel modelo probit ordenado como el de regresión por intervalos asumen la existenciade una variable de salud continua que subyace latente bajo la variable observable desalud auto-percibida. La distribución de dicha variable latente se supone normal enambos casos.

El modelo probit ordenado es el más básico de los dos procedimientos. Estableceuna equivalencia entre la variable continua (latente) y la variable discreta (observ-able) como sigue: cuanto mayor es el valor latente de la salud, más probabilidadtendrá el individuo de escoger una categoría más elevada de salud auto-percibida.Basándose en la función de distribución de la normal N(0; 1), se obtienen los valoresde salud que establecen la frontera entre distintas categorías (umbrales). Las predic-ciones que obtenemos de la regresión no pueden ser empleadas directamente comopesos de calidad, ya que no están expresadas en unidades naturales de la medida, esdecir, entre 0 y 1. Por tanto, es necesario emplear un tipo de re-escalamiento (porejemplo, el propuesto por van Doorslaer y Jones, 2003), para obtener las estima-ciones �nales.

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El método de regresión por intervalos es uno de los más empleados en los últimosaños. Fue propuesto por van Doorslaer y Jones (2003), y está basada en una ideasimilar al modelo probit ordenado. En esta ocasión, los autores proponen estimar apriori el valor de los umbrales, lo cual permite estimar posteriormente la varianza dela salud latente, en lugar de suponer que es igual a la unidad, tal y como estableceel modelo probit ordenado. La originalidad se este modelo se basa en la estimaciónde los umbrales a partir de información externa (ej. una base de datos diferente enla que podamos observar una variable continua de calidad de vida). Además, comolos umbrales vienen ya medidos en unidades naturales (entre 0 y 1), las prediccionesque genera la regresión pueden tomarse directamente como pesos de calidad, sinnecesidad de un proceso de re-escalamiento.

Los métodos de escalamiento anteriores tienen como denominador común lahipótesis de que la salud latente en la población puede representarse mediante unadistribución normal. Esta hipótesis elimina la posibilidad de encontrar asimetríasen la distribución de salud. Sin embargo, la existencia de asimetrías se re�eja con-tinuamente en los datos de salud de las poblaciones. Este hecho es especialmenterelevante si se pretende realizar un análisis sobre estados de salud en países desar-rollados, donde una gran proporción de la población reporta estar en un buen oexcelente estado de salud.

El nuevo método que propongo considera que la salud latente sigue una distribu-ción log-normal, introduciendo así posibles asimetrías. Bajo esta nueva hipótesis,se construyen los modelos de probit ordenado y regresión por intervalos. Los datosempleados para evaluar dichos métodos son obtenidos de la Enquesta Catalana deSalut, 2006, que permite obtener tanto variables continuas (las tarifas derivadasdel cuestionario EuroQol, basadas en los métodos de Escala Visual Análoga y deIntercambio Temporal) como categóricas (Salud Auto-Percibida), referidas a salud.Esto nos permite observar qué métodos estiman con mayor exactitud la distribucióncontinua de salud. Los nuevos modelos (que asumen log-normalidad) ajustan mejorla distribución de las tarifas que aquellos que asumen normalidad. En particular,el modelo de regresión probit ordenado se acerca más a la aproximación lineal demínimos cuadrados ordinarios.

En la segunda parte de la tesis ("Health e¤ects"), analizo las pérdidas de saludoriginadas por accidentes graves (no mortales) de trá�co, en términos de calidadde vida. Me centro en aquellas personas que han sufrido un accidente grave detrá�co hasta un año antes, y que ya se encuentran en sus domicilios habituales,reincorporados en su mayoría a su vida habitual. Lo que quiero observar en mianálisis es si la calidad de vida del afectado se ha restablecido completamente. Esdecir, si el accidentado está viviendo en el mismo estado de salud en el que viviríasi no hubiera sufrido el accidente; o, por el contrario, las lesiones le han producidounas secuelas que aún no han desaparecido, después de un año. Si es cierto estoúltimo, estaremos ante una importante consecuencia de los accidentes de trá�co, yserá imprescindible considerarla para analizar la verdadera extensión de la carga delas lesiones.

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En el primer capítulo que corresponde a esta parte (capítulo 2), empleo los datosde la Encuesta de Discapacidades, De�ciencias y Estados de Salud elaborada por elINE en 1999. El desarrollo de este capítulo tiene dos secciones diferentes. Por unaparte, es necesario tratar los datos, ya que la encuesta re�eja únicamente resultadoscategóricos sobre salud auto-percibida. Para poder expresar los resultados en tér-minos de pesos de calidad de vida, utilizo los métodos de escalamiento que sugieroen la primera sección. Obtengo así las tarifas derivadas del cuestionario EuroQol,basadas en los métodos de Escala Visual Analógica y de Intercambio Temporal.

Por otra parte, la metodología para evaluar los efectos de las lesiones por acci-dente de trá�co se basa en la literatura sobre �efectos de tratamiento�, en particularel método de estimación propuesto por Abadie en 2005. Las pérdidas de salud secomputan evaluando las diferencias medias de salud entre los accidentados y ungrupo de comparación. La composición de tal grupo comparativo debe plantearsecon suma cautela. Las personas que sufren una colisión vial ni son seleccionadasaleatoriamente, ni forman un grupo de población con características completamentedeterminadas. Es razonable pensar que existe un componente aleatorio en verseenvuelto en un accidente de trá�co (por ejemplo, cuando la colisión se produce demanera súbita provocada por otra persona). Sin embargo, en los datos se puedenobservar una serie de factores recurrentes en la población accidentada: mayor pro-porción de hombres, con edades entre 16 y 35 años, mayor número de fumadoreshabituales (antes del accidente), entre otros. Por tanto, considero que es necesarioestablecer ponderaciones para ajustar las medias entre los individuos que formaránel grupo de comparación, antes de estimar las diferencias.

Obtengo en el estudio que los heridos por colisiones viales pierden alrededor deun 7% de su calidad de vida. Las estimaciones son bastante similares, independi-entemente de la métrica utilizada, lo que señala robustez en los resultados. Parainterpretar correctamente este resultado, conviene señalar que la calidad media devida se sitúa en torno a 0.8. Si tomamos en representación de la población al in-dividuo medio, una disminución del 7% de su calidad de vida implica que, ante lapregunta de SAP "¿cómo valora su estado de salud, en general?", el individuo pasede responder "Bueno" a elegir la categoría "Regular". Además de representar estouna disminución importante de la calidad de vida, el problema adquiere una dimen-sión mayor a causa del elevado número de personas accidentadas. Como ejemploilustrativo, solo durante 2007 cerca de 20,000 personas sufrieron heridas de gravedaden accidentes de trá�co en España (datos de la DGT).

Un análisis parecido se efectúa en el Capítulo 3, esta vez usando los datos prove-nientes de la Encuesta Catalana de Salud. En España, Cataluña es una de lasregiones más afectadas por lesiones originadas por accidentes de circulación, alcan-zando cada año cifras en torno a los 4,000 afectados. La situación se agrava cuandoobservamos que cerca del 50% de los lesionados son jóvenes de entre 15 y 34 añosde edad, lo que representa una cifra crítica de pérdidas de salud.

Al contar dicha encuesta catalana con variables continuas, no es necesario re-alizar un primer paso de cardinalización. Calculo nuevamente el efecto medio de losaccidentes en la salud de los afectados, en términos de pérdidas en la calidad de vida.

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En los resultados se observan pequeñas variaciones respecto de la métrica empleada.Por ejemplo, basándome en las tarifas obtenidas con la Escala Visual Analógica,estimo un decrecimiento de la calidad de vida de 0:1328 en media. Si obtengo laestimación mediante la tarifa creada por el método del Intercambio Temporal, eldecrecimiento medio de la salud es igual a 0:1296. Los extremos de los intervalosde con�anza (al 95%) de las tarifas son estrictamente negativos. Este resultadohace evidente la existencia de una reducción en la calidad de vida del individuo,incluso un año después del accidente. Finalmente, para enfatizar el resultado, cal-culo la proporción de salud que, en media, los individuos han perdido, con respectoal estado de salud que, en media, los afectados tendrían si no hubieran sufrido elaccidente. Obtengo que, en términos esperados, hasta un año después del accidentelos afectados viven con un 14% menos de salud.

En el Capítulo 4 se realiza una re�exión sobre los métodos orientados a estimarpérdidas de salud debidas a causas externas en general, y a accidentes de trá�coen particular. Se analizan los principales factores que son necesarios en el proceso.En un primer punto discuto sobre la elección de una métrica adecuada con la queexpresar los resultados. En la segunda sección analizo la manera de re�ejar la cal-idad de vida que pierde el afectado. A continuación repaso de manera exhaustivalos criterios necesarios para realizar una combinación correcta de cantidad y calidadde vida. En este punto analizo los principales supuestos que emplean numerososautores, y señalo los posibles sesgos que tales procedimientos pudieran originar.En la cuarta sección, discuto los distintos puntos de vista bajo los cuales se puedecontemplar el hecho de agregar los resultados entre distintos grupos de individuos(fallecidos/heridos, jóvenes/viejos, etc.). Finalizo este capítulo con varias sugeren-cias para el tratamiento de datos, y las principales conclusiones.

La tercera parte de la tesis es de carácter teórico. En los anteriores capítuloshe trabajado con pesos de calidad de estados de salud, sin llegar a combinar dichospesos con la duración de un estado de salud, lo cual me hubiera permitido expresarlos resultados, por ejemplo, en términos de Años de Vida Ajustados por la Calidad(AVACs). El motivo de mi decisión puede sin duda verse re�ejado en esta últimaparte de la tesis. En ella, re�exiono acerca de la base teórica sobre la cual se asientael concepto de AVACs. Parece razonable pensar que las personas podemos establecerun orden entre los diferentes estados de salud concretos. Sin embargo, cuando yatenemos en cuenta la duración de tales estados de salud, ¿es inmediato el pensarque también podemos establecer un orden entre estados de salud que perdurandistinto número de años? Por ejemplo, si tenemos que elegir entre vivir en saludcasi perfecta sólo 5 días, y vivir en un estado de salud lamentable (h) durante 5meses, quizás elijamos la segunda propuesta, ya que un horizonte de vida de tansolo 5 días parece demasiado escaso. Si a continuación nos dieran a elegir entre vivir5 meses en el estado h y vivir 7 meses en ese mismo estado, es posible que anteestas dos alternativas, en las que no podemos variar el estado de salud, pre�ramosla segunda. Ahora bien, una vez sabemos que vamos a vivir 7 meses en el estado h,imaginemos que nos ofrecieran cambiarlo por vivir solo 5 días, pero con un estadode salud casi perfecto. Ante la anterior elección, la base teórica que fundamenta la

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métrica de los AVACs diría: el individuo escogerá vivir 7 meses en el estado h: Sinembargo, ¿quién puede estar seguro de no aceptar la otra alternativa? Puede existirun punto a partir del cual vivir en ciertos estados de salud nos resulte insoportable.¿Cuánto in�uye la duración de la vida, y cuánto in�uye la calidad?

El Capítulo 5 establece una caracterización axiomática de las preferencias indi-viduales sobre estados crónicos de salud, bajo el supuesto de certeza. A diferencia delos modelos establecidos por otros autores, tal procedimiento establece una máximaduración de vida, y no asume que las preferencias individuales tengan la propiedadde transitividad. El modelo permite la existencia de un orden entre los diferentesestados de salud, sin tener en cuenta su duración (orden atemporal). También esposible, partiendo del modelo, derivar el factor de descuento de una manera endó-gena, así como estimar la tendencia de las tasas de descuento en los casos en losque no se asume transitividad de las preferencias. La conocida estructura de AVACsse establece como caso particular de nuestro modelo, si admitimos el supuesto detransitividad.

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Bibliography

[1] Abadie, A. 2005. Semiparametric Di¤erence-in-Di¤erences Estimators. Reviewof Economic Studies 72: 1-19.

[2] DGT. 1993. Aproximación al estudio de las secuelas de los accidentes de trá�co.Estudio piloto (1992-1993). Madrid: Dirección General de Trá�co, Ministeriodel Interior.

[3] DGT. 1994. Análisis de la morbilidad de los heridos en accidente de trá�coingresados en los centros hospitalarios de Madrid y Barcelona (1993-1994). 1994.Madrid: Dirección General de Trá�co, Ministerio del Interior.

[4] DGT. Marzo 2005. Estudio nacional multicéntrico sobre morbilidad derivadade los accidentes de trá�co. Madrid: Dirección General de Trá�co, Ministeriodel Interior.

[5] Dolan, P., Gudex, C., Kind, P., and Williams, A. 1996. Valuing health states:a comparison of methods. Journal of Helath Economics 15(2): 209-231.

[6] Generalitat de Catalunya. 2007. Enquesta de salut de Catalunya 2006(ESCA06). Departament de Salut.

[7] INE (Instituto Nacional de Estadística). 1999. Encuesta sobre discapacidades,de�ciencias y estado de salud (EDDES). Madrid.

[8] Mathers, C. D. and Loncar, D. 2006. Projections of global mortality and burdenof disease from 2002 to 2030. PLoS Medicine 3(11):e442.

[9] Mayou, R., Bryant, B. and Duthie, R. 1993. Psychiatric consequences of roadtra¢ c accidents. The British Medical Journal 307: 647-651.

[10] Ministerio de Sanidad y Consumo. Grupo de trabajo sobre la medida del im-pacto sobre la salud de los accidentes de trá�co en España. 2004. Estudio de laMortalidad a 30 días por accidente de trá�co (EMAT-30). Madrid: DirecciónGeneral de Salud Pública, Ministerio de Sanidad y Consumo.

[11] Murray, C. and Lopez, A.D. 1996. The global burden of disease: an assessmentof mortality and disability from diseases, injuries, and risk factors in 1990 andprojected to 2020. Global Burden of Disease and Injury Series, Vol.1.

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[12] O�Donnell, M., Creamer, M., Elliott, P., Atkin, C. and Kossmann, T. 2005.Determinants of Quality of Life and Role-Related Disability After Injury: Im-pact of Acute Psychological Responses. The Journal of Trauma 59 (2005): 1328�1335.

[13] OMS. 1946. Preamble to the Constitution of the World Health Organization asadopted by the International Health Conference. Nueva York, 19-22 June 1946;signed on 22 July 1946 by the representatives of 61 States. O¢ cial Records ofthe World Health Organization, 2: 100.

[14] OMS. 2003. World report on road tra¢ c injury prevention. Environmental Bur-den of Disease Series, 1.

[15] OMS. 2004. World Health Day: Road safety is no accident!. WHO Press Re-leases, April 7. Ginebra / París.

[16] Peiró-Pérez, R., Seguí-Gómez, M., Pérez-González, C., Miralles-Espí, M.,López-Maside, A. and Benavides, F. G. 2006. Lesiones por trá�co, de ocio ydomésticas y laborales. Descripción de la situación en España. Gaceta Sanitaria20, Supl.1: 32-40.

[17] Pérez-González, C., Cirera, E., Borrell, C. and Plasència, A. 2006. Motor vehiclecrash fatalities at 30 days in Spain. Gaceta Sanitaria 20(2): 108-15.

[18] Redondo, J. L., Luna, J.D., Jiménez, J. J., Lardelli, P. and Gálvez, R. 2000.Variabilidad geográ�ca de la gravedad de los accidentes de trá�co en España.Gaceta Sanitaria 14(1): 16-22.

[19] Van Doorslaer, E. and Jones, A. 2003. Inequalities in self-reported health: vali-dation of a new approach to measurement. Journal of Health Economics 22(1):61-87.

[20] Williams, A. 1999. Calculating the Global Burden of Disease: Time for a Strate-gic Reappraisal?. Health Economics 8: 1-8.

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Introduction

The analysis of health states registered a very signi�cant upturn in the second halfof the 20th century. Basically, three major guidelines can be outlined: the establish-ment of a new concept of health, which incorporates elements concerning quality-of-life, beyond the simple lack of illness; the new method of classifying health statesbased on the individual preferences, in terms of utility; and the increasing interestin the analysis of health problems at a population level.

The foundation of the Word Health Organization (WHO) in 1948 representsthe �rst attempt to re�ect the major health problems at population levels. TheWHO de�ned health as follows: �Health is not only the absence of in�rmity anddisease, but also a state of physical, mental and social well-being�. This broadde�nition captures essential elements of quality of life, which underlies most humanhealth metrics. Based on this de�nition, it was also clear that life expectancy ormortality-based measures were no longer been considered adequate as measures ofa population�s health.

Health analysis at individual level also experiments a similar change of directionas those focused on population levels. Quality-Adjusted Life Years (QALYs) wasgoing to be, by far, the most widely used metric in health economics. It was intro-duced by the York Health Economics School in the sixties, and since then, has beenaccepted and used worldwide. This metric combines quality and quantity of life, inorder to represent individual health states in units of time, taking as reference theso called �perfect health�state. From the beginning, it had an utilitarist interpre-tation, in the sense that a QALY was de�ned as the utility an agent derived frombeing one year in perfect health. From here, the scienti�c QALY literature evolvedin order to adequate it to that individual interpretation (the introduction of uncer-tainty, the valuation of health states at di¤erent moments in time, chronic versusnon-chronic situations, etc). Furthermore, the aggregation problem was identi�ed asa major one: all traditional welfare aggregation problems stand here in a prominentway. Nonetheless, in practice, QALY indices are inferred from the population, andthe valuation of a representative agent is used to aggregate across individuals.

As a result of the novel health metrics, and always under the new health frame-work, numerous studies have been performed, aimed at re�ecting the actual healthproblems in targeted populations. The Global Burden of Disease Study (GBD; seeMurray and Lopez, 1996) can be considered as a representative one. The GBD studywas arranged by the WHO in 1992, at request of the World Bank. It signi�ed anambitious project, since it consisted of an exhaustive compilation of information onthe extent of disability, morbidity or death, at a global and regional level. The GBDrepresented the �rst attempt to re�ect the major health problems in the world.

One of the most signi�cant results that were derived from the GBD is the elab-oration of a ranking (referred to 1990) that re�ects the ten leading causes of healthlosses in the global population. In addition, the GBD projected di¤erent healthscenarios from 1990 to 2020, in order to estimate the forthcoming worldwide major

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health problems. Posterior studies (see Mathers and Loncar, 2006) updated theseevaluations, by estimating the major mortality causes by 2030.

In the GBD study we observe that the three leading causes of health losses in2020 are suggested to be ischemic heart diseases, unipolar major depression and roadtra¢ c accidents, contrary to the leading causes of health lost in 1990. Furthermore,road tra¢ c accidents is the one that raises the most in relative rank respect to 1990.If I analyze in more detail the data about tra¢ c accidents, it is shocking to observethat it is the fourth leading cause of health loses in developed regions in 1990, andthe main cause of ill-health and premature death for men aged 15-44 worldwide,surpassing other causes as could be war, suicide, hearth diseases or malnutrition.In addition, following Mathers and Loncar (2006), road crashes are expected to bethe main origin of the projected 40% increase in global deaths resulting from injurybetween 2002 and 2030.

This is a problem that has always been undervalued, since the prevention oftra¢ c accidents has never meant a priority for global public health. However, themagnitude of this �global public health crisis�(WHO, 2004) requires an exhaustivestudy about the main factors that could lead to tra¢ c accidents and the developmentof e¤ective actions to prevent them.

If I focus my attention in Spain, I �nd that the number of deaths caused by tra¢ caccidents is higher than in other countries as United Kingdom, Italy, Portugal orGermany, in spite of the fact that the number of road crashes is lower.4 Similarly toother countries, road tra¢ c accidents a¤ect young people more signi�cantly, causingmore than half of the deaths for those aged 15-24. Therefore, it is obvious that inSpain tra¢ c accidents is also a major health problem that requires not only socialand political will but also e¤ective action.

Di¤erent reasons can be found to explain this public health problem. The keypoint is that major risk factors are entirely identi�able: defects in road design,infrastructures situation, lack of police control, vehicle factors, behavior of drivers(excess of speed, alcohol and other drugs, fatigue, not using helmets or seat-belts...),de�ciencies in the chain of medical care from urgent sanitary attention to rehabil-itation, etc. (see WHO, 2004). Therefore, it is obvious that road safety is a re-sponsibility that must be shared between citizens, government, police, professionals,industry and media.

The possibility of developing or introducing new measures for reducing the hightoll of mortality in roads, makes crucial the following questions: is it worth to investin improving emergency services? Or is it worth to invest in more sophisticatedcomputer systems, and to increase the tra¢ c controls, in order to properly managethe recent penalty-point system? Hence the evaluation of the costs and bene�ts ofsuch novel instruments is essential.

Thus, I face the following problem: how can I quantify the bene�ts in health ofreducing the accident rate in roads? Two di¤erent approaches can be found: (1)

4See statistical tables form CARE (Community Road Accident Database).

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Analyze the impact of the policy interventions on population health, and (2) analyzethe general impact of road crashes on population health, in order to evaluate ex-post the bene�ts of their reduction. The choice between both options is not trivial,since it determines the methodology and metrics to be used. The main advantage ofthe �rst approach �the most generalized option in cost-e¤ectiveness analysis- is theimmediate application of the results for the adoption of a stance by policy-makers.The second approach �to focus on evaluating the size of the problem -, has a majorrelevance, since in the context of injury prevention and control, the availabilityand/or quality of data is not as favorable as in other health environments. For thisreason, in the present work the second option is selected, that is, the main objectiveis the estimation of the total toll that deaths, injuries and sequelae derived fromtra¢ c accidents cause over population.

Several earlier studies attempt to provide some description of the impact of roadcrashes on the Spanish population health. I highlight the following ones: EMAT-30(2004); Pérez et al. (2006); Peiró-Pérez et al. (2006); Redondo et al. (2000); DGT(Dirección General de Trá�co) Estudio Piloto (1993); DGT Análisis de Morbilidaden Madrid y Barcelona (1994); DGT Estudio nacional multicéntrico (2005).

The Estudio Piloto (1993) was arranged by the DGT in Valladolid in 1993. Thisstudy represents the �rs step towards the attempt of measuring the characteristicsof the nonfatal injuries. Data regarding the people injured by road crashes were ob-tained from the DGT and local hospitals. Moreover, a follow-up survey was arrangedfor obtaining data about the evolution of morbidity, including several questions re-lated to the quality of life and functional incapacity. This is a totally descriptivestudy.

A further step would be done by the Morbidity Analysis in Madrid and Barcelona(DGT Análisis de Morbilidad en Madrid y Barcelona, 1994). This work focuseson the morbidity in Barcelona and Madrid, 1993-1994, with data from hospitaladmissions and discharges. The measures employed in the analysis are, mainly,the Average Time of Recovery, the existence of work-related disabilities, and theexistence of permanent disabilities. This study also performs an economic estimationof the costs associated to these targeted health states.

In 2000 Redondo et al. publish a general analysis of the consequences or roadcrashes in Spain. This study is also written in a descriptive nature, and it is based ondata from the DGT in the period 1985-1994. The mortality and morbidity measuresthat are used in this work are: mortality indices and injury rates. Indicators of theseriousness of road crashes are: population densities and number of motor vehiclesand licenses.

In the so-called �Estudio Nacional Multicéntrico�, the DGT analyzes the injuredpeople in the period 2000-2004. More speci�cally, a sample of 2,180 injured individ-uals who have been attended in hospitals in Spain is selected, and it is performeda follow-up to each individual, during the subsequent four years following the roadcrash. Morbidity is described by establishing comparisons with data relating to2001, and the seriousness of the injuries are characterized by means of the Injury

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Severity Score (ISS) and the Social Pain Index (Índice de Dolor Social, IDS).

The studies arranged by the EMAT-30 (Motor Vehicle crash fatalities at 30 daysin Spain, or Estudio de la Mortalidad a 30 días por Accidentes de Trá�co) andPérez et al., (2006), perform the �rst analysis based on the CMBDAH database5 ata country level. Data from this source are compared with data form the DGT, 2001.These works perform a descriptive analysis of mortality by socioeconomic variables,besides a description of the morbidity of those injured, by means of the ISS.

In Peiró-Pérez et al. (2006), a brief but exhaustive epidemiological description ofthe road crashes in 2002 can be found, based on the results obtained from the EMAT-30. The signi�cancy of this work consists on the use of several complementarysources of information: mortality data from INE, CMBDAH and DGT, whereasmorbidity data are obtained from DGT, CMBDAH,6 RCSCV and ENS03.7 Themorbidity is measured by the ISS.

The Studies commented above have a common descriptive nature, and the mor-bidity measures try to estimate the seriousness of the injuries, either re�ecting thedegree of functional limitation of the injured individuals, or attending to the mor-tality risk or life threat. Nonetheless, several authors have provided the existence ofimportant psychological consequences following a morbid precess, as can be stressor depression (see O�Donnell et al., 2005), specially those following a road crash (seeMayou et al., 1993); therefore, the psychological dimension of the problem shouldnot be obviated.

The di¢ culties above mentioned lead me to suggest a new perspective, accordingto the new health framework de�ned by the WHO: that is, to analyze the impact offatal and nonfatal injuries on the quality of life of the injured individuals, not onlyattending to the physical damage that the injury caused, but also contemplating thepossible psychological consequences, as well as the potential impact on the well-beingof those a¤ected. These three dimensions (physical, emotional and well-being) de�nethe concept of �individual health state�, that will be used in the di¤erent essays.

Once the methodological framework has been established, it should be remindedthat the goodness of the empirical analyses is conditioned on the quality of data thatsupports the study. The increasing interest regarding the population health states,as well as the progressive computerization of data, have promoted the creation ofdi¤erent data sources, as those concerning hospital discharges, road crashes, publicand private health surveys, etc., beside the computerization and availability of thosealready existing databases.

However, certain health measures cannot be easily included in general popula-tion surveys. For instance, the analysis of health measures is usually derived fromthe respondents�assessment of her own health status, and that piece of informationabout self-assessed health (SAH) is usually obtained from an easy question such as:

5Conjunto Mínimo y Básico de datos de las Altas Hospitalarias.6Red Centinela Sanitaria de la Comunidad Valenciana.7Encuesta Nacional de Salud de 2003.

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"In your opinion, how is your health in general?", where respondents must chooseone between several categories, typically ranging from "excellent" to "very poor".SAHmeasures are straightforwardly understandable by the general population, whatstands for an advantage. Nevertheless, the use ordinal measures as SAH is not ap-propriate in contexts of mean-based measures (e.g. analysis of health inequalities,measuring health e¤ects, etc.). Moreover, the subjective character of these metricsdoes not allow for aggregating the responses over the population, since the het-erogeneity of the preferences would be ignored. Thus, a wide variety of methodshave been developed for deriving quality weights from categorical health measuresas SAH responses (see van Doorslaer and Jones 2003).

The main objective of this thesis dissertation consists on analyzing the valuationof health states from di¤erent perspectives. The �rst part of the dissertation showsmore empirical analyses. Several econometric models have been applied, seeking theevaluation of chronic health losses, in terms of quality of life lost, reported by thosewho have su¤ered from a serious road crash, which have caused limitations in dailyactivities. Data have been obtained from general population health surveys. Mostof them describe the health state of the individuals by means of SAH questions.Thus, some scaling methods have been used. In the second part of the thesis, Ifocus on the valuation of health states, from an axiomatic perspective, re�ecting onthe theoretical basis of the measures used in the �rst part.

In detail:

The �rst chapter analyzes di¤erent scaling approaches (ordered probit/logitmodel and interval regression model, both under normality). A new procedureis suggested, that is attaching a lognormal distribution, that allows the introductionof skewness in the distribution of health. Di¤erent scaling procedures have beencompared, with data obtained from the Catalan Health Survey (CHS, 2006). Thevalidity of the scaling approaches is assessed by measuring to what extent the healthvalues derived from categorical health variables suit the actual health values. Twodi¤erent health tari¤s have been used for each procedure (VAS tari¤ and TTO tari¤), so that the results are robust to the selection of a metric. In general, modelsunder lognormality outperform the other approaches.

The objective of the second chapter is to evaluate the e¤ect of a road crashon the health-related quality of life of injured people. A new approach based onthe cardinalization of di¤erent categorical measures of ill-health, such as TTO andVAS indices, is suggested and used for assessing the robustness of the results. Themethodology is based on the existing literature about treatment e¤ects. My maincontribution focuses on evaluating the chronic loss oh health, that would allow toproperly estimate the health losses in quality-of-life terms.

A similar analysis is performed in chapter 3. The main goal of this chapter isto estimate the QoL lost due to nonfatal road tra¢ c injuries in Catalonia, Spain,between the years 2002 and 2006. My contribution to the literature focuses inevaluating the chronic loss of health, that would allow to estimate properly thehealth losses in QALYs. Data are obtained from the CHS for the years 2002 and

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2006, and di¤erent health metrics will be used for assessing the robustness of theresults. Since the CHS includes health indices, it is not necessary to implement acardinalization process.

Chapter 4 re�ects on the di¤erent parameters to be followed in the evaluationof treatments and prevention strategies: the proper de�nition of health e¤ect atindividual and aggregate levels; the correct selection of a health metric; the accurateestimation of the short-term e¤ect (direct health gain/loss) and long-term e¤ect(total of health gain/loss throughout the life of the individual) that injuries mayproduce; the suitable selection and management of databases. This review chapterfocuses on the particular topic of road accidents, but the analysis can be extendedto any sort of injury.

Chapter 5 provides an axiomatic characterization of individual preferences overchronic health states, under certainty. My main contribution focuses on establishinga maximum life horizon, besides relaxing the assumption of transitivity of prefer-ences, which is usually incorporated to the modelling of health preferences. Themodel of Quality-Adjusted Life Years (QALYs) is embraced in my model of prefer-ences, if transitivity is assumed.

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Bibliography

[1] DGT. 1993. Aproximación al estudio de las secuelas de los accidentes de trá�co.Estudio piloto (1992-1993). Madrid: Dirección General de Trá�co, Ministerio delInterior.

[2] DGT. 1994. Análisis de la morbilidad de los heridos en accidente de trá�coingresados en los centros hospitalarios de Madrid y Barcelona (1993-1994). 1994.Madrid: Dirección General de Trá�co, Ministerio del Interior.

[3] DGT. Marzo 2005. Estudio nacional multicéntrico sobre morbilidad derivada delos accidentes de trá�co. Madrid: Dirección General de Trá�co, Ministerio delInterior.

[4] Dolan, P., Gudex, C., Kind, P., and Williams, A. 1996. Valuing health states: acomparison of methods. Journal of Helath Economics 15(2): 209-231.

[5] Generalitat de Catalunya. 2007. Enquesta de salut de Catalunya 2006 (ESCA06).Departament de Salut.

[6] INE (Instituto Nacional de Estadística). 1999. Encuesta sobre discapacidades,de�ciencias y estado de salud (EDDES). Madrid.

[7] Mathers, C. D. and Loncar, D. 2006. Projections of global mortality and burdenof disease from 2002 to 2030. PLoS Medicine 3(11):e442

[8] Mayou, R., Bryant, B. and Duthie, R. 1993. Psychiatric consequences of roadtra¢ c accidents. The British Medical Journal 307: 647-651.

[9] Ministerio de Sanidad y Consumo. Grupo de trabajo sobre la medida del impactosobre la salud de los accidentes de trá�co en España. 2004. Estudio de la Mortali-dad a 30 días por accidente de trá�co (EMAT-30). Madrid: Dirección General deSalud Pública, Ministerio de Sanidad y Consumo.

[10] Murray, C. and Lopez, A.D. 1996. The global burden of disease: an assessmentof mortality and disability from diseases, injuries, and risk factors in 1990 andprojected to 2020. Global Burden of Disease and Injury Series, Vol.1.

[11] O�Donnell, M., Creamer, M., Elliott, P., Atkin, C. and Kossmann, T. 2005.Determinants of Quality of Life and Role-Related Disability After Injury: Impactof Acute Psychological Responses. The Journal of Trauma 59 (2005): 1328 �1335.

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[12] Peiró-Pérez, R., Seguí-Gómez, M., Pérez-González, C., Miralles-Espí, M.,López-Maside, A. and Benavides, F. G. 2006. Lesiones por trá�co, de ocio ydomésticas y laborales. Descripción de la situación en España. Gaceta Sanitaria20, Supl.1: 32-40.

[13] Pérez-González, C., Cirera, E., Borrell, C. and Plasència, A. 2006. Motor vehiclecrash fatalities at 30 days in Spain. Gaceta Sanitaria 20(2): 108-15.

[14] Redondo, J. L., Luna, J.D., Jiménez, J. J., Lardelli, P. and Gálvez, R. 2000.Variabilidad geográ�ca de la gravedad de los accidentes de trá�co en España.Gaceta Sanitaria 14(1): 16-22.

[15] Van Doorslaer, E. and Jones, A. 2003. Inequalities in self-reported health: val-idation of a new approach to measurement. Journal of Health Economics 22(1):61-87.

[16] WHO. 1946. Preamble to the Constitution of the World Health Organization asadopted by the International Health Conference. Nueva York, 19-22 June 1946;signed on 22 July 1946 by the representatives of 61 States. O¢ cial Records of theWorld Health Organization, 2: 100.

[17] WHO. 2003. World report on road tra¢ c injury prevention. EnvironmentalBurden of Disease Series, 1.

[18] WHO. 2004. World Health Day: Road safety is no accident!. WHO Press Re-leases, April 7. Ginebra / París.

[19] Williams, A. 1999. Calculating the Global Burden of Disease: Time for a Strate-gic Reappraisal?. Health Economics 8: 1-8.

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Part I

Scaling Methods

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Chapter 1

Scaling Methods for CategoricalSelf-Assessed Health Measures

1.1 Introduction

The estimation of a quality weight related to a particular health state is the basis

of an extensive area of health-related studies. Fundamentally, quality weights are

required for the computation of health-related quality of life (HRQoL) measures,

that represent an essential tool in cost-e¤ectiveness and cost-utility analysis. The

use of these weights is also desirable in other health issues, as measuring health

levels for populations, estimating quality adjustments of life expectancy, or analyzing

inequalities and inequities in health, among others.

Theoretically, quality of life associated to health states is considered as a con-

tinuum, with maximum and minimum values, that admits a complete order. Thus,

health states can be represented in a 0 -1 scale, where 0 represents the worst health

state and 1 the best health state. In practice, however, the information about

the health state of individuals are often derived from general health surveys. More

concretely, from the respondent�s assessment of her own health status, typically mea-

sured on an ordinal scale. Thus, a wide variety of methods have been developed for

deriving quality weights from categorical health measures. This paper compares al-

ternative procedures designed to impose cardinality on the ordinal health responses,

and suggests a new methodology.

Most studies in this area deal with the cardinalization of the so-called self-

assessed health (SAH). This piece of information is usually obtained from a question

such as: "In your opinion, how is your health in general?", where respondents must

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Chapter 1 Scaling Methods for Categorical Self-Assessed Health Measures

choose one between several categories, typically ranging from "excellent" to "very

poor". SAH measures present numerous advantages. First, they are one of the most

commonly used indicators in socioeconomic and epidemiological surveys. Second,

they o¤er a summary of the general health state of the respondents. Third, they

have shown a good performance at predicting future mortality and morbidity (Idler,

1997).

The usual procedures assume the existence of a latent, continuous but unobserv-

able health variable (y�) with a normal distribution. This framework can embrace

well-known approaches as the estimation of ordered-probit regressions (Groot, 2000)

or combining the distribution of observed SAH with external information (the in-

terval regression approach, in Van Doorslaer and Jones, 2003). Ordered-probit re-

gressions have the requirement of re-scaling to compute quality weights. The use

of external information allows to identify the scale of y� without having any scaling

or identi�cation problems, but it requires the use of additional assumptions. The

interval regression approach is found superior to the ordered-probit approach (Van

Doorslaer and Jones, 2003; Lauridsen et al., 2004), and is one of the most widespread

methods of scaling (Lecluyse and Cleemput, 2006).

The major drawback of those approaches is that they rule out any skewness in

the distribution of the latent health variable y�. This fact is specially important

for the analysis of health measures in developed countries, where a large proportion

of the general population report good health. One possible strategy is to use the

standard lognormal distribution rather than the standard normal distribution. The

shape of the health distribution is captured better, but the estimation will require

an ex-post re-scaling. Wagsta¤ and Van Doorslaer (1994), assigned to every cat-

egory of SAH a value that equals the midpoints of the intervals corresponding to

the standard lognormal distribution. However, this method fails to introduce the

required continuity in health scales. As far as I know, no other approach under

log-normality has been suggested.

In this work I propose methods for scaling SAH measures, considering that the

latent health variable is log-normally distributed. These new methods are compared

to the existing procedures, which contemplate values of health as normally distrib-

uted. Data from the Catalonia Health Survey 2006 (CHS) are used to provide the

results. In this survey every respondent reports a categorical SAH evaluation. Also,

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Chapter 1 Scaling Methods for Categorical Self-Assessed Health Measures

utility values can be derived by means of the EQ-5D descriptive system provided by

the survey. Thus, I can take advantage of having actual continuous health values in

this study. The validity of the scaling approaches is assessed by measuring to what

extent the health values derived from SAH suit the actual health values, available

in the survey.

The paper is structured as follows: Section 2 summarizes the di¤erent scaling

methodologies: ordered probit model and interval regressions, both under normal

and log-normal distribution. In Section 3 I present the data. Section 4 presents the

results and compares the performance of di¤erent models. Section 5 summarizes

the main conclusions and provides a discussion about the validity of the scaling

approaches.

1.2 Methodology

This section describes the di¤erent scaling methods that are compared in this paper.

All of them assume the existence of a continuous latent health variable y� underlying

the categorical SAH variable. I divide them into two groups, depending on the

distribution properties that are assigned to y� (normal or lognormal distribution).

The objective of these procedures is to estimate the quality weight associated to

every respondent i (wi), conditioning on the self-assessed health value of individual

i, SAHi, and on a vector of socioeconomic variables for individual i, xi:

I will compare the characteristics of these continuous health values as well as the

regression-based measures obtained from these actual values, with the descriptive

performance of the scaling methods.

1.2.1 A standard normal latent health variable

Suppose that SAH has J categories, with category 1 corresponding to the worst

health and J corresponding to the best. The SAH stated by individual i; and

her/his true health state are recorded as SAHi and y�i , respectively. Then, y�i and

SAHi are related as follows:

SAHi = j i¤ �j�1 < y�i � �j , j = 1; 2:::J (1.1)

where y� 2 (�0; �J ], and �j, j = 1; 2; :::; J stand for the thresholds between

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Chapter 1 Scaling Methods for Categorical Self-Assessed Health Measures

categories of SAH, with �j�1 � �j: Note that the thresholds are constant for indi-viduals. Depending on each methodology, the support of y� may vary. In general, I

am interested in obtaining a continuous health index for every respondent wi 2 [0; 1]:The usual scaling methods (the ordered probit model and the interval or grouped

data regression model) are summarized below.

Ordered Probit model (OP+N)

Now the latent health variable y� can take any real value (that is, �0 = �1 and

�1 = +1), and for each individual it is assumed to be a function of a vector ofsocioeconomic variables xi:

y�i = x0i�+ "i; with "i � N(0; 1)

Predictions of the linear index, E[y�i jxi], can then be used as a measure of indi-vidual health and, after appropriate re-scaling, as �quality weights�or utility proxies.

I use one of the re-scaling methods proposed by van Doorslaer and Jones (2003),

that do not require the availability of a continuous health variable. Let y0i = x0ib�and let ymax and ymin the largest and smallest individual predictions, respectively.

Then the re-scaled values (wi), that represent quality weights, can be calculated as:

wi =y0i � yminymax � ymin (1.2)

Interval Regression (IR+N)

This method provides an alternative to (OP+N) when the threshold values (�j) are

directly observed. In many cases, the thresholds are not observed, but they can

be obtained from external information. The methodology proposed by Doorslaer

and Jones (2003) for establishing the ��s consists in combining the distribution of

observed SAH with external information on the distribution of a generic measure

of health utilities y, ranging from 0 to 1. The relationship between y�(latent),

SAH (available at the current data) and y (obtained from external information), is

assumed to be as follows: the higher the value of y�, the more likely the individual is

to report a higher category in SAH; and a higher value in y. For such a connection to

be correct, it is necessary to assume that the reported variables have rank properties:

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Chapter 1 Scaling Methods for Categorical Self-Assessed Health Measures

the qth-quantile of the distribution of y will correspond to the qth-quantile of the

distribution of y�, and this will also correspond to the qth-quantile of the distribution

of SAH. The model results:

SAHi = j i¤ �j�1 < yi � �j; j = 1; 2:::J

with

yi = x0i� + "i; "i � N(0; �2)

where it is set that �0 = 0; �J = 1 and �j � �j+1:Since ��s are known, the variance of the error term can be estimated. Also the

predicted values from the interval regression are measured in the same units that

the thresholds, avoiding an ex-post re-scaling.

The previous model establishes that yi � N(x0i�; �2); and yi 2 [0; 1]: Under theseassumptions, the variance in the distribution of yi is forced to be small enough, in

order to de�ne values inside the 0 � 1 interval. Although this is not an appealingassumption, it allows the estimated values to be expressed in the required units.

Moreover, the variance is restricted, but not completely determined, as in ordered

probit models.

Figure 1.1 illustrates the relationship between the di¤erent measures (for sim-

plicity, I take J = 5).

yi ∼ N(x'iβ ,    )2σ

SAH = 1 SAH = 2

SAH= 3

SAH= 4SAH = 5

P(SAHi = 1) =

= P(0< y < µ1)

µ1µ0 = 0 µ2 µ3 µ4 µ5 = 1

Figure 1.1. Relation of SAH and y� under normality

A slight variation with respect to Doorslaer and Jones (2003) for determining

the thresholds is introduced in this study.1 Since the main objective of this method

is injecting continuity into the distribution function of health, the threshold values

1The interpolating method is also used by Lecluyse and Cleemput (2006).

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Chapter 1 Scaling Methods for Categorical Self-Assessed Health Measures

result from interpolating the closest quantiles. More formally, let Gj represent the

cumulative relative frequency of the j -th category of SAH, and F (�) be the empiricaldistribution function (EDF) of the continuous health variable y (variables SAH and

y may be obtained from di¤erent samples). Let F�1(�) be the inverse of F (�). Idenote the set of actual values of F (�) as ImF . Now de�ne (for j = 1; 2; :::; J � 1):Glj = fg 2 ImF such that g < Gj and @g0 2 ImF with g < g0 < Gjg ;Guj = fg 2 ImF such that g > Gj and @g0 2 ImF with g > g0 > Gjg ; andqlj = F

�1(Glj); quj = F

�1(Guj )

Therefore, the threshold �j is obtained with the expression:

�j = qlj +

�quj � qlj

��Gj �GljGuj �Glj

(1.3)

Since the thresholds are derived form self-assessed valuations, it is necessary to

analyze if they change among di¤erent subgroups of population, e.g. male, younger,

etc. (the so-called cut-point shift). If so, the estimated thresholds should be condi-

tioned on each group.

Finally, the estimated quality weights are given by:

wi = E[yijxi]

1.2.2 A standard lognormal latent health variable

It is well-known that the health of a general population sample has a very skewed

distribution, with the great majority of respondents reporting their health in higher

levels. To ensure that the latent health variable is skewed in the appropriate direc-

tion, I rede�ne the true health of the individual in the range (�1; 0], and assumethat h� = �y� has a standard lognormal distribution. The new variable h� is de-creasing in health, so that represents the latent "ill-health" of the individual. Then,

respecting the notation in (1.1), h�i and SAHi are related as follows:

SAHi = j i¤ � �j < h�i � ��j�1 , j = 1; 2; :::; J

where y� 2 (�0; �J ], �0 = �1; �J = 0.The procedures described above (ordered probit and interval regression models)

are now reinterpreted in terms of h�.

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Chapter 1 Scaling Methods for Categorical Self-Assessed Health Measures

Ordered Probit model (OP+LN)

The latent ill-health variable for individual i is assumed to be a function of a vector

of socioeconomic variables xi as follows:

ln(h�i ) = x0i + "i , with "i � N(0; 1), or

h�i = ex0i � e"i , with "i � N(0; 1)

Thus the expression:

h0i = E [y�i jxi] = �E [h�i jxi] = �ex

0ib E [e"ijxi] = �ex0ib e1=2

gives us the predictions of health values in a continuous scale (�1; 0]. For

obtaining health indices or quality weights, I perform the same re-scaling method

that I used under the assumption of normality, shown in equation (1.2):

wi =h0i � hminhmax � hmin (1.4)

Interval Regression (IR+LN)

Since the connection between yi and SAHi is due to represent the latent variable,

an adaptation is needed.

I denote hi = 1�yi , a new health variable now interpreted in terms of ill-health.If the values of the generic measure y yields in the range [0; 1], the connection

between the variables holds as Table 1.1 shows:

health ill-healthSAH y hJ (�J�1; 1] [0; 1� �J�1)...

......

2 (�1; �2] [1� �2; 1� �1)1 [0; �1] [1� �1; 1]

[0; 1] [0; 1]

Table 1.1. Connection of di¤erent health variables

Let me call �j = 1� �J�j; for j = 1; 2; :::; J � 1; with �0 = 0; �J = 1:

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Chapter 1 Scaling Methods for Categorical Self-Assessed Health Measures

The model turns out to be:

SAHi = j i¤ �j�1 � hi < �j; j = 1; 2:::J

where

ln(hi) = x0i� + "i; "i � N(0; �2)

The values of �j; j = 1; 2; :::; J�1 can be obtained in a similar way to (1.3), beingh and SAH (with the categories reversed) the variables to be interpolated. However,

due to the "ceiling e¤ect" of health valuations (high proportion of observations with

yi = 1), the higher categories of SAH may possibly be linked to hi = 0. In that

case no interpolation is feasible, and those categories should be merged. In order

to avoid this possible drawback, I obtained the thresholds directly from the ��s

values computed in equation (1.3), by applying the de�nition of �j = 1 � �J�j;j = 1; 2; :::; J � 1. With such thresholds the same continuity that was inducedinto the empirical distribution function of y is also considered into the EDF of h,

what establishes consistency between (IR+N) and (IR+LN). Also, the is no need of

reducing the number of categories of SAH, what would mean a loss of information.

Figure 1.2 illustrates the context of (IR+LN) for the typical case of J = 5:

µ1 µ2 µ3 µ4 µ5 = 1

yi SAH = 1SAH = 2

SAH = 3SAH= 4 SAH= 5

1­ µ11­ µ21­ µ31­ µ4 1­ µ0 = 11­ µ 5 = 0

SAH = 1SAH = 2

SAH = 3SAH= 4SAH= 5

hi = 1 ­ yi

hi ∼ LN(xi'd , σ2)

P(SAHi = 5) =

= P(0< hi < 1­ µ4)

µ0 = 0

1­ µ11­ µ21­ µ31­ µ4 1­ µ0 = 11­ µ 5 = 0

Figure 1.2. Relation of health and ill-health measures under lognormality

As I discussed in (IR+N), the possible existence of cut-point shift should be

determined. The unconditional predictions are computed straightforward (here, I

do not need of rescaling, since �2 can be identi�ed):

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Chapter 1 Scaling Methods for Categorical Self-Assessed Health Measures

wi = E [hijxi] = Ehex

0i�e"ijxi

i= ex

0i�E [e"ijxi] = 1� ex

0i�e�

2=2

1.3 Data

My source of data is the Catalonia Health Survey 2006 (CHS). The data were

collected throughout the year 2006, and comprise a total of 18; 126 individuals. The

survey includes questions on the state of health, the habits of life (including feeding,

physical exercise and tobacco and alcohol consumption), and the utilization of the

health services-managed by the regional government.

Several measures of the health state are provided by the CHS : �rst, a numeric

self-evaluation of the health state (SAH question); second, the EQ-5D descriptive

system; and third, the EuroQol visual analogue scale or EQ VAS (The EuroQol

Group, 1990). The EQ-5D descriptive system comprises the following 5 dimensions:

mobility, self-care, usual activities, pain/discomfort and anxiety/depression. Each

dimension has 3 levels: no problems, some problems, severe problems. A total of

243 possible health states is de�ned in this way. In order to translate these variables

to a particular score of health status, a �preferences tari¤�is needed.2

Two tari¤s for these scores have been computed for Spain: the VAS tari¤ (based

on the EQ VAS), by Badia et al. (1997), and the TTO tari¤ (based on the temporal

equivalence method), by Badia et al. (2001). Both scores allow for negative values,

that is, health states worse than death, what confuses the measurement of health

e¤ects. One criterion for overcoming these controversies consists on changing the

negative values to zero (e.g. Burström et al., 2003; Zozaya et al., 2005). A di¤erent

method is based on re-scaling the scores to the interval (0,1), based on the minimal

and maximal values obtained in the tari¤ (related to health states 33333 and 11111,

respectively), as in Busschbach et al. (1999). None of both is, in principle, preferable

to the other, but they can lead to di¤erent results. My analysis is performed by

using the re-scaling method, reducing in that way the number of observations at the

bottom of the distribution. Both tari¤s are used as health measures (y =VAS tari¤ ,

y =TTO tari¤ ), in order to control for the robustness of the results. The regression

procedures explained in previous sections are used to approximate these tari¤s by

2See Cutler and Richardson (1997) and Torrance (1986).

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Chapter 1 Scaling Methods for Categorical Self-Assessed Health Measures

the response category of SAH, conditioning on several socioeconomic factors. If the

approximation is good enough, in situations where health tari¤s are not available,

the scores obtained from these regression models (w) could be adopted as quality

weights.

For practical reasons, the analysis is performed over the population aged 15

or higher. From the CHS, I dropped 2; 342 observations (corresponding to children

aged under 16), 76 (missing values or inconsistent answers) and 60 individuals whose

answers in the relevant variables were not considered trustworthy by the interviewer.

The �nal size for the sample is 15; 648 individuals. Sampling weights are not used

in the analysis.

I consider a wide range of factors that can a¤ect the self-valuation of the health

state of an individual: age-gender groups, activity status (employed, unemployed,

unable, retired, student, houseworker), educational level (no studies, primary, sec-

ondary, superior), marital status (single, married, widow, separated or divorced),

household size, if born in a foreign country, existence of a chronic illness (epilepsy,

cholesterol,...), existence of some de�ciency (mental, visual,...), if sleeps 8 hours or

more, if practices sports regularly, Body Mass Index (BMI: less than 18, between

18 and 25, and higher than 25), if heavy smoker (in the present or in the past), if a

hard-drinking individual. The variable related to income has not been included as

a regressor (6; 373 missing values); an indicator of the social class of the respondent

(high, medium, low) has been taken as a proxy.

1.4 Results

Before giving estimates for the continuous health measures, I explore whether in-

terval boundaries di¤er greatly across demographic groups. Data are grouped by

gender and age category. SAH is mapped into VAS tari¤ and TTO tari¤, as it is

detailed in (IR+N). Figures 1.3 and 1.4 illustrate the results:3

3Figures regarding TTO tari¤ show similar results. For simplicity, only �gures regarding VAStari¤ are reported.

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Chapter 1 Scaling Methods for Categorical Self-Assessed Health Measures

VAS tariffs and SAH by age groupsmen

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

15­24 25­34 35­44 45­54 55­64 65­74 75+

age groups

SAH= 1 SAH = 2 SAH = 3 SAH = 4 SAH = 5

Figure 1.3. Thresholds by age category. Women.

VAS tariffs and SAH by age groupsmen

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

15­24 25­34 35­44 45­54 55­64 65­74 75+

age groups

SAH= 1 SAH = 2 SAH = 3 SAH = 4 SAH = 5

Figure 1.4. Thresholds by age category. Men

Figures 1.3 and 1.4 show that subjective thresholds do not di¤er signi�cantly

among the subpopulations. They specially tend to be constant in populations aged

25-74. This pattern is also observed in di¤erent samples, by Van Doorslaer and

Jones (2003). Thus the interval regression approach is unlikely to be sensitive to

making the interval boundaries age�sex speci�c, so the response-category cut-point

shift is ignored hereafter.

Table 1.2 andTable 1.3 show summary statistics for TTO tari¤ andVAS tari¤

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Chapter 1 Scaling Methods for Categorical Self-Assessed Health Measures

by SAH category, respectively. The last rows show the upper bounds of intervals

corresponding to (IR+N) and (IR+LN)

Emp. cum. freq. (%) Upper bounds

SAH N health ill-health mean sd (IR+N) (IR+LN)

poor 863 0.0552 1.0000 0.5587 0.2427 0.5436 1.0000

fair 3,131 0.2552 0.9448 0.8178 0.1818 0.9006 0.4564

good 6,971 0.7007 0.7448 0.9546 0.0883 0.9713 0.0994

very good 3,522 0.9258 0.2993 0.9791 0.0630 0.9929 0.0287

excellent 1,161 1.0000 0.0742 0.9888 0.0467 1.0000 0.0071

Table 1.2. Summary statistics of TTO tari¤ by categories of SAH and upper bounds in IR

Emp. cum. freq. (%) Upper bounds

SAH N health ill-health mean sd (IR+N) (IR+LN)

poor 863 0.0552 1.0000 0.4255 0.2216 0.4131 1.0000

fair 3,131 0.2552 0.9448 0.7058 0.2139 0.7675 0.5869

good 6,971 0.7007 0.7448 0.9064 0.1403 0.9019 0.2325

very good 3,522 0.9258 0.2993 0.9545 0.1058 0.9757 0.0981

excellent 1,161 1.0000 0.0742 0.9746 0.0808 1.0000 0.0243

Table 1.3. Summary statistics of VAS tari¤ by categories of SAH and upper bounds in IR

The most chosen category of SAH is the one in the middle, "good health"; how-

ever, the continuous variables are very skewed to better health valuations. Standard

deviations of VAS tari¤ and TTO tari¤ also show an increase in low categories of

SAH.

The upper bounds of the thresholds are interpreted as follows: for instance, re-

ferring to the methodology of (IR+N) for the VAS tari¤ in Table 1.3, an individual

who reports the worst category of health (SAH = "poor") will be assumed to have

a VAS tari¤ that belongs to the interval [0, 0.4131]. Similarly, the values for the re-

maining SAH categories are (0.4131, 0.7657] for the �fair�category, (0.7657, 0.9019]

for the �good�category, (0.9019, 0.9757] for the "very good" level and (0.9757, 1]

for the �excellent�category. Similarly the continuous health valuations can be in-

terpreted as a measure of "ill-health": the thresholds corresponding to the (IR+LN)

approach show that an individual who reports the lowest amount of ill-health (SAH

= "excellent") will be assumed to have a continuous ill-health valuation (derived

from VAS tari¤) belonging to the interval [0, 0243], etc. Figure 1.5 illustrates the

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Chapter 1 Scaling Methods for Categorical Self-Assessed Health Measures

procedure for obtaining the thresholds and their interpretation.

0

.1

.2

.3

.4

.5

.6

.7

.8

.9

1

VA

S ta

riff

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1Empirical Cumulative Frequencies

0

.1

.2

.3

.4

.5

.6

.7

.8

.9

1

TTO

 tarif

f

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1Empirical Cumulative Frequencies

0

.1

.2

.3

.4

.5

.6

.7

.8

.9

1

TTO

 tarif

f for

 ill­h

ealth

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1Empirical Cumulative Frequencies

0

.1

.2

.3

.4

.5

.6

.7

.8

.9

1

VAS

 tarif

f for

 ill­h

ealth

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1Empirical Cumulative Frequencies

SAH = 1 SAH = 2 SAH = 3 SAH = 4

SAH = 5

0.5436

0.90060.97130.9929

0.00710.02870.0994

0.4565

SAH = 1

SAH = 2SAH = 3

SAH = 4SAH =5

0.4131

0.7675

0.90190.9757

SAH = 1SAH = 2

SAH = 3 SAH = 4

SAH = 5

0.0243

0.0981

0.2325

0.5869

SAH = 1

SAH = 2SAH = 3

SAH = 4SAH =5

health ill­health

TTO tariff

VA

S tariff

Figure 1.5. Estimated health and ill-health intervals for TTO tari¤ and VAS tari¤

Notice that the values corresponding to the TTO tari¤ are higher than those

corresponding to the VAS tari¤.

I display a comparison of the descriptive performance of scaling methods with

the regressions based on actual health valuations (TTO tari¤ and VAS tari¤, re-

spectively). The purpose is to examine to what extent the TTO and VAS tari¤s can

be approximated by the predicted values of the scaling approaches. The following

measures are contrasted:(i) Actual VAS tari¤ / TTO tari¤(ii) OLS regression of actual VAS/TTO tari¤ on xi(iii) (IR+N)(iv) (OP+N) re-scaled by (1.2)(v) (IR+LN)(vi) (OP+LN) re-scaled by (1.4)

Although the tari¤s in (i) are considered as continuous variables, the actual

quantities that appear in the survey constitute a reduced selection of values. For

instance, the formation of these tari¤s allow for 243 di¤erent values, but only 149

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Chapter 1 Scaling Methods for Categorical Self-Assessed Health Measures

of them are assigned to individuals in the survey. These tari¤s also present the

negative aspect of the existence of a �ceiling e¤ect� (a value of health equal to

one is assigned to the majority of the individuals, 57.1% in CHS for both tari¤s),

what makes di¢ cult the comparison of health outcomes along di¤erent sub-groups

of population. An OLS regression on actual tari¤ values can be seen as a proper

interpretation of continuity for those tari¤s. Yet the extended use of the assumption

of normality in the distribution of health (e.g. Van Doorslaer and Jones, 2003;

Lauridsen et al., 2004) may be a too restrictive assumption. The predictions in (v)

and (vi) allow for considering a skewed distribution of health valuations.

Variable mean sd min p(25) p(50) p(75) max

Actual TTO 0.9135 0.1591 0.0000 0.8951 1.0000 1.0000 1.0000OLS actual TTO 0.9135 0.1003 0.4957 0.8914 0.9554 0.9741 1.0321OLS re-scaled 0.7789 0.1870 0.0000 0.7378 0.8570 0.8919 1.0000

(OP+N) 0.5759 0.1863 0.0000 0.4629 0.5978 0.7086 1.0000(IR+N) 0.8979 0.0786 0.5836 0.8704 0.9255 0.9507 1.0119

(OP+LN) 0.8908 0.1213 0.0000 0.8721 0.9342 0.9643 1.0000(IR+LN) 0.8905 0.1007 0.1862 0.8723 0.9260 0.9532 0.9865

Table 1.4. Descriptive statistics of actual and predicted TTO tari¤s.

Tables 1.4 and 1.5 show some descriptive statistics of actual and predicted

TTO tari¤s and VAS tari¤s, respectively. The regression results are presented in

Tables 1.6 and 1.7 (TTO index) and Tables 1.8 and 1.9 (VAS index), in the

Appendix.

Concerning to the TTO tari¤, Table 1.4 shows that the non-rescaled OLS pre-

dictions on actual values approximates properly the observed scores. The negative

aspect of this methodology is the fact of reporting predictions higher than 1. It

also fails to represent the lower tari¤s, since this model assigns a minimum value

of 0.4957. However, re-scaling these predictions does not yield to a better estima-

tion of the actual tari¤. Upon the assumption of normality, predictions from the

interval regression (IR+N) outperform those obtained by the ordered probit model

(OP+N). This result accords well with the statements reported by other authors

(Van Doorslaer and Jones, 2003; Lauridsen et al., 2004). But the methods that

are based on the log-normality of the latent health variable approximate better the

actual TTO tari¤, specially for lowest values. The slight di¤erences for higher val-

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Chapter 1 Scaling Methods for Categorical Self-Assessed Health Measures

ues of the variables is understandable, because of the cumulative estimation errors.

These methods also retain the standard deviation of the actual values.

Variable mean sd min p(25) p(50) p(75) max

Actual VAS 0.8557 0.2067 0.0000 0.7574 1.0000 1.0000 1.0000OLS actual VAS 0.8557 0.1368 0.3365 0.8103 0.9034 0.9387 1.0468OLS re-scaled 0.7309 0.1925 0.0000 0.6670 0.7981 0.8478 1.0000

(OP+N) 0.5759 0.1863 0.0000 0.4629 0.5978 0.7086 1.0000(IR+N) 0.8060 0.1009 0.4356 0.7597 0.8335 0.8762 0.9802

(OP+LN) 0.8908 0.1213 0.0000 0.8721 0.9342 0.9643 1.0000(IR+LN) 0.7926 0.1266 0.1085 0.7468 0.8288 0.8808 0.9567

Table 1.5. Descriptive statistics of actual and predicted VAS tari¤s.

Similar conclusions can be obtained in relation to the VAS tari¤ predictions

(Table 1.5). In this case, predictions from (OP+LN) seems to approach the actual

values even better than the OLS.

Figure 1.6 illustrates the approximation of the scaling methods to the actual

values, by representing jointly the empirical cumulative frequency of each method,

for both tari¤s. The predicted values for OLS (say, byi) have been taken as a baseline.In order to assess the goodness of the approximations, I de�ne the area [byi�2b�; byi+2b�], where b� stands for the standard error of the predictions in (ii). Each scalingmethod is represented together with the distribution of the predicted values for

OLS as well as that con�dence interval of 95%. The illustration clearly supports the

results commented above.

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Health

TTO

0

.2

.4

.6

.8

1

Empi

rical

 Cum

ulat

ive 

Freq

uenc

ies

.4 .6 .8 1TTO tariff

[CI(95%) OLSCI(95%)] IR+N

(TTO) IR+N

0

.2

.4

.6

.8

1

Em

piric

al C

umul

ativ

e Fr

eque

ncie

s

0 .2 .4 .6 .8 1TTO tariff

[CI(95%) OLSCI(95%)] OP+N

(TTO) OP+N

VAS

0

.2

.4

.6

.8

1

Em

piric

al C

umul

ativ

e Fr

eque

ncie

s

.2 .4 .6 .8 1VAS tariff

[CI(95%) OLSCI(95%)] IR+N

(VAS) IR+N

0

.2

.4

.6

.8

1

Em

piric

al C

umul

ativ

e Fr

eque

ncie

s

0 .2 .4 .6 .8 1VAS tariff

[CI(95%) OLSCI(95%)] OP+N

(VAS) OP+N

Ill-health

TTO

0

.2

.4

.6

.8

1

Empi

rical

 Cum

ulat

ive 

Freq

uenc

ies

.2 .4 .6 .8 1TTO tariff

[CI(95%) OLSCI(95%)] IR+LN

(TTO) IR+LN

0

.2

.4

.6

.8

1

Em

piric

al C

umul

ativ

e Fr

eque

ncie

s

0 .2 .4 .6 .8 1TTO tariff

[CI(95%) OLSCI(95%)] OP+LN

(TTO) OP+LN

VAS

0

.2

.4

.6

.8

1

Empi

rical

 Cum

ulat

ive 

Freq

uenc

ies

0 .2 .4 .6 .8 1VAS tariff

[CI(95%) OLSCI(95%)] IR+LN

(VAS) IR+LN

0

.2

.4

.6

.8

1

Empi

rical

 Cum

ulat

ive 

Freq

uenc

ies

0 .2 .4 .6 .8 1TTO tariff

[CI(95%) OLSCI(95%)] OP+LN

(VAS) OP+LN

Figure 1.6. Empirical cumulative frequency for di¤erent scaling methods

Finally, it is worth to say that using the IR model applies only if there is no

continuous health weight in a database, and thus the thresholds are obtained from

external information. If so, it is neccessary to assume that the population from both

samples are highly comparable. On the contrary, we could be bringing some bias

on the health measures. If it is not possible to �nd that external information from

a proper survey, the OP model shall be the optimal scalig method.

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Chapter 1 Scaling Methods for Categorical Self-Assessed Health Measures

1.5 Conclusions

The lack of continuous health measures is a major drawback in health analyses over

broad populations. On the contrary, general surveys usually include self-assessed

health questions, where the respondents must choose among di¤erent health levels

or categories. The use of these categorical responses to approximate a continuous

health variable is an usual procedure in health studies. The most common ap-

proaches (ordered probit/logit model and interval regression model) assume that

health is an unobservable latent variable that is normally distributed. However, this

is a rough assumption, since many studies have reported skewness in the distribution

of self-assessed health (that is, a great majority of the population reporting good

health), what is consistent with the idea of a skewed distribution of latent health.

In the present study I suggest a new procedure: to assume that health values are

lognormally distributed.

The scaling methodologies discussed above have been compared. Data has been

obtained from the Catalan Health Survey, taking advantage of having actual contin-

uous health values as well as SAH questions. The validity of the scaling approaches

is assessed by measuring to what extent the health values derived from SAH suit

the actual health values. In order to ensure robustness to the selection of a metric,

I use two di¤erent health tari¤s for each procedure (VAS tari¤ and TTO tari¤ ).

In general, models under lognormality outperform the other approaches. In

particular, the (OP+N) model is clearly surpassed by the others. The Interval

Regression model under normality (suggested by Van Doorslaer and Jones, 2003,

and probably the most used in recent years), approximates the actual health tari¤s

in a similar way to the same model under lognormality; however, the latter seems to

match better the lower values. Surprisingly, the (OP+LN) procedure is the one that

better models the distribution of health, specially if the VAS tari¤ is used. It is also

the closest to the OLS predictions. As a drawback, it is important to notice that

(IR+N) and (IR+LN) are developed under the most ideal scenario: the thresholds

between categories have been directly derived from actual data, whereas they are

assumed to be obtained from external information. Therefore, using (OP+LN), we

are omitting the possible bias associated to combining di¤erent sources of data, if

the two sources do not arise from the same population.

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Chapter 1 Scaling Methods for Categorical Self-Assessed Health Measures

Introducing cardinality in health valuations is nowadays a challenging task. Car-

dinalization is an intrinsic problem even at the de�nition of HRQoL valuations. As

Busschbach et al. (1999) stated, whatever method is used for the evaluation of

health states (VAS, TTO, Standard Gamble), the responses must be assumed to

have interval properties rather than ratio properties; otherwise, the empirical order

cannot be extended to additional health states. For instance, VAS is introduced

to the respondent as a thermometer, what somehow entails the idea of continuity;

however, many surveys report that a high percentage of respondents choose scores

ending in 0 (about 81% or respondents in CHS). This suggests that individuals tend

to use the thermometer as a combination of a numerical and a rating scale.

If de�ning HRQoL measures with cardinal properties from (presumably) con-

tinuous variables is a challenging task, then, obtaining them from ordinal variables

is even more complicated. Assigning a numerical valuation for a category only

masks the ordinal relationship between categories (an exhaustive discussion about

this topic can be found in Kind, 2003). If the main goal of an analysis is obtaining

quality weights from health states, regression methods are, therefore, a powerful

tool as scaling procedures. The results obtained in this paper can provide a new

benchmark for the proper cardinalization of health measures.

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1.6 Appendix

OLS OP+N IR+N OP+LN IR+LNmale 15-25 -0.010 0.079 -0.003 -0.079 -0.089

(2.87)** (1.60) (1.24) (1.60) (1.76)male 35-45 -0.002 -0.179 -0.005 0.179 0.182

(0.71) (4.57)** (1.98)* (4.57)** (4.61)**male 45-55 -0.004 -0.374 -0.012 0.374 0.368

(1.28) (8.78)** (4.17)** (8.78)** (8.73)**male 55-65 -0.006 -0.450 -0.017 0.450 0.439

(1.23) (9.14)** (4.15)** (9.14)** (9.10)**male 65-75 -0.004 -0.428 -0.014 0.428 0.410

(0.45) (6.43)** (1.81) (6.43)** (6.41)**male 75+ -0.029 -0.510 -0.024 0.510 0.496

(2.83)** (7.30)** (2.68)** (7.30)** (7.40)**female 15-25 -0.010 -0.000 -0.004 0.000 -0.007

(2.64)** (0.01) (1.26) (0.01) (0.14)female 25-35 -0.012 -0.126 -0.007 0.126 0.124

(4.18)** (3.17)** (2.98)** (3.17)** (3.10)**female 35-45 -0.022 -0.292 -0.015 0.292 0.285

(6.21)** (6.74)** (5.09)** (6.74)** (6.60)**female 45-55 -0.033 -0.518 -0.028 0.518 0.502

(7.72)** (11.27)** (7.73)** (11.27)** (11.12)**female 55-65 -0.052 -0.679 -0.047 0.679 0.653

(9.02)** (12.82)** (8.81)** (12.82)** (12.70)**female 65-75 -0.057 -0.718 -0.052 0.718 0.691

(7.05)** (11.21)** (6.68)** (11.21)** (11.22)**female 75+ -0.106 -0.701 -0.056 0.701 0.676

(10.89)** (10.33)** (6.34)** (10.33)** (10.44)**high social class 0.009 0.119 0.008 -0.119 -0.117

(3.42)** (4.69)** (4.24)** (4.69)** (4.67)**medium social class -0.000 0.022 0.000 -0.022 -0.023

(0.12) (1.02) (0.23) (1.02) (1.12)household size -0.001 -0.000 0.001 0.000 0.000

(0.67) (0.01) (1.20) (0.01) (0.04)alcohol -0.003 0.030 0.004 -0.030 -0.032

(0.84) (0.70) (1.42) (0.70) (0.75)heavy smoker -0.002 -0.111 -0.006 0.111 0.110

(1.21) (5.25)** (3.88)** (5.25)** (5.31)**sleeps +8h 0.003 0.071 0.006 -0.071 -0.069

(2.50)* (9.12)** (6.69)** (9.12)** (9.16)**sports -0.000 0.135 0.006 -0.135 -0.142

(0.08) (5.14)** (3.64)** (5.14)** (5.37)**Observations 15648 15648 15648 15648 15648R-squared 0.40

Robust t statistics in parentheses. * signi�cant at 5%; ** signi�cant at 1Table 1.6. Regression coe¢ cients in procedures for converting SAH to TTO tari¤

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Chapter 1 Scaling Methods for Categorical Self-Assessed Health Measures

OLS OP+N IR+N OP+LN IR+LNBMI < 18 -0.026 -0.205 -0.027 0.205 0.189

(2.83)** (2.70)** (3.72)** (2.70)** (2.55)*BMI > 25 -0.006 -0.123 -0.007 0.123 0.120

(2.71)** (6.42)** (4.06)** (6.42)** (6.46)**chronic illness -0.035 -0.590 -0.028 0.590 0.588

(26.16)** (26.58)** (24.06)** (26.58)** (26.01)**de�ciences 0.146 0.752 0.106 -0.752 -0.720

(32.56)** (25.09)** (23.44)** (25.09)** (25.49)**unemployed -0.021 -0.194 -0.021 0.194 0.179

(4.33)** (4.05)** (4.74)** (4.05)** (3.82)**unable -0.154 -0.941 -0.150 0.941 0.867

(14.30)** (15.05)** (13.57)** (15.05)** (15.72)**retired -0.003 -0.207 -0.016 0.207 0.197

(0.49) (4.59)** (2.63)** (4.59)** (4.62)**student 0.003 0.117 0.000 -0.117 -0.123

(1.11) (2.49)* (0.07) (2.49)* (2.52)*houseworker -0.002 -0.115 -0.011 0.115 0.110

(0.55) (3.04)** (2.68)** (3.04)** (3.04)**other -0.083 -0.454 -0.037 0.454 0.426

(1.71) (2.49)* (1.19) (2.49)* (2.60)**no studies -0.030 -0.111 -0.020 0.111 0.105

(5.98)** (3.44)** (4.47)** (3.44)** (3.42)**secondary studies 0.006 0.129 0.010 -0.129 -0.122

(2.25)* (5.17)** (4.38)** (5.17)** (4.99)**superior studies 0.009 0.269 0.016 -0.269 -0.259

(3.16)** (8.99)** (6.44)** (8.99)** (8.80)**married -0.003 0.016 -0.004 -0.016 -0.021

(1.00) (0.61) (2.12)* (0.61) (0.81)widow -0.027 0.061 0.001 -0.061 -0.064

(3.77)** (1.33) (0.17) (1.33) (1.45)separed or divorced -0.018 0.029 -0.008 -0.029 -0.033

(3.21)** (0.58) (1.71) (0.58) (0.67)foreigner -0.003 -0.016 -0.002 0.016 0.013

(0.93) (0.45) (0.85) (0.45) (0.37)Constant 0.695 0.709 -2.004

(53.16)** (58.72)** (22.07)**Observations 15648 15648 15648 15648 15648R-squared 0.40

Robust t statistics in parentheses * signi�cant at 5%; ** signi�cant at 1Table 1.7. Regression coe¢ cients in procedures for converting SAH to TTO tari¤

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OLS OP+N IR+N OP+LN IR+LNmale 15-25 -0.008 0.079 0.000 -0.079 -0.088

(1.68) (1.60) (0.10) (1.60) (2.04)*male 35-45 -0.005 -0.179 -0.012 0.179 0.155

(1.07) (4.57)** (3.44)** (4.57)** (4.76)**male 45-55 -0.008 -0.374 -0.028 0.374 0.302

(1.69) (8.78)** (6.59)** (8.78)** (8.87)**male 55-65 -0.017 -0.450 -0.035 0.450 0.352

(2.69)** (9.14)** (6.55)** (9.14)** (9.29)**male 65-75 -0.011 -0.428 -0.032 0.428 0.329

(1.05) (6.43)** (3.51)** (6.43)** (6.92)**male 75+ -0.052 -0.510 -0.045 0.510 0.388

(4.40)** (7.30)** (4.32)** (7.30)** (7.93)**female 15-25 -0.015 -0.000 -0.003 0.000 -0.014

(2.68)** (0.01) (0.71) (0.01) (0.32)female 25-35 -0.019 -0.126 -0.012 0.126 0.099

(4.26)** (3.17)** (3.29)** (3.17)** (2.97)**female 35-45 -0.037 -0.292 -0.026 0.292 0.227

(7.34)** (6.74)** (6.21)** (6.74)** (6.46)**female 45-55 -0.054 -0.518 -0.048 0.518 0.392

(9.28)** (11.27)** (9.77)** (11.27)** (10.97)**female 55-65 -0.085 -0.679 -0.071 0.679 0.496

(11.41)** (12.82)** (10.81)** (12.82)** (12.60)**female 65-75 -0.088 -0.718 -0.077 0.718 0.521

(9.01)** (11.21)** (8.58)** (11.21)** (11.42)**female 75+ -0.140 -0.701 -0.079 0.701 0.508

(12.77)** (10.33)** (7.92)** (10.33)** (10.71)**high social class 0.014 0.119 0.012 -0.119 -0.090

(4.23)** (4.69)** (4.64)** (4.69)** (4.50)**medium social class -0.000 0.022 0.001 -0.022 -0.020

(0.12) (1.02) (0.49) (1.02) (1.23)household size -0.000 -0.000 0.001 0.000 0.002

(0.19) (0.01) (0.73) (0.01) (0.27)alcohol -0.008 0.030 0.004 -0.030 -0.023

(1.59) (0.70) (1.06) (0.70) (0.67)heavy smoker -0.007 -0.111 -0.010 0.111 0.088

(2.51)* (5.25)** (4.63)** (5.25)** (5.37)**sleeps +8h 0.006 0.071 0.009 -0.071 -0.051

(4.93)** (9.12)** (7.77)** (9.12)** (9.31)**sports -0.000 0.135 0.011 -0.135 -0.120

(0.13) (5.14)** (4.46)** (5.14)** (5.55)**Observations 15648 15648 15648 15648 15648R-squared 0.44

Robust t statistics in parentheses * signi�cant at 5%; ** signi�cant at 1%Table 1.8. Regression coe¢ cients in procedures for converting SAH to VAS tari¤

48

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Chapter 1 Scaling Methods for Categorical Self-Assessed Health Measures

OLS OP+N IR+N OP+LN IR+LNBMI < 18 -0.027 -0.205 -0.032 0.205 0.127

(2.64)** (2.70)** (3.58)** (2.70)** (2.19)*BMI > 25 -0.010 -0.123 -0.012 0.123 0.093

(3.47)** (6.42)** (5.22)** (6.42)** (6.53)**chronic illness -0.070 -0.590 -0.049 0.590 0.479

(33.68)** (26.58)** (27.08)** (26.58)** (24.91)**de�ciences 0.188 0.752 0.123 -0.752 -0.482

(35.44)** (25.09)** (24.98)** (25.09)** (24.67)**unemployed -0.034 -0.194 -0.027 0.194 0.120

(5.45)** (4.05)** (4.80)** (4.05)** (3.32)**unable -0.188 -0.941 -0.165 0.941 0.572

(16.90)** (15.05)** (14.30)** (15.05)** (15.45)**retired -0.016 -0.207 -0.024 0.207 0.140

(2.08)* (4.59)** (3.41)** (4.59)** (4.66)**student 0.005 0.117 0.005 -0.117 -0.113

(1.00) (2.49)* (1.46) (2.49)* (2.71)**houseworker -0.011 -0.115 -0.014 0.115 0.075

(1.99)* (3.04)** (2.91)** (3.04)** (2.82)**other -0.081 -0.454 -0.052 0.454 0.316

(1.55) (2.49)* (1.54) (2.49)* (2.95)**no studies -0.033 -0.111 -0.022 0.111 0.067

(5.79)** (3.44)** (4.26)** (3.44)** (3.10)**secondary studies 0.009 0.129 0.015 -0.129 -0.087

(2.67)** (5.17)** (4.98)** (5.17)** (4.69)**superior studies 0.015 0.269 0.026 -0.269 -0.197

(3.81)** (8.99)** (7.98)** (8.99)** (8.56)**married -0.002 0.016 -0.003 -0.016 -0.025

(0.45) (0.61) (1.05) (0.61) (1.22)widow -0.029 0.061 0.004 -0.061 -0.056

(3.59)** (1.33) (0.50) (1.33) (1.77)separed or divorced -0.021 0.029 -0.005 -0.029 -0.040

(2.88)** (0.58) (0.81) (0.58) (1.06)foreigner -0.008 -0.016 -0.002 0.016 0.007

(1.97)* (0.45) (0.62) (0.45) (0.25)Constant 0.581 0.597 -1.408

(37.88)** (43.49)** (21.06)**Observations 15648 15648 15648 15648 15648R-squared 0.44

Robust t statistics in parentheses * signi�cant at 5%; ** signi�cant at 1%Table 1.9. Regression coe¢ cients in procedures for converting SAH to VAS tari¤

49

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Bibliography

[1] Badia, X., Roset, M. and Herdman, M. 1997. The Spanish VAS tari¤ based

on valuation of EQ-5D health states from the general population. In: EuroQol

Plenary meeting, ed: Rabin, R. E., Busschbach, J.J.V., Charro, F.Th. de, Essink-

Bot, M.L. and Bonsel, G.J. 2-3 October. Discussion papers. Centre for Health

Policy & Law, Erasmus University, Rotterdam.

[2] Badia, X., Roset, M., Herdman, M. and Kind, P. 2001. A comparison of United

Kingdom and Spanish general population time trade-o¤ values for EQ-5D health

states. Medical Decision Making 21: 7-16.

[3] Burström, K., Johannesson, M. and Diderichsen, F. 2003. The value of the

change in health in Sweden 1980/81 to 1996/97. Health Economics 12: 637-654.

[4] Busschbach, J.J.V., McDonnell, J., Essink-Bot, M.L. and van Hout, B.A. 1999.

Estimating parametric relationships between health description and health valu-

ation with an application to the EuroQoL EQ-5D. Journal of Health Economics

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[5] Cutler, D. M. and Richardson, E. 1997. Measuring the health of the US popula-

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[6] Groot, W. 2000. Adaptation and scale of reference bias in self-assessments of

quality of life. Journal of Health Economics 19: 403�420.

[7] Idler, E. and Benyamini, Y. 1997. Self-rated health and mortality: a review of

twenty-seven community studies. Journal of Health and Social Behavior 38(1):

21-37.

[8] Kind, P. 2003. Using standardised measures of health-related quality of life:

Critical issues for users and developers. Quality of Life Research 12: 519�521.

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Chapter 1 BIBLIOGRAPHY

[9] Lauridsen, J., Christiansen, T. and Häkkinen, U. 2004. Measuring inequality in

self-reported health - discussion of a recently suggested approach using Finnish

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[10] Lecluyse, A. and Cleemput, I. 2006. Making health continuous: implications of

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[11] The EuroQol Group. 1990. EuroQol - a new facility for the measurement of

health-related quality of life. Health Policy 16(3): 199-208.

[12] Torrance, G. W. 1986. Measurement of health state utilities for economic ap-

praisal. Journal of Health Economics 5, 1-30.

[13] Van Doorslaer, E., Wagsta¤, A., Bleichrodt, H., Calonge, S., Gerdtham, U.G.,

Ger�n, M., Geurts, J., Gross, L., Häkkinen, U., Leu, R.E., O�Donnell, O., Prop-

per, C., Pu¤er, F., Rodríguez, M., Sundberg, G. and Winkelhake, O. 1997.

Income-related inequalities in health: some international comparisons. Journal

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[14] Van Doorslaer, E. and Jones, A. 2003. Inequalities in self-reported health: val-

idation of a new approach to measurement. Journal of Health Economics 22:

61-87.

[15] Wagsta¤, A. and van Doorslaer, E. 1994. Measuring inequalities in health in the

presence of multiple-category morbidity indicators. Health Economics 3: 281-291.

[16] Zozaya, N., Oliva, J. and Osuna, R. 2005. Measuring Changes in Health Capital.

FEDEA �DT 2005-15

51

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Part II

Health E¤ects

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Chapter 2

Quality of Life Lost Due to RoadCrashes

2.1 Introduction

The objective of this work is to estimate the chronic loss of health following a

road crash. The methodology is based on the de�nition of comparison groups, by

using the existing literature regarding treatment e¤ects. The main contribution of

this paper is the evaluation of health losses due to diseases in terms of quality of

life. Moreover, this paper develops a di¤erent method for scaling categorical health

measures, a powerful tool in health-related analysis.

The selection of the topic "road crashes" is not pointless. In 2001, injuries

represented 12% of the global burden of disease (WHO, 2001). The category of

injuries worldwide is dominated by those incurred in road crashes. In 2004, over 50%

of deaths caused by road crashes were associated to young adults in the age range

of 15�44 years, and tra¢ c injuries were the second-leading cause of death worldwide

among both children aged 5�14 years, and young people aged 15�29 years (WHO,

2004). In addition, road crashes are expected to be the main origin of the projected

40% increase in global deaths resulting from injury between 2002 and 2030 (WHO,

2007).

In Spain tra¢ c accidents are also a major health problem. The tendency is de-

creasing, but still in 2006 the number of deaths by road tra¢ c injuries (RTIs, here-

after) reached 4,144 individuals (data from INE, 2006). Similarly to other countries,

RTIs a¤ect young people more than any other group, causing more than half of the

deaths for those aged 15-24 (see Figure 2.1).

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Chapter 2 Quality of Life Lost Due to Road Crashes

Deaths following RTIs (% over total of deaths)

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

0­4 5­14 15­19 20­24 25­29 30­34 35­39 40­44 45­49 50+

age intervals

Men

Women

Figure 2.1. Deaths by RTIs over all causes of death. Spain, 2006 (Source: INE)

Counts of the (absolute or relative) number of deaths have been one of the

primary instruments for quantifying the burden of illnesses. However, as the World

Health Organization (WHO) de�ned in 1946, the idea of health "is not only the

absence of in�rmity and disease, but also a state of physical, mental and social well-

being�(WHO, 1946). This broad de�nition captures essential elements of quality

of life, which underlies most human health metrics. Based on this de�nition, it

is also clear that life expectancy or mortality-based measures are no longer being

considered adequate as measures of a population�s health.

Currently, measures of disability and health-related quality of life are becoming

important, even essential parameters in the evaluation of treatment and prevention

strategies for reducing the burden of injury (see, e.g., Seguí-Gomez and MacKen-

zie, 2003). Studies in such a context are performed throughout the evaluation of

cost-e¤ectiveness ratios, that are obtained by taking the cost of the treatment and

dividing it by the health gains (Gold et al., 1996). The cost of the treatment is

calculated in monetary terms, and there exists a general agreement about the com-

putational methodology. However, evaluating changes in health (gains or losses)

requires a thoughtful analysis of several key features.

Let me illustrate this idea in the context of evaluating health losses due to a

road crash.1 Following the path that Gold et al. (1996) establish, I consider the life

1The design of the methodology we develope later implies that it only makes sense in an ag-gregate context, that is, it must be framed in a context of evaluating average losses for targetedgroups of population. However, for simplicity purposes, the following explanation will refer to theevaluation of health losses for a single individual.

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Chapter 2 Quality of Life Lost Due to Road Crashes

path of an individual as a continuous function that represents frequent changes in

health-related quality of life (QoL). Let me call Y (i; t) the function that describes

the health status of individual i at time t: Now imagine that the individual i su¤ers

from a tra¢ c accident at time T . Let me represent the health state of this individual

under two possible scenarios: in case the accident did not happen, and in case it

did (let me call these health paths as Y0(i; t) and Y1(i; t), respectively). The loss

of health (evaluated at time T + 1) would coincide to the di¤erence Y0(i; T + 1)�Y1(i; T +1). However, the potential health status Y0(i; T +1) is always unidenti�ed,

since it is impossible to know what the state of health of the individual would

have been had the accident not occurred. The problem is how to approximate this

unknown potential health status.

Many authors consider the health state prior to the accident (pre-injury status),

evaluated at time T�1; as a proxy of the potential health state, that is, Y0(i; T+1) 'Y0(i; T�1). And yet, a problem related to the lack of data could arise at this point. Ifthe studies deal with institutionalized individuals, that is, if the treatment is de�ned

over targeted subpopulation with well-known health state (e.g. cancer treatments,

e¤ectiveness of dialysis programs, etc.), it is plausible to obtain proper information

about the pre-injury status of the patients. Specially di¢ cult is the analysis of

injuries in prevention control (burning, road crashes, falls, poisoning, etc.), since

the pre-injury status of the individual is completely unknown. Given the lack of

pre-injury measures, most studies in this area consider the pre-injury health state

as "perfect health".

A di¤erent strategy for approximating the potential health state of the injured

people remains on obtaining information from other people, rather than the injured

individual per se. Following My illustration, let�s imagine that we can �nd informa-

tion about the health state of an individual (j) who has not su¤ered a road tra¢ c

crash, and that j is highly comparable to the injured individual, since they share

several characteristics (such as age, gender, education, etc.). Call the health state

of individual j Y (j; t): I can approximate Y0(i; T + 1) by means of Y (j; T + 1); but

the results could still present some bias (see Figure 2.2).

The approaches suggested previously (pre-injury status and comparison groups)

are highly connected, and can be easily combined. In fact, the use of comparison

groups to approximate the pre-injury status is the most common choice nowadays.

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Chapter 2 Quality of Life Lost Due to Road Crashes

Indeed, the use of population norms that provide some benchmark against which

to compare pre-injury status is often particularly important to the study of trauma

outcomes (MacKenzie, 2001)

In this work I estimate the chronic loss of health (in quality of life terms) that is

due to a road crash, for those who su¤er the road crash. The methodology is based

on the de�nition of comparison groups, by using the existing literature concerning

treatment e¤ects. In Section 2 the methodology is described, starting with the

cardinalization of categorical variables, and following with the estimation of the

direct loss of health. Section 3 describes the data used for the analysis. In Section

4 I present the main results, and several robustness checks. Section 5 concludes.

T

Y0(i,t)

Y1(i,t)

Y(j,t)

tT­1 T+1

QoL

Health loss(evaluated at time T+1)

If Y0(i,T+1)= Y0(i,T­1)

If Y0(i,T+1)= Y(j,T+1)

Figure 2.2. Possible bias at estimating potential health e¤ect in terms of pre-injury statusand comparison group

2.2 Methodology

2.2.1 Measurement of health

A wide variety of metrics are used to quantify the burden of illnesses and injuries

to population (an exhaustive description of these measures can be found in Seguí-

Gomez and MacKenzie (2003), MacKenzie (2001) or Sturgis et al. (2001), among

others). In general terms, I can talk about two di¤erent sort of measures, depending

on how I approach the health status.

Measures in the �rst group focus on the impact of the injury over the general

health state of the individual, developing a variety of indices or metrics that de�ne

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Chapter 2 Quality of Life Lost Due to Road Crashes

"health". Measures as Visual Analogue Scale (V AS), Self-Assessed Health (SAH),

Euroquol �ve-dimensional index (EQindex or V AS preferences tari¤ ), Time-Tradeo¤

tari¤ (TTO preferences tari¤ ), Short-Form Healh Survey (SF-36 ) or Health Utility

Index (HUI ) can be placed within such an approach. These metrics are commonly

used in cost-e¤ectiveness analysis of medical treatments, since they re�ect the qual-

ity of health states both from a physical and psychological aspect. Those measures

are, generally, being preference-based. Consequently, they can combine the e¤ect

of both death and nonfatal consequences into a summary measure which typically

ranges from 0 (representing death) to 1 (representing optimal health) and where

any number re�ects the relative preference for a particular health state. However,

it must be taken into account that most of them re�ect self-reported health states.

Previous characteristic can complicate, on the one hand, interpersonal comparisons

among subjects (and therefore the consistency of aggregation procedures), and, on

the other hand, secure data from some targeted groups of population, such as chil-

dren, the elderly or the unconscious.

Metrics in the second group aim to estimate the seriousness of the injuries, either

re�ecting the degree of functional limitation of the injured individuals (Functional

Capacity Index (FCI ), Disability weights, etc.), or attending to the mortality risk

or life threat (Abbreviated Injury Scale (AIS), Injury Severity Score (ISS), ICD-9

Injury Severity Score (ICISS), Anatomic Pro�le Score (APS), etc.). These sorts

of metrics are considered as objective, since they can be observed from the medical

point of view; are easy to obtain, and examine in detail the characteristics of the

concrete injury. Nonetheless, not all metrics in this group have been clearly validated

(Schluter et al., 2005). Moreover they present some other disadvantages: they do not

allow for heterogeneity, problems with comorbidities, and not taking into account

the psychological dimension.

Of the scales that have been reviewed, those that belong to the second group

are the ones most commonly used to asses health losses due to injuries. However,

several studies suggest that an individual�s injury and acute psychological responses

are strongly linked. Hence, both play important roles in determining quality of

life and disability outcomes (e.g. O�Donnell et al., 2005). Although measures of

severity in the second group provide some understanding of the relative seriousness

of injuries in terms of threat to life and resource utilization, they still fall short

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Chapter 2 Quality of Life Lost Due to Road Crashes

in measuring the long-term impact of nonfatal injuries on the person, his or her

family, and the society at large. These considerations have challenged the �eld to

move beyond counting injuries by severity alone to measuring their direct impact

on health-related quality of life.

In the present work I approach the problem from a quality-of-life perspective,

that is: I analyze the impact of fatal and nonfatal injuries on the quality of life of

the injured individuals, not only attending to the physical damage that the injury

caused, but also contemplating the possible psychological consequences, as well as

the potential impact on the well-being of those a¤ected. In order to check the

robustness of the results, the analysis is performed by using di¤erent quality-related

health state scores (V AS tari¤ and TTO tari¤ ), that are obtained by applying

the Spanish EQ-5D index tari¤s (see Badia et al., 1995 and Badia et al., 2001).

Both scores allow negative values, that is, health states worse than death, what

may create some confusion in the measurement of health e¤ects. One criterion to

overcome these controversies is to change the negative values to zero (e.g. Burström

et al. 2003; Zozaya et al. 2005). A di¤erent method (see Busschbach et al., 1999)

re-scales the scores to the interval (0,1), based on the minimal and maximal values

obtained in the tari¤ (related to health states 33333 and 11111, respectively). None

of both is, in principle, preferable to the other, but they can lead to di¤erent results.

My analysis is performed by using both criteria for each measure. I denote the

outcomes as V ASz; V ASr; TTOz and TTOr , depending on the tari¤ (V AS tari¤

or TTO tari¤ ) and the adopted criterion (to change negative values to zero or to

re-scale them).

2.2.2 Cardinalization of SAH

The loss of health is derived from the respondent�s assessment of her own health

status. That piece of information about self-assessed health will be obtained from

the categorical variable SAH : "In your opinion, how is your health in general?",

where respondents must choose one of the following categories: "very good", "good",

"fair", "bad" or "very bad". Since categorical measures of health are one of the most

commonly used indicators in socioeconomic surveys, a wide variety of methods were

developed with the aim of dealing with the cardinalization of ordinal health measures

(e.g. Van Doorslaer and Wagsta¤, 1994; Cutler and Richardson, 1997; Groot, 2000).

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Chapter 2 Quality of Life Lost Due to Road Crashes

In this study I adopt the interval regression model, stated by Van Doorslaer and

Jones (2003). This model is shown to outperform other econometric approaches, in

terms of validity and ability to mimic the distribution of scaling health measures.

This methodology combines the distribution of observed SAH with external

information on the distribution of a generic measure of health y, in order to con-

struct a continuous standardized latent health variable. The crucial idea that lies

beneath the selected methodology (interval regression) remains on considering the

true health state of an individual i as a latent, continuous but unobservable variable

(y�i ), that can take on any real value. The relationship between the true health state

of individual i (y�i ) and the self-reported health variables (SAHi and yi) is assumed

to be as follows: the higher the value of y�i , the more likely the individual is to

report a higher category in SAHi; and a higher value in yi. For such a connection

to be correct, it is necessary to assume that there is a stable mapping from y�i to

yi that determines SAHi, and that this applies for all individuals in both samples.

This statement implies that the reported variables have rank properties; that is,

the qth-quantile of the distribution of y will correspond to the qth-quantile of the

distribution of SAH.

I divide the range of y and y� into �ve intervals, each one corresponding to a

di¤erent value of SAH :

SAH i= j if �j�1< y�i< �j; j = 1; 2; 3; 4; 5 (2.1)

SAH i= j if �j�1< yi< �j; j = 1; 2; 3; 4; 5 (2.2)

where it is set that �0 = �1; �5 = +1; �0 = 0; �5 = 1; �j � �j+1; �j � �j+1 and y�iis assumed to be a linear function of a vector of socioeconomic factors Xi

y�i= X i� + ui; with ui � N(0; �2) (2.3)

Expressions (2.1) and (2.3) represent the well-known ordered probit model, and

(2.2) will allow me to use a nonparametric approach to estimate the (re-scaled)

thresholds of the model, by using the cumulative frequency of observations for each

category of SAH to �nd the quantiles of the empirical distribution function for

y: Since I have set the thresholds, this allows me to identify the variance of the

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Chapter 2 Quality of Life Lost Due to Road Crashes

error term b�2 and hence, the scale of y� without having any scaling or identi�cationproblems (Van Doorslaer and Jones, 2003).

A variation of the methodology explained above will be used in this study. It

is well-known that the health of a general population sample has a very skewed

distribution, with the great majority of respondents reporting their health in higher

levels. To ensure that the latent health variable is skewed in the appropriate direc-

tion, I rede�ne the true health of the individual in a range (�1; 0], and assume thath�i = �y�i has a standard lognormal distribution. The new variable h�i is decreasingin health, so that represents the latent "ill-health" of the individual. Since the con-

nection between y and SAH is due to represent the latent variable, an adaptation

is needed.

Let me denote h = 1 � y , and de�ne SAH ih as a new variable where the

ordering of the self-assessed health categories has been reversed, now interpreted in

terms of ill-health. If the values of the generic measure y yields in the range [0; 1],

the connection between the variables holds as Table 2.1 shows:

health ill-healthSAH y y� SAH ih h h�

1 [0; �1] (�1; �1] 5 [1� �1; 1] [��1;+1]2 ]�1; �2] [�1; �2] 4 [1� �2; 1� �1] [��2;��1]3 ]�2; �3] [�2; �3] 3 [1� �3; 1� �2] [��3;��2]4 ]�3; �4] [�3; �4] 2 [1� �4; 1� �3] [��4;��3]5 ]�4; 1] [�4; 0] 1 [0; 1� �4] [0;��4]

Table 2.1. Relationship among health and ill-health variables

Let �0 = 0; �1 = 1 � �4; �2 = 1 � �3; �3 = 1 � �2; �4 = 1 � �1 and �5 = 1: Themethodology assumes that the latent true ill-health h�can be represented by h in a

0� 1 scale, and the thresholds of the intervals determining SAH ih (�j; j = 1::4) are

obtained from external information and thus, are observable.

Therefore, the model becomes:

SAH ihi = j i¤ �j�1< hi< �j; j = 1; 2; 3; 4; 5

log (hi) = X i� + ui; with ui� N(0; �2) (2.4)

My aim is to estimate the average health valuation in a continuous 0 -1 scale, for

each individual by conditioning on Xi. Noticing that exp(ui) � lognormal(0; �2); Iobtain the expression:

H(i) = E [hijxi]� exp�Xib�� � exp �b�2=2� ;

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Chapter 2 Quality of Life Lost Due to Road Crashes

where H(i) captures the estimated average value of ill-health, ranging from 0 to

1, associated to the observable characteristics of individual i:

In order to evaluate the robustness of this methodology, the thresholds are de-

termined in terms of di¤erent generic health measures obtained from external data.

I use TTOz; TTOr; V ASz and V ASr as the continuous self-assessed measures.

2.2.3 Evaluation of health losses

The analysis of health losses due to RTIs can be performed using the treatment

e¤ects literature. In this context, the "treatment" is interpreted as the occurrence

of a road crash that causes severe injuries to the a¤ected individuals. Some notation

is useful at this point. LetDi indicate whether individual i had a road crash (Di = 1)

or not (Di = 0). Let H(i) represent the health status2 for individual i. This health

state is measured after the road crash takes place.

Following Rubin (1974) and Heckman (1990), causality is de�ned in terms of

potential outcomes. H0 (i) is the outcome that individual i would attain if he had not

been a¤ected by the treatment. Equivalently, H1 (i) is the outcome that individual

i would realize if he had received the treatment. Individual causal e¤ects cannot

be calculated since only one of these potential outcomes is observed for a given

individual at a given time period. Thus, the evaluation literature analyzes average

measures of the e¤ect of the treatment. In this paper I focus on the average loss

of health as a result of a road crash, for those who had an accident. This quantity

is known as the average treatment e¤ect on the treated (ATET ) and is written as

follows:

ATET = E [H1 (i)�H0 (i) jDi= 1]= E [H1 (i) jDi= 1]�E [H0 (i) jDi= 1]

The ATET cannot be identi�ed using observational data since H0 (i) is only

observed for those targeted by Di = 0. A suitable solution is to approximate the

average health state that injured people would have had in the absence of the road

crash (potential health status) by the average health state observed in a comparable

group of people that have not had an accident. As I mentioned in the Introduction,

2The concept "health status" could be interpreted broadly. In this case, we consider H(i) as acontinuous measure of ill-health, ranging from 0 (absence of ill-health or perfect health) to 1 (fullill-health).

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Chapter 2 Quality of Life Lost Due to Road Crashes

data show that tra¢ c crashes are not random, but they are more likely to happen to

people with particular traits (for instance men aged 15-29). Therefore the average

health of injured (a¤ected group, hereafter) and non-injured (comparison group,

hereafter) individuals cannot be unconditionally compared. Thus, the validity of this

approximation is likely to be higher once di¤erences in the distribution of observed

individual characteristics are controlled for.

Let Z(i) be a vector including information relative to individual i that is a

priori thought to in�uence his probability of su¤ering a road crash. Under this

approximation the ATET can be expressed as follows:3

ATET = E [HjZ;D = 1]�E [HjZ; D = 0] ; (2.5)

where H = D �H1 + (1�D) �H0 is the observed health status of the individuals.The power of this estimator to identify the ATET relies on the so-called �selec-

tion on observables�restriction, that can be formally written as:

ASSUMPTION 1: E [H0jZ;D = 1] = E [H0jZ; D = 0]

This condition states that the average health status that those who su¤ered a

road crash (D = 1) would have attained had the road crash not occurred, conditional

on observable characteristics Z, is equal to the average health status of those who

did not su¤er an accident (D = 0) conditional on observable characteristics Z. In

other words: the e¤ect of events other than the road crash do not contaminate the

causal analysis. Furthermore, Assumption 1 implicates that unobserved individual

characteristics do not a¤ect the causal analysis, or its overall average impact is equal

for both a¤ected and comparison group.

Figure 2.3 illustrates the assumption.3Hereafter the individual argument will dropped out to simplify notation.

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Chapter 2 Quality of Life Lost Due to Road Crashes

t

H0|D=1

H1|D=1

H0|D=0

t­1 t+1

QoL

Health loss(evaluated at time T+1)

If H0|Z,D=1 ~ H0|Z,D=0

Road crash

ATET

Figure 2.3. Estimating health e¤ect under selection on unobservables

Abadie (2005) develops a simple two-step procedure to estimate the ATET us-

ing the di¤erence-in-di¤erences estimator. In Abadie (2005), the unique element

required to estimate the ATET is the conditional probability of receiving the treat-

ment, also called propensity score. This procedure is now adapted to the situation

where I only have cross-section data for the post-treatment period. Since identi-

�cation is attained after conditioning on covariates, it is required that for a given

value of each covariate there is some fraction of the population in the pre-treatment

period to be used as controls.4

ASSUMPTION 2: P (D = 1) > 0 and with probability one P (D = 1jZ) < 1.

In a similar way to Abadie, I establish the following lemma:

Lemma 1. If Assumptions 1 and 2 hold, then E [H1 �H0jZ;D = 1] = E [� �HjZ],where

� =D � P (D = 1jZ)

P (D = 1jZ) � (1� P (D = 1jZ))

Proof. For simplicity, let me call w = P (D = 1jZ). Then:4Assumption 2 is a well-known condition for identi�cation of the average impact on the treated

under selection on covariates (see, e.g. Heckman et al., 1997).

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Chapter 2 Quality of Life Lost Due to Road Crashes

E [� �HjZ] =

= E [� �H j Z;D = 1] �P (D = 1jZ)+E [� �HjZ;D = 0] �P (D = 0jZ)

= E

�D � w

w � (1� w) �HjZ;D = 1��w + E

�D � w

w � (1� w) �HjZ;D = 0��(1� w)

= E [HjZ;D = 1]�E [HjZ;D = 0]

= E [D �H1 + (1�D) �H0jZ;D = 1]�E [D �H1 + (1�D) �H0jZ;D = 0]

= E [H1jZ;D = 1]�E [H0jZ;D = 0]

Under Assumption 1, the previous expression can be written as:

E [H1 �H0 j Z;D = 1] ;

that estimates the ATET for those values of Z such that 0 < P (D = 1jZ) < 1.

The previous Lemma allows me to express the ATET as follows:

E [H1 �H0jD = 1] =ZE [H1 �H0jZ;D = 1] dP (ZjD = 1) (2.6)

=

ZE [� �HjZ] dP (ZjD = 1)

= E

�� �H � P (D = 1jZ)

P (D = 1)

�= E

�H

P (D = 1)� D � P (D = 1jZ)1� P (D = 1jZ)

�Equation (2.6) suggests a simple two-step method to estimate the ATET under

Assumptions 1 and 2. First, the conditional probabilities are estimated using a

probit model and the �tted values of P (D = 1jZ) are calculated for each individualin the sample. Second, the �tted values are plugged into the sample analog of

equation (2.6). Then, a simple weighted average of the outcome variable recovers

the ATET. Finally, the asymptotic variance of the estimator is also calculated,

following the procedure developed in Abadie (2005) for the conventional di¤erence-

in-di¤erences estimator, now adapted for the selection on observables case.

2.3 Data and variable de�nitions

The analysis is performed with data collected from diverse sources of information:

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Chapter 2 Quality of Life Lost Due to Road Crashes

For estimating the impact of RTIs on population health, I use the survey about

diseases, disabilities and health states (Encuesta de Discapacidades, De�ciencias y

Estados de Salud), arranged by the Spanish National Institute of Statistics (INE,

1999). The survey includes 70,402 households (about 217,760 individuals), selected

with a probability proportional to the size of each region. The weighting factor of the

survey is 175, that is, each observation in the sample represents, on average, about

175 individuals of the general population. It implies that subsamples that include

less than 25 observations must be taken with caution, since they can contain sam-

pling errors. The survey is divided into two sections: Diseases and Disabilities Unit

(Módulo de Discapacidades y De�ciencias), and Health Unit (Módulo de Salud, MS

hereafter). The data for my investigation come from the MS. In that unit an indi-

vidual in each household is randomly chosen - in total: 69,555 individuals; however,

840 observations from Ceuta and Melilla were dropped. The interviewed is con-

fronted with a battery of questions related to health habits, as well as demographic

and socioeconomic information.

I consider a wide range of factors that can a¤ect the self-valuation of the health

state of an individual (some observations are dropped because of missing values in

some of the regressors): age, gender, location of residence, existence of a chronic

illness (bronchitis, allergy, diabetes, hypertension, heart injury, arthritis, epilepsy,

cholesterol, ulcer, hernia, cardiovascular diseases, anaemias), existence of disorders

(mental, visual, auditory, articulation, bones, nervous system, visceral), if the in-

dividual is taking some medicines, if she had some accident (other than tra¢ c ac-

cident), sleeps more than 6 hours, practices sports, BMI, smoking, marital status,

studies, income, household size, population size, and nationality. For practical rea-

sons, the analysis is performed over the population aged 15 or higher. The �nal

sample size is 53; 303 individuals.

Two questions in MS have been selected to target those seriously injured due

to tra¢ c accidents. These questions state as follows: "During the last 12 months,

have you su¤ered from a tra¢ c accident that has prevented you from performing

any usual activity?" (Yes/No), and "How has this tra¢ c accident in�uenced in your

daily life" (Seriously/ Quite a lot /Slightly). From a total of 900 individuals who give

an a¢ rmative answer to the �rst question, I select those who answered "Seriously"

(149) or "Quite a lot" (178) in the latter.

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Average characteristics for key variables are given in Table 2.2. A Wilcoxon-

Mann-Whitney test is used to compare the means of each variable for injured and

non-injured (the null hypothesis is equality of means). We can observe that people

who had a tra¢ c crash (D = 1) di¤ers considerably from people who belong to the

comparison group (D = 0). In order to emphasize these di¤erences, I calculate the

di¤erence ratio between both groups (e.g. the injured group includes a percentage of

(52�46)=46) �100 = +13% more male than the comparison group. Thus, I �nd thatthe group of injured people includes a higher proportion of males (+13%), aged 16-

35 (+58%), and present more unhealthy habits: smokers (+55%), consumption of

alcoholic drinks in labour days (+25%) and weekends (+16%), and less people who

sleep more than 6 hours (�2%). Furthermore, on average income is slightly lowerfor those in the injured group (�2%), and the highest level of education completeddi¤ers mainly by the higher proportion in secondary studies (+37%) in contrast to

a lower proportion of superior studies (�28%) and no studies (�17%). Given thesedi¤erences in the distribution of observed individual characteristics, it is necessary

to control for them, with the aim of obtaining a valid estimate of the ATET.

Injured Non-injured Di¤. Sign.(D=1) (D=0) test

N 327 52,802male 52 46 -6.63 **age 44 (20.7) 50 (20.1) 6.47 ***

16 - 25 22.0 12.2 -9.83 ***26 - 35 21.7 15.4 -6.27 ***36 - 45 13.5 14.3 0.7946 - 55 9.2 12.9 3.70 **56 - 65 8.9 14.0 5.13 ***66 - 75 15.6 17.9 2.3075 + 9.2 13.3 4.17 **

income 101,184 102,881 1,697 **(63,759) (64,087)

smoker 44.0 28.4 -15.64 **alcohol lab. days 5.8 4.7 -1.15alcohol wkds 25.1 21.6 -3.48education

less than primary 19.3 23.3 3.98 *primary 30.9 33.6 2.67

secondary 39.8 29.1 -10.67 ***more than secondary 10.1 14.1 4.02 **

(Standard deviation in brackets)Table 2.2. Average characteristics for a¤ected and comparison groups.* sign. at 10% ** sign. at 5% *** sign. at 1%

The required external information is obtained from the Catalan health surveys

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Chapter 2 Quality of Life Lost Due to Road Crashes

Enquesta de Salut de Catalunya 2002 (ESCA02 hereafter) and Enquesta de Salut de

Catalunya 2006 (ESCA06 hereafter), arranged by the Catalan government (Gener-

alitat de Catalunya). A total of 8,400 individuals (in the former) and 18,126 indi-

viduals (in the latter) were selected for the surveys, which include di¤erent health

measures as V AS, EQ-5D and SAH. From these variables, three cardinal health

measures could be obtained: V AS (directly from the survey),VAS tari¤ and TTO

tari¤ (estimated from EQ-5D). These measures are used to estimate the health

e¤ect. In the ESCA02 I dropped 1,420 observations from the sample: 1,401 proxy-

respondent interviews (related to children aged under 15 or impairments) and 19

observations because either V AS or SAH were not reported. A total of 2,247 ob-

servations (2,200 corresponding to children aged under 15 and 47 missing values of

SAH or V AS, respectively) were dropped from the ESCA06.

Finally, observations presenting inconsistencies were discarded. Those have been

detected based on the values provided by the variables V AS and SAH. Thus, several

individuals reported "excellent" health or V AS close to 1, but negative values for

the tari¤s. Similarly, some individuals reported "bad" health or V AS close to 0, but

tari¤ values close to 1. The �nal sample sizes are 7,081 in the ESCA02 and 15,875

in the ESCA06

It is important to notice that the SAH variable included in both surveys is not

identical to the SAH variable incorporated intoMS. The dissimilarity lies in the �ve

possible answers given to the respondents: the category �very bad�is not available

in either ESCA02 or ESCA06, but "excellent�is incorporated. In order to de�ne a

single health index, the construction of SAH containing 4 categories is performed

(the new variable will be called SAH4), following the approach adopted by several

authors (e.g. Lindley and Lorgelly, 2003; Hernández-Quevedo et al., 2005; García

and López, 2004). The collapsed categorizations are summarized in Table 2.3. As

I did with the SAH; let me de�ne SAH4ih as a new variable where the ordering

of the self-assessed health categories has been reversed, now interpreted in terms of

ill-health. Similarly, I denote yih = 1� y; for y 2 fTTOz; TTOr; V ASz; V ASrg :

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Chapter 2 Quality of Life Lost Due to Road Crashes

SAHSAH4 ESCA02/06 MS

1 Bad Very badBad

2 Fair Fair3 Good Good4 Very good Very good

Excellent

Table 2.3. De�nition of SAH4

2.4 Results

Table 2.4 shows the characteristics of the thresholds obtained in ESCA02 and

ESCA06 :

thresholds (ill-health)�0 �1 �2 �3 �4

ESCA02 V ASzih 0 0.09 0.23 0.62 1V ASrih 0 0.08 0.22 0.58 1TTOzih 0 0.04 0.13 0.72 1TTOrih 0 0.02 0.08 0.43 1

ESCA06 V ASzih 0 0.11 0.25 0.63 1V ASrih 0 0.10 0.23 0.59 1TTOzih 0 0.05 0.16 0.75 1TTOrih 0 0.03 0.10 0.46 1

Table 2.4. Thresholds in ESCA02 and ESCA06

Observe that �1 is considerably small both for the V AS and TTO tari¤s. This

is a direct consequence of the "ceiling e¤ect" of these scores: a value of health = 1

is assigned to the majority of the people, and this results in a value of ill-health =0.

Indeed, the interpolation used for estimating the thresholds avoids that �1 = 0 for

these metrics.

In both samples I observe that the thresholds are signi�cantly independent from

gender and age. The values should be interpreted as follows: for instance, referring

to V ASz in the ESCA02, an individual who reports the worst category of health

(SAH4ih = 4) is assumed to have a V ASzih level that belongs to the interval (0.62,

1]. Similarly, the values for the remaining SAH4ih categories are (0.23, 0.62] for

the �fair�category, (0.09, 0.23] for the �good�category and [0, 0.09] for the �very

good�and �excellent�categories (low amount of ill-health or SAH4ih = 1).

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Chapter 2 Quality of Life Lost Due to Road Crashes

The speci�cation for intervals is implemented into similar regression models. The

characteristics of the regressors as well as the parameter estimates of the interval

regression model are found in the Appendix. The health status of each individual

is controlled for a wide range of socioeconomic variables, and most of the coe¢ cients

are signi�cant (CI 5%). The McKelvey and Zavoina5 pseudo-R2 is computed for each

model, and rounds 0.48, indicating that these predictors account for approximately

48% of the variability in the latent outcome variable. On average, 63% of the

estimated health tari¤s lay into the correct interval (settled by the reported answer

to the SAH question). A Regression Error Speci�cation Test (RESET test)6 has

been applied to each interval and probit regression model, and none of them shows

evidence of mis-speci�cation.

It is important to remark that the value of health is highly linked to the self-

perception of health status, rather than the actual health status per se. A positive

coe¢ cient means that an individual has a higher value of latent ill-health and is more

likely to report a lower category of self-assessed health. The regressors have been

built so that the reference individual is a woman aged 25-35, who lives in Galicia,

married, employed, completed higher education, who did not su¤er an injury during

the last 12 months, no chronic illness, non-smoker, sleeps less than 6 hours per

day, does not make any physical exercise and has a proper BMI (does not show

underweight or obesity).7

As it was expected, the ill-health decreases with income, level of education,

absence of chronic illness, and absence of injuries or limitations. Besides, those

that sleep more than 6 hours per day or exercise have partial e¤ects lower than

1, which means that ill-health decreases with them. Students are healthier than

any other employment condition, married and widowers are more likely to report a

lower category of SAH ih (and thus higher value of true health) than single people.

5The McKelvey and Zavoina pseudo-R2 is an attempt to measure model �t as the proportionof variance accounted for: var(h)/[var(h) + var(u)].

6RESET test is popular means of diagnostic for correctness of functional form. I test: H0 : = 0against the alternative H1 : 6= 0; in log(hi) = Xi�+ yi + error, where yi is generated by takingpowers of the predicted values \log(hi) in (2.4). A failure to reject H0 says the test has not beenable to detect any misspeci�cation.

7In order to allow for some variability in the e¤ect of a road crash in health, several interactions(e.g. with gender, age, education, labor status) were introduced in the preliminar models; sinceany interaction was signi�cant, and they did not modify the results, they were �nally dropped.

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Related to regions, Galicia shows the poorest levels of health. The results also

provide evidence about the decline of quality of life as age increases.

For evaluating the propensity score I perform a logit model that include as co-

variates every variable that could a¤ect the probability of having a road crash, most

of them already included in the interval regression (gender, age, region, if smoker,

etc.) and some additional variables as: if drinks alcohol, if pregnant, etc. I must

take special care for not including causal-e¤ect reversals into the regression. The

characteristics of the injured people are recorded up to one year after the accident, so

that they could be re�ecting the consequences of a road crash rather than the prob-

ability of su¤ering it. These sort of variables could introduce an additional problem,

that is the endogeneity in the regression, what could reduce the estimated e¤ect

of the treatment. Taking this fact into consideration, the individual characteristics

that are likely to be a consequence rather than a factor related to the propensity to

have an accident, are dropped from the regression. For instance, the current labour

status, number of hours of sleep, BMI, among others.

It interesting to stress the main objective of the logit regression. From equation

(2.6) I can write:

ATET=Ecomp [w �H]� Eaff [H]

where Eaff [�] = E [�jD = 1] ; Ecomp [�] = E [�jD = 0] and w = P (D=1jZ)1�P (D=1jZ) �

P (D=0)P (D=1)

.

Thus, the probit model balances the samples of comparison and a¤ected groups,

by introducing a weight for each individual in the comparison group. Table 2.5

illustrates this idea. Some average characteristics have been already shown in Table

2.2, and the table includes the Wilcoxon-Mann-Whitney test on the equality of

means for the a¤ected group and the weighted comparison group:

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Chapter 2 Quality of Life Lost Due to Road Crashes

Z Eaff [z] Ecomp [z] Ecomp [w � z] Signmale 52 46 52 *age 44 (20.7) 50 (20.1) 44 (27.9)

16 - 25 22.0 12.2 23.3 **26 - 35 21.7 15.4 18.7 *36 - 45 13.5 14.3 13.346 - 55 9.2 12.9 10.7 *56 - 65 8.9 14.0 10.9 *66 - 75 15.6 17.9 13.775 + 9.2 13.3 9.3 *

income 101,184 102,881 101,011(63,759) (64,087) (101,977)

smoker 44.0 28.4 43.8 **alcohol lab. days 5.8 4.7 5.7alcohol wkds 25.1 21.6 24.9educationless than primary 19.3 23.3 19.1

primary 30.9 33.6 30.6secondary 39.8 29.1 40 *

more than secondary 10.1 14.1 10.3 *(Standard deviation in brackets)* Sign. at 10% ** Sign. at 5% *** Sign. at 1%Table 2.5. Descriptive statistics for a¤ected groups, comparisongroups and comparison groups with adjustment

The average health e¤ect under "selection of observables" is estimated in terms

of decrease in health. The standard errors and con�dence intervals are computed by

bootstrapping. The number of iterations is 1,500, and the bias-corrected estimate

has been considered, assuming that standard errors are normally distributed. It can

be observed that the e¤ects di¤er depending on the metric in which the ratio is

expressed. The results of the estimation and the con�dence interval are illustrated

in Figure 2.4. For a better comprehension, the results are expressed in terms

of decrease in health, instead of increase in bad health. On average terms, I can

talk about a decrease in health from 0:039 (TTO tariff , after re-scaling, with

the thresholds given by the ESCA02 ) to 0:061 (TTO tariff; changing negative

values to zero, with the thresholds obtained from the ESCA06 ). Once a particular

metric is �xed, the health e¤ects that are estimated from the surveys do not di¤er

signi�cantly, maybe slightly lower those corresponding to the ESCA06. For every

health measure, the con�dence interval embraces values strictly negative, what gives

evidence to the existence of a reduction in quality of life for those injured by a road

tra¢ c crash.

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Chapter 2 Quality of Life Lost Due to Road Crashes

­0.08

­0.06

­0.08­0.08

­0.09

­0.06

­0.08­0.07

­0.03­0.02

­0.02 ­0.02­0.03

­0.02 ­0.02 ­0.02

­0.06

­0.04

­0.06 ­0.05­0.06

­0.04

­0.05 ­0.05

­0.10­0.09

­0.08­0.07­0.06­0.05

­0.04­0.03­0.02

­0.010.00

TTOz TTOr VASz VASr TTOz TTOr VASz VASrQ

oL lo

stESCA02 ESCA06

Figure 2.4. ATET by di¤erent thresholds and health measures.

The average e¤ect can be analyzed for di¤erent population groups. Figure 2.5

shows the ATET evaluated for di¤erent age groups (only for those metrics were the

intervals were signi�cant). Notice that the ATET increases with age, and so does

the con�dence interval (probably caused by the smaller sample size).

­0.05­0.03

­0.06 ­0.06 ­0.05­0.03

­0.11

­0.08

­0.11 ­0.11­0.11

­0.09

­0.16

­0.11

­0.15­0.14

­0.12

­0.01 ­0.01 ­0.01 ­0.01 ­0.01 ­0.01 ­0.01 ­0.01­0.02 ­0.01 ­0.02

­0.16

0.00­0.03

0.00 0.00

­0.02 ­0.02­0.02

­0.03­0.02 ­0.04 ­0.04 ­0.04 ­0.02

­0.06­0.04

­0.06 ­0.06­0.06

­0.05

­0.09

­0.06­0.08 ­0.07

­0.10

­0.07

­0.18

­0.16­0.14

­0.12­0.10

­0.08­0.06

­0.04­0.02

0.00

VASz VASr TTOz TTOr TTOz TTOr VASz VASr TTOz TTOr TTOz TTOr VASz VASr TTOz TTOr TTOz TTOr

QoL

 lost

ESCA02 ESCA06

Aged 15­35 Aged 35­55 Aged 55 +

ESCA02 ESCA06 ESCA02 ESCA06

Figure 2.5. ATET by di¤erent thresholds, health measures and age-intervals.

Finally I compute the loss of health separately for men and women (see Figure

2.6). I can observe that the loss of health is, on average, signi�cantly higher for

men. This fact could be explained by factors related to driving behavior, as higher

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Chapter 2 Quality of Life Lost Due to Road Crashes

speed, more use of highways, less use of security measures.

­0.09

­0.06

­0.09­0.09

­0.10

­0.07

­0.09­0.09

­0.10

­0.07

­0.10­0.09

­0.10

­0.07

­0.09­0.09

­0.03­0.02 ­0.02 ­0.02

­0.03­0.02

­0.02 ­0.02­0.01 ­0.01

0.00 0.00­0.01 ­0.01

0.00 0.00

­0.06

­0.04

­0.06 ­0.06­0.06

­0.04

­0.06 ­0.05 ­0.05

­0.04

­0.05 ­0.05­0.06

­0.04­0.05 ­0.05

­0.12

­0.10

­0.08

­0.06

­0.04

­0.02

0.00

0.02VASz VASr TTOz TTOr VASz VASr TTOz TTOr VASz VASr TTOz TTOr VASz VASr TTOz TTOr

QoL

 lost

ESCA02 ESCA06 ESCA02 ESCA06

MEN WOMEN

Figure 2.6. ATET by di¤erent thresholds, health measures and gender.

The di¤erences between simple averages of health for a¤ected and comparison

group have been computed (Table 2.6). The results di¤er from the estimated

ATET, what supports the validity of the hypothesis about the existence of selection

on observables. In order to highlight the real impact of the total loss of health on

individuals�health state, I compute the following rate:

�H =Eaff [H]� Ecomp [w �H]

Ecomp [w �H]=

ATET

Eaff [H]� ATET�H indicates the proportion of health that on average individuals have lost due

to a road crash, with respect to the health state that, on average, individuals would

have had if the accident had not happened, estimated by using adjusted comparison

groups. The con�dence interval of �H is also re-scaled. The results are shown in

Table 2.6.

ATET Eaff [H]�Ecomp [H] �H CI(�H)ESCA02 V ASz -0.056 -0.037 -7.10% [-10.09% , -3.12%]

V ASr -0.052 -0.035 -6.55% [-9.35% , -2.87%]TTOz -0.056 -0.038 -6.61% [-9.35% , -3.53%]TTOr -0.037 -0.026 -4.15% [-6.09% , -2.16%]

ESCA06 V ASz -0.053 -0.035 -6.97% [-7.89% , -2.84%]V ASr -0.050 -0.033 -6.41% [-9.82% , -2.55%]TTOz -0.061 -0.041 -7.37% [-10.42% , -1.21%]TTOr -0.041 -0.028 -4.63% [-9.36% , -3.48%]

Table 2.6: di¤erent estimates for health e¤ects

It is worth to illustrate the meaning of �H. I use the results I obtained from one

of the health utility indices that are computed: VAS tari¤ (zero). Figure 2.7 shows

73

Page 76: Scaling Methods, Health Preferences and Health Effects

Chapter 2 Quality of Life Lost Due to Road Crashes

the empirical density function of QoL de�ned by this health tari¤. t0; t1; :::; t4 stand

for the estimated thresholds. For a representative individual whose QoL is in the

average, RTIs makes QoL drop from 0.766 to 0.712, that also turns into dropping

one category in SAH response.

02

46

Den

sity

t1 t2 t3t0 t4

0 .2 .4 .6 .8 1VAS tariff

~ death ~ perfect health

Figure 2.7. Empirical density function of

VASzero

2.5 Conclusions

The fact that road crashes represent an alarming threat to health has been reported

by most of studies that deal with injuries, causes of death or the evaluation of

the burden of diseases. The application of di¤erent policies aimed at reducing the

magnitude of the problem is essential. The e¤ectiveness of these policies should be

estimated carefully, allowing for making a distinction among the di¤erent outcomes

they could yield: a reduction of the number of crashes, fatalities and severity of the

nonfatal injuries. In this context, Bishai et al. (2006) demonstrated that observed

patterns in rich countries show only a decline in fatalities, but no decline of crashes

or injuries. Improvements in emergency transport, trauma care and passenger pro-

tection devices may be the mediating factor for that better survival.

Several countries as Germany, Great Britain and Denmark started up in 2002

what was called �intelligent emergency system�, which combines information tech-

nology and communications for reducing the time for emergency vehicles to reach

the crash scene (a minimum time of assistance was �xed: 12 min. in Germany, 8

in Great Britain and 5 in Denmark). Such measures have leaded to considerably

74

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Chapter 2 Quality of Life Lost Due to Road Crashes

reducing the probability of death subsequent to the tra¢ c accident. The possibility

of introducing these measures has already been considered in Spain, where quick

emergency response is decisive, since 35% of deaths occur beyond four hours af-

ter the crash. On February 2004 the RACC8 published a study which re�ects the

fact that emergency resources in Spain (e.g. SAMUR, provision of helicopters for

emergencies), are adequate enough for introducing such intelligent system; what the

country evidences is the lack of a more advanced coordination, as well as a large

investment for supporting this infrastructure.

Improvements in occupant protection devices also yield to decreasing death rates.

For instance, establishing the use of mandatory seat belts on car passengers in Spain

(June 1992) led to a permanent reduction in fatalities (ranging from -15% to -18%,

depending on alternative models�estimates) and seriousness on injuries, but it did

not involve a reduction in the number of crashes (García-Ferrer et al., 2007).

Besides the investment in trauma care systems and tra¢ c safety measures, it is

determinant the success of the government to enforce the tra¢ c laws. The attitude of

drivers and passengers might get into the habit of new regulations (e.g. mandatory

seat belts, maximum blood alcohol level, etc.), but enforcement e¤orts are essential

to achieve this goal (e.g. speed cameras, police controls, drivers education, etc.), in

particular at a starting point.

A large investment is needed for supporting both the implementation of new

measures and the enforcement resources. Hence the evaluation of the costs and

bene�ts of such novel instruments is essential. In order to pursue this task, and for

allowing a comparison among analysis of di¤erent measures, we should express the

total toll of deaths, injuries and sequelae derived from tra¢ c accidents in a simple

metric, that could estimate the total loss of health that could be avoided.

Databases are becoming more complete. CARE (European Road Accident Data-

base), IRTAD (International Road Tra¢ c and Accident Database) or CCIS (Co-

operative Crash Injury Study) are examples of the improvement in the data col-

lection, and they include a wide set of variables related to road crashes that some

decades ago were ignored. However, there is still much to do before there is a com-

plete set of data that comprises all valuable information (details of the accident,

joint with description of the health state of the injured individuals, etc.). Mean-

8Reial Automòbil Club de Catalunya

75

Page 78: Scaling Methods, Health Preferences and Health Effects

Chapter 2 Quality of Life Lost Due to Road Crashes

while, the short-term objective consists of obtaining the best estimation of health

losses under the limitation of the lack of available data.

Several measures have been developed in this direction. To start with, monitoring

health-related quality of life can be enhanced by establishing equivalences between

cardinal and categorical health variables, since the former are the preferred measures

for cost-e¤ectiveness analysis, but the latter is more frequently enclosed in surveys.

Furthermore, overcoming typical assumptions, as could be considering health states

as chronic or pre-injury health status as perfect health, can be considered as a

great step forward. For instance, given the lack of pre-injury measures, the use of

appropriately de�ned comparison groups should be crucial for the study of trauma

outcomes.

The permanent e¤ect that RTIs causes in health must be pointed out. In the

present study I have analyzed the health status of individuals up to one year after

the road crash. Signi�cant decreases in QoL have been observed, that are robust to

changes in data or in the measure de�nitions. Results have shown that the QoL of

people seriously injured by a tra¢ c crash decreases on a rate of 6.23%9. Therefore

it is plausible to talk about health e¤ects in QoL produced by a tra¢ c crash, what

should be taking into account at evaluating the impact on the injured individuals.

Up to my knowledge few studies evaluate health losses due to nonfatal RTIs

in QoL terms. Redelmeier and Weinstein (1999) estimate that RTIs report a loss

of health of 0:127 QoL. Sullivan et al. in 2003 estimated the morbidity caused by

RTIs in 0:356: Both estimations are signi�cantly higher than 0:051 (my result). The

reason is that these authors consider the baseline quality of life for calculating the

decrement due to injury as 1.00 (this is, non-injured are always in perfect health).

Also, Sullivan et al. (2003) do not express the result in preference-based metrics, so

that it cannot be extended to a policy or social framework. More recently, Nyman et

al. (2008) computes the health lost following a nonfatal road crash as 0:061. These

authors do not assume take "perfect health" as baseline; however, they consider

road crashes as stochastic occurrences, contrary to my main hypothesis. Besides,

my work extends upon previous research on the fact that it establishes equivalences

between cardinal and categorical health variables; it has immediate applications

9The average of the rates obtained from the di¤erent metrics has been taken as the representative�gure.

76

Page 79: Scaling Methods, Health Preferences and Health Effects

Chapter 2 Quality of Life Lost Due to Road Crashes

for policy evaluations by cost-e¤ectiveness; and computes the health decrements in

terms of utility weights.

My research has limitations, mainly derived from the source of data. Due to

the lack of available information, continuous measures of health have been partially

obtained from external data. Despite the validity of the model, it may have intro-

duced some bias, derived from di¤erent self-perceptions. Furthermore, both surveys

are administered to non-institutionalized population, so that the analysis cannot be

performed for those individuals, maybe the most seriously injured, that still remain

in trauma centers. Following Seguí-Gómez, for those individuals that are hospital-

ized due to a motor vehicle crash, about 90% of these patients are discharged from

hospital to home in less than one year after the crash. Therefore the selection that

could be derived form this matter may not be signi�cant. There is also missed infor-

mation regarding possible RTIs occurred in the past (more than one year previous

to the survey), that may be a¤ecting the actual health state of the individual but

is not observed. Thus, my results could be interpreted as a lower bound of the real

e¤ect, and hence they bring to light the relevance of the impact of road crashes in

health-related quality of life.

77

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Chapter 2 Quality of Life Lost Due to Road Crashes

2.6 Appendix

Interval regression models (dependent variable: health indices)

78

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Chapter 2 Quality of Life Lost Due to Road Crashes

ESCA02

ESCA06

TTOtari¤

VAStari¤

TTOtari¤

VAStari¤

zero

rescaled

zero

rescaled

zero

rescaled

zero

rescaled

male1525

-0.095

-0.095

-0.078

-0.078

-0.096

-0.096

-0.067

-0.067

(4.97)***

(4.90)***

(5.35)***

(5.34)***

(5.12)***

(5.05)***

(5.22)***

(5.21)***

male2535

0.013

0.014

0.006

0.007

0.011

0.012

0.007

0.007

(-0.81)

(-0.84)

(-0.52)

(-0.53)

(-0.71)

(-0.74)

(-0.62)

(-0.62)

male3545

0.079

0.078

0.062

0.062

0.079

0.078

0.054

0.054

(4.80)***

(4.68)***

(4.97)***

(4.94)***

(4.90)***

(4.77)***

(4.89)***

(4.86)***

male4555

0.142

0.14

0.115

0.114

0.144

0.142

0.099

0.098

(7.88)***

(7.62)***

(8.51)***

(8.45)***

(8.18)***

(7.90)***

(8.26)***

(8.20)***

male5565

0.161

0.157

0.131

0.131

0.164

0.16

0.112

0.112

(8.19)***

(7.84)***

(9.17)***

(9.09)***

(8.65)***

(8.26)***

(8.79)***

(8.71)***

male6575

0.05

0.043

0.056

0.055

0.06

0.052

0.043

0.042

(2.17)**

(1.79)*

(3.39)***

(3.30)***

(2.71)***

(2.28)**

(2.92)***

(2.83)***

male7585

0.065

0.057

0.066

0.065

0.074

0.066

0.053

0.051

(2.39)**

(2.01)**

(3.44)***

(3.35)***

(2.87)***

(2.44)**

(3.02)***

(2.92)***

male85m

0.099

0.097

0.089

0.088

0.105

0.102

0.074

0.074

(1.95)*

(1.81)*

(2.57)**

(2.53)**

(2.20)**

(2.04)**

(2.34)**

(2.30)**

female1525

-0.082

-0.082

-0.065

-0.065

-0.082

-0.082

-0.057

-0.057

(4.22)***

(4.18)***

(4.40)***

(4.39)***

(4.30)***

(4.25)***

(4.34)***

(4.33)***

female3545

0.076

0.075

0.064

0.063

0.078

0.077

0.054

0.054

(4.79)***

(4.65)***

(5.27)***

(5.24)***

(5.00)***

(4.85)***

(5.09)***

(5.07)***

female4555

0.181

0.181

0.141

0.141

0.179

0.18

0.123

0.123

(9.98)***

(9.82)***

(10.54)***

(10.50)***

(10.23)***

(10.05)***

(10.34)***

(10.30)***

female5565

0.214

0.216

0.163

0.163

0.211

0.213

0.143

0.143

(10.93)***

(10.74)***

(11.46)***

(11.42)***

(11.18)***

(10.97)***

(11.26)***

(11.22)***

female6575

0.156

0.155

0.124

0.123

0.156

0.156

0.107

0.107

(7.46)***

(7.20)***

(8.22)***

(8.16)***

(7.81)***

(7.52)***

(7.93)***

(7.87)***

female7585

0.163

0.162

0.127

0.126

0.162

0.161

0.11

0.11

(6.68)***

(6.40)***

(7.40)***

(7.33)***

(7.01)***

(6.71)***

(7.11)***

(7.04)***

female85m

0.194

0.195

0.151

0.151

0.192

0.193

0.132

0.131

(5.16)***

(4.91)***

(5.89)***

(5.83)***

(5.48)***

(5.20)***

(5.60)***

(5.53)***

79

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Chapter 2 Quality of Life Lost Due to Road Crashes

ESCA02

ESCA06

TTOtari¤

VAStari¤

TTOtari¤

VAStari¤

zero

rescaled

zero

rescaled

zero

rescaled

zero

rescaled

Andalucia

-0.248

-0.256

-0.179

-0.18

-0.238

-0.246

-0.161

-0.162

(17.63)***

(17.57)***

(17.99)***

(17.99)***

(17.78)***

(17.72)***

(17.89)***

(17.88)***

Aragon

-0.177

-0.184

-0.126

-0.127

-0.169

-0.176

-0.114

-0.115

(8.97)***

(9.04)***

(8.97)***

(8.99)***

(8.96)***

(9.03)***

(9.00)***

(9.02)***

Asturias

-0.066

-0.069

-0.047

-0.048

-0.063

-0.066

-0.043

-0.043

(2.97)***

(2.97)***

(2.98)***

(2.98)***

(2.98)***

(2.97)***

(2.98)***

(2.98)***

Canarias

-0.082

-0.085

-0.06

-0.061

-0.079

-0.082

-0.054

-0.055

(3.96)***

(4.00)***

(4.11)***

(4.12)***

(4.00)***

(4.04)***

(4.08)***

(4.09)***

Cantabria

-0.16

-0.166

-0.114

-0.115

-0.152

-0.158

-0.103

-0.104

(6.43)***

(6.48)***

(6.36)***

(6.37)***

(6.40)***

(6.45)***

(6.40)***

(6.41)***

CLM

-0.188

-0.196

-0.134

-0.135

-0.179

-0.187

-0.121

-0.122

(11.18)***

(11.28)***

(11.20)***

(11.23)***

(11.16)***

(11.27)***

(11.24)***

(11.26)***

CYL

-0.164

-0.171

-0.116

-0.117

-0.156

-0.162

-0.105

-0.106

(11.24)***

(11.31)***

(11.12)***

(11.14)***

(11.18)***

(11.26)***

(11.19)***

(11.21)***

Catalunya

-0.167

-0.173

-0.12

-0.121

-0.16

-0.166

-0.108

-0.109

(10.82)***

(10.82)***

(10.92)***

(10.92)***

(10.87)***

(10.86)***

(10.90)***

(10.90)***

CV

-0.243

-0.25

-0.175

-0.176

-0.233

-0.241

-0.157

-0.158

(14.60)***

(14.57)***

(14.68)***

(14.68)***

(14.65)***

(14.63)***

(14.67)***

(14.67)***

Extremadura

-0.168

-0.171

-0.123

-0.124

-0.163

-0.166

-0.11

-0.11

(7.66)***

(7.51)***

(7.94)***

(7.92)***

(7.82)***

(7.65)***

(7.82)***

(7.80)***

Baleares

-0.195

-0.2

-0.14

-0.14

-0.187

-0.192

-0.126

-0.126

(7.87)***

(7.86)***

(7.79)***

(7.79)***

(7.85)***

(7.85)***

(7.82)***

(7.82)***

LaRioja

-0.165

-0.17

-0.118

-0.119

-0.158

-0.163

-0.106

-0.107

(4.64)***

(4.67)***

(4.65)***

(4.66)***

(4.64)***

(4.67)***

(4.66)***

(4.67)***

Madrid

-0.16

-0.165

-0.115

-0.116

-0.153

-0.159

-0.104

-0.104

(9.01)***

(9.06)***

(9.03)***

(9.05)***

(9.01)***

(9.07)***

(9.05)***

(9.07)***

Murcia

-0.192

-0.197

-0.14

-0.141

-0.185

-0.191

-0.125

-0.126

(8.06)***

(7.99)***

(8.22)***

(8.21)***

(8.15)***

(8.07)***

(8.16)***

(8.15)***

Navarra

-0.155

-0.16

-0.111

-0.112

-0.148

-0.154

-0.1

-0.101

(6.34)***

(6.42)***

(6.27)***

(6.29)***

(6.29)***

(6.38)***

(6.32)***

(6.34)***

P.Vasco

-0.1

-0.105

-0.071

-0.071

-0.095

-0.1

-0.064

-0.065

(4.78)***

(4.85)***

(4.69)***

(4.71)***

(4.73)***

(4.81)***

(4.74)***

(4.77)***

80

Page 83: Scaling Methods, Health Preferences and Health Effects

Chapter 2 Quality of Life Lost Due to Road Crashes

ESCA02

ESCA06

TTOtari¤

VAStari¤

TTOtari¤

VAStari¤

zero

rescaled

zero

rescaled

zero

rescaled

zero

rescaled

bronchitis

0.345

0.366

0.232

0.235

0.321

0.341

0.215

0.217

(23.22)***

(23.19)***

(23.30)***

(23.30)***

(23.27)***

(23.25)***

(23.30)***

(23.30)***

allergy

0.058

0.06

0.043

0.043

0.056

0.058

0.038

0.039

(5.44)***

(5.40)***

(5.61)***

(5.60)***

(5.51)***

(5.47)***

(5.55)***

(5.55)***

epilepsy

0.343

0.361

0.231

0.234

0.319

0.338

0.213

0.216

(6.75)***

(6.68)***

(6.79)***

(6.78)***

(6.79)***

(6.72)***

(6.76)***

(6.75)***

diabetes

0.244

0.258

0.162

0.164

0.226

0.24

0.151

0.152

(15.18)***

(15.15)***

(15.34)***

(15.34)***

(15.26)***

(15.23)***

(15.31)***

(15.31)***

hypert.

0.047

0.05

0.031

0.031

0.044

0.046

0.029

0.029

(4.19)***

(4.22)***

(4.08)***

(4.09)***

(4.15)***

(4.18)***

(4.12)***

(4.14)***

heartinj

0.325

0.344

0.211

0.214

0.298

0.318

0.197

0.2

(20.91)***

(20.78)***

(20.77)***

(20.75)***

(20.93)***

(20.79)***

(20.79)***

(20.77)***

cholesterol0.099

0.104

0.067

0.068

0.093

0.098

0.062

0.062

(8.07)***

(8.08)***

(8.10)***

(8.11)***

(8.08)***

(8.09)***

(8.10)***

(8.11)***

arthritis

0.395

0.405

0.273

0.274

0.374

0.385

0.248

0.249

(38.12)***

(37.71)***

(38.81)***

(38.74)***

(38.54)***

(38.09)***

(38.54)***

(38.46)***

ulcer

0.205

0.215

0.141

0.143

0.193

0.203

0.129

0.13

(13.47)***

(13.37)***

(13.69)***

(13.68)***

(13.59)***

(13.49)***

(13.61)***

(13.59)***

hernia

0.168

0.176

0.114

0.116

0.157

0.166

0.105

0.106

(10.74)***

(10.71)***

(10.90)***

(10.89)***

(10.81)***

(10.78)***

(10.85)***

(10.85)***

cardiovasc

0.18

0.189

0.119

0.12

0.167

0.176

0.11

0.111

(15.63)***

(15.63)***

(15.42)***

(15.42)***

(15.57)***

(15.58)***

(15.50)***

(15.50)***

anaemias

0.212

0.23

0.145

0.148

0.196

0.215

0.135

0.137

(7.89)***

(8.04)***

(8.00)***

(8.04)***

(7.87)***

(8.05)***

(8.02)***

(8.06)***

other

0.338

0.354

0.235

0.237

0.319

0.335

0.214

0.216

(22.64)***

(22.49)***

(23.22)***

(23.20)***

(22.89)***

(22.74)***

(23.03)***

(23.00)***

81

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Chapter 2 Quality of Life Lost Due to Road Crashes

ESCA02

ESCA06

TTOtari¤

VAStari¤

TTOtari¤

VAStari¤

zero

rescaled

zero

rescaled

zero

rescaled

zero

rescaled

mentaldef.

0.414

0.449

0.276

0.281

0.38

0.416

0.258

0.263

(12.42)***

(12.43)***

(12.47)***

(12.48)***

(12.42)***

(12.46)***

(12.47)***

(12.47)***

visualdef.

0.084

0.089

0.052

0.053

0.076

0.081

0.05

0.05

(4.04)***

(4.00)***

(3.80)***

(3.80)***

(3.97)***

(3.94)***

(3.86)***

(3.86)***

auditorydef.

0.019

0.021

0.014

0.014

0.018

0.02

0.013

0.013

(-0.96)

(-0.97)

(-1.03)

(-1.03)

(-0.99)

(-0.99)

(-1.01)

(-1.02)

articul.def.

0.132

0.145

0.097

0.099

0.125

0.138

0.089

0.09

(1.67)*

(1.67)*

(1.83)*

(1.83)*

(1.72)*

(1.72)*

(1.78)*

(1.78)*

bonesdef.

0.373

0.404

0.234

0.238

0.335

0.367

0.222

0.226

(21.17)***

(21.28)***

(20.43)***

(20.46)***

(20.87)***

(21.05)***

(20.71)***

(20.74)***

nervousdef.

0.551

0.609

0.357

0.365

0.497

0.557

0.339

0.347

(15.40)***

(15.51)***

(15.04)***

(15.07)***

(15.22)***

(15.38)***

(15.19)***

(15.21)***

visceraldef.

0.406

0.452

0.257

0.264

0.363

0.409

0.246

0.252

(12.38)***

(12.39)***

(11.79)***

(11.80)***

(12.15)***

(12.21)***

(11.98)***

(11.98)***

otherdef.

0.288

0.314

0.183

0.186

0.26

0.286

0.173

0.177

(10.57)***

(10.62)***

(10.26)***

(10.28)***

(10.44)***

(10.52)***

(10.38)***

(10.39)***

roadcrash

0.2

0.205

0.147

0.147

0.194

0.199

0.131

0.132

(4.09)***

(4.05)***

(4.24)***

(4.23)***

(4.15)***

(4.11)***

(4.19)***

(4.18)***

otherinj

0.164

0.175

0.109

0.111

0.151

0.163

0.102

0.103

(7.14)***

(7.26)***

(6.94)***

(6.97)***

(7.04)***

(7.17)***

(7.04)***

(7.07)***

limitations

0.085

0.087

0.059

0.06

0.08

0.083

0.054

0.054

(3.45)***

(3.40)***

(3.56)***

(3.55)***

(3.50)***

(3.45)***

(3.51)***

(3.50)***

sleep+6h

-0.185

-0.195

-0.122

-0.124

-0.17

-0.181

-0.113

-0.115

(13.68)***

(13.74)***

(13.40)***

(13.42)***

(13.57)***

(13.66)***

(13.52)***

(13.54)***

exerciseft

-0.137

-0.138

-0.103

-0.103

-0.134

-0.136

-0.091

-0.091

(15.07)***

(15.01)***

(15.27)***

(15.26)***

(15.16)***

(15.11)***

(15.21)***

(15.20)***

exercisewrk

-0.05

-0.051

-0.035

-0.035

-0.047

-0.049

-0.031

-0.032

(5.11)***

(5.19)***

(4.84)***

(4.86)***

(4.99)***

(5.09)***

(4.95)***

(4.97)***

82

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Chapter 2 Quality of Life Lost Due to Road Crashes

ESCA02ESCA02

ESCA06

TTOtari¤

VAStari¤

TTOtari¤

VAStari¤

zero

rescaled

zero

rescaled

zero

rescaled

zero

rescaled

BMIinfra

0.095

0.099

0.071

0.071

0.092

0.096

0.063

0.064

(3.58)***

(3.62)***

(3.63)***

(3.64)***

(3.58)***

(3.63)***

(3.63)***

(3.64)***

BMIsupra

0.014

0.014

0.012

0.012

0.015

0.014

0.01

0.01

(1.86)*

(1.77)*

(2.25)**

(2.24)**

(2.02)**

(1.93)*

(2.12)**

(2.10)**

medicines

0.313

0.314

0.237

0.236

0.308

0.309

0.208

0.208

(38.86)***

(38.17)***

(39.94)***

(39.81)***

(39.48)***

(38.78)***

(39.50)***

(39.34)***

smoker

0.021

0.021

0.017

0.017

0.021

0.021

0.014

0.014

(2.66)***

(2.63)***

(2.83)***

(2.83)***

(2.73)***

(2.69)***

(2.77)***

(2.77)***

married

0.001

0.003

00

00.002

00.001

(-0.14)

(-0.29)

(-0.04)

(-0.01)

(-0.03)

(-0.19)

(-0.05)

(-0.09)

widow

-0.144

-0.152

-0.097

-0.098

-0.134

-0.142

-0.089

-0.09

(9.18)***

(9.32)***

(8.84)***

(8.87)***

(9.01)***

(9.18)***

(8.99)***

(9.02)***

sepordiv

0.085

0.089

0.059

0.06

0.08

0.085

0.054

0.055

(3.82)***

(3.92)***

(3.72)***

(3.74)***

(3.75)***

(3.86)***

(3.78)***

(3.81)***

nostuds

0.332

0.339

0.242

0.243

0.321

0.328

0.216

0.217

(21.99)***

(21.84)***

(22.28)***

(22.25)***

(22.15)***

(22.00)***

(22.18)***

(22.15)***

primstuds

0.196

0.197

0.152

0.152

0.195

0.195

0.133

0.133

(16.28)***

(16.03)***

(16.93)***

(16.89)***

(16.59)***

(16.33)***

(16.69)***

(16.64)***

secndstuds

0.088

0.089

0.071

0.071

0.089

0.089

0.061

0.061

(8.20)***

(8.08)***

(8.67)***

(8.64)***

(8.40)***

(8.27)***

(8.50)***

(8.47)***

83

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Chapter 2 Quality of Life Lost Due to Road Crashes

ESCA02

ESCA06

TTOtari¤

VAStari¤

TTOtari¤

VAStari¤

zero

rescaled

zero

rescaled

zero

rescaled

zero

rescaled

unemployed

0.044

0.043

0.034

0.034

0.044

0.043

0.03

0.03

(3.18)***

(3.08)***

(3.34)***

(3.31)***

(3.27)***

(3.16)***

(3.26)***

(3.24)***

unable

0.342

0.356

0.232

0.234

0.32

0.335

0.213

0.214

(8.09)***

(7.80)***

(8.28)***

(8.20)***

(8.25)***

(7.93)***

(8.14)***

(8.06)***

retired

0.118

0.119

0.085

0.085

0.114

0.115

0.076

0.076

(7.42)***

(7.24)***

(7.66)***

(7.62)***

(7.55)***

(7.37)***

(7.54)***

(7.49)***

housekeeper

0.074

0.073

0.057

0.057

0.074

0.073

0.05

0.049

(5.71)***

(5.46)***

(6.03)***

(5.98)***

(5.90)***

(5.64)***

(5.87)***

(5.81)***

student

-0.068

-0.068

-0.055

-0.055

-0.068

-0.068

-0.047

-0.047

(4.03)***

(4.01)***

(4.29)***

(4.28)***

(4.12)***

(4.10)***

(4.21)***

(4.20)***

other

0.107

0.108

0.078

0.078

0.103

0.105

0.069

0.069

(4.61)***

(4.47)***

(4.85)***

(4.81)***

(4.73)***

(4.58)***

(4.74)***

(4.70)***

logincome

-0.105

-0.106

-0.077

-0.077

-0.102

-0.103

-0.069

-0.069

(14.79)***

(14.59)***

(15.09)***

(15.06)***

(14.97)***

(14.76)***

(14.96)***

(14.92)***

househsize

0.01

0.01

0.007

0.007

0.009

0.01

0.006

0.007

(3.35)***

(3.45)***

(3.35)***

(3.37)***

(3.31)***

(3.43)***

(3.39)***

(3.41)***

municipsize

-0.028

-0.027

-0.02

-0.02

-0.027

-0.026

-0.018

-0.017

(3.86)***

(3.71)***

(3.81)***

(3.78)***

(3.90)***

(3.74)***

(3.79)***

(3.76)***

nation

0.079

0.079

0.061

0.061

0.078

0.078

0.053

0.053

(2.29)**

(2.26)**

(2.35)**

(2.35)**

(2.32)**

(2.29)**

(2.33)**

(2.32)**

Constant

-1.584

-2.061

-1.212

-1.281

-1.426

-1.902

-1.148

-1.218

(17.04)***

(21.59)***

(17.99)***

(18.94)***

(15.95)***

(20.69)***

(19.04)***

(20.11)***

Obs

53129

53129

53129

53129

53129

53129

53129

53129

%�t

62%

62%

64%

64%

63%

64%

64%

64%

pseudo-R2

0.49

0.49

0.48

0.48

0.49

0.48

0.48

0.47

Robustzstatisticsinparentheses

*signi�cantat10%;**signi�cantat5%;***signi�cantat1%

84

Page 87: Scaling Methods, Health Preferences and Health Effects

Bibliography

[1] Abadie, A. 2005. Semiparametric Di¤erence-in-Di¤erences Estimators. Review

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[2] Badia, X., Fernández, E. and Segura, A. 1995. In�uence of sociodemographic

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[9] García-Ferrer, A., de Juan, A. and Poncela, P. 2007. The relationship between

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[12] Gold, M. R., Siegel, J.E., Russell, L.B. and Weinstein, M.C. 1996. Cost E¤ec-

tiveness in Health and Medicine. Nueva York: Oxford University Press.

[13] Groot, W. 2000. Adaptation and scale of reference bias in self-assessments of

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[14] Heckman, J. 1990. Varities of Selection Bias. The American Economic Review

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[15] Hernández-Quevedo, C., Jones, AM. and Rice, N. (mimeo). 2005. Reporting

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[17] INE (Instituto nacional de Estadística). 2006. Death Statistic according to

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[20] Nyman, J. A., Barleen, N. A. and Kirdruang, P. 2008. Quality-Adjusted Life

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[21] O�Donnell, M., Creamer, M., Elliott, P., Atkin, C. and Kossmann, T. 2005.

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[22] Parkin, D. and Devlin, N. 2006. Is there a case for visual analogue scale valua-

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[23] Redelmeier, D. A. and Weinstein, M. C. 1999. Cost-E¤ectiveness of Regulations

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[24] Rubin, D.B. 1974. Estimation Causal E¤ects of Treatments in Randomized and

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[25] Schluter, P., Neale, R., Scott, D., Luchter, S. and McClure, R. 2005. Validating

the Functional Capacity Index: A Comparison of Predicted versus Observed Total

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[26] Seguí-Gomez, M. and MacKenzie, E. 2003. Measuring the public health impact

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[27] Sturgis, P., Thomas, R., Purdon, S., Bridgwood, A. and Dodd, T. 2001. Com-

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[28] Sullivan, P. W., Follin, S. L and Nichol, M. B. 2003. Transitioning the second-

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[29] Van Doorslaer, E. andWagsta¤, A. 1994. Measuring inequalities in health in the

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Chapter 2 BIBLIOGRAPHY

[31] WHO. 1946. Preamble to the Constitution of the World Health Organization as

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[32] WHO. 2001. Mental health: new understanding, new hope. In: The world health

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[33] WHO. 2004. World report on road tra¢ c injury prevention. Geneva.

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FEDEA �DT 2005-15.

88

Page 91: Scaling Methods, Health Preferences and Health Effects

Chapter 3

QALYs Lost Due to Road Crashesin Catalonia, 2002-2006

3.1 Introduction

Road tra¢ c injuries (RTIs) are a major cause of mortality and morbidity worldwide.

In 2004, over 50% of deaths caused by road crashes was associated with young adults

in the age range of 15�44 years, and RTIs meant the second-leading cause of death

worldwide among both children aged 5�14 years, and young people aged 15�29 years

(see WHO, 2001). RTIs are even considered the only public health problem for which

society and decision-makers still accept death and disability among young people

on a large scale (Mohan, 2003). Moller (2005) showed that RTIs have important

consequences in terms of signi�cant di¤erences in the labour-market outcomes (e.g.

lower average employment rate). The WHO claimed in 3007 that, if the present

trend continues, road tra¢ c crashes will represent the main reason of the projected

40% increase in global deaths resulting from injury between 2002 and 2030.

In Spain, Catalonia is one of most a¤ected regions byRTIs. In 2006, 470 residents

in Catalonia died as a consequence of fatal RTIs, 11% of the total of deaths by

this cause in Spain (Figure 3.1 and Figure 3.2 illustrate this fact). The major

drawback, however, should be found in those who have been seriously injured, over

4000 individuals per year. In spite of the downward trend, which becomes more

�at from year 2000, still in 2004 these �gures represented 18% of the total. This

situation is aggravated by the fact that around 50% of injured are aged 15 -34, what

89

Page 92: Scaling Methods, Health Preferences and Health Effects

Chapter 3 QALYs Lost Due to Road Crashes in Catalonia, 2002-2006

represents a critical loss of health.

Deaths by RTIs. Catalan population

13% 14% 14% 14% 15% 15% 15% 14% 13% 12% 11%

819881 860 850 811

761

549470

756

925

616

0%

5%

10%

15%

20%

25%

1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

% o

ver d

eath

s by

 RTI

s in

 Spa

in

0100200300400500

6007008009001000

num

ber o

f dea

ths

% over total of deaths by RTIs in SpainDeaths

Figure 3.1. Deaths by RTIs in Catalonia. 1996-2006Source: Cubí-Mollá and Herrero (2008)

Injured by RT crashes. Catalan population

24% 24%22%

19%16% 16% 16% 15%

18%

7561

407541834305

8391

4058

8182

4504

5954

0%

5%

10%

15%

20%

25%

1996 1997 1998 1999 2000 2001 2002 2003 2004

% o

ver t

otal

 of n

on­f

atal

 RT 

Inju

red 

inSp

ain

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

num

ber o

f inj

ured

% over total of non­fatal RT Injured in SpainInjured

Figure 3.2. Injured by RTIs in Catalonia. 1996-2004Source: Cubí-Mollá and Herrero (2008)

It is clear that Figures 3.1 and 3.2 present a rough view of the problem.

Although counts on deaths and injured are essential parameters in the evaluation of

the burden of RTIs, these measures do not provide an answer to the crucial question:

how much health is actually lost due to RTIs? This query should be answered not

only for those who die by RTIs, but also for those who are seriously injured by them.

90

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Chapter 3 QALYs Lost Due to Road Crashes in Catalonia, 2002-2006

In previous literature few studies have aimed to estimate the loss of health due to

nonfatal RTIs. Roughly speaking, I could classify these studies into two groups,

depending on the interpretation they give to "loss of health".

In the �rst group the seriousness of RTIs is measured as the risk of death. The

usual methodology in these studies (e.g. DGT, 2005; Pérez González et al., 2006) is

based on the Abbreviated Injury Scale (AIS), that is considered the "gold standard",

or their derivatives (ISS, MAIS, etc.). All these measures express the loss of health

attending to the functional disability of injured people. This procedure presents

three major drawbacks: (1) it ignores the evident heterogeneity of the evaluated

population; (2) psychological consequences of the RTIs are not reported; and (3)

it is not possible to analyze the real impact of the RTIs in the daily living of the

injured individuals.

The second group embraces studies that deal with measures of Quality of Life

(QoL). In economic studies, it is usual to compute the number of Quality-Adjusted

Life Years (QALYs) lost due to RTIs. This metric presents the advantage of com-

bining the QoL of the injured individuals with his/her remaining life duration. The

dissimilarities among these studies lie mainly on the way of evaluating the QoL. A

wide range of techniques have been developed with this goal, among which I un-

derscore (for being the most commonly used in the context of RTIs): Functional

Capacity Index (FCI), Injury Impairment Scale (IIS), Medical Outcomes 36-Items

Short Form Health Survey (SF-36), EuroQoL index (EQ-5D index) or VAS tari¤

, Time Trade-O¤ tari¤s (TTO tari¤s) and Health Utilities Index (HUI). All these

metrics can be expressed in a continuous 0-1 scale, and are aimed to represent utility

losses attributed to nonfatal injuries.

In the context of RTIs the most common approach is to base the study on metrics

belonging to the �rst group. This is a direct consequence of the di¢ culties that arise

at evaluating health losses in injuries. The ideal is to estimate the chronic sequelae

that a tra¢ c crash can have on the a¤ected, and to evaluate the impact of these

sequelae in their daily living. However, this is a challenging task, since it is hard to

obtain a complete set of data that comprises all the required information (details

of the accident, together with description of the health state of injured people in

future periods of time, etc.).

The main goal of this work is to estimate the QoL lost due to nonfatal RTIs in

91

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Chapter 3 QALYs Lost Due to Road Crashes in Catalonia, 2002-2006

Catalonia, Spain, between the years 2002 and 2006. My contribution to the literature

focuses in evaluating the chronic loss of health that would allow to estimate properly

the health losses in QALYs. Data are obtained from the Catalonia Health Survey for

the years 2002 and 2006. The methodology is based on the existing literature about

treatment e¤ects. Di¤erent health metrics will be used for assessing the robustness

of the results.

Section 2 describes the methodology that will be applied for the estimation of

the loss of health. Section 3 describes the data and variables used for the analy-

sis. Section 4 provides the main results. Finally, Section 5 summarizes the main

conclusions, including a discussion on the context of RTIs.

3.2 Methodology

Let me illustrate this idea in the context of evaluating health losses due to a road

crash.1 Following the path that Gold et al. (1996) establish, I consider the life

path of an individual as a continuous function that represents frequent changes in

health-related QoL. Let me call Y (i; t) the function that describes the health status

of individual i at time t.2 Now imagine that the individual i has a tra¢ c accident at

time T that has reported serious damage to him. Let me represent the health state

of an individual under two possible scenarios: in case the accident did not happen,

and in case it did (let me call these health paths as Y0(i; t) and Y1(i; t), respectively).

Both life paths are represented in Figure 3.3 with a decreasing trend, caused by

the age factor, and linear, for simplicity.

The days immediately after the road crash are considered critical for injured

people. The survivor is expected to recover gradually, but maybe not achieving the

previous health state. Let T + 1 be the instant where the a¤ected has restored

his/her health at maximum. Following Seguí-Gómez, for those individuals that

are hospitalized due to a RTIs, about 90% of these patients are discharged from

1The design of the methodology I develope later implies that it only makes sense in an aggregatecontext, that is, it must be framed in a context of evaluating average losses for targeted groups ofpopulation. However, for simplicity purposes, the following explanation will refer to the evaluationof health losses for a unique individual.

2In this section the concept "health status" can be interpreted broadly. For my analysis, Iconsider Y (i; t) as a continuous measure of health, ranging from 0 (worst health state) to 1 (besthealth state)

92

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Chapter 3 QALYs Lost Due to Road Crashes in Catalonia, 2002-2006

hospital to home in less than one year after the crash. Therefore T + 1 could be

placed between T and one year after T . The di¤erence Y0(i; T + 1)� Y1(i; T + 1)

would indicate the loss of health due to RTIs, evaluated one year after the road

crash, that is the amount I want to estimate. Figure 3.3 illustrates the idea.

Qualityof life

time

recover

Y0(i,t)

Y1(i,t)

loss of health

T T+1

Figure 3.3. Hypothesis: existence of health losses

Obviously, the problem is how to approximate the unknown potential health sta-

tus, that is, the health state of this individual under the unreal unrealistic scenario

in which the accident did not happen. A typical strategy for approximating this po-

tential health state consists on obtaining information from other people, rather than

the injured individual per se. In other words: imagine that I can �nd information

(dated prior to the accident) regarding the health state of an individual or a group

of people who did not su¤ered a road tra¢ c crash; assume that the individual (or

group of people) is highly comparable to the injured one, since they coincide in sev-

eral observed factors (such as age, gender, studies, etc.). Therefore, the health state

of that individual or comparison group can be taken as a proxy for the potential

health state of the victim.

If I analyze deeply the nature of RTIs, the strategy of setting a comparison

group seems quite reasonable. People a¤ected by RTIs can be neither considered

as randomly selected, nor as a perfectly targeted population. The existence of some

random component cannot be denied, mainly related to the occurrence or non-

occurrence of a road crash, rather than the seriousness of the injures. For instance,

being involved into a crash caused by a di¤erent individual, or an unexpected punc-

ture on the road. However, many factors that in�uence the existence of RTIs (as can

be wearing seat-belts, airbags, driving carefully, not being drunk, using the pedes-

trian crossing, etc.) are chosen by the individual. In fact, data show that the group

of people that have a road crash includes higher proportion of male, aged 16-35,

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have unhealthy habits as smoking or consumption of alcoholic drinks, among other

features. These characteristics, that may a¤ect both health status and the proba-

bility of being injured by a road crash, are observable by us. I could think about

the existence of unobservable factors, such as the degree of risk aversion, the driving

ability, and so on, that could a¤ect also the health state as well as the probability

of having a road crash. For the results that I obtain in this paper to be correct, it

is necessary to assume that 1) there are no unobservable factors a¤ecting both the

outcome and the probability of having a crash, or 2) if unobservable factors exist,

these can be captured by the observable ones (e.g. risk-loving behavior is usually

related to consumption of alcohol), or 3) if unobservable factors exist and are not

re�ected by the observables, its overall average impact is equal for both the a¤ected

and the comparison group.

The rest of this section explains in more detail the methodology suggested above.

Following Rubin (1974) and Heckman (1990), causality is de�ned in terms of

potential outcomes. Some new notation is useful at this point. I use the same

formulation as before, but time is dropped from the health function, since every

health state will be only measured after the road crash takes place. The variable

Y0 (i) is now the outcome that individual i would attain had the road crash not

happened. Equivalently, the variable Y1 (i) is the outcome that individual i would

have if he had su¤ered RTIs. Let Di indicate whether individual i had a road crash

(Di = 1) or not (Di = 0).

In this paper I focus on the average loss of health as a result of a road crash, for

those who had an accident. This quantity is known as the average treatment e¤ect

on the treated (ATET ) and, with the new notation, it can be written as follows:

ATET = E [Y1 (i)�Y 0 (i) jDi= 1]= E [Y1 (i) jDi= 1]�E [Y0 (i) jDi= 1]

The ATET cannot be identi�ed using observational data since Y0 (i) is only ob-

served for those who did not have an accident. A suitable solution is to approximate

the average potential health status by the average health state observed in a compa-

rable group of people that did not have an accident. As it was discussed before, the

average health of injured (a¤ected group, hereafter) and non-injured (comparison

group, hereafter) individuals cannot be unconditionally compared. Thus, the valid-

ity of this approximation is likely to be higher once di¤erences in the distribution of

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observed individual characteristics are controlled for.

Let Z(i) be a vector including information relative to individual i that is a priori

thought to a¤ect his probability of su¤ering a road crash. Under this approximation

the ATET can be expressed as follows:3

ATET = E [Y jZ;D = 1]�E [Y jZ; D = 0] ;

where Y = D � Y1 + (1�D) � Y0 is the observed health status of the individuals.This di¤erence in averages identi�es the ATET as long as the so-called �selection

on observables�restriction holds, which can be expressed as:

ASSUMPTION 1: E [Y0jZ;D = 1] = E [Y0jZ; D = 0]

This condition states that the average health status of those who have been

injured by a road crash, had the road crash not occurred, is equal to the average

health state of those who did not have an accident, and share the same values of

the variables in Z. In other words: the e¤ect of events other than the road crash do

not contaminate the causal analysis.

Abadie (2005) develops a simple two-step procedure to identify the ATET using

the di¤erence-in-di¤erences estimator. In Abadie (2005), the basic element needed

to estimate the ATET is the conditional probability of receiving the treatment, also

called propensity score. This procedure is now adapted to the situation where I only

have data for the post-treatment period, that is, to the selection on observables case.

Since identi�cation is attained after conditioning on covariates, it is required that

for a given value of the covariates there is some fraction of the population that have

not been a¤ected to be used as a comparison group.4

ASSUMPTION 2: P (D = 1) > 0 and with probability one P (D = 1jZ) < 1.

In a similar way to Abadie, it can be proved that the ATET can be expressed

as follows:

E [Y1 � Y0jD = 1] = E

�Y

P (D = 1)� D � P (D = 1jZ)1� P (D = 1jZ)

�(3.1)

= Eaff [Y ]�Ecomp [w � Y ] ;3Hereafter the individual argument will dropped out to simplify notation.4Assumption 2 is a well-known condition for identi�cation of the average impact on the treated

under selection on covariates (see, e.g. Heckman et al., 1997).

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where Eaff [�] = E [�jD = 1] ; Ecomp [�] = E [�jD = 0] and w = P (D=1jZ)1�P (D=1jZ) �

P (D=0)P (D=1)

.

Equation (3.1) suggests a simple two-step method to estimate the ATET under

Assumptions 1 and 2. First, the conditional probabilities are estimated using a

probit/logit model and the �tted values of P (D = 1jZ) are calculated for eachindividual in the sample. Second, the �tted values are plugged into the sample analog

of equation (3.1). Then, a simple weighted average of the outcome variable recovers

the ATET. Finally, the asymptotic variance of the estimator is also calculated,

following the procedure developed in Abadie (2005) for the conventional di¤rence-

in-di¤erences estimator, now adapted for the selection on observables case.

3.3 Data and variable de�nitions

My primary source of data is the Catalonia Health Survey (CHS) for the years

2002 and 2006 (CHS02 and CHS06, hereafter). The data for the �rst survey was

collected from October of 2001 to April of 2002, with a sample of 8,400 people living

in Catalonia (6,343,818 inhabitants in 2001). The data from the second survey was

collected throughout the year 2006, and comprises a total of 18,126 individuals. The

surveys included questions on the state of health, the habits of life�including feeding,

physical exercise and tobacco and alcohol consumption�and the use of the health

services managed by the regional government. The surveys are similarly designed

and their results are highly comparable.

CHS provides several measures of the health state: �rst, a numeric self-evaluation

of the health state; second, a detailed valuation of �ve dimensions determining the

health state; and third, a visual score of the health state by means of a graphical

�thermometer�. In order to translate these variables into a particular score of health

status, a �preferences tari¤�is needed.5 Two tari¤s for these scores have been com-

puted for Spain: the VAS tari¤ (based on a visual analogue scale; see Badia et al.,

1995) and the TTO tari¤ (based on the temporal equivalence method; see Badia et

al., 2001). Both scores allow for negative values, that is, health states worse than

death, what confuses the measurement of health e¤ects. In this paper I follow the

criterion for overcoming these controversies stated by [?], which consists on changing

5See Cutler and Richardson (1997) and Torrance (1986)

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the negative values to zero. The health state scores obtained from these tari¤s can

be adopted as the weights for the calculation of the QALYs.

For targeting those seriously injured due to tra¢ c accidents, I select those ques-

tions related to road crashes and limitations that are stated in the same words in

both samples. The �rst question is included in the surveys into a section called

"Accidents", that indicates that "Questions in this section are related to accidents

that had caused any restriction in the usual activities for the individual, and/or that

had required medical assistance". The question (Q1) states as follows: "During the

last 12 months, have you had a tra¢ c accident, being occupant or pedestrian?"

(Yes/No).

A second battery of questions has been used to assess the seriousness of the

RTIs, for those who answered "Yes" to (Q1) (answers are Yes/No): (Q2) "Dur-

ing the last 12 months, have you had any impediment and/or di¢ culty for work-

ing/studying/housekeeping, due to some chronic health problem (that endured or is

expected to endure three months or more)?" and (Q3) "During the last 12 months,

have you had to restrict or reduce your usual activities (e.g. go walking, sports, go

shopping, etc.) due to some chronic health problem?". From a total of 279 and 443

individuals who give an a¢ rmative answer to (Q1) in CHS02 and CHS06, respec-

tively, I select those who answer "Yes" in (Q2) or (Q3), 51 respondents in CHS02

and 105 respondents in CHS06. The size of the targeted population amounts to 156

individuals. The remaining individuals (228 in CHS02 and 338 in CHS06 ) were

dropped from the sample, with the intention of establishing a comparison group

that includes those who have not had a road crash.

The issue of selecting variables that could a¤ect the probability of having a

road crash has been a specially di¢ cult task. The information should be included in

both surveys, and both questions and possible answers should not di¤er signi�cantly

from one survey to the other. I must take special care for not including causal-e¤ect

reversals into the regression. The characteristics of injured individuals are recorded

up to one year after the accident, so that they could be re�ecting the consequences

of a road crash rather than the probability of su¤ering it. These sort of variables

could introduce an added problem, that is the endogeneity in the regression, what

could bias the e¤ect of RTIs. The individual characteristics that are likely to be a

consequence rather that a factor related to the propensity to have an accident, are

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Chapter 3 QALYs Lost Due to Road Crashes in Catalonia, 2002-2006

not included in the regression. For instance: the current labour market status, the

existence of chronic illnesses such as anxiety or depression, or the physical activities

of the individual, among others.

Factors as age and gender (interacting) are crucial for estimating the probability

of being seriously injured by a road crash. I have considered additional factors

as: sleeping less than 8 hours; if the person is a usual smoker; if he/she usually

consumes alcohol; marital status; level of education; household size; if the individual

has regular checkups and get temporal vaccination (taking preventative measures

could be a proxy of the existence some risk aversion and sensible behavior), and the

birth region (what could re�ect the level of exposure to tra¢ c, mainly on holiday

dates). Finally, the analysis includes a dummy variable to indicate the survey to

whom the data belongs.

For practical reasons, the analysis is performed over the population aged 15 or

higher. From CHS02 I dropped 1; 330 observations (corresponding to children aged

under 16) and 148 (missing values or inconsistent answers). From CHS06 I dropped

2; 315 observations (corresponding to children aged under 16), 75 (missing values

or inconsistent answers) and 95 individuals whose answers in the relevant variables

were not considered trustworthy by the interviewer. The �nal size for the sample

is 6; 491 individuals in CHS02 and 15; 333 individuals in CHS06, a total of 21; 888

observations.

Averages for the key variables are given in Table 3.1. In the group composed

by people that had a tra¢ c crash (D = 1). A Wilcoxon-Mann-Whitney test is

used to compare the means of each variable for injured and non-injured (the null

hypothesis is equality of means). I �nd higher proportion of males, people aged

16-35, and present unhealthy habits: smokers, consumption of alcoholic drinks and

less people sleep more than 8 hours. It is interesting to see among the injured

individuals a higher proportion of people aged 56-65 than among the non-injured

ones. Also, the a¤ected group has a higher proportion of individuals with secondary

or superior educational level, and a slightly higher proportion of people born in other

municipalities in Catalonia. The proportion of people living in couple is signi�cantly

lower for those who have been injured. I �nd signi�cant the average diferences in age,

smoking habit, household size and living together with the couple. These average

di¤erences on observed individual characteristics give evidence of the necessity of

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Chapter 3 QALYs Lost Due to Road Crashes in Catalonia, 2002-2006

controlling for them, if we want to obtain a consistent estimate of the ATET.

Injured Non-injured WMW test Sign.(D=1) (D=0) Prob > jzj

N 144 21,680male 53 48.9 0.2747age 38 (17.8) 47.2 (19.1) 0.0000 ***

16 - 25 26.4 12.5 0.0000 ***26 - 35 25 18.6 0.0473 **36 - 45 15.3 18.1 0.375446 - 55 10.4 16.0 0.0706 *56 - 65 13.9 13.0 0.752766 - 75 5.6 11.3 0.0298 **75 + 3.5 10.6 0.0057 ***

smoker 41 27.4 0.0003 ***alcohol 65.3 63.9 0.3463sleep +8h 40.3 44.2 0.3463second. or sup. education 65.3 60.1 0.2049household size 3.3 (1.18) 3.1 (1.26) 0.0536 *living in couple 42.4 59.3 0.0000 ***Place of birth

Catalonia 72.9 70.1 0.4521rest of Spain 20.8 23.6 0.4338

foreigner 6.3 6.3 0.9603(Standard deviation in brackets)Table 1: average characteristics for the a¤ected group and comparison group.* sign. at 10% ** sign. at 5% *** sign. at 1%.

3.4 Results

In order to estimate the propensity score I perform a logit model that include as

covariates the variables de�ned in the previous section. A total of 182 observations

have been dropped, in order to obtain a common support for both groups. I perform

a RESET test, a convenient way of testing the speci�cation of the model. The

RESET test gives a F-test statistic of 2:29 (p = 0:1305), and so there is no evidence

of mis-speci�cation. The results are shown in Table 3.2. Almost every variable

where age and gender interact is signi�cant, showing that young men aged 16-25

are the more prone to have a serious road crash. Smoking signi�cantly increases the

probability of an accident. In the last row the odds-ratios are shown.

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Coe¢ cient Sig. Std. Err. Odds-ratioAge and gender (reference class: male, aged 16-25)

male, aged 26-35 -0.966 *** 0.339 0.38male, aged 36-45 -0.993 ** 0.389 0.37male, aged 46-55 -1.342 *** 0.463 0.26male, aged 56-65 -0.887 ** 0.455 0.41male, aged 66-75 -1.848 *** 00.674 0.16male, aged 76+ -2.558 ** 1.064 0.08

female, aged 16-25 -0.883 ** 0.361 0.41female, aged 26-35 -0.470 0.323 0.63female, aged 36-45 -1.301 *** 0.442 0.27female, aged 46-55 -1.464 *** 0.491 0.23female, aged 56-65 -0.905 ** 0.473 0.40female, aged 66-75 -1.563 *** 0.587 0.21female, aged 76+ -1.952 *** 0.648 0.14

CHS02 0.098 0.182 1.10Secondary or superior education 0.230 0.204 1.26

If regular checkups and vaccination 0.128 0.199 1.14Sleep +8 h -0.263 0.175 0.77Alcohol -0.027 0.355 0.97Smoking 0.401 ** 0.181 1.49

Household size 0.011 0.073 1.01Place of birth (r.c: Catalonia)

rest of Spain 0.169 0.233 1.18foreign country -0.052 0.355 0.95

Marital status (r.c: single)lives with a couple -0.381 * 0.249 0.68

separated/divorced/widow 0.124 0.366 1.13constant -4.070 *** 0.374 0.02

* signi�cant at 10%; ** signi�cant at 5%; *** signi�cant at 1%Table 2: Results of the logit regression

I compute the average health e¤ect under "selection of observables" in terms

of decrease in health. It can be observed that the e¤ect di¤ers slightly depending

on the metric in which the ratio is expressed. On average terms, I �nd a decrease

in health of 0:1328 (VAS tari¤, CI: [�0:3096;�0:0077]) to 0:1296 (TTO tari¤ , CI:[�0:3020;�0:0058]): With a con�dence of 95%, the con�dence interval referred toboth tari¤s embraces strictly negative values. This is evidence of the existence of a

reduction in quality of life for those injured by a road tra¢ c crash.

The di¤erences between sample averages of health for a¤ected and comparison

group have been computed (Table 3.3). The results di¤er from the estimated

ATET, that is, P (D = 1jZ) 6= P (D = 1). This re�ects that the evaluation of

RTIs shall not be approached as a stochastic occurrence. In order to highlight the

real impact of the total loss of health on individuals�health state, I compute the

following rate:

�Y =Eaff [Y ]� Ecomp [w � Y ]

Ecomp [w � Y ]=

ATET

Eaff [Y ]� ATET

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Chapter 3 QALYs Lost Due to Road Crashes in Catalonia, 2002-2006

�Y indicates the proportion of health that on average individuals have lost due

to a road crash, with respect to the health state that, on average, individuals would

have had if the accident had not happened, estimated by using adjusted comparison

groups. The con�dence interval of �Y is also re-scaled. The results are shown in

Table 3.

ATET Eaff [Y ]�Ecomp [Y ] �Y CI(�Y )VAS tari¤ -0.1328 -0.1027 -14.73% [-34.36% , -0.85%]TTO tari¤ -0.1296 -0.0987 -14.64% [-34.11% , -0.65%]

Table 3: di¤erent estimates for health e¤ects

The rate �Y is perhaps the most informative result. It re�ects that, on average,

an individual that has su¤ered/su¤ers RTIs is living in a 14% lower health status.

This should bring to light the relevance of the impact of road crashes in health-

related quality of life.

3.5 Conclusions

The results of this paper show that for a successful RTIs prevention strategy, drivers

should be more aware of the risks and change their behavior. Certainly, if I analyze

the more recent data (CHS06 ), I observe a decrease in the use of security resources:

5.23% of motorcycle drivers fail to wear the helmet in urban areas, 2.44% fail to

wear it on roads. Also, concerning to the seat belts: 9.30% of copilots fail to use

them in urban areas, and 1.96% on road; specially important is the use of seat belts

by other occupants (back seats): 14.9% fail to wear them on road and 24.45% fail

to use it in urban areas.

The attitude of drivers and passengers might get into the habit of new regu-

lations. In 1997 the Swedish Parliament introduced a �Vision Zero� policy that

requires that fatalities and serious injurious are reduced to zero by 2020. This is a

signi�cant step change in transport policy at the European level and may soon be

followed by Switzerland. Whether Vision Zero might be broadly adopted as best

practice across Europe or not is currently under discussion in the European Road

Safety Commission.

Vision Zero accepts that preventing all accidents is unattainable and unrealistic.

Therefore, its goal is to manage them so that they do not cause serious health

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impairments. It is explicitly based on the idea that road safety is also a matter

of ethics and human values (non-monetary aspects). It also claims against the

concept of �road safety� from the unique point of view of costs and bene�ts, and

recommends an extensive use of published information on the cost-e¤ectiveness of

road safety interventions.

The application of di¤erent policies in order to reduce the magnitude of the

problem is essential. The e¤ectiveness of these policies should be estimated carefully,

making a distinction among the di¤erent outcomes they could yield: reduction of the

number of crashes and fatalities, but also a reduction of the number and severity

of RTIs. Thus, the e¤ectiveness of policies aimed at reducing the seriousness of

tra¢ c injuries (e.g. improvements in emergency transport, trauma care , passenger

protection devices), under the �Vision Zero�, should consider the decrease in the

quality of life following a road crash, even after the recover. This work proposes new

approaches to evaluate quality-of-life weights, to assess the problem by computing

Quality-Adjusted Years of Life (QALYs) Lost.

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Bibliography

[1] Abadie, A. 2005. Semiparametric Di¤erence-in-Di¤erences Estimators. Review

of Economic Studies 72: 1-19.

[2] Badia, X., Fernández, E. and Segura, A. 1995. In�uence of sociodemographic

and health status variables on valuation of health states in a Spanish population.

European Journal of Public Health 5: 87-93.

[3] Badia, X., Roset, M., Herdman, M. and Kind, P. 2001. A comparison of United

Kingdom and Spanish general population time trade-o¤ values for EQ-5D health

states. Medical Decision Making 21: 7-16.

[4] Burström, K., Johannesson, M. and Diderichsen, F. 2003. The value of the

change in health in Sweden 1980/81 to 1996/97. Health Economics 12: 637-654.

[5] Busschbach, J.J.V., McDonnell, J., Essink-Bot, M.L. and van Hout, B.A. 1999.

Estimating parametric relationships between health description and health valu-

ation with an application to the EuroQol EQ-5D. Journal of Health Economics

18: 551-571.

[6] Cubí-Mollá, P. and Herrero, C. 2008. Evaluación de riesgos y del impacto de

los accidentes de trá�co sobre la salud de la población española. 1996-2004. Eds:

Fundación BBVA, Bilbao.

[7] Cutler, D. and Richardson, E. 1997. Measuring the health of the United States

Population. Brookings Papers on Economic Activity: Microeconomics: 217-271.

[8] DGT. Marzo 2005. Estudio nacional multicéntrico sobre morbilidad derivada de

los accidentes de trá�co. Madrid: Dirección General de Trá�co, Ministerio del

Interior.

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Chapter 3 BIBLIOGRAPHY

[9] Gold, M. R., Siegel, J.E., Russell, L.B. and Weinstein, M.C. 1996. Cost E¤ec-

tiveness in Health and Medicine. Nueva York: Oxford University Press.

[10] Heckman, J. 1990. Varities of Selection Bias. The American Economic Review

80(2): 313-318.

[11] Mohan, D. 2003. Road tra¢ c injuries - a neglected pandemic. Bulletin of the

World Health Organization 81(9).

[12] Moller Dano, Anne. 2005. Road injuries and long-run e¤ects on income and

employment. Health Economics 14: 955�970

[13] Pérez-González, C., Cirera, E., Borrell, C. and Plasència, A. 2006. Motor vehicle

crash fatalities at 30 days in Spain. Gaceta Sanitaria 20(2): 108-15.

[14] Rubin, D. B. 1974. Estimation Causal E¤ects of Treatments in Randomized

and Nonrandomized Studies. Journal of Educational Psicology 66: 688-701.

[15] Torrance, G. W. 1986. Measurement of health state utilities for economic ap-

praisal. Journal of Health Economics 5: 1-30.

[16] WHO. 2004. World report on road tra¢ c injury prevention. Geneva.

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[18] Zozaya, N., Oliva, J. and Osuna, R. 2005. Measuring Changes in Health Capital.

FEDEA �DT 2005-15.

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Chapter 4

Estimating Health E¤ects

4.1 Introduction

1Currently, measures of disability and health-related quality of life are becoming

important, even essential, parameters in the evaluation of treatment and prevention

strategies for reducing the burden of injury. The estimation of the �health e¤ect�

induced by these policies should incorporate several important aspects: the proper

de�nition of health e¤ect, at individual and aggregate levels; the correct selection

of a health metric; the accurate estimation of the short-term e¤ect (direct health

gain/loss) and long-term e¤ect (total gain/loss of health throughout the life of the

individual) that injuries may produce; and the suitable selection and management

of databases. This review article focuses on the particular topic of road crashes, but

the analysis can be extended to any sort of injury.

In 2001, injuries represented 12% of the global burden of disease (WHO, 2001).

The category of injuries worldwide is dominated by those incurred in road crashes.

In 2004, over 50% of deaths caused by road crashes were associated with young

adults in the age range of 15�44 years, and tra¢ c injuries were the second-leading

cause of death worldwide among both children aged 5�14 years, and young people

aged 15�29 years (WHO, 2004). In addition, road crashes are expected to be the

main reason of the projected 40% increase in global deaths resulting from injury

between 2002 and 2030 (WHO, 2007).

Evaluation of policy or clinical interventions are essential aspects of injury pre-

1Part of this section was published in: Estimating health e¤ects in quality-of-life terms: healthlosses following road crashes. Patricia Cubí-Mollá. Expert Review of Pharmacoeconomics &Outcomes Research 2008 8:5, 471-477

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Chapter 4 Estimating Health E¤ects

vention, oriented towards reducing the burden of injury. Evaluations of these treat-

ments or policies are usually performed through the estimation of cost�e¤ectiveness

ratios (a discussion regarding di¤erent methodologies for evaluating health interven-

tions, cost�utility analysis and cost�bene�t, can be found in Nord, 1999), which are

obtained by taking the cost of the treatment and dividing it by the �health gains�

the treatment produces (Gold et al., 1996).

The cost of the treatment is calculated in monetary terms. Some standards

must be adopted in order to make di¤erent studies comparable. The Panel on Cost�

E¤ectiveness in Health and Medicine agreed some guiding principles referred to the

assignment of costs (see Manning, 1999), by adopting a society perspective (e.g., to

include costs of the healthcare sector, costs of the individual and broader society

costs).

The concept of �health gain�has experienced a signi�cant development during

the past few decades. Most of the analyses concerning policy interventions, in the

case of road injuries, interpret �health gain�as a reduction in the number of crashes,

fatalities and people injured, expressed in terms of absolute �gures (Bishai et al.,

2006) or relative risks (Pérez et al., 2007). Elvik presents a good review of the

evaluation of policy interventions (Elvik, 2004).

Selecting one or another of the aforementioned ways of interpreting health gains

is highly linked to the di¢ culties in properly estimating several dimensions related

to the evolution of the individuals a¤ected. Let me mention their pre-injury status,

their health status after the crash, their evolution and the �nal or chronic health

status observed in the a¤ected individuals.

In dealing with the pre-injury status, researchers do not normally deal with insti-

tutionalized individuals, that is, this is not a situation where policy interventions are

de�ned over targeted subpopulations with a well-known health state (for instance,

with cancer treatments or e¤ectiveness of dialysis programs). In those cases, it is

plausible to obtain proper information about the pre-injury status of the patients.

In the context of injuries (burning, road crashes, falls or poisoning, among others),

nonetheless, it is especially di¢ cult to analyze the e¤ectiveness of prevention control

since the pre-injury status of the individual is usually unknown.

As a consequence, to properly analyze the e¤ectiveness of prevention control in

road crashes we have to deal with �ve fundamental problems:

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� The selection of an adequate metric to evaluate health status;

� The need to select a method of properly estimating pre-injury health status,

direct health e¤ect of an injury and post-injury chronic loss of health of any plausible

individual a¤ected by a crash;

�We should decide if it is plausible to aggregate individual results in some number

expressing the e¤ects on the population, considering two dimensions: adding health losses

of individuals with the same order of seriousness (fatal/nonfatal; see e.g. Elvik, 1995) , or

even the rationality of expressing the total loss of health for nonfatally and fatally injured

individuals into a single �gure (Hofstetter and Hammitt, 2001);

�We must carefully choose the source of information, testing for the completeness

and reliability of the data;

� The lack of a gold-standard methodology requires the application of some cri-

teria that could assess the validity of the results.

In dealing with the �rst problem, a metric to evaluate population general health

status is needed (Seguí-Gómez and MacKenzie, 2003). In 1946, the World Health

Organization (WHO) de�ned health as: �. . . not only the absence of in�rmity and

disease, but also a state of physical, mental and social well-being�(see WHO, 1948).

This broad de�nition captured essential elements of quality of life, which underlie

most human health metrics (Bishai and Hyder, 2006). Based on this de�nition, it is

clear that life expectancy or mortality-based measures are no longer considered as

adequate as measures of a population�s health.

Once a metric is established, analyzing health e¤ects requires to explore some

aspects in detail. First, how can we estimate the pre-injury health status? Can

we presume it, or should we establish a comparison group? Second, how can we

capture the chronic health loss of the people injured? How long a period after the

tra¢ c crash should we de�ne for considering the damage as having a chronic e¤ect?

Closely related to that point, it is also crucial to establish how to measure the total

health e¤ect throughout the life of individuals. In other words, how to combine

direct health e¤ects with life expectancy. Shall we assume that the people a¤ected

would retain a constant health loss along the rest of their life path? Finally, we

shall consider whether the accident may also have e¤ects on the life expectancy of

the individuals injured in the crash.

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The lack of information concerning the previous question has led to researchers

to use some speci�c simplifying assumptions. For instance, it is quite common

to consider pre-injury status as one of �perfect health� and the immediate post-

injury status as a chronic one, when estimating the health e¤ects of injuries (e.g.

Elvik, 1995; Bishai and Hyder, 2006). The life expectancy of the a¤ected people

is usually obtained from external information, and is taken as a �xed amount for

men and women, without controlling for other crucial factors such as age or region

(Bishai and Hyder, 2006). Moreover, when calculating the e¤ects of road crashes

on injured individuals, it is usually assumed that the accident does not change their

life expectancy (e.g. Seguí-Gómez et al., 2002). Nonetheless, all these simplifying

assumptions allow a rough approximation of the actual magnitude of the injury

e¤ect on health.

The possibility of improving the previous rough approach is linked to the avail-

ability of more extensive and reliable databases. Although they are certainly im-

proving, we are far from achieving a complete set of data that contains all the

information required for an accurate analysis.

This paper explores some ways of answering previous questions, through improv-

ing the extra-simpli�ed analysis, and aims to address:

� The most appropriate metrics to quantify a health status;

� How to estimate the direct health e¤ect of an injury;

� How to estimate the e¤ect on the full life path of a particular individual;

� How to estimate the burden of some particular type of injuries on the population

health;

� How to obtain the best estimation of health losses, which is always limited by the

availability and quality of data.

Table 4.1 summarizes the current problems in evaluating health e¤ects, and the

usual proposed solutions. These will be discussed in the following sections.

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Problems Usually proposed solutionsUnknown potential health state � Estimation of pre-injury status

� Use of comparison groupsCombining direct health status � Post injury health state: chronic health statewith life expectancy � Life expectancy: does not change after the injury

Aggregating health e¤ects among � Introduce age-weighting factorsdi¤erent groups of individuals � Interpret health values as indices

Incomplete databases � Use di¤erent sources of data as complementary.Lack of a gold-standard � Then-tests (control for a response shift)methodology � Follow some criteria of validityTable 4.1. current problems in evaluating health e¤ects, and the usual proposed solutions

4.2 Appropriate metrics

A wide variety of metrics are used to quantify the burden of illnesses and injuries

to population (an exhaustive description of these measures can be found in Seguí-

Gómez and MacKenzie, 2003; MacKenzie, 2001; Sturgis et al., 2001). In general

terms, let me classify the di¤erent sorts of measures into two groups, depending on

the way they approach the problem: either (i) estimating the amount of good health

or (ii) assessing the degree of functional limitation. It is important to underline that

each of these two groups can include measures with di¤erent characteristics: (a)

health status measures, which do not indicate preferences for health states; and (b)

preference-based measures, which are de�ned by means of preference-based methods,

such as time trade-o¤ or standard gamble.

The literature suggests that non preference-based measures should not be applied

in the context of decision analysis or economic evaluations (e.g., cost�utility analyses;

see Seguí-Gómez and MacKenzie, 2003).

Measures in the �rst group (i) focus on the impact of the injury over the general

health state of the individual, comprising a variety of indices or metrics that de�ne

�health�. Health status measures (a), including Visual Analogue Scale, Self-Assessed

Health, Euroqol �ve-dimensional descriptive system or Short-Form Health Survey

(SF-36); and preference-based measures (b), such as Health Utility Index (HUI-3,

the current version), Quality of Well-Being, or Euroquol �ve-dimensional index, can

be placed within such an approach. These metrics re�ect the quality of health states,

both from a physical and a psychological aspect. The preference-based measures can

combine the e¤ect of death and nonfatal consequences into a summary measure that

typically ranges from zero (representing death) to one (representing optimal health),

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where any number re�ects the relative preference for particular health states. In-

stead of self-reported scores, these metrics provide community values for the health

states. Previous characteristics can complicate, on the one hand, interpersonal com-

parisons among subjects (and therefore the consistency of aggregation procedures),

and, on the other hand, secure data from some targeted groups of population, such

as children, the elderly or the unconscious.

Metrics in the second group (ii) aim to estimate the seriousness of the injuries.

The preference-based measures (a) attempt to re�ect the degree of functional limita-

tion of the people injured (e.g., Functional Capacity Index and Disability weights).

The nonpreference measures (b) quantify the seriousness of the injuries by attending

to the mortality risk or life threat (e.g., Abbreviated Injury Scale, Injury Severity

Score, ICD-9 Injury Severity Score and Anatomic Pro�le Score). These sorts of met-

rics are considered as external to the patients and are constructed from the clinicians�

and researchers�points of view; they are easy to obtain, and examine the character-

istics of the concrete injury in detail. Nonetheless, not all metrics in this group have

been clearly validated (Schluter et al., 2005). Moreover, they present some other

disadvantages, since they do not allow for heterogeneity, they have problems with

comorbidities and they do not take into account the psychological dimension.

Of the scales that have been reviewed, those that belong to the second group

are the ones most commonly used to assess health losses due to injuries. However,

several studies suggest that an injury and acute psychological responses are strongly

linked, and so both play important roles in determining quality-of-life and disability

outcomes (O�Donnell et al., 2005). Although measures of severity in the second

group provide some understanding of the relative seriousness of injuries in terms of

threat to life, they still fall short in measuring resource utilization and the long-term

impact of nonfatal injuries on the person, his or her family, and the society at large.

These considerations have challenged the �eld to move beyond counting injuries by

severity alone to measuring their direct impact on health-related quality of life.

However, the use of health-state outcomes as a method for describing the con-

sequences of tra¢ c injuries from diverse perspectives (such as the e¤ects for health

at the individual level, at the aggregate level, for public health or at the decision-

making process), must be performed carefully. The wide set of alternatives that have

been already mentioned (such as selection of health measures and ways of combining

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quality of life to length) demand an assessment of the validity of the results. In 1995

Elvik summarizes the di¤erent criteria of validity that could be set at measuring the

consequences of tra¢ c injuries:

� Statistical validity (errors and variations of mean values);

� Internal validity (logical descriptions of health states, especially at self-assessment);

� Theoretical validity (coverage of all dimensions of health, and their contribution

to the general health state, as well as coherency with the medical theory);

� External validity (agreement of results between di¤erent indices or studies);

� Practical relevance (the extent to which the results can be applied to decision-

making).

4.3 Evaluating the direct health e¤ect

Once the health metric has been selected, the direct health e¤ect of a road crash

must be estimated. Now imagine that an individual has a tra¢ c accident, and

we can evaluate his or her post-injury health status. Let me imagine the health

state of this individual under the unrealistic scenario in which the accident did not

happen (potential health status). The actual loss of health would equal the di¤erence

between the values at post-injury and potential health states.

The post-injury health status is assumed to be well-known by the analyst. If

the chosen metric corresponds to the second group (seriousness of the injuries), it

is relatively easy to obtain trauma-care information from hospital databases, such

as hospital discharge registers. If the aim remains to measure the quality of life

(QoL), health-related surveys are the most common choice. Both sources may be

complementary, in that some speci�c surveys can embrace questions from which

measures such as AIS or ISS can be deduced and, moreover, it is increasingly common

in hospitals, residences and trauma centers to distribute a questionnaire to patients

regarding their QoL. The selection of the source in�uences the time that is taken

for evaluating the health e¤ect. The survivor is expected to recover gradually, but

maybe not achieve his/her previous health state. Thus, in some cases data from

hospitals may report health states at a point in time previous to when the a¤ected

individual has restored their health to a maximum. By using those data as a proxy

of the �nal chronic health state of the individual, the health impact of the injury

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can be biased.

What I call potential health status is unidenti�ed, since it is impossible to know

what the state of health of the individual would have been had the accident not

occurred. The problem is how to approximate this unknown potential health status.

Many authors consider the health state prior to the accident (pre-injury status)

as a proxy of the potential health state. Nonetheless, even under such an assumption,

the evaluation of a pre-injury health status could be complicated, due to the lack

of data. Consequently, the majority of earliest studies in this area considered the

pre-injury health state as that of �perfect health�(Elvik, 1995; Bishai and Hyder,

2006). However, under that assumption, only a rough approximation to the actual

magnitude of the loss due to the injury was obtained.

A di¤erent and more recent strategy for approximating the potential health state

of the people injured consists of obtaining information from other people, rather than

the injured individual per se. In other words: imagine that we can �nd information

(dated prior to the accident) regarding the health state of an individual or a group of

people who did not su¤er a road tra¢ c crash; assume that the individual (or group

of people) is highly comparable to the injured one, since they coincide in several

factors (such as age, gender or usual daily activities). Therefore, the health state of

that individual or comparison group can be taken as a proxy of the potential health

state of the victim.

The approaches suggested previously (pre-injury status and comparison group)

are highly connected, and can be easily combined. In fact, the use of comparison

groups to approximate the pre-injury status is the most common choice nowadays

(see MacKenzie, 2001). This methodology is mainly based on the use of population

norms that provide some benchmark against which to compare pre-injury outcomes.

This methodology is improving, and norms are calculated for groups of the popula-

tion with di¤erent characteristics among them, becoming narrow partitions of the

total population. Nowadays, the recommendation is to use di¤erent health baselines

for men and women as well as for di¤erent age-groups.

I must remark that the selection of a comparison group should be performed

carefully. As I mentioned in the introduction, data show that tra¢ c crashes are

not random, but are more likely to happen to people with particular attributes

(e.g., men aged 15�29 years). Therefore the health state of the comparison group

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Chapter 4 Estimating Health E¤ects

and the post-injury health state of the victim cannot be unconditionally compared.

In the case where extra information about the people injured is available (e.g.,

socioeconomic variables), the comparison group can be de�ned quite accurately by

using statistical techniques, usually embraced in the literature of �treatment e¤ects�

(Challmers, 1997). The more partitioned the comparison groups are de�ned, the

more accurate the estimation becomes.

4.4 Combining values for health with life exten-sion

Once I have chosen the method for measuring the quality of a concrete health state

and the health e¤ect valuations have been determined, I must deal with the question

of how to estimate the total health e¤ect throughout the life of the individual. In

order to do so, it is commonly considered a generic age-health pro�le for any a¤ected

individual, representing the valuation of his or her life from birth to death (Gold

et al., 2002). The descending shape of the curve is based on the rationality of

deterioration of health with aging. By considering a continuous metric for QoL,

which ranges from zero (death) to one (perfect health), the area under the curve

from any two points in life t to t�stands for the total valuation of the health state

over that portion of his or her lifetime (t; t�) (see Figure 4.1).

Figure 4.1. Possible bias in the evaluation of nonfatal injures.

As an example, I can assume that an individual su¤ers a major injury in time

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T . When computing the health e¤ects of that injury, most cost�e¤ectiveness studies

implicitly make the following assumptions:

� The post-injury health state is a chronic health state (both curves are parallel).

� Life expectancy does not change because of the injury.

The use of previous assumptions implies a rough approximation to the actual

magnitude of the injury. Figures 4.1 & 4.2 illustrate the cases of a nonfatal and a

fatal road crash, respectively. The largest irregular curve de�ned from birth to death

represents the potential health status of an individual, from birth to life expectancy.

An injury occurs at time T , which deteriorates the health of the individual. In the

scenario of a nonfatal injury, the valuation of the post-injury status (w1) remains

decreasing from T to death. If the accident causes death, the post injury status

falls to zero. The value w0 stands for the estimated potential health status. In both

�gures, I assume that the potential health valuation is known, and thus, the value

w1 � w0 represents the direct health e¤ect, presumably unbiased.The area between the curves that represent the potential health state and the

post-injury health state (the horizontal axis in the case of fatal crashes), would

represent the true health losses due to the collision, from the moment the accident

happens up to the individual�s death (in nonfatal crashes), and from the moment the

accident happens up to his or her life expectancy, in the case of a fatal crash. Under

the two assumptions, the estimated health status turns into a constant function �

the potential health pro�le equals w0 , and the post-injury status equals w1 , from

T to life expectancy. The area of the rectangle de�ned by these constant functions

is an estimation of the health loss.

The di¤erence between both areas does not clearly indicate whether the health

e¤ect is biased or not. Indeed, the e¤ect of the injury may imply a �change of

level�in health (i.e., the handicap induces a constant decrease by age at the same

rate it would have decreased had the accident not occurred). Thus, the conventional

methodology would be likewise compelling, by adjusting for the proper direct health

e¤ect. However, in the case of fatal injuries, where the estimated injury status is

taken as the null function (Figure 4.2), the two assumptions lead to overestimating

the actual health loss.

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Figure 4.2. Possible bias in the evaluation of fatal injures.

4.5 Aggregating health e¤ects among individuals

Quality-adjusted health measures can be interpreted as utilities or as health indices.

The utilitarian interpretation identi�es the aggregation problem as a major one: all

traditional welfare aggregation problems stand here in a prominent way. Under the

extra-welfarist interpretation, however, the metrics are interpreted as health indices

rather than health utilities, solving the aggregation problem.

Even taking into account that aggregation procedures are linked to the selected

instrument, there is still a lack of consensus regarding the form of combining results

from di¤erent groups of population.

For instance, there is a debate on the use of the age-weighting function originally

proposed in the Global Burden of Disease study 1996, still most widely applied in

disability-adjusted life years calculations. Also, the so-called �fair innings�argument

(see Williams, 1996) claims that everybody should enjoy the healthiest life possible,

but until a certain age (70�75 years). Other general discrepancies can be found

when talking about aggregating the e¤ects of mortality and morbidity into a single

�gure (Elvik, 1995). Furthermore, it is worth mentioning the �worst-o¤�rst�criteria

and the notion of double jeopardy (the idea that disabled people are disadvantaged

twice in aggregate data). An exhaustive discussion of these distributional and ethical

considerations can be found in (Hofstetter and Hammitt, 2001).

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4.6 Data

Besides the previously mentioned theoretical di¢ culties, another major issue is as-

sociated with the lack of available data.

Mortality and morbidity data are usually collected from diverse sources of infor-

mation, such as police databases, hospital discharge registers, forensic reports, health

surveys and insurance companies. These databases should be taken as complemen-

tary, not only because some of them only provide partial data, but also because

of di¤erences in the data collection methodology, which may lead to di¤erences in

results (Pérez et al., 2006).

If I focus on the evaluation of QoL lost due to nonfatal crashes, having access

to proper data is still a major issue. The days immediately after the road crash are

considered critical for the injured. The survivor is expected to recover gradually,

but not necessarily achieve his or her previous health state. The ideal is to estimate

the chronic sequelae that a tra¢ c crash can produce in those a¤ected, and to eval-

uate the impact of these sequelae in their daily living. In the most advantageous

scenario, the post-injury health status can be obtained directly from the people

injured. However, some authors claim that adaptation to a moderately disabling

chronic illness is associated with a response shift (Yardley and Dibb, 2007). There-

fore, the recommendation is to use then-tests to collect this information (Ahmed

and Mayo, 2005).

Besides the di¢ culties of obtaining state-independent health measures, there is

still the problem of estimating the potential health state from external comparison

groups. These comparative health values are mostly obtained from Health Surveys.

However, health information is usually only available at an ordinal level (see Van-

Doorslaer and Jones, 2003), with questions such as: �In your opinion, how is your

health in general?�where respondents must choose, for instance, one category out

of �excellent, very good, good, fair or poor�. Such questions do not provide the

cardinal health scale needed for estimating the generic life path.

Since categorical measures of health are one of the most commonly used indica-

tors in socioeconomic surveys, a wide variety of methods has been developed with

the aim of dealing with the proper cardinal counterparts of ordinal health measures

(e.g. Van Doorslaer and Jones, 2003; Groot, 2000).

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Related to the estimation of health losses, databases are becoming more com-

plete. The European Road Accident Database, the International Road Tra¢ c and

Accident Database and the Co-operative Crash Injury Study are examples of the

improvement in the data collection, and they include a wide set of variables related

to road crashes that some decades ago were ignored. However, there is still much to

do in order to have a complete set of data that comprises all required information

(e.g., details of the accident and description of the health state of the injured indi-

vidual, and panel data). Meanwhile, the short-term objective consists of obtaining

the best estimation of health losses under the limitation of the lack of available data.

4.7 Concluding remarks

Currently, measures of disability and health-related quality of life are becoming

important, even essential, parameters in the evaluation of treatment and prevention

strategies for reducing the burden of injury. Hence, the evaluation of the costs and

bene�ts of such novel instruments is essential. In order to pursue this task, and to

allow a comparison among analyses of di¤erent measures, I should express the total

toll of deaths, injuries and sequelae derived from tra¢ c accidents in a simple metric,

which could estimate the total level of avoidable loss-of-health.

Several measures have been developed in this direction. For a start, monitoring

health-related quality of life can be enhanced by establishing equivalences between

cardinal and categorical health variables, since the former are the preferred measures

for cost�e¤ectiveness analysis, but the latter are more frequently enclosed in surveys.

Furthermore, overcoming typical assumptions, such as considering health states as

chronic health states, or pre-injury health status as perfect health, can be considered

as a giant step forward. For instance, given the lack of pre-injury measures, the use

of appropriately de�ned comparison groups should be crucial for the study of trauma

outcomes. Also, panel data can help in post-crash health status.

Both govermental and nongovernmental institutions are showing an increasing

interest in the prevention of road-tra¢ c injuries and deaths. It is important to

commence a systematic and accurate collection of data related to road crashes and

health states. These factors will allow for an improvement in the estimation of

health e¤ects, which will lead to an improved knowledge of the signi�cance of the

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Chapter 4 Estimating Health E¤ects

problem.

It is also possible, and desirable, that health and road policies will base their

decisions on cost�e¤ectiveness analysis in the near future. Finally, and perhaps

most crucially, it is possible that we will witness great changes in attitude of the

general population, as we gradually become aware of the actual risk related to road

crashes.

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Part III

Health Preferences

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Chapter 5

Generalizing Quality-Adjusted LifeYears in a Context of Certainty

5.1 Introduction

The analysis of individual preferences for health states and health pro�les over time

is the basis of the theory that entails the concept of QALYs. The most general

approach deals with complete sets of health pro�les under uncertainty (e.g. see Ble-

ichrodt and Gafni, 1996). There exists a widespread literature about the appropriate

characterization of preferences over lotteries over health outcomes, rather than over

health outcomes themselves. Chronic health states are a particular case of health

pro�les. Several studies have also addressed the estimation of preferences framed

in the space of lotteries de�ned over chronic health states (e.g. Bleichrodt at al.,

1997).

The simplest scenario dealing with chronic health states (under the assumption

of certainty), has usually been focused as a particular case of the scenarios above

mentioned. One possible disadvantage is that the methodologies are sustained by

assumptions that have been stated over spaces that have been di¤erently de�ned;

thus they may be misleading some aspects that could be linked to these kind of

preferences.

In this paper, we restrict my attention to individual preferences over chronic

health states under certainty. That is, let the pair (x; t) designate the situation

where a person lives for t years in health state x and then dies. Following the

classical notation, if the pair (y; s) is not strictly preferred to the pair (x; t); then I

write (x; t) % (y; s):

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The de�nition of an utility function is an usual tool for estimating preferences.

It has often been criticized, but it is still the prevailing theory (Bleichrodt et al.,

2007). This utility approach holds if there exists a function U from the set of chronic

health states to the real numbers, called the utility function, such that:

(x; t) % (y; s) i¤ U(x; t) � U(y; s):

In addition, the classical structure of QALYs for evaluating chronic health status

is the following: U(x; t) = u(x)D(t); where u(x) is a continuous, increasing function

that represents the atemporal preferences over health states, and D(t) stands for a

discounting factor. These assumptions lead to the usual model of preferences:

(x; t) % (y; s) i¤ u(x) � D(s)

D(t)u(y) (5.1)

Assuming the existence of such a function, many studies are aimed to characterize

the form of D(t), by introducing di¤erent requirements to the preferences structure.

Let as review some of them:1

Bleichrodt and Gafni (1996) study the intertemporal preferences under certain

assumptions (completeness and transitivity of %, and cardinally coordinate inde-pendency, impatience and stacionarity of health pro�les). For the particular set of

chronic health states under certainty, the authors allow for two di¤erent models of

discounting factor (the models are exclusive): D(t) = t and D(t) = 1 � �t; where0 < � < 1:

Also Bleichrodt, Wakker and Johannesson (1997) provide a characterization of

QALYs by means of risk neutrality. They state that if condition 0 holds, and for

every health state risk neutrality holds for life years, then the preferences sustain the

relation (5.1) with D(t) = t:The authors claim also that the previous assumptions

imply both utility independence and constant proportional trade-o¤s.2

However, some contradictions arise from empirical evaluation of preferences. In

descriptive analysis, it is common to �nd preference reversals, what can be derived

from framing e¤ects, but can also be a sign of intransitivity. The latter justi�cation

1A more detailed review about the discounting factor can be found in Fredericket al., 2002.2The constant proportional tradeo¤s assumption holds if, for all health states x and y, there

exists a positive number q such that U(x; t) = U(y; qt) for all life durations t (Bleichrodt et al.,1997)

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is extremely signi�cant, since the absence of transitivity questions the existence of

an utility function that could represent the intertemporal preferences.

The objective of this paper is to perform an axiomatic characterization of indi-

vidual preferences over chronic health states, under certainty. I establish a maximum

life horizon TM , and do not assume transitivity for the preferences. I study under

what conditions the following model holds:

(x; t) % (y; s) i¤ u(x) � �(s; t)u(y) (5.2)

where �(�; t) is increasing, with �(0; t) = 0, and �(s; t) = 1=�(t; s):If transitivity does not hold, �(s; t) is not a separable function. Therefore it is

impossible to deduce any common utility function U(x; t) for every (x; t): Thus, if

I am analyzing the individual preferences between (x; t) and (y; s); the evaluation

depends on (i) the atemporal valuation of x, (ii) the duration of x, and (iii) the

duration of the alternative health state y. This means that chronic health states are

only comparable by pairs, and thus it is not possible to establish a ranking.

The model (5.2) presents several advantages. First, it does not contradict the

usual assumption about the existence of an atemporal order for health states, that

is, it allows for transitivity in the atemporal preferences over health states. Second,

the model allows for deriving the discount factor in an endogenous way, and it is

also possible to estimate the trend of discount rates in the case where transitivity is

not assumed. Third, the basis of this analysis is independent from uncertainty, and

thus the model does not induce the usual dilemmas about revealing preferences in

contexts of uncertainty (Tsuchiya and Dolan, 2005).

Notice also that the models revised above are particular cases of (5.2). In general,

if the preference relation % is assumed to be transitive, the discount factors revisedabove, correspond to the particular case in my theorem where transitivity applies,

that is, �(r; t) = �(r; s)�(s; t): This fact is specially relevant since the results have

been derived from assumptions that do not coincide to the usual ones. Thus, the

model (5.2) can be interpreted as a genuine generalization of the QALY models.

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Chapter 5 Generalizing Quality-Adjusted Life Years in a Context of Certainty

5.2 A general theorem for time preferences overchronic health states

5.2.1 Main De�nition

Let H be the set of all heath states strictly better than "death". I assume that H is

a connected metric space, and let �H be a topology de�ned over H. I assume that

�H is a separable topological space.3

I interpret H as an (atemporal) outcome space. Life-time will lay inside the

interval T := (0; TM), where T M denotes the maximum age considered for that

individual.

Thus, the preferences of the decision maker are de�ned over the space

:= H � T

An element (x; t) designates the situation in which the individual lives in the

chronic heath state x for t years, followed by death. As usual, I metrize T by the

Euclidean metric, and by the product metric. Note that if 0 2 T were considered,

then the pair (x; 0) would be equivalent to "immediate death", for every x 2 H ,

and thus (x; 0) v (y; 0) and (x; t) � (y; 0) for any x, y 2 H, t 2 T:4 Since this isan special case that presents no problem at comparing with other outcomes, I omit

the possibility of t = 0.

Therefore, I interpret the statement (x; t) % (y; s) as meaning that the individualprefers to live in a heath state x for t years until death, to live in a heath state y

for s years until death.

For each t 2 T , by the tth-time projection of %, I mean the binary relation %t onH de�ned as x %t y i¤ (x; t) % (y; t):Therefore, %t is interpreted as the preferencerelation of the individual over chronic health states that endure t years.

I consider the de�nition of Time Preference stated by Ok and Masatlioglu in

2007:3The structural assumptions of connection and separability of the set of health states have been

exploited by many authors (see Bleichrodt and Gafni, 1996). Note also that the health state of"perfect health" is contained in H

4These results related to t = 0 are based on the "Condition 0", stated by Bleichrodt et al.,1997.

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De�nition 5.1. For any metric space H, a binary relation % on is said to be a

time preference on if it satis�es the following conditions:

(i) % is complete and continuous,

(ii) %t is complete and transitive,

(iii) %t=%t0 for each t, t0 2 T

The completeness and continuity of a time preference % are standard require-

ments. This is a more general de�nition of time preference since it doe not consider

transitivity of %, since this allows for preference cycles that may arise solely due tochanges in life duration. The third requirement can be seen as a direct application of

the property of "utility independence of quality and quantity of life" under certainty

(see Bleichrodt et al. 1997).

5.2.2 Axioms for Time Preferences

All axioms are imposed on %.

(A1) (Time Sensitivity). For any x ; y 2 H, and t 2 T , there exists an s 2 T suchthat (x; t) % (y; s):

This axiom says that any chronic health state would be undesirable if the time

that the individual is living in it is su¢ ciently small. This assumption is quite

reasonable, since the proximity to death could play a signi�cant role in the statement

of preferences. In other words, even if health state y were far better than x (imagine

y � "almost in perfect health", and x �"almost dead"), it is rational to considerthat it could exist an s 2 T small enough (e.g. s �"two hours") so that (y; s) cannot be preferred anymore.

(A2) (Outcome Sensitivity). For any x 2 H, and t, s 2 T , there exists y 2 H;such that (x; t) % (y; s):

This axiom says that any life-time lenght would be undesirable if the health state

that the individual is living in it is su¢ ciently poor. Again, this assert is founded

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on the fact that proximity to death could play a signi�cant role in the statement

of preferences. Observe that this axiom also implies the non existence of a minimal

element on H (by taking t = s), but it does allow the existence of a maximal.

(A3) (Monotonicity). For any x, y, z 2 H, and s, t, r 2 T , if r � t and z %t x,then

(x; t) % (y; s) implies that (z; r) % (y; s)

Moreover, if either % or %t holds strictly, then (z; r) � (y; s):

The previous axiom re�ects the idea that % is increasing in both time and qualityof health states.5

Note that (A3) implies clearly a symmetric result: if (x; t) % (y; s), y %t z ands � r, then (x; t) % (z; r):

(A4) (Separability). For any x, y, z, w 2 H and s1; s2; t1; t2 2 T , then

(x; t1) � (y; s1)

(z; t1) � (w; s1) imply that (z; t2) � (w; s2)

(x; t2) � (y; s2)

In order to illustrate this property, let s1 < t1 (by (A3) it follows that y �tx;w �t z; s2 < t2). Suppose that the individual is disposed to lose t1 � s1 years oflife, in order of living in health state y instead of x, and the same quantity of years,

in order of living in health state w instead of z. Imagine that the individual is also

disposed to lose t2� s2 years of life, in order of living in health state y instead of x:So, if the agent�s evaluation of "time" is independent of the improvement of health,

then the number of years that she is willing to renounce for living in health state w

instead of z should be t2 � s2:This is a separability condition analogues of which are widely used in certain

branches of decision theory. In the theory of additive utility representation this

property is called either the Reidemaister condition (Debreu, 1960; Krantz et al.,

5Although the monotonicity in duration is a structural assumption in QALY models, someauthors have found violations of this property (see Sutherland et al.,1982; Dolan, 1996, amongothers)

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Chapter 5 Generalizing Quality-Adjusted Life Years in a Context of Certainty

1971; Wakker, 1988) or the corresponding tradeo¤s condition (Keeney and Rai¤a,

1976.). It is also commonly used in non-expected utility theory (Wakker, 1994;

Abdellaoui, 2002).

Figure 5.1 illustrates this axiom.

t2 s2

0

1

H

T1s1t1

x

y

z

w

Figure 5.1. Separability (A4)

To be able to fully separate the evaluation of time and prizes, I need an additional

independence requirement:

(A5) (Path independence). For any x, y, z, w 2 H and t1; t2; t3 2 T , then

(x; t1) � (y; t2)

(z; t1) � (w; t2) imply that (x; t2) � (z; t3)

(y; t2) � (w; t3)

For illustrating the meaning of this axiom, let us take t1 > t2 > t3, andH = (0; 1].

The hypothesis (x; t1) � (y; t2) and (y; t2) � (w; t3) can be interpreted as follows: ifthe individual is living in a health state x for t1 years, then the individual is disposed

to decrease the life time in t1� t2 years, and subsequently decreasing it in t2� t3, ifshe is given in return a total increase (in quality of life) of (w� y)+ (y�x) = w�xunits. Similarly, by taking the inequalities (z; t1) � (w; t2) and (x; t2) � (?; t3), theindividual is disposed to decrease the life time in the same amount and steps as

before, if she is given in return a total increase (in quality of life) of (?�x)+(w�z)units. (A5) sets that ? = z; what guarantees that the aggregate increase in health

is indepentent on the order in which the health states are selected.

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Figure 5.2 provides a visual illustration of (A5).

x yt1 t2

z w

t1 t2

t2

t3

t2

t3

Figure 5.2. Path Independence (A5)

(A6) (Monotonicity in outcomes). %t is strictly increasing on H:

See that (A6) provides the possibility of characterizing H as a nonempty, open

interval in R, where higher numbers correspond to better health states. Thus, I canestablish a bijection between H and the interval (0; 1], where 0 and 1 stand for the

health state "death" and "perfect health", respectively. Without lack of generality,

I normalize the maximum amount of time allowed, TM , to 1; that is, T = (0; 1).

(A7) For any s; t 2 T; with s � t; there exists x 2 H such that (x; t) � (1; s):

Note that (A3) implies the uniqueness of that x; another direct implication of

(A3) and (A7) is that for any w 2 (x; 1], (w; t) � (1; s):For any s; t 2 (0; 1), with s � t; I de�ne hst as the health state in (0; 1] such that

any health state better than hst is preferred in t to any health state in s :

hst 2 H such that 8x 2 H with x �t hst, then (x; t) � (y; s);8y 2 H

Note that (A7) assures the existence of that element. I extend hst to T 2; by

setting hst = 1 if s � t: Notice also that (A3) and (A7) imply that hst is unique,increasing in s; decreasing in t:

A last axiom is required for establishing coherency for hst: This assumption

guarantees that "Condition 0" holds, and assures continuity of hst as a function of

(s; t) :

(A8) hst is a continuous function of s and t; and hst ! 0 if s! 0

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Figure 5.3 shows the partition that hst establishes in (0; 1]. Figure 5.4 illus-

trates the behaviour of hst when s changes.

s t0

1

H

T1

Strictly preferred toany health state in s

sth

Figure 5.3. De�nition of hst

s1 t0

1

H

T1s2s3

tsh1

tsh2

tsh3

Figure 5.4. Changes in hst

5.2.3 Main Result

In the context de�ned in this section, I obtain the following characterization of

health preferences over chronic health states.

Theorem 2. Let H be a connected metric space. Let �H be a topology de�ned over

H, and assume that �H is separable. Let % be a binary relation on = H � T : % isa time preference on that satis�es the properties (A1)-(A8) if, and only if, there

exist a continuous and surjective map u : H ! (0; 1] that represents the atemporal

preferences %t over H, and a continuous map � : T 2 ! R++ such that, for all x, y2 H and s, t 2 T ,

(x; t) % (y; s) i¤ u(x) � �(s; t)u(y) (5.3)

while (i) �(�; t) is increasing, with �(0; t) = 0,and (ii) �(s; t) = 1=�(t; s):

Moreover, the discount factor �(s; t) takes the following shape:

�(s; t) =u(hst)

u(hts)=

�u(hst) , if s < t1

u(hts), if s � t

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where hst 2 H such that 8x 2 H with x �t hst, then (x; t) � (y; s);8y 2 H:

Proof. (See Apendix 1)

The theorem claims that for an individual to establish preferences over chronic

health states (x; t) and (y; s), it must be considered the utility of the health states in

comparison, u(x) and u(y);as well as the importance of life-time for the individual

in question, �(s; t): Moreover, the discount factor between s and t depends on a

particular health stated called hst: Throughout the proof of the theorem, for s �t, hst is found to be the health state that enduring t years (and then death), is

equivalent to live in perfect health for s years, until death. That is,

(perfect health, s) � (hst; t) (5.4)

In the present study, it is assumed the existence of hst (assumption A7). However,

the existence of such a health state is the key of the well-known Time-Tradeo¤

procedure (TTO) for estimating quality weights. For instance, let x stand for "being

deaf". The TTO procedure estimates the number of years (say, t) that makes

indi¤erent (living 10 years while being deaf) and (living t years in perfect health): In

my terminology, I write (x; 10) � (1; t): The relative utility of being deaf, u(x)=u(1),is usually assessed through the ratio D(t)=D(10): Under this scenario, the theorem

states that (x; 10) � (1; t) is equivalent to the expression u(x) = �(t; 10)u(1), that

is, u(x)=u(1) can be estimated by terms of �(t; 10): Notice that, by de�nition, the

map �(t; s) embraces D(t)=D(s) as a particular case.

Notice that the de�nition of �(s; t) is consistent with the result of the theorem.

If s is close enough to 0; by (A8) also hst ! 0 (s << t), and since �(s; t) = u(hst); in

the limit �(0; t) = u(h0t)! u(0) = 0: Then any health state is t is better than any

health state in s if s! 0 (immediate death); because u(x) � �(s; t)u(y)! 0: Also,

if t tends to 0; in the limit �(s; 0) = 1=u(h0s)! +1: This shows that the property�(s; t) = 1=�(t; s) also holds in the limits. Figure 5.5 illustrates the behavior of the

discount factor for every pair of time lenghts (s; t) in the already normalized time

dimension T � T = (0; 1)� (0; 1).

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s

t

0

1

1

s = t

s < th(s,t) < 1

h(s,t) = 1

T

T

h(s,t)à 0

h(s,t)à +inf

sà 0

tà 0s > t

h(s,t) > 1

Figure 5.5. Scheme of the values for the discount factor

5.2.4 Transitive time preferences over health outcomes

In this section I characterize transitive preferences by means of Theorem. For sim-

plicity, in this section I do not normalize the time dimension T = (0; TM) to (0; 1);

what allows me to consider that 1 2 T:Firstly I state the following property (the proof is straightforward):

Proposition 5.1. A time preference that can be represented as in Theorem 2.1 is

transitive if and only if �(r; t) = �(r; s)�(s; t)

For instance, see that if r < s < t; transitivity holds i¤ u(hrt) = u(hrs)u(hst)

Corollary 3. Let H be a connected metric space. Let �H be a topology de�ned over

H, and assume that �H is separable. Let % be a binary relation on = H � T: %is a transitive time preference on that satis�es the properties (A1)-(A8) if, and

only if, there exist a continuous and surjective map u : H ! (0; 1] that represents

the atemporal preferences &tover H, and an increasing, continuous map d : T !(0;+1) such that d(0) = 0, d(1) = 1; and for all x, y 2 H and s, t 2 T ,

(x; t) % (y; s) i¤ d(t)u(x) � d(s)u(y)

Moreover, the discount factor d(k) takes the following shape:

d(k) =u(hk1)

u(h1k)=

�u(hk1) , if k < 11

u(h1k), if k � 1

�where hst 2 H such that 8x 2 H with x �t hst, then (x; t) � (y; s);8y 2 H:

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Notice that the corollary frames the de�nition of QALYs, since it allows for the

existence of a map U(x; t) that establishes the preferences over chronic health states

as follows: (x; t) % (y; s) i¤ U(x; t) � U(y; s): Moreover, since the corollary sets

a double implication, the existence of such a map implies that the time preference

over health states must satisfy the axioms (A1)-(A8). Also, in the corollary, it

is necessary to set a �xed amount of time (I have chosen t = 1), since the results

are obtained in comparative terms to that time. In the particular case of quality

weights obtained by the TTO methodology (see example in (5.4)), the set amount

of time is t = "10 years".

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5.3 Appendix

For any topological spaces X and Y , I denote the set of all homeomorphisms that

map X onto Y by Hom(X; Y ), but I write Hom(X) for Hom(X;X).

Given the preference relation % (R), I denote as � (I) the symmetric part of

% (R).The proof is based on the result provided by Jarczyk et al. (1994), I have

followed the thread established by Ok and Masatlioglu (2007). For this purpose,

it is neccessary to adjust the current preference model to �t in the one settled by

these authors. The major dissimilarity between both models lies in the lack of a

maximal element in the outcome space, in Ok and Masatlioglu�s framework. The

existence of a maximal element in the set of health states (perfect health) con�icts

the second axiom stated in Ok and Masatlioglu (2007); that is, in the context of

health states, it cannot be assured the existence of z 2 H such that (z; s) % (x; t),for any x 2 H; s; t 2 T:This drawback is overcome as follows: I de�ne a new set of outcomes H, that

embraces H, in addition to imaginary health states, strictly better than "perfect

health". I also de�ne the time preferences over H �T , such that if these preferencesare restricted to the inicial set H, they coincide to % : I proof that the axioms (A1)-(A8) can be generalized to the new space H �T: Under this arti�cial framework, Ican follow the procedure established in Ok and Masatlioglu (2007). Once the result

is obtained for the health-time space H �T; I restrict it to the inicial H � T:

5.3.1 De�nition of preferences over H �T

I de�ne a more general set that embraces H; that will be called H: In order to clarify

ideas, the chart below summarizes the di¤erent preferences my work is dealing with:

H : %t �t �tH � T : % � �

H: Rt Pt ItH� T : R P I

Let (H;�H) be a connected and separable metric space, and % a time preferenceon = H � (0; TM) that satis�es (A1)-(A8). Without loss of generality, I takeH = (0; 1] and TM = 1

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I de�ne the set H =H [ (1; 2):Every new element x in (1; 2) stands for imaginaryhealth state.

Now, given s; t 2 (0; 1), I de�ne a function st(y) that will help me to de�ne thepreferences over health states in H as follows:6

x; y 2 (0; 2)If x 2 (0; 1] and y 2 (0; 1] ! (x; t)R(y; s) i¤ (x; t) % (y; s)If x 2 [1; 2) and y 2 [1; 2) ! (x; t)R(y; s) i¤ (2� y; t) % (2� x; s)If x 2 (0; 1) and y 2 (1; 2) ! (x; t)R(y; s) i¤ x � hst(y)

hst(y) must hold the following requirements:

(R1) hst : (0; 2)! (0; 2)

(R2) hst(hts) = hst and hst(2� hst) = 2� hts:

(R3) hst is continuous, bijective and strictly increasing in (0; 2)

(R4) �1hst = hts

(R5) hst is continuous and increasing in hst

(R6) hst(y)! 0 if s! 0, and st(y)! 2 if t! 0

(R7) If s 6= t, then hst has no �xed point into (0; 2)

(R8) hst must hold that for any y, z, w 2 H and r; s; t 2 T , then (w; t) � (y; s)and (w; s) � (z; r) imply that hts(z) = hsr(y)

(R9) hst must hold that for any x; y; z; w 2 H and r; s 2 T such that (x; r) � (y; s)and (z; r) � (w; s); then hst(y) = 2� z i¤hst(x) = 2� w

See that Requirements (R1) and (R3) imply that, in the limit, hst(0) = 0

and hst(2) = 2: Properties of hst and (R5) imply that hst is increasing in s and

decreasing in t:

Requirements (R8) and (R9) are necessary for extending the axiom about path

independence (A5) from H to H. The interpretation of these requisites is similar

to the interpretation of (A5), but it also involves health states in (1; 2): Figures

5.6 and 5.7 provide a graphycal illustration of these properties.6The de�nition implies that if x 2 (1; 2) and y 2 (0; 1), then (x; t)R(y; s) i¤ y � hts(x)

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? yr s

z w

r s

s

t

s

t

x yr s

z wr s

s

t

s

t2­w 2­z

Figure 5.6. (R8) Figure 5.7. (R9)

It is relatively easy to �nd functions that support the requirements (R1)-(R7).

Yet thus far it has not been demonstrated the existence of a function that also

holds (R8) and (R9), that is considered a rather reasonable property. I must state

a conjecture dealing with the existence of hst :

Conjecture 4. A function that holds requirements (R1)-(R9) exists.

Hereafter the argument h in hst will dropped out to simplify notation. I must

keep in mind that the expression st = rq implies that hst = hrq, and not s =

r; r = q:

Figures 5.8 and 5.9 provide a graphical illustration of a potential function st:

Notice that, by construction, st can be partitioned into three areas:

st : H �! H(0; hts] �! (0; hst]

(hts; 2� hst] �! (hst; 2� hts](2� hst; 2) �! (2� hts; 2)

Notice that requirements (R2) and (R3) imply that for every y 2 (hts; 2 � hst],st(y) � y if s � t and st(y) � y if s � t: Note also that if s = t then st = Id:

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10

2

H

2

1

y

Yst

s>t s=t

s<t

hts

2­hts

Yst(y)

hs’t’

ht’s’

hst

2­hs’t’

Ys’t’

y’

Ys’t’(y’)

ht’s’hst

H

Figure 5.8. Some possible indiference curves

10

2

2

1

s>t s=t

s<t

hs2t

Ys1t Ys2t

hs1t

s2 < s1

Yst1

Yst2

ht1sht2s

t2 < t1

H

H

Figure 5.9. Changes in st

By construction, H does not contain either its minimal or its maximal. Also by

de�nition H is a connected metric space, and the topology de�ned over H , �H ,

maintains the separability assumed for �H : Figure 5.10 illustrates the casuistry of

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the de�nition of R (the reader should be cautious about the bounderies in each set).

1 20

y

x

1

2

(2­y,t)�(2­x,s)

x ³ Yst(y)(x,t)�(y,s)

(x,t) R (y,s) iff

y £ Yts(x)

Figure 5.10. Casuistry of the de�nition of R

The preferences over H (Rt) are induced by the preferences over H�T (R) asfollows:

x; y 2 (0; 2)If x 2 (0; 1] and y 2 (0; 1] ! xRty i¤ x � yIf x 2 [1; 2) and y 2 [1; 2) ! xRty i¤ 2� y � 2� xIf x 2 (0; 1) and y 2 (1; 2) ! xRty i¤ x � y

Note that for any value of x and y, xRty i¤ x � y: Thus the preferences Rt couldbe summarized as follows:

xRty i¤ x � y;

what coincides with the natural preferences on (0; 2). Thus, Rt is complete and

transitive, and Rt = Rs for each t; s 2 (0; 1):

H Rt Pt It(0; 2) � > =

Let me recall the de�nition of time preferences that is stated in this work :

"For any metric space H, a binary relation % on is said to be a time preference

on if it satis�es the following conditions: (i) % is complete and continuous, (ii)

%t is complete and transitive, and (iii) %t=%t0 for each t, t0 2 T . For proving thatfor the metric space H ; the binary relation R is time preference on H �T; properties(ii) and (iii) have been already shown. It is only necessary to prove (i), that is, R

is complete and continuous.

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The completeness of R can be stated straightforward the de�nition and the

completeness of % :The characterization of continuity I am going to use is the following: R is contin-

uous if it has the closed graph property, that is: lim am R lim bm holds for famg; fbmgsuch that am R bm 8mFigure 5.11 summarizes the analysis of continuity of R that I establish:

1 20

y

x

1

2

......

4

3

2

1

5

B

A

C

D

Figure 5.11. Casuistry of the analysis of continuity

Since % is assumed to be continuous, it can be easily proved that R inherits thisproperty in the interior of the sets B and D (de�ned in Figure 5.10), where the

preferences R rely on the preferences % : If sequences are taken such that the limitlies in the interior of A and C sets, it is also true that the closed graph property

hold, since the set f(x; t)j(x; t)R(y; s)g is de�ned as the hipograph (epigraph) of acontinuous function st. The values of possible discontinuity can be found at the

frontiers of the sets. I distinguish among �ve di¤erent situations (as it is illustrated

in Figure 5.10).

case 1

f(xm; tm)g ! (1; t); xm 2 (1; 2);8mf(ym; sm)g ! (y; s); ym; y 2 (0; 1);8m ; with (xm; tm)R (ym; sm) 8m

I have to show that (1; t)R(y; s), that in this case is equivalent to (1; t) % (y; s):

The fact that (xm; tm)R (ym; sm) implies that ym � tmsm(xm);8m: In the limit,y � ts(1): If t � s; (R2) and hst = 1 imply that ts(1) = ts(hst) = hts: Byde�nition of hts; y � hts implies that (1; t) % (y; s): If t > s; axioms (A1) and(A2) lead to (1; t) % (y; s). Therefore the continuity holds.�

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case 2

f(xm; tm)g ! (x; t); xm; x 2 (1; 2);8mf(ym; sm)g ! (1; s); ym 2 (0; 1);8m ; with (xm; tm)R (ym; sm) 8m

I have to show that (x; t)R(1; s); that in this case is equivalent to (1; t) % (2�x; s)

The fact that (xm; tm)R (ym; sm) implies that ym � tmsm(xm);8m: In the limit,1 � ts(x). If t � s; (R2) implies that 1 = 2 � hst = ts(2 � hts), and thusts(2 � hts) � ts(x): By (R3) 2 � hts � x, and so 2 � x � hts:By de�nitonof hts; I obtain that (1; t) % (2� x; s): If t > s; since 1 > 2� x; axioms (A1)and (A2) lead to (1; t) % (y; s)�

case 3

f(xm; tm)g ! (1; t); xm 2 (0; 1);8mf(ym; sm)g ! (y; s); ym; y 2 (1; 2);8m ; with (xm; tm)R (ym; sm) 8m

I have to show that (2� y; t) % (1; s)

(xm; tm)R (ym; sm)! xm � smtm(ym)! 1 � st(y): If t > s; hts = 1 and by (R2)st(2� hst) = 1 � st(y): Now applying (R3) I obtain 2� hst � y; and thus(2 � y; t) % (1; s): See that in this case the condition t � s does not apply:

y 2 (1; 2) ! y � hts ! st(y) � st(hts) = hst = 1 if t � s; what contradictsthe hypothesis.�

case 4

f(xm; tm)g ! (x; t); xm; x 2 (0; 1);8mf(ym; sm)g ! (1; s); ym 2 (1; 2);8m ; with (xm; tm)R (ym; sm) 8m

(xm; tm)R (ym; sm) $ xm � smtm(ym) ! x � st(1): If t > s, hts = 1; and

(R2) implies that x � st(1) = st(hts) = hst: By de�nition of hts; x � hst

involves that (x; t) % (1; s);or (x; t)R (1; s) : If t � s; by (R2) x � st(1) =

st(2� hst) = 2� hts ! hts � 2� x > 1; what is a contradiction. Thereforethe case t � s is not possible under the hypothesis. �

case 5

It can be easily shown that whatever the approach to (1; 1) is, the preferences keep

continuity and converge to (1; t)R(1; s)�

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5.3.2 Generalization of the axioms over H �T

Once I have properly de�ned the time preferences over the new set H, I analyse

whether the properties that the axioms establish for elements of H, can be gen-

eralised to any element of H, in the framework of the new time preferences. The

extended properties that will refer to H are called: (B1) as the one that generalizes

(A1); (B2) extending the result of (A2); etc. (A7) and (A8) deal with de�ni-

tions and properties of H, so that no extension is plausible. The axiom (A5) will

be extended to H after stating some results that do not depend on that axiom, but

they simplify a lot the proof.

(B1) (Time sensitivity). For any x; y 2 (0; 2), and t 2 (0; 1), there exists an s2 (0; 1) such that (x; t)R(y; s):

Proof. If I select x; y 2 (0; 1], (B1) is nothing but the property stated by (A1)�If x; y 2 [1; 2); then 2 � x; 2 � y 2 (0; 1]; and (A1) implies that given t 2 (0; 1)

there exists s 2 (0; 1) such that (2� y; t)R(2� x; s); that is, (x; t)R(y; s)�If x 2 [1; 2) and y 2 (0; 1] : simply by taking s = t; (x; t)R(y; t)�Finally, let me consider the situation where x 2 (0; 1]; y 2 [1; 2), for a given

t 2 (0; 1):The existence of an s 2 (0; 1) such that (x; t)R(y; s) is equivalent to theexistence of an s 2 (0; 1) such that x � st(y): But this is trivial, since by (R6)

we can �nd s 2 (0; 1) small enough such that x � st(y) > 0; that is the s that

provides the result.�

(B2) (Outcome sensitivity). For any x 2 (0; 2), and t, s 2 (0; 1), there exists

y; w 2 (0; 2); such that (w; s)R(x; t)R(y; s):

Proof. If x 2 (0; 1]; (R3) and (R4) assure the existence of w � ts(x); and by

de�nition of the preferences in H , I have that (w; s)R(x; t):The existence of y such

that (x; t)R(y; s) is directly implied by (A2).

If x 2 (1; 2), 2 � x 2 (0; 1); and thus (A2) implies the existence of z 2 (0; 1]such that (2�x; s)R(z; t); that is, (2�x; s) % (2� (2� z); t) or, what is equivalent,(2 � z; s)R(x; t): Calling w := 2 � z; I found w such that (w; s)R(x; t): Finally, lety 2 (0; 1] such that y � ts(x): This implies that (x; t)R(y; s)�

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(B3) (Monotonicity). For any x, y, z 2 (0; 2), and s, t, r 2 (0; 1), if r � t and

zRtx, then

(x; t)R(y; s) implies that (z; r)R(y; s)

Moreover, if either R or Rt holds strictly, then (z; t)P (y; s):

Proof. I assume that (x; t)R(y; s); r � t; z � x:I have to show that (z; r)R(y; s)

case 1 x; y 2 (0; 1]

If also z 2 (0; 1]; (B3) coincides with (A3) �If z 2 (1; 2) and r < s; then by (R3) and (R2) rs(z) > rs(1) = rs(hsr) = hrs:

Since hrs is increasing in r; hrs � hts; and the de�nition of hrs and the fact that

(x; t)R(y; s) imply that hts � y, and so rs(z) > y:If z 2 (1; 2) and r � s; then (R3) and (R7) also imply that rs(z) > rs(1) >

rs(y) > y:

Thus, I have obtained thatrs(z) > y; what is equivalent to say that (z; r)P (y; s):�Finally, in the present case it is always true that (z; r)P (y; t), so that the second

part of (B3) is immediate.�

Proof.

case 2 x; y 2 [1; 2)

(x; t)R(y; s) ! (2 � y; t) % (2 � x; s): Also z � x ! 2 � x � 2 � z: Since2�x; 2� y; 2� z 2 (0; 1]; then (A3) implies that (2� y; t) % (2� z; s): Since r � t;applying (A3) again, (2 � y; r) % (2 � z; s);that is equivalent to the expression(z; r)R(y; s)�Finally, if z > x or r > s, by (A3) then (z; r)P (y; s)�

Proof.

case 3 x 2 (1; 2); y 2 (0; 1)

First, note that x 2 (1; 2); (x; t)R(y; s) imply that y � ts(x):

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Since z � x; x 2 (1; 2); then z 2 (1; 2); thus, I have to show that y � rs(z)And that result is immediate by (R3): r � t! rs(z) � ts(z) � ts(x) � y �

Since st(y) is strictly increasing in y; the second part of (B3) is immediate.�

Proof.

case 4 x 2 (0; 1); y 2 (1; 2)

Firstly note that the previous condition ((x; t)R(y; s)) hold only if x � st(y)If z 2 (0; 1]: since r � t; z � x and st is decreasing in t (R5): z � x � st(y) �

sr(y), so z � sr(y) what is equivalent to say that (z; r)R(y; s)�If z 2 (1; 2): the previous result shows that if (x; t)R(y; s) and 1 � x , then

(1; t)R(y; s):Since z � 1; r � t; I am now located in case 2, and thus (z; r)R(y; s)�

Before enunciating (B4), let me state the following properties:

Property 1 For each s; t 2 (0; 1); the expression (hst; t)I(hts; s) holds.

Property 2 For any s; t; r; q 2 (0; 1); if (hst; r)I(hts; q), then hst = hqr:

Proof. If s � t, then hts = 1 and (hst; t)I(1; s) holds. Also, (hst; r)I(1; q) and hst � 1implies that hst = hqr:

If s > t, then hst = 1 and (1; t)I(hts; s) holds. Also, (1; r)I(hts; q) and hts � 1

implies that hts = hrq:

(B4) (Separability). For any x, y, z, w 2 (0; 2) and s1; s2; t1; t2 2 (0; 1), then

(x; t1)I(y; s1)

(z; t1)I(w; s1) imply that (z; t2)I(w; s2)

(x; t2)I(y; s2)

Proof. The chart above summarizes all possible cases that can be found, depending

on the values of x; y; z; w: I mark "1" if the value lies in (0; 1], and "2" if it lies in

(1; 2). Note that some cases can embrace di¤erent situations. Note also that the

cases (1; 2; 2; 1) and (2; 1; 1; 2) are not contemplated, since they are not sustainable

under the hypothesis.

case: 1 2 3 4 5 6 7 8 9x 1 2 1 1 1 2 1 2 1 2 1 2 2 2y 1 2 1 1 2 1 1 2 2 1 2 1 2 2z 1 2 1 2 1 1 2 1 1 2 2 2 2 1w 1 2 2 1 1 1 2 1 2 1 2 2 1 2

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case 1 x; y; z; w 2 (0; 1] : by (A4)�

case 2 x; y; z; w 2 (1; 2) :

(x; t1)I(y; s1)! (2� y; t1)I(2� x; s1)(z; t1)I(w; s1)! (2� w; t1)I(2� z; s1)(x; t2)I(y; s2)! (2� y; t2)I(2� x; s2)

By case 1: (2� w; t2)I(2� z; s2), (z; t2)I(w; s2)�

case 3 x; y; z 2 (0; 1]; w 2 (1; 2) :

(z; t1)I(w; s1)! z = s1t1(w): By Property 1, (hs1t1 ; t1)I(ht1s1 ; s1); and applying

case 1:

(hs1t1 ; t1)I(1; s1)(x; t1)I(y; s1) )(x; t2)I(y; s2)

(hs1t1 ; t2)I(1; s2)

Then hs2t2 = hs1t1 ) s2t2 = s1t1 ) w = s1t1(z) = s2t2(z)) (z; t2)I(w; s2)�

case 4 x;w; z 2 (0; 1]; y 2 (1; 2) :

(x; t1)I(y; s1)) x = s1t1(y)(x; t2)I(y; s2)) x = s2t2(y)

) s1t1(y) = s2t2(y) ) s1t1(�) = s2t2(�) ) hs1t1 = hs2t2 and thus, applying

case 1 and Property 1:

(hs1t1 ; t2)I(ht1s1 ; s2)(hs1t1 ; t1)I(ht1s1 ; s1) )(z; t1)I(w; s1)

(z; t2)I(w; s2)�

Cases 3 and 4 are illustrated in Figure 5.12

t1 s2

0

2

T1s1

1

t2

y y

x x

w

z

t1 s2

0

2

T1s1

1

t2

w

z

y

x

y

x

Case 3 Case 4

hs1t1 hs2t2 hs1t1 hs2t2

Ħ Ħ

Figure 5.12. Cases 3 and 4 in the proof of (B4)

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case 5 x; y 2 (0; 1]; w; z 2 (1; 2) :

(w; s1)I(z; t1) ! (2� z; s1)I(2� w; t1)(x; t1)I(y; s1)(x; t2)I(y; s2)

By case 1: (2� z; s2)I(2� w; t2)) (w; s2)I(z; t2)�

case 6 x; y 2 (1; 2); w; z 2 (0; 1] :

(w; s1)I(z; t1)(x; t1)I(y; s1) ! (2� y; t1)I(2� x; s1)(x; t2)I(y; s2) ! (2� y; t2)I(2� x; s2)

By case 1: (w; s2)I(z; t2)�

case 7 x; z 2 (0; 1]; y; w 2 (1; 2) :

(x; t1)I(y; s1)! x = s1t1(y)(x; t2)I(y; s2)! x = s2t2(y)

) hs1t1 = hs2t2

Since (z; t1)I(w; s1)! z = s1t1(w) = s2t2(w); and thus (z; t2)I(w; s2)�

case 8 x 2 (0; 1]; y; z; w 2 (1; 2) :

(w; s1)I(z; t1)! (2� z; s1)I(2� w; t1)

Applying case 4, I obtain (2� z; s2)I(2� w; t2); (w; s2)I(z; t2)�

case 9 x; y; z 2 (1; 2); w 2 (0; 1]

Since 2� x; 2� y 2 (0; 1); applying case 3�

(B6) (Open set). For any x; y 2 H, x Pty; exists z 2 H such that x PtzPty

Proof. Since H= (0; 2) and x Pty $ x > y, the property holds straightforward.

(A7) and (A8) deal with de�nitions and properties of H, so that no extention

is plausible.

The axioms (B1)-(B6) describe properties for the time preferences R over H�Tthat almost coincide with the properties established by Ok and Masatlioglu (2007).

The only di¤erence lies in the idea that now R is �increasing� in time, whereas it

is "decreasing" in time in the prize-time space established by Ok and Masatlioglu

(2007). That dissimilarity is re�ected in (B3), and the subsequent demonstrations

diverge minimally from Ok�s ones.

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5.3.3 Proof of Theorem 2.1.

Proof of ")"

Claim 5. For any x, y 2 H, s,t 2 T such that (x; t)P (y; s), then there exist z; w 2H such that

(i) (x; t)P (z; t)P (y; s) and

(ii) (x; t)P (w; s)P (y; s).

Proof. (i) I de�ne the sets:

A:= fw 2 H : (x; t)P (w; t)g, B:= fw 2 H : (w; t)P (y; s)g.By (B6), A can be rewritten as A = fw 2 (0; 2) : x > wg = (w; 2), and it�s clear

that A is an non-empty, open set. B 6= ; since x 2 B by hypothesis, and it is opensince R is lower semicontinuous7.

H � A [B:I take any w 2H = (0; 2). By completeness, either w � x, or x > w. If w �

x;w =2 A, and the fact that (x; t)P (y; s), w � x; implies by (B3) that (w; t)P (y; s);what means that w 2 B: If x > w, it is immediate that w 2 A�Since it is trivial that also A [B �H, then A [B = H:However, H is connected by hypothesis, so that it must happen that A\B 6= ;:

Taking z 2 A \B; I have that (x; t)P (z; t)P (y; s)�:(ii) It can be proven similar to (i)

Claim 6. For any y 2 H, s, t 2 T , there exists a unique x 2 H such that (x; t)I(y; s)

Proof. (1) Existence

Given y 2H , s, t 2 T , by (B2), 9 w 2 H such that (w; t)R(y; s). Therefore, by(B3), the set

A := fw : (w; t)R(y; s) g

is non empty: I take an increasing V 2 Hom(H;R) that represents the preferencesin H:

inf V (A) 6= �1 :

If inf V (A) = �1; then for any z 2H, 9w 2 A (that is, (w; t)R(y; s)) such

that V (w) < V (z); that is, z > w: By (B3), (z; t)P (y; s): Therefore, for any z 2H,(z; t)P (y; s); what violates (B2)�

7 i.e., the set f(w;et) 2H: (y; s)R(w;et)g is closed, and project in et:147

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Since inf V (A) > �1; by surjectivity of V I can de�ne x := V �1(inf V (A));

V (x) = inf V (A):

x 2 A:(By way of contradiction): If x =2 A, then (y; s)P (x; t). By Claim 1, 9z 2H such

that (y; s)P (z; t)P (x; t).

Since x =2 A and V is bijective, I have that V (x) = inf(V (A)) =2 V (A):By de�nition of x, I can �nd a sequence fxmg � A1 such that V (x1) > V (x2) >

::: > V (xm) >..., with fV (xm)gm �! V (x) = inf V (A)

Note that V (xm) > V (x)! xm > x

By hypothesis, (z; t)P (x; t), i.e., V (z) > V (x), what implies that there exists

an m0 such that V (z) > V (xm0) > V (x): From that I obtain that z > xm0 : Since

(y; s)P (z; t); applying (B3) I derive that (y; s)P (xm0 ; t); that implies that xm0 =2 A,what is a contradiction.

Therefore, by completeness of R, I have that (x; t)R(y; s), that is, x 2 A�(x; t)I(y; s) :

(By way of contradiction): If (x; t)P (y; s) , then Claim 1 implies that 9z 2H suchthat (x; t)P (z; t)P (y; s) . This implies that z 2 A and x > z, so that V (x) > V (z),but this is a contradiction, since V (x) is assumed to be inf V (A).�(2) Uniqueness

Assume that (w; t)I(y; s) and (z; t)I(y; s). By completeness of Rt (�):-If z > w ! By (B3), (z; t)P (y; s), what is a contradiction.

-If w > z ! By (B3), (w; t)P (y; s), what is a contradiction.

) w = z �

Claim 7. For any t 2 T and x, y 2 H, with y � x, there exists an s � t such that(y; s)I(x; t).

Proof. If y = x, then taking s = t �If y > x, I de�ne the set A := fr 2 T : (x; t)R(y; r)g.A 6= ; by (B1), so I can de�ne s := supA:Lower semicontinuity of R implies that f(z; r) 2 H� T : (x; t)R(z; r)g is closed,

and thus supA 2 A(y; s) I (x; t):

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(By way of contradiction): If (x; t)P (y; s), then, upper semicontinuity of R im-

plies that f(z; r) 2H;�T : (x; t)P (z; r)g is open, and so fr0 2 T : (x; t)P (y; r0)g isopen. Then, 9r > s such that (x; t)P (y; r), what is a contradiction with the fact

that s = supA:

s � t :(By way of contradiction): If s > t, then (x; t)I(y; s) and y > x imply, by (B3),

that (y; t)P (y; s). Since s > t, (B3) implies that (y; t)P (y; t), what is an absurd.�

For any s, t 2 T , I de�ne the self-map:

@st : H �! Hy �! @st(y) = y such that (y; s)I(y; t)

By Claim 2, @st is well-de�ned for any s; t 2 T: Notice that, by construction,if x 2 (0; 1) and y 2 (1; 2); then (y; s)I(x; t) i¤ @st(y) = x = st(y); similarly, forx 2 (1; 2) and y 2 (0; 1); then (y; s)I(x; t) i¤ @ts(x) = y = ts(x).

Claim 8. @st is a bijection, and @�1st = @ts

Proof. @st is injective:If @st(y) = @st(x) =: z ! (y; s)I(z; t)

(x; s)I(z; t)(Claim2)������!

y = x�@st is surjective:Let x 2H. By Claim 2, there exists y 2H such that (y; s)I(x; t); and thus

@st(y) = x �The property @st = @ts is immediate�

The following properties report further information of these self-maps.

P.1 @st is strictly increasing in H

Proof. Let s; t 2 T , and let x; y 2H such that x > y: I will show that @st(x) > @st(y)Since x > y, (B6) implies that 9z : x > z > y. By Claim 2, 9x; y; z :

(x; s)I(x; t); (z; s)I(z; t); (y; s)I(y; t) (see that by de�nition, x = @st(x); y = @st(y)and z = @st(z))If z > x; since (z; s)I(z; t); (B3)would imply that (z; s)P (x; t), and using the fact

that x > z; applying (B3) again I obtain that (x; s)P (x; t); that is a contradiction.

Therefore x � z

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Similarly, it can be shown that z � y:Since Rt (�) is complete and transitive,thus @st(x) = x � y = @st(y):Finally, see that if x = y, then by Claim 4, x = y, what is a contradiction.�

P.2 @st is continuous in H

Proof. By Claim 4 and P1�

P.3 @st is increasing in s

Proof. Let s1; s2 2 T such that s1 > s2, and let y 2H: I will show that @s1;t(y) >@s2;t(y)

Let y1 = @s1;t(y)! (y; s1)I(y1; t)Let y2 = @s2;t(y)! (y; s2)I(y2; t)

If y2 � y1; thenn(y; s2)I(y2; t)s1 > s2

o(B3)���!

(y; s1)P (y2; t); and using the fact that

y2 � y1; by (B3) I obtain that (y; s1)P (y1; t), which is a contradiction�

P.4 @st is decreasing in t

Proof. By Claim 4 and P.3 �

Now I have the elements to proof the extension of (A5) in H to H (property

B5·)

(B5) (Path independence). For any x, y, z, w 2 (0; 2) and t1; t2; t3 2 (0; 1), then

(x; t1)I(y; t2)

(z; t1)I(w; t2) imply that (x; t2)I(z; t3)

(y; t2)I(w; t3)

The demonstration is organized in a similar to the proof of (B4). The chart

above summarizes all possible cases that can be found, depending on the values of

x; y; z; w: I mark "1" if the value lies in (0; 1], and "2" if it lies in (1; 2). Cases that

embrace di¤erent situations correspond to hyphothesis that can be rewritten in a

similar way (e.g. using (x; t)I(y; s) () (2 � y; t)I(2 � x; s); etc.). Note that the

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cases (1; 2; 2; 1) and (2; 1; 1; 2) are not contemplated, since they are not sustainable

under the hypothesis.

case: 1 2 3 4 5 6 7 8x 1 2 1 2 2 1 1 2 1 2 1 2 1 2y 1 2 1 2 1 1 2 2 2 1 1 2 2 1z 1 2 1 2 1 2 2 1 1 2 2 1 1 2w 1 2 2 1 1 1 2 2 1 2 2 1 2 1

Proof.

case 1 x, y, z, w 2 (0; 1]: By (A5) �

case 2 x, y, z, w 2 (1; 2):

(x; t1)I(y; t2) ! (2� y; t1)I(2� x; t2)(z; t1)I(w; t2) ! (2� w; t1)I(2� z; t2)(y; t2)I(w; t3) ! (2� w; t2)I(2� y; t3)By Claim 3, and given 2 � w; 2 � y 2H and t1 2 T; there exists s 2 T such

that (2� y; t1)I(2� w; s): Since 2� x, 2� y, 2� z, 2� w 2 (0; 1); case 1 sets that(2 � x; t1)I(2 � z; s): Now applying (B4) I obtain that (2 � x; t3)I(2 � z; t2); thatis, (z; t3)I(x; t2) �

case 3 x, y; z 2 (0; 1]; w 2 (1; 2):

By hypothesis (x; t1)I(y; t2): Claim 3 implies that given z; x and t2; there exists

s 2 T such that (x; t2)I(z; s): By (R8), (x; t1)I(y; t2) and (x; t2)I(z; s) imply thatt1t2(z) = t2s(y): Since t1t2(z) = w = t2t3(y); then t2t3(y) = t2s(y): If s < t3;

then ht2t3 < ht2s and t2t3(y) < t2s(y); what is a contradiction. Similarly I can

state that s > t3 contradicts the fact that t2t3(y) = t2s(y): Therefore s = t3 and

(x; t2)I(z; t3)�

case 4 y, z; w 2 (0; 1]; x 2 (1; 2):

(x; t1)I(y; t2) ! t2t1(y) = x(z; t1)I(w; t2) ! (z; t1) � (w; t2)(y; t2)I(w; t3) ! (y; t2) � (w; t3)By (R8), (y; t2) � (w; t3) and (z; t1) � (w; t2) imply that x = t2t1(y) = t3t2(z);

and thus t3t2(z) = x, i.e., (x; t2)I(z; t3)�

case 5 x; z; w 2 (0; 1]; y 2 (1; 2):

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Reorganizing the preferences embraced by the hyphothesis I obtain:

(z; t1)I(w; t2); (x; t1)I(y; t2); (w; t3)I(y; t2): By Claim 3, there exists s 2 T suchthat (w; t2)I(y; s): Applying case 3 and after that (B4), I have that:

(z; t1)I(w; t2)(x; t1)I(y; t2) ! (z; t2)I(x; s)(w; t2)I(y; s) (w; t2)I(y; s) ! (z; t3)I(x; t2)�

(w; t3)I(y; t2)

case 6 x; y 2 (0; 1]; z; w 2 (1; 2)

By hypothesis (z; t1)I(w; t2); that is equivalent to (2� w; t1) � (2� z; t2); with2� w; 2� z 2 H: Then, by (R9):

(x; t1)I(y; t2)(2� w; t1)I(2� z; t2)

23(y) = w! 23(x) = z; that is, (z; t3)I(x; t2)�

case 7 x; z 2 (0; 1]; y; w 2 (1; 2)

By Claim 4, given z; t2; t3; there exists v 2H such that (z; t3) � (v; t2): Then

applying (R9):(z; t1)I(w; t2) ! w = 12(z)(y; t2)I(w; t3) ! (2� w; t2) � (2� y; t3) !

(v; t2) � (z; t3)12(v) = y = 12(x): Thus,

v = x�

case 8 x; z 2 (1; 2); y; w 2 (0; 1]: Claim 3 states the existence of s, and by applyingcase 7:

(x; t1)I(y; t2) ! (2� x; t2)I(2� y; t1)(z; t1)I(w; t2) ! (2� z; t2)I(2� w; t1) ! (2� z; t2)I(2� x; s)

(2� w; t1)I(2� y; s)Now, by (B4):(2� z; t2)I(2� x; s)(2� w; t1)I(2� y; s)(2� w; t2)I(2� y; t3)

! (2� z; t2)I(2� x; t3); that is, (x; t2)I(z; t3)�

Claim 9. For any increasing V 2 Hom(H,R) (that represents the preferences inH), the map:

X : T 2� H �! R(s; t; x) �! X(s; t;x) = V (@st(x))

is continuous, increasing in s, decreasing in t, and strictly increasing in H.

Proof. By P1, P3 and P4, it can be easily shown thatX is increasing in s, decreasing

in t and strictly increasing in H.

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X is continuous:

I will show that for any sequence fsm, tm, ymg, with sm, tm 2 T , ym 2 H, I havethat lim supX(sm; tm; ym) = lim infX(sm; tm; ym) = X(lim sm; lim tm; lim ym):

Let fsmg, ftmg be any sequences of elements in T , with �nite limits s and t,respectively, and let fymg be any sequence of elements in H with �nite limit y.I de�ne s� := inf sm, t� := inf tm, s� := sup sm, t� := sup tm, and note that

s�; t�; s� and t�are �nite, since the sequences converge.

Let Vm := V (ym) 2 R, and V0 := V (y) 2 R: Since the set fV0;V1; V2; :::gis compact, the continuity of V �1 implies that the set Y := fy; y1; y2;...g is alsocompact. Thus, by Weierstrass theorem,(

V : H �! Rcontinuous in HY compact in �H

)=) 9maxV (Y )

9minV (Y )Let y� := V �1(maxV (Y )), and y� := V �1(minV (Y )); with y�; y� 2 H.By the properties of X yet seen, I have that: X(s�; t�; y�) � X(sm; tm; ym) �

X(s�; t�; y): Thus, X(sm; tm; ym) 2 [X(s�; t�; y�); X(s

�; t�; y�)]; what implies that

lim supX(sm; tm; ym) and lim infX(sm; tm; ym) yield inside the interval, and thus

they are �nite numbers.

Let � := lim supX(sm; tm; ym) < +1:I will show that � = X(s; t; y):

Clearly, there exists a strictly increasing subsequence fmkg of natural numberssuch that X(smk

; tmk; ymk

) �! � = lim supX(sm; tm; y):

@ vsm1 tm1 ym1 �! @sm1 ;tm1 (ym1) =: z1 �! X(sm1 ; tm1 ; ym1)sm2 tm2 ym2 �! @sm2 ;tm2 (ym2) =: z2 �! X(sm2 ; tm2; ym2)...

......

......

...smk

tmkymk

�! @smk ;tmk (ymk) =: zk �! X(smk

; tmk; ymk

)...

......

......

...# # # # # #s t y 99K99K @st(y) z =: V �1(�) L99 �

V �1 is continuous, so fzkgk �! z =: V �1(�): By construction, (ymk; smk

)I(zk; tmk),

and continuity of the preferences imply that, in the limit, (y; s)I(z; t); so that

z = @st(y), what proves that � = V (@st(y)) = X(s; t; y):One can similarly show that lim infX(sm; tm; ym) = X(s; t; y); so that I conclude

that X is continuous.

Let v be a function that represents the atemporal preferences on H = (0; 1];

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giving to any health state a value anchored in (�1; b], where b is a �nite realnumber. Note that v(1) = b:

v : (0; 1] �! (�1; b]x �! v(x)

Let V be a continuous, increasing function that represents the atemporal pref-

erences on H= (0; 2), de�ned as follows:

V : H �! Rx 2 (0; 1] �! v(x)x 2 (1; 2) �! 2b� v(2� x)

For any s; t 2 T , I de�ne:

fst := V � @st � V �1

V �1 @st Vfst : R �! H �! H �! R

n �! x �! x �! mm m

V (x) = n (x; s)I(x; t) V (x) = m

Note: fst transforms the utility of a health state x when it endures s -the utility

is referred to the case when every health state is assumed to long s-, into the "utility

referred to the case when every health state is assumed to long t" of a health state

x that endures t, being (x; s) and (x; t) equivalent.

Claim 10. fst 2 Hom(R) and f�1st = fts, for any s, t 2 T . Moreover, if s > t, thenfst � idR:

Proof. fst is a continuous bijection because of Claim 4 and Claim 5

By Claim 4, f�1st = (V � @st � V �1)�1 = V � @�1st � V �1 = V � @ts � V �1 = fts .

Then, since fts is continuous, it follows that fst 2 Hom(R):Now let x be any health state in H. Since (x; t)I(x; t), Claim 2 implies that

@tt(x) = x, so that @tt = idH for any t 2 T , and therefore ftt = idRLet t, s 2 T be such that t < s, take any n 2 R and let V �1(n) = x. Then, by

(P.3) and Claim 5 I have that:

fst(n) = f�1ts (n) = (V �@st�V �1)(n) = V (@st(V �1(n))) = V (@st(x)) � V (@tt(x)) =

ftt(n) = n

Thus, fst � idR�

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Now I de�ne

� := f(s; t) 2 T 2 : @st 6= idHg

Note that (t; t) =2 �

If � = ; : then @st(x) = x, this is, (x; s) � (x; t);8x 2 H, s, t 2 T . In particular,(x; s) � (x; t);8x 2 H, s, t 2 T . Therefore the theorem would simplify a lot, since

the time-dimension does not a¤ect the preferences over atemporal health states, and

thus (x; t) % (y; s) i¤ x %t y i¤ v(x) � v(y) i¤ u(x) := ev(x) � 1 � ev(y) = �(s; t)u(y):The values set by function u can be embraced into a (0; 1] interval by setting b = 0,

that is, v : (0; 1]! (�1; 0] �Thus, I assume in what follows that � 6= ;:

� � can be rewritten as:

� := f(s; t) 2 T 2 : Fix(@st) = ;g

Proof. If (x; s)I(x; t) then let y 2 H and y = @st(y):Then, by (B4),(y; s)I(y; t)(x; s)I(x; t)(x; s)I(x; s)

=) (y; s)I(y; s);

what means that yIsy, that is, y = y; so that y = @st(y)Since this result holds for any y, this would imply that @st = idH, and thus

(s; t) =2 ��

Hence,

F := ffst : (s; t) 2 �g � A := ff 2 Hom(R) : Fix(f) = ;g

Claim 11. For any x, y, z, w 2 H, and s1, s2, t1, t2 2 T ,

(x; t1)I(y; s1)(z; t1)I(w; s1) �! (x; s2)I(z; t2)(y; s2)I(w; t2)

Proof. By Claim 3, and given y, w 2 H and s1 2 T , there exists s 2 T such that(w; s)I(y; s1): Applying (B5) and after that (B4), I have that:

(x; t1)I(y; s1)(z; t1)I(w; s1) �! (x; s1)I(z; s)(y; s1)I(w; s) (y; s1)I(w; s) �! (x; s2)I(z; t2)

(y; s2)I(w; t2)

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� Now I show that hF ; �i is an abelian group:Take any (s1; t1); (s2; t2) 2 �, x 2 H.@s1t1(@s2t2(x)) = @s1t1(x2) = z = @s2t2(x1) = @s2t2(@s1t1(x))*x2 = @s2t2(x) ! (x; s2)I(x2; t2)z = @s1t1(x2) ! (x2; s1)I(z; t1) Claim7�����! (x1; s2)I(z; t2) �! z = @s2t2(x1)x1 = @s1t1(x) ! (x; s1)I(x1; t1)

Given such result, let fs1t1 ; fs2t2 2 hF ; �i, and:fs1t1� fs2t2 = V �@s1t1 �V �1 �V �@s2t2 �V �1 = V �@s2t2 �@s1t1 �V �1 = fs2t2 �fs1t1So hF ; �i is an abelian group.�

� Fix an arbitrary (s�; t�) 2 �, with s� > t�, and de�ne:

g := fs�;t�

By Claim 6, g � idR: Since (s�; t�) 2 �, I have that g > idR:

� Now I need to introduce some further terminology. Let n 2 f2; 3; 4:::g, anddenote the generic element of T 2n by (si; ti)ni=1. For any x 2 H, I say that (si; ti)ni=1is an x-cycle if there exists y1; ::; yn 2 H such that:(x; t1)I(y1; s1); (y1; t2)I(y2; s2); :::; (yn�1; tn)I(x; sn)

Claim 12. For any n 2 f2; 3; 4:::g, and x 2 H, if (si; ti)ni=1 is an x-cycle, then(si; ti)

ni=1 is a z-cycle, for any z 2 H.

Proof. Take any x 2 H, n � 2, and an x-cycle (si; ti)ni=1. Then:(x; t1)I(y1; s1); (y1; t2)I(y2; s2); :::; (yn�1; tn)I(x; sn)

Now take any element of H, say z:

By Claim 2, given (si)ni=1; (ti)ni=1, there exist fwigni=1; such that

(z; t1)I(w1; s1); (w1; t2)I(w2; s2); :::; (wn�1; tn)I(wn; sn):

I have to show that wnIsz

By Claim 3, given x and z, there exist s, t 2 T such that (x; t)I(z; s): Then,(z; t1)I(w1; s1)(x; t1)I(y1; s1) Claim 7�����! (y1; t)I(w1; s)(x; t)I(z; s) (w1; t2)I(w2; s2) Claim 7�����! (y2; t)I(w2; s)

(y1; t2)I(y2; s2) � � �: : :

(yn�1; t)I(wn�1; s)(wn�1; tn)I(wn; sn) Claim 7�����! (x; t)I(wn; s)(yn�1; tn)I(x; sn)

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Since I know that (x; t)I(z; s), by applying Claim 2, it follows that z = wn�

Observe that, if (si; ti)ni=1 is an x-cycle, so that:

(x; t1)I(y1; s1); (y1; t2)I(y2; s2); :::; (yn�1; tn)I(x; sn); with y1; ::; yn 2H, I can write:y1 = @t1s1(x)y2 = @t2s2(y1)...

...yn�1 = @tn�1sn�1(yn�2)x = @tnsn(yn�1)

Therefore, x = (@tnsn � @tn�1sn�1 � ::: � @t1s1)(x): Thus, Claim 8 simply says that

if (@tnsn � @tn�1sn�1 � ::: � @t1s1)(x) = x for some x 2 H, then (@tnsn � @tn�1sn�1 � ::: �@t1s1)(z) = z for any z 2H, that is, @tnsn � @tn�1sn�1 � ::: � @t1s1 = idH:

� If f 2 hF ; �i and Fix(f) 6= ;, then f = idR :Let f 2 hF ; �i: Then, there exist n 2 N; (si; ti)ni=1 2 � such that f = fs1t1 � ::: �

fsntn.

- If n = 1: then f = fs1t1 and Fix(fs1t1) 6= ; by de�nition.�- If n � 2 : assume that 9n 2 R such that f(n) = n; and let x := V �1(n): Then:f(n) = (fs1t1 � ::: � fsntn)(n) = n () (V � @s1t1 � ::: � @sntn � V �1)(n) = n ()

V ((@s1t1 � ::: � @sntn)(x)) = V (x): The injectivity of V implies that (@s1t1 � ::: �@sntn)(x) = x, and by Claim 8 I have that @tnsn � @tn�1sn�1 � ::: � @t1s1 = idH, so

f = V � idH � V �1 = idR�

This is, the only member of hF ; �i that has a �xed point is idR:� QF(n) := [ff(n) : f 2 hF ; �ig = R :Take any n 2 R. Let x := V �1(n), and for any m 2 R;m 6= n, let y := V �1(m)

(see that, by V 2 Hom(H;R), n 6= m implies that x 6= y). By Claim 2, there exists

(s; t) 2 � such that (y; t)I(x; s), that is, y = @st (x):Then, fst(n) = V (@st(V �1(n))) = V (@st(x)) = V (y) = mTherefore, I have demonstrated that given n 2 R, for anym 2 R;m 6= n, 9(s; t) 2

� such that fst(n) = m: This implies that [ff(n) : f 2 hF ; �ig = R� fng: Finally,since idR 2 hF ; �i, idR(n) = n and then QF(n) := [ff(n) : f 2 hF ; �ig = R�

� If I denote as LF(n) the set of all limit points of QF(n), the previous propertyimplies that LF(n) = f�1;+1g�

I have veri�ed the requirements of Theorem A, that are:

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F is a nonempty subset of A := ff 2 Hom(R) : Fix(f) = ;gF contains a map g 2 A with g > idRhF ; �i is an abelian groupThe only member of hF ; �i that has a �xed point is idRThere is some n 2 R such that LF(n) = f�1;+1gTherefore, Theorem A implies that

�g(f) := supfm

n: (m;n) 2 Z� N and gm < fng 6= ;, for all f 2 hF ; �i

Moreover, there exists a continuous bijection F : R �! R such that

F � f � F = �g(f) for all f 2 F

� Now I de�ne:

' : � �! R(s; t) �! '(s; t) = �g(fst)

Theorem A implies that there exists a continuous bijection F : R �! R suchthat F (fst(n))� F (n) = '(s; t) for all (n; (s; t)) 2 R� �:

Claim 13. For any (s; t) 2 �, I have(i) '(s; t) = �'(t; s)(ii) (t� s)'(s; t) < 0

Proof. (i) '(s; t) = �g(fst) = F (fst(n)) � F (n) for any n 2 R, in particular forf�1st (n) 2 R; '(s; t) = F (fst(n)) � F (n) = F (fst(f

�1st (n))) � F (f�1st (n)) = F (n) �

F (fts(n)) = � '(t; s)�(ii) First, assume that s > t. Then, by using the de�nition of g as well as Claim

6, I have that g > idR > fts. Thus, for any (m;n) 2 Z+ � N, gm > idR > fnts, whatimplies that �g(fts) := supfm

n: (m;n) 2 Z � N and gm < fng is < 0: Therefore,

'(t; s) < 0 and s > t make that '(s; t)(t� s) = �'(t; s)(t� s) < 0.Now assume that s < t. Claim 6 implies that fst < idR, so gm > idR > fnst and

thus �g(fst) < 0, what implies that '(s; t) (t� s) < 0�

� F is strictly increasing:

Take any n 2 R, and let m := g(n) = fs�t�(a), s� > t�.

F (m) = F (g(n)) = F (n) + '(s�; t�) > F (n) by Claim 9. Note also that m =

g(n) > n since g > idR: Therefore, there exist n, m 2 R, m > n; with F (m) > F (n);

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what implies that F cannot be strictly decreasing. Finally, since F is a bijection, I

have that F is strictly increasing.�

�I also extend ' to T 2, by setting:

' : T 2 �! R� �! '(�) = �g(fst)

T 2 � � �! 0

� See also that if (s; t) =2 �, then @st = idH, so that fst = idR, and '(s; t) =

F (fst(n))�F (n) = F (n)�F (n) = 0: Thus, I can say that '(s; t) = F (fst(n))�F (n)for all s, t 2 T . It also holds that '(s; t) = � '(t; s), for all s, t 2 T�

� ' is continuous, since F is continuous.� ' is increasing in its �rst component:Let s1 < s2. Let n 2 R, and let x := V �1(n). By de�nition, ' (s1; t) =

F (fs1t(n)) � F (n); ' (s2; t) = F (fs2t(n)) � F (n): Claim 5 and de�nition of V im-

ply that fst(n) = V (@st(V �1(n))) = V (@st(x)) is increasing in s, so that: s1 < s2

) fs1t < fs2t ) F (fs1t) < F (fs2t)) ' (s1; t) < ' (s2; t)�

� Now de�ne

W : H �! Rx �! W (x) = (F � V )(x)

See that W is a continuous bijection and xRty () W (x) = F (V (x)) �F (V (y)) = W (y), since F and V are continuous and increasing functions. Therefore,

W represents the preferences of H �

� Finally, letU := eW : H �! (0;+1)� := e' : T 2 �! (0;+1)

See that �(s; t) = e'(s;t) = e�'(t;s) = 1�(t;s)

� At this point, I have demonstrated that:

9U : H �! (0;+1)(continuoussurjectiverepresents Rt

)

9� : T 2 �! (0;+1)(continuous�(�; t) increasing�(s; t) = 1=�(t; s)

)

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Chapter 5 Generalizing Quality-Adjusted Life Years in a Context of Certainty

Claim 14. For all x; y 2H and s; t 2 T; (x; t)R(y; s) i¤ U(x) � �(s; t)U(y)

Proof. 1. For any (x; t); (y; s) 2 H�T , (x; t)P (y; s) i¤ (x; t)P (@st(y); t):

" =) ": (By way of contradiction): Assume that (x; t)P (y; s), and (@st(y); t)R(x; t):

If (@st(y); t)I(x; t), then by Claim 2, @st(y) = x; what implies that (x; t)I(y; s),what is a contradiction.

If (@st(y); t)P (x; t), then @st(y)Ptx and (y; s)I(@st(y); t), so that (B3) impliesthat (y; s)P (x; t); what is a contradiction.�

" (= ": Assume that (x; t)P (@st(y); t). Then xPt@st(y) , (y; s)I(@st(y); t),and (B3) implies that (x; t)P (y; s)�

2. (x; t)I(y; s) i¤ (x; t)I(@st(y); t):

This is clear since (x; t)I(y; s) () x = @st(y); and taking @st(y), I obtainthat xIt@st(y) and (x; t)I(@st(y); t).�

I have obtained that:

(x; t)nPI

o(y; s) i¤ (x; t)

nPI

o(@st(y); t)

Since U represents Rt;

(x; t)nPI

o(@st(y); t) i¤ x

nPI

o@st(y) i¤

U(x)

U(y)

n>=

o U(@st(y))U(y)

Now, let n := V (y). Then:

U(@st(y))U(y)

= expfW (@st(y))�W (y))= expfF (V (@st(y)))� F (V (y))g= expfF (V (@st(V �1(n))))� F (V (V �1(n)))g= expfF (fst(n))� F (n)g= expf'(s; t)g= �(s; t)

I have demonstrated that for any x; y 2 H, s; t 2 T;

(x; t)nPI

o(y; s) i¤ U(x)

n>=

o�(s; t)U(y)

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Chapter 5 Generalizing Quality-Adjusted Life Years in a Context of Certainty

The result obtained in Claim10 is valid for those elements in H= (0; 2): In par-

ticular, the result also holds for those elements in H �H, that represent true healthstates. Here it is necessary to rede�ne the functions over H.

Recall that v is a function that represents the atemporal preferences on H =

(0; 1]; giving to any health state x a value v(x) 2 (�1; b]; where b is a �nite realnumber. Since F 2 Hom(R); I can �x b := F�1(0): Now, given that F is strictly

increasing, it is possible to consider the following partition:

F : R �! R(�1; b] �! (�1; 0](b;+1) �! (0;+1)

Therefore, given any x 2 H = (0; 1]; I �nd that U(x) = eW (x) = eF (V (x)) =

eF (v(x)) 2 (0; 1]:Thus, I de�ne:

u := U jH : H �! (0; 1]x �! u(x)

And the result of Claim 10 can be stated as follows:

(x; t)n ��o(y; s) i¤ u(x)

n>=

o�(s; t)u(y)

It only remains to prove that �(0; t) = 0 :

First I �x t 2 T , and choose " 2 (0; 1): Let x := u�1("); y := 1: (Note that u issurjective, and this guarantees the existence of x).

By (A1), 9s 2 T such that (x; t) % (1; s), that is, " = u(x) � u(1)�(s; t) =

1 � �(s; t) = �(s; t):Then, for any arbitrary " 2 (0; 1), �xing t 2 T , there exists an s 2 T such that

�(s; t) � ": Since �(�; t) is increasing, I must have that �(0; t) = 0 �

Proof of "(":

I assume that:

% is a binary relation on H � T

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Chapter 5 Generalizing Quality-Adjusted Life Years in a Context of Certainty

9u : H �! (0; 1)

(continuoussurjectiverepresents &t

)

9� : T 2 �! R++

8<:continuous�(�; t) increasing�(0; t) = 0�(s; t) = 1=�(t; s)

9=;such that (x; t) % (y; s) i¤ u(x) � �(s; t)u(y) for all x; y 2 H and s; t 2 T

I have to demonstrate that % is a time preference on H � T that satis�es (A1)-(A8).

� % is a time preference:

It is obvious that % is complete and continuous.By �(s; t) = 1=�(t; s); it follows that �(t; t) = 1; so x %t y i¤ u(x) � u(y):

Therefore, u represents %t for any t 2 T; what implies that %t �%t0 for any t,t0 2 T . Since it is clear that %tis then transitive and complete, I have all therequirements for % to be a time preference�

� (A1): Given any x; y 2 H, and t 2 T;since � is continuous and �(0; t) = 0; itassures the existence of s 2 T such that u(x)

u(y)� �(s; t), that is, the existence of s 2 T

such that (x; t) % (y; s).�

� (A2): Given any x 2 H, and t; s 2 T; since u is surjective, it assures the

existence of y 2 H such that u(y) � u(x)�(s;t)

; that is, the existence of y 2 H such that

(x; t) % (y; s).

� (A3): Assume that (x; t) % (y; s), z %t x and r � t:Then, u(z) � �(s; t)u(y) >�(s; r)u(y) since �(s; �) is decreasing, and that implies that (z; r) % (y; s).�

� % satis�es (A4) and (A5) obviously.�

� (A6): By surjectivity of u.�

� (A7): Given s; t 2 T; s � t; it can be easily shown that �(s; t) � 1. By

surjectivity of u, there exists x 2 H such that u(x) = �(s; t) = �(s; t)u(1), in other

words, there exists x 2 H such that (x; t) � (1; s)�

� (A8) Firstly note that the health state hst is well de�ned:"hst 2 H such that 8x 2 H with x �t hst, then (x; t) � (y; s);8y 2 H"If s < t: (A7) implies the existence of z such that (z; t) � (1; s), that is,

u(z) = �(s; t): Take any x 2 (0; 1] with x �t z, that is, u(x) > u(z). Then

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Chapter 5 Generalizing Quality-Adjusted Life Years in a Context of Certainty

u(x)u(z)

= u(x)�(s;t)

> 1 � u(y) for any y 2 H: The existence of hst is proved by takinghst = z:

If s � t: suppose that there exists z 2 H such that 8x 2 H with x �t z,then (x; t) � (y; s);8y 2 H: See that (x; t) � (1; s) leads to the inequality u(x) >

�(s; t)u(1) = �(s; t) � 1: That result denies the existence of an element x 2 H such

that (x; t) � (y; s);8y 2 H; what can be expressed by taking z = 1, that is, hst = 1:The previous analysis gives me an alternative de�nition of hst as a map on (s; t):

hst =

�u�1 (�(s; t)) , if s < t1 , if s � t

�The continuity of hst is clearly deduced from that. Finally, if s! 0; �(s; t)! 0

and thus u�1 (�(s; t))! 0�

5.3.4 Proof of corollary 2.1.

Proof. I de�ned : T ! (0;+1)

k ! d(k) = �(k; 1)

By the properties of map � stated by Theorem 1, I easily obtain that d is continu-

ous, increasing, with d(1) = 1, d(0) = 0, and since �(1; k) = 1=d(k), then transitivity

of % and Proposition 1 imply that �(s; t) = �(s; 1)�(1; t) = d(s)=d(t):Therefore, (x; t) % (y; s) i¤ u(x) � �(s; t)u(y) = (d(s)=d(t))u(y), and thus,

(x; t) % (y; s) i¤ d(t)u(x) � d(s)u(y):

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