1 Age-Period-Cohort Analysis: New Models, Methods, and Empirical Analyses Kenneth C. Land, Ph.D....

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Age-Period-Cohort Analysis: New Models, Methods, and

Empirical Analyses

Kenneth C. Land, Ph.D.John Franklin Crowell Professor of

Sociology and DemographyDuke University

PresentationIndiana University

April 15, 2011

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Famous quote from George E. P. Box, Emeritus Professor of Statistics, University of Wisconsin at Madison:

“All statistical models are wrong, but some are useful.”

Ken Land’s Version:

“All statistical models are wrong, but some have better statistical properties than others – which may make them useful.”

GUIDING PRINCIPLE FOR THIS WORK

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Organization

Briefly Review the Early Literature on Cohort Analysis and the Age-Period-Cohort (APC) Identification Problem

Describe Models & Methods Developed Recently for APC Analysis for Three Research Designs, with Empirical Applications:

1) APC Analysis of Age-by-Time Period Tables of Rates 2) APC Analysis of Microdata from Repeated Cross-Section Surveys3) Cohort Analysis of Accelerated Longitudinal Panel Designs

Conclusion

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Part I: The Early Literature on Cohort Analysis and the Age-Period-Cohort (APC) Identification Problem

Why cohort analysis?

See the abstract from Norman Ryder’s classic article:

Ryder, Norman B. 1965. The Cohort as A Concept in the Study of Social Change. American Sociological Review 30:843-861.

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And what is the APC identification problem?

See the abstract from the classic Mason et al. article:

Mason, Karen Oppenheim, William M. Mason, H. H. Winsborough, W. Kenneth Poole. 1973. Some Methodological Issues in Cohort Analysis of Archival Data. American Sociological Review 38:242-258.

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These two articles were particularly important in framing the literature on cohort analysis in sociology, demography, and the social sciences over the past five decades:

Ryder (1965) argued that cohort membership could be as important in determining behavior as other social structural features such as socioeconomic status.

Mason et al. (1973) specified the APC multiple classification/accounting model and defined the identification problem therein.

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The Mason et al. (1973) article, in particular, spawned a large methodological literature, beginning with Norval Glenn’s critique:

Glenn, N. D. (1976). Cohort Analysts’ Futile Quest: Statistical Attempts to Separate Age, Period, and Cohort Effects. American Sociological Review 41:900–905.

and Mason et al.’s (1976) reply:

Mason, W. M., K. O. Mason, and H. H. Winsborough. (1976). Reply to Glenn. American Sociological Review 41:904-905.

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The Mason et al. reply continued with Bill Mason’s work with Stephen Fienberg:

Fienberg, Stephen E. and William M. Mason. 1978. "Identification and Estimation of Age-Period-Cohort Models in the Analysis of Discrete Archival Data." Sociological Methodology 8:1-67,

which culminated in their 1985 edited volume:

Fienberg, Stephen E. and William M. Mason, Eds. 1985. Cohort Analysis in Social Research. New York: Springer-Verlag,

a defining volume on the methodological literature on APC analysis in the social sciences as of about 25 years ago.

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New approaches and critiques thereof continued over the years; see, e.g., an article applying a Bayesian statistics approach:

Saski, M., & Suzuki, T. (1987). Changes in Religious Commitment in the United States, Holland, and Japan. American Journal of Sociology 92:1055–1076,

and the critique:

Glenn, N. D. (1987). A Caution About Mechanical Solutions to the Identification Problem in Cohort Analysis: A Comment on Sasaki and Suzuki. American Journal of Sociology 95:754–761.

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For additional material on these and related contributions to the literature on cohort analysis, see the following three reviews:

Mason, William M. and N. H. Wolfinger. 2002. “Cohort Analysis.” Pp. 151-228 in International Encyclopedia of the Social and Behavioral Sciences. New York: Elsevier.

Glenn, Norval D. 2005. Cohort Analysis. 2nd edition. Thousand Oaks: Sage.

Yang, Yang. 2007. “Age/Period/Cohort Distinctions.” Pp. 20-22 in Encyclopedia of Health and Aging. Kyriakos S. Markides (ed). Sage Publications.

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Where does this literature on cohort analysis leave us today?

If a researcher has a temporally-ordered dataset and wants to tease out its age, period, and cohort components, how should he/she proceed?

Are there any methodological guidelines that can be recommended?

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There are some guidelines – and cautions, e.g., in Glenn (2005).

But can more be done with new statistical models and methods? Perhaps, but any new method must meet the criteria laid down by Glenn (2005: 20) that it may prove useful:

“if it yields approximately correct estimates ‘more often than not,’

if researchers carefully assess the credibility of the estimates by using theory and side information, and

if they keep their conclusions about the effects tentative.”

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Generally, however, the problem with much of the extant literature is a deficiency of useful guidelines on how to conduct an APC analysis. Rather, the literature often leads a researcher to conclude either that:

it is impossible to obtain meaningful estimates of thedistinct contributions of age, time period, and cohort tothe study of social change,

or that:

the conduct of an APC analysis is an esoteric art that isbest left to a few skilled methodologists.

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Yang and Land and co-authors have bravely taken on Glenn’s challenge and have developed new approaches for APC analysis that are less esoteric and can be used by researchers.

These new approaches are bound together as members of the class of Generalized Linear Mixed Models (GLMMs), models that allow linear and nonlinear exponential family links and mixed (both fixed and random) effects.

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Part II: First Research Design: APC Analysis of Age-by-Time Period Tables of Rates or Proportions

References for Part II:Fu, W. J. 2000. “Ridge Estimator in Singular Design with Application to

Age-Period-Cohort Analysis of Disease Rates.” Communications in Statistics--Theory and Methods 29:263-278.

Yang Yang, Wenjiang J. Fu, and Kenneth C. Land. 2004. “A Methodological Comparison of Age-Period-Cohort Models: The Intrinsic Estimator and Conventional Generalized Linear Models.” Sociological Methodology 34:75-110.

Yang Yang, Sam Schulhofer-Wohl, Wenjiang J. Fu, and Kenneth C. Land. 2008. “The Intrinsic Estimator for Age-Period-Cohort Analysis: What It Is and How To Use It.” American Journal of Sociology 114(May): 1697-1736.

Yang Yang. 2008. “Trends in U.S. Adult Chronic Disease Mortality, 1960-1999: Age, Period, and Cohort Variations.” Demography 45(May):387-416.

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Data Structure: Tabular Rate Data

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Example: Lung Cancer Death Rates for U.S. Adult Females,

1960 – 1999 Analyzed in Yang (2008)

Deaths per 100,000 PopulationAge Period

1960 - 64 1965 - 69 1970 - 74 1975 - 79 1980 - 84 1985 - 89 1990 - 94 1995 - 9920 - 24 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.125 - 29 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.230 - 34 0.8 0.9 1.0 0.9 0.8 0.8 0.9 0.835 - 39 2.3 3.0 3.6 3.5 3.3 2.7 2.9 3.040 - 44 5.1 7.1 9.1 10.5 9.9 8.9 7.5 7.745 - 49 8.6 12.9 18.1 22.2 23.9 23.1 20.9 17.050 - 54 12.5 19.4 28.9 36.9 44.2 47.2 44.9 39.055 - 59 16.1 25.5 40.1 53.1 69.0 78.3 81.8 74.260 - 64 19.9 28.8 46.6 69.2 92.6 115.2 127.3 125.165 - 69 24.5 33.9 51.3 78.6 114.7 145.5 172.6 180.070 - 74 29.2 38.4 52.8 77.7 120.2 168.3 208.2 233.575 - 79 34.0 41.8 56.1 76.1 111.4 162.0 219.4 251.680 - 84 36.9 45.8 57.3 75.3 102.5 141.1 199.8 249.685 - 89 39.8 48.6 59.7 75.2 96.9 120.9 164.5 214.890 - 94 34.2 43.1 60.6 73.6 91.8 108.8 136.3 166.2

95 - 125+ 26.5 44.2 51.0 68.9 82.7 104.1 120.0 132.8All 10.3 14.7 21.3 28.9 38.3 47.9 57.0 61.7

Source: CDC/NCHS Multiple Cause of Death File

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Part II: First Research Design: APC Accounting/Multiple Classification Model

The Algebra of the APC Identification ProblemLinear Model Specification:

(1)

– Mij denotes the observed occurrence/exposure rate of deaths for the i-th age group for i = 1,…,a age groups at the j-th time period for j = 1,…, p time periods of observed data

– Dij denotes the number of deaths in the ij-th group, Pij denotes the size of the estimated population in the ij-th group

– μ denotes the intercept or adjusted mean

– αi denotes the i-th row age effect or the coefficient for the i-th age group

– βj denotes the j-th column period effect or the coefficient for the j-th time period

– γk denotes the k-th cohort effect or the coefficient for the k-th cohort for k = 1,…,(a+p-1) cohorts, with k=a-i+j

– εij denotes the random errors with expectation E(εij ) = 0

– Fixed effect GLIM reparameterization: , or setting one of each of the categories as the reference group.

ijkjiijijij PDM /

0 kkjjii

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The Algebra of the APC Identification ProblemAlternative Specifications In the Generalized Linear Models (GLM)

Class:Simple Linear Models

where Yij is the expected outcome in cell (i, j) that is assumed to be normally distributed or equivalently the error term is assumed to be normally distributed with a mean of 0 and variance σ2;

Log-Linear Models

log(Eij) = log(Pij) + μ + αi + βj + γk

where Eij denotes the expected number of events in cell (i,j) that is assumed to be distributed as a Poisson variate, and log(Pij) is the log of the exposure Pij

Logistic Models

where θij is the log odds of event and mij is the probability of event in cell (i,j).

kjiij

ijij m

m

1log

ijkjiijY

ij

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The Algebra of APC Identification ProblemLeast-squares regression in matrix form:

(2)

Identification Problem:

(3)

The solution to these normal equations does not exist because the

Design matrix X is singular with 1 less than full rank (one column can be

written as a linear combination of the others); this is due to the identity:

Period = Age + Cohort

thus, (XTX)-1 does not exist

Tpapab ),...,,,...,,,...,( 211111

XbY

YXXXb TT 1)(ˆ

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Conventional Solutions to APC Identification Problem

Constrained Coefficients GLIM (CGLIM) Estimator Impose one or more equality constraints on the coefficients of the coefficient

vector in (2) in order to just-identify (one equality constraint) or over-identify (two or more constraints) the mod

Proxy Variables/Age-Period-Cohort Characteristic (APCC) Approach Use one or more proxy variables as surrogates for the age, period, or cohort

coefficients (see O'Brien, R.M. 2000. "Age Period Cohort Characteristic Models." Social Science Research 29:123-139);

Nonlinear Parametric (Algebraic) Transformation Approach Define a nonlinear parametric function of one of the age, period, or cohort

variables so that its relationship to others is nonlinear.

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Limitations of Conventional Solutions to APC Identification Problem

Proxy Variables Approach the analyst may not want to assume that all of the variation associated with

the A, P, or C dimensions is fully accounted for by a proxy variable;

Nonlinear Parametric (Algebraic) Transformation Approach it may not be evident what nonlinear function should be defined for the

effects of age, period, or cohort;

Constrained Coefficients GLIM (CGLIM) Estimator it is the most widely used of the three approaches, but suffers from some

major problems summarized below.

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Limitations of Conventional Solutions to APC Identification Problem

Constrained Coefficients GLIM (CGLIM) Estimator: the analyst desires to employ the flexibility of the APC accounting model

with its individual effect coefficients for each of the A, P, or C categories; the analyst needs to rely on prior or external information to find

constraints that hardly exists or can be well verified; different choices of identifying constraints can produce widely different

estimates of patterns of change across the A, P, and C categories of the analysis;

all just-identified CGLIM models will produce the same levels of goodness-of-fit to the data, making it impossible to use model fit as the criterion for selecting the best constrained model.

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So, what can be done? Some Guidelines for Estimating APC Models for Tables of Rates or Proportions

Step 1: Descriptive data analyses using graphics

Step 2: Model specification testsObjectives: to provide qualitative understanding of patterns of age, or period, or

cohort variations, or two-way age by period and age by cohort variations;

to ascertain whether the data are sufficiently well described by any single factor or two-way combination of the A, P, and C dimensions or if it is necessary to include all three.

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Step 1: Graphical analyses: Female Lung Cancer Example

from Yang (2008)

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Step 2: Model selection proceduresExamples from Yang et al. (2004) and Yang (2008)

Table 1. Goodness-of-Fit Statistics for Age-Period-Cohort Log Linear Models of U.S. Adult Mortality

Female

Cause of Models A AP AC APC* Death DF 112 105 90 84

Deviance 695527 40443 72089 18903 Total AIC 695751 40653 72269 19071

BIC 695763 40664 72279 19080 Deviance 782210 52225 18638 9243

Heart Disease AIC 782434 52435 18818 9411 BIC 782446 52446 18827 9420 Deviance 655622 12660 25967 1480

Stroke AIC 655846 12870 26147 1648 BIC 655858 12881 26157 1657 Deviance 320050 42126 5296 245

Lung Cancer AIC 320274 42336 5476 413 BIC 320286 42347 5486 422 Deviance 9748 7403 1553 512

Breast Cancer AIC 9972 7613 1733 680 BIC 9984 7625 1743 689

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Guidelines for Estimating APC Models of Rates or Proportions

If the foregoing descriptive analyses suggest that only one or two of the A, P, and C dimensions is operative, then the analysis can proceed with a reduced model (2) that omits one or two dimensions and there is no identification problem.

If, however, these analyses suggest that all three dimensions are at work,

then Yang et al. (2004, 2008) recommend:

Step 3: Apply the Intrinsic Estimator (IE).

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What is the Intrinsic Estimator (IE)?It is a new method of estimation that yields a unique solution to the

model (2) and is the unique estimable function of both the linear and nonlinear components of the APC model determined by the Moore-Penrose generalized inverse. It achieves model identification with minimal assumptions.

Why is the IE useful? The basic idea of the IE is to remove the influence of the design

matrix (which is fixed by the number of age and period groups and not related to the outcome observations Yij) on coefficient estimates. This constraint produces estimates that have desirable statistical properties.

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Some preliminary matrix algebra concepts:

Let A be a matrix of dimension q by d (q rows and d columns), let x be a column vector of dimension d, and y a column vector of dimension q.

For a set of linear equations Ax = y, the set of vectors x0 of (real) numbers such that Ax0 = 0 is called the null space of the matrix A. When a matrix A is rank deficient (has linearly dependent columns), the dimension of the null space is at least one. In this case, if we have Ax = y, then we also have A(x + x0) = y. When A is rank deficient, the equation Ax = y has an infinite set of solutions, which differ by an element of the null space (if vectors x1 and x2 are solutions, then A(x1 – x2) = 0 and the vector x1 – x2 is in the null space). When A is rank deficient, there always is a well-defined solution whose projection on the null space is zero; this solution corresponds to the generalized inverse of A.

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The Intrinsic Estimator (IE): Algebraic Definition The linear dependency between A, P, and C in model (2) is

mathematically equivalent to:(4)

which defines the null space for model (2) where the eigenvector B0 of eigenvalue of 0 is fixed by the design matrix X:

00 XB

0

00 ~

~

B

BB

TCPAB ),,,0(~

0

2

1)1(,,

2

11

aa

aA

)1(

2

1,,1

2

1p

ppP

2

)2(,,2

1pa

papa

C

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The Intrinsic Estimator (IE): Algebraic Definition

Parameter vector orthogonal decomposition:

(5)

(6)

where is the projection of b to the non-null space of X

and t is a real number, tB0 is in the null space of X and represents

trends of linear constraints – Different equality constraints used byCGLIM estimators, such as b1 and b2, yield different values of t.

00 tBbb

bPb proj0

b2 b0 b1

0 B0 tB0

bBBIb T )( 000

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The Intrinsic Estimator (IE) Method: Algebraic DefinitionFrom the infinite number of estimators of b in model (2):

(7)

the IE B estimates the parameter vector b0 corresponding to t = 0:

(8)

The IE is the special estimator that uniquely determines the age, period, and cohort effects in the parameter subspace defined by b0 :

(9)

0ˆ tBBb

bBBIB T ˆ)( 00

XBXBtXBXBtBBXbX 0)(ˆ00

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The Intrinsic Estimator (IE) Method: Desirable statistical properties (Yang et al. 2004, 2008):

1) Estimability: Yang et al. (2004) established that the IE satisfies the Kupper et al. (1985) condition for estimability, namely

where where lT is a constraint vector (of appropriate dimension) that defines a linear function lTb of b.

Reference:Kupper, L.L., J.M. Janis, A. Karmous, and B.G. Greenberg. 1985.

“Statistical Age-Period-Cohort Analysis: A Review and Critique.” Journal of Chronic Disease 38:811-830.

00 Bl T

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Proof: Note that

Estimable functions are desirable as statistical estimators because they are linear functions of the unidentified parameter vector that can be estimated without bias, i.e., they have unbiased estimators.

0)( 0000000000 BBBBBBBBBIBl TTT

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Yang et al. (2004) also proved independently of the Kupper et al. (1985) estimability condition that the IE has the following two properties:

2) Unbiasedness: For a fixed number of time periods of data, it is an unbiased estimator of the special

parameterization (or linear function) b0 of b. 3) Relative efficiency: For a fixed number of time periods

of data, it has a smaller variance than any CGLIM estimators.

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3) Asymptotic consistency: This properties derive largely from the fact that the length of the eigenvector B0 decreases with increasing numbers of time periods of data, and, in fact, converges to zero as the number of periods of data increases without bound.

Therefore, for any two estimators:

and

where t1 and t2 are nonzero and correspond to different identifying constraints, as the number of time periods in an APC analysis increases, the difference between these two estimators decreases towards zero, and, in fact, that the estimators converge toward the IE B.

011̂ BtBb

022ˆ BtBb

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4) Monte Carlo Simulation: Numerical simulation demonstrations of the foregoing statistical properties were given in Yang et al. (2008); one example is reproduced on the following slide.

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Simulation Results of the IE and CGLIM Estimators: True Cohort Effects = 0 Age Effect: Mean Estimates

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

a1 a2 a3 a4 a5 a6 a7 a8 a9Age

Lo

g c

oe

f

True effect IECGLIM_p CGLIM_c

Age Effect: MSE

0.0

5.0

10.0

15.0

20.0

25.0

30.0

a1 a2 a3 a4 a5 a6 a7 a8 a9Age

Period Effect: Mean Estimates

-0.5

-0.3

-0.1

0.1

0.3

0.5

p1 p2 p3 p4 p5Period

Lo

g c

oe

f

Period Effect: MSE

0.0

2.0

4.0

6.0

8.0

p1 p2 p3 p4 p5Period

Cohort Effect: Mean Estimates

-2.0

-1.0

0.0

1.0

2.0

c1 c3 c5 c7 c9 c11 c13Cohort

Lo

g c

oe

f

Cohort Effect: MSE

0

100

200

300

400

c1 c3 c5 c7 c9 c11 c13Cohort

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Based on these statistical properties, Yang et al. (2008) also showed how the IE can be used in an asymptotic t-test to evaluate a substantively informed equality constraint on the APC accounting model with respect to whether the estimated coefficient vector that results therefrom is (statistically) estimable, that is, within sampling error of meeting the Kupper et al. condition for estimability.

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The Intrinsic Estimator (IE) Method: Computation Software

Two programs for calculating the IE are available for use in popular statistical packages:

1) a S-Plus/R program

and

2) a Stata Ado File

(both referenced in Yang et al., 2008)

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Example: Intrinsic Estimates of Age, Period, and Cohort Effects of Lung Cancer Mortality by Sex (Yang 2008)

Age Effect

-6.0

-4.0

-2.0

0.0

2.0

Age

Lo

g c

oe

ffic

ien

t

Male

Female

Period Effect

-2.0

-1.0

0.0

1.0

2.0

Year

Male

Female

Cohort Effect

-3.0

-2.0

-1.0

0.0

1.0

2.0

Cohort

Lo

g c

oe

ffic

ien

t

Male

Female

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Some Recent Empirical Applications of the Intrinsic Estimator:

Schwadel, P. 2011. “Age, period, and cohort effects on religious activities and beliefs”, Social Science Research 40:181-192.

Unknown Author. 2011. “Age, Period, and Cohort Effects on Social Capital and Voting.” Social Forces

90:forthcoming.

Winkler, Richelle L., Jennifer Huck, and Keith Warnke. 2009. “Deer hunter demography: An age-period-cohort

approach to population projections.” Paper presented at the Population Association of America Annual Meeting, Detroit, MI, April 30, 2009.

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The Intrinsic Estimator (IE): Conclusion

Is the Intrinsic Estimator a “final” or “universal” solution to the APC “conundrum”?

No. There will never be such a solution. The APC identification problem is one of structural under-identification in linear or generalized linear models for which there can only be partial solutions.

But the IE has been shown to be a useful approach to the identification and estimation of the APC accounting model that

• has desirable mathematical and statistical properties; and• has passed both case studies and simulation tests of model

validation.

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Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys

References for Part III:

Yang, Yang. 2006. Bayesian Inference for Hierarchical Age-Period-Cohort Models of Repeated Cross-Section Survey Data. Sociological Methodology 36:39-74.

Yang Yang and Kenneth C. Land. 2006. A Mixed Models Approach to the Age-Period-Cohort Analysis of Repeated Cross-Section Surveys, With an Application to Data on Trends in Verbal Test Scores. Sociological Methodology 36:75-98.

Yang Yang and Kenneth C. Land. 2008. Age-Period-Cohort Analysis of Repeated Cross-Section Surveys: Fixed or Random Effects? Sociological Methods and Research 36(February):297-326.

Yang, Yang 2008. “Social Inequalities in Happiness in the United States, 1972 to 2004: An Age-Period-Cohort Analysis.” American Sociological Review 73(April): 204-226.


References for Part III, Continued:

Yang Yang, Steven M. Frenk, and Kenneth C. Land. 2010. “Assessing the Significance of Cohort and Period Effects in Hierarchical

Age-Period-Cohort Models.” Revision of a paper presented at the American Sociological Association Annual Meeting, San Francisco, CA, August 2009.

Zheng, Hui, Yang Yang, and Kenneth C. Land. 2011. “Heteroscedastic Regression in Hierarchical Age-Period-Cohort Models, With Applications to the Study of Self-Reported Health. Revision of a paper presented at the American Sociological Association Annual Meeting, Atlanta, GA, August 2010.

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Data Structure: Individual-level Data in an Age-by-Period Array

Period j

Age i

nij >1

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Approach to the Identification Problem

Many researchers previously have assumed that the APC identification problem for age-by-time period tables of rates transfers over directly to this research design.

But note that this research design yields individual-level data, i.e., microdata on the ages and other characteristics of individuals in the samples.

Proposal: Use different temporal groupings for the A, P, and C dimensions to break the linear dependency:

Single year of ageTime periods correspond to years in which the surveys are

conductedCohorts can be defined either by five- or ten-year intervals that

are conventional in demography or by application of a substantive classification (e.g., War babies, Baby Boomers, Baby Busters, etc.).

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Example: Two-way Cross-Classified Data Structure in the GSS: Number of Observations by Cohort and Period in the Verbal Ability Data (Yang and Land 2006)

Year (K)

Cohort (J) 1974 1976 1978 1982 1984 1987 1988 1989 1990 1991 1993 1994 1996 1998 2000 Total

1890 12 18 8 0 0 0 0 0 0 0 0 0 0 0 0 38 1895 31 25 19 19 6 0 0 0 0 0 0 0 0 0 0 100 1900 62 52 49 27 18 17 13 11 5 2 0 0 0 0 0 256 1905 88 69 68 43 38 23 11 12 11 11 15 15 10 0 0 414

1910 77 89 69 75 50 48 34 27 25 29 13 31 27 18 8 620 1915 109 111 84 100 81 81 42 36 37 41 37 60 39 24 27 909

1920 115 104 112 110 73 97 60 53 40 56 55 85 59 32 37 1088 1925 113 108 106 131 99 92 52 53 53 40 50 84 81 68 52 1182

1930 129 92 90 111 81 95 47 54 43 62 43 86 72 45 64 1114 1935 130 106 108 112 80 101 39 59 44 37 58 101 100 61 64 1200

1940 119 140 130 127 100 142 49 74 49 65 58 134 117 65 78 1447 1945 179 161 184 163 133 143 98 84 85 74 85 168 161 104 85 1907

1950 179 180 197 199 170 185 101 94 95 111 99 173 169 101 111 2164 1955 89 151 180 260 162 219 102 117 106 118 127 198 213 149 145 2336

1960 0 8 59 175 186 190 109 121 102 118 103 231 208 161 147 1918 1965 0 0 0 38 75 161 101 86 76 91 111 182 188 157 111 1377

1970 0 0 0 0 0 29 32 48 55 77 81 157 188 116 145 928 1975 0 0 0 0 0 0 0 0 0 1 23 59 128 84 107 402

1980 0 0 0 0 0 0 0 0 0 0 0 0 4 34 62 100 Total 1432 1414 1463 1690 1352 1623 890 929 826 933 958 1764 1764 1219 1243 19500

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This Data Structure illustrates that: respondents are nested in and cross-classified

simultaneously by the two higher-level social contexts defined by time period and birth cohort,

individual members of any birth cohort can be interviewed in multiples replications of the survey, and

individual respondents in any particular wave of the survey can be drawn from multiple birth cohorts.

Key Points:

1) this approach builds on the recognition that age is an intrinsically individual-level property that individuals carry with them and that varies from period to period;

2) by comparison an individual’s cohort is fixed, as is the time period of a particular survey, and both cohort and period are contexts within which individuals mature and age and experience certain events.

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Further Questions:

Is there evidence for clustering effects of random errors, due to the facts that:

• individuals surveyed in the same year may be subject to similar unmeasured events that influence their outcomes, and

• members of the same birth cohort may be subject to similar unmeasured events that influence their outcomes?

How can this random variability be modeled and explained?

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Method: Apply Hierarchical Age-Period-Cohort (HAPC) Models

These models generally are members of what statisticians call mixed (fixed and random) effects models; in the social sciences, these models typically are called hierarchical linear models (HLM).

The mixed models may be linear mixed effects (LMM) models or, more generally, allow for nonlinear link functions, in which case they are generalized linear mixed models (GLMM).

A form of HLMs applicable to cross-classified data of the form shown above is the class of cross-classified random effects models (CCREM).

Objective: Model the level-two heterogeneity to: Assess the possibility that individuals within the same periods and

cohorts could share unobserved random variance; Explain the level-two variance by contextual characteristics of time

periods and birth cohorts.

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Application 1 – A HAPC-LMM of General Social Survey (GSS) Data on Verbal Test Scores: 1974 – 2006

The Initial Papers:

Alwin, D. 1991. “Family of Origin and Cohort Differences in Verbal Ability.” American Sociological Review 56:625-38.

Glenn, N.D. 1994 “Television Watching, Newspaper Reading, and Cohort Differences in Verbal Ability.” Sociology of Education 67:216-30.

The debate in the American Sociological Review:

Wilson, J.A. and W.R. Gove. 1999. "The Intercohort Decline in Verbal Ability: Does It Exist?" and reply to Glenn and Alwin & McCammon. ASR 64:253-266, 287-302.

Glenn, N.D. 1999. “Further Discussion of the Evidence for An Intercohort Decline in Education-Adjusted Vocabulary.” ASR 64:267-71.

Alwin, D.F. and R.J. McCammon. 1999. “Aging Versus Cohort Interpretations of Intercohort Differences in GSS Vocabulary Scores.” ASR 64:272-86.

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Research Questions What are the distinct age, period, and cohort components of

change in verbal ability in the U.S.? How can period and/or cohort level heterogeneity be explained by

period and/or cohort characteristics?

Analytic Method Apply the HAPC-CCREM to estimate

• fixed effects of age and other individual level and level-two covariates,

• random effects of period and cohort and variance components

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Because the WORDSUM outcome variable has a relatively bell-shaped sample frequency distribution, it is reasonable to use a HAPC model specification that includes a conventional normal-errors regression model. Specifically, Yang and Land (2006: 87) specified the cross-classified random effects model (CCREM):

Level-1 or “Within-Cell” Model:

WORDSUMijk = β0jk + β1AGEijk + β2AGE2ijk + β3EDUCATIONijk + β4FEMALEijk +

β5BLACKijk + eijk , eijk ~ N(0, σ2 ) (1)

Level-2 or “Between-Cell” Model:

β0jk = γ0 + u0j + v0k, u0j ~ N(0, τu), v0k ~ N(0, τv) (2)

Combined Model:

WORDSUMijk = γ0 + β1AGEijk + β2AGE2ijk + β3EDUCATIONijk + β4FEMALEijk +

β5BLACKijk + u0j + v0k + eijk (3)

for i = 1, 2, …, njk individuals cross-classified within cohort j and period k;j = 1, …, 20 birth cohorts; k = 1, …, 17 time periods (survey years)

57

Table 2. HAPC Models of the GSS WORDSUM Data, 1974-2006 Fixed Effects coefficient se t ratio p value INTERCEPT 6.175 0.055 112.50 < .000 AGE 0.026 0.015 1.71 .087 AGE2 -0.057 0.005 -11.87 < .000 FEMALE 0.229 0.024 9.49 < .000 BLACK -1.030 0.034 -30.07 < .000 EDUCATION 0.366 0.004 86.57 < .000 Random Effects Cohort coefficient se t ratio p value

1894 -0.210 0.142 -1.48 0.140 1895 -0.114 0.123 -0.93 0.353 1900 -0.051 0.104 -0.49 0.625 1905 -0.294 0.090 -3.27 0.001 1910 0.021 0.081 0.26 0.797 1915 0.163 0.073 2.22 0.027 1920 -0.079 0.068 -1.15 0.249 1925 0.083 0.068 1.23 0.220 1930 0.001 0.067 0.01 0.990 1935 0.068 0.064 1.06 0.289 1940 0.240 0.061 3.91 < .000 1945 0.447 0.060 7.50 < .000 1950 0.184 0.059 3.10 0.002 1955 -0.035 0.061 -0.57 0.568 1960 0.002 0.065 0.04 0.970 1965 -0.157 0.071 -2.20 0.028 1970 -0.135 0.080 -1.70 0.090 1975 -0.001 0.092 -0.01 0.990 1980 0.062 0.112 0.55 0.583 1985 -0.195 0.146 -1.34 0.180

58

Period 1974 0.033 0.043 0.77 0.442 1976 0.060 0.043 1.41 0.158 1978 -0.002 0.042 -0.04 0.967 1982 -0.014 0.040 -0.36 0.718 1984 0.016 0.042 0.37 0.709 1987 -0.061 0.040 -1.52 0.129 1988 -0.128 0.046 -2.76 0.006 1989 -0.061 0.046 -1.34 0.182 1990 0.020 0.047 0.43 0.670 1991 0.042 0.046 0.92 0.358 1993 -0.004 0.045 -0.09 0.926 1994 0.019 0.039 0.49 0.623 1996 -0.060 0.039 -1.52 0.128 1998 0.044 0.043 1.02 0.306 2000 0.005 0.043 0.11 0.915 2004 0.038 0.043 0.88 0.381 2006 0.052 0.045 1.16 0.247

Variance Components variance se z value p value Cohort 0.034 0.013 2.56 .010 Period 0.005 0.003 1.49 .135 Individual 3.116 0.030 104.87 < .000 Deviance 87707.2 AIC 87713.2

59

Figure 1. Estimated Cohort and Period Effects and 95 Percent Confidence Bounds for GSS Verbal Ability Model

5.00

6.00

7.00

Verb

al T

est S

core

Cohort

5.00

6.00

7.00

Verb

al T

est S

core

Period

Cohort Effect j0̂ Period Effect k0̂

60

To further test whether the birth cohort and time period effects – as a whole – make statistically significant contributions to explained variance in an outcome variable, a general linear hypothesis may be applied.

Specifically, one can either:

1) examine the statistical significance of the variance components (an asymptotic t-test for LMMs), or

2) use an F test to test the hypothesis of the presence of random effect.

The sampling distribution of F statistic is exact in LMMs when the random effects are independently distributed as normal random variables.

This F-test statistic is preferred over the z-score when the sample sizes for random effects are small. The statistical theory for such tests has been developed in a very general LMM context by E. Demidenko (Mixed Models: Theory and Applications. Wiley, 2004).

61

In the present case, for the CCREM-HAPC model of Equations (1)-(3), there are only two sets of random effect coefficient that are estimated, namely, the set of residual random effects of cohort j, u0j, and the set of residual random effects of period k, v0k. Each of these sets of random coefficients is assumed to be independently, normally distributed with mean 0 and variances τu and τv, respectively.

Thus, for a CCREM-HAPC model with random intercepts of the form of Equations (1)-(3), the exact F-test amounts to testing null hypotheses for the relevance either of the birth cohort random effects:

H0: τu = 0, vs. Ha: τu > 0

or the time period effects:

H0: τv = 0, vs. Ha: τv > 0.

Alternatively, one can test for the joint relevance of both the cohort and period effects:

H0: τu = τv = 0, vs. Ha: τu > 0 or τv > 0

62

Table 3. F-tests for the Presence of Random Effects, GSS WORDSUM Data

Cohort Effects τu = 0 vs. τu > 0

Period Effects τv = 0 vs. τv > 0

Cohort and Period Effects τu = τv = 0 vs. τu or τv > 0

OLSS 69,377 69,377 69,377

minS 68,696 69,268 68,558

R 25 22 42

M 5 5 5 NT 22,042 22,042 22,042

)/()( min mrSSOLS 34.05 6.41 22.14

)/(min rNS T 3.12 3.15 3.12

F 10.9 2.03 7.096 f0.95(r – m, NT – r) 1.571 1.623 1.411

63


BACK TO THE DEBATE ON TRENDS IN VERBAL ABILITY: So, who is right, Alwin and Glenn or Wilson and Gove?

The results of the HAPC analyses show: significant random variance components that reside in all three

levels of the APC data: individuals nested within cohorts and periods; quadratic age effects that are not explained away by controlling for

the effects of key individual characteristics, namely, education, sex and race, and for period and cohort effects;

significant contextual effects of cohorts and periods on verbal ability, but this is mainly a cohort story; and

strong effects of cohort characteristics: cohorts that have a larger proportion of daily newspaper readers are better off in their verbal ability; more hours of TV watching per day tend to undermine average cohort verbal ability.

Bottom Line: Alwin and Glenn are more right than Wilson and Gove.

64


Extensions of HAPC Modeling:– Fixed Effects vs. Mixed Effects Model– A Full Bayesian HAPC Model– Generalized Linear Mixed Models (GLMM)

65


Fixed Effects vs. Mixed Effects Model:

The HAPC-CCREM approach illustrated above uses a mixed (fixed and random) effects model with a random effects specification for the level-2 (time period and cohort) contextual variables.

Alternative: fixed effects specification for the level-2 variables in which ones uses dummy (indicator) variables to record the cohort and the time period of the survey.

The comparison seems especially pertinent when the number of replications of the survey is relatively small—say 3 to 5.

66


Fixed Effects vs. Mixed Effects Model:The estimates of cohort and time period effects from a fixed effects

model for the GSS data are quite similar in pattern to those from the random effects model (Yang and Land 2008).

The mixed effects model is preferred to the fixed effect model: It avoids potential model specification error by not using the

assumption of the fixed effect model that the indicator/dummy variables representing the fixed cohort and periods effects fully account for all of the group effects;

It allows group level covariates to be incorporated into the model and explicitly models cohort characteristics and period events to test explanatory hypotheses;

For unbalanced research designs (designs in which there are unequal numbers of respondents in the cells), such as one typically has in repeated cross-section survey designs, a random effect model for the level-2 variables generally is more statistically efficient.

67


A Full Bayesian HAPC Model:Limitations of HAPC Modeling Using REML-EB Estimation

• Small numbers of cohorts (J) and periods (K)

• Unbalanced data

• Inaccurate REML estimates of variance-covariance components

• Inaccurate EB estimates of fixed effects regression coefficients

A Remedy: Bayesian Model Estimation (Yang 2006)• A full Bayesian approach, by definition, ensures that inferences about

every parameter fully account for the uncertainty associated with all others.

68


Application 2: A HAPC-GLMM of American National Election Survey (ANES) Data on Voting Turnout in U.S. Presidential Elections, 1952-2004 (Yang, Frenk, and Land 2010)The GLMM Family of Models:

• Normal outcome: Linear mixed models using Gaussian link• Binomial outcome: Logistic mixed models using logit link• Ordinal or nominal outcome: Ordinal logistic mixed models• Count outcome: Poisson mixed models using log link• Count outcome with dispersion: Negative Binomial mixed

models

REML-EB Estimation: Use, e.g., SAS PROC GLIMMIXED

69

Application 2: A HAPC-GLMM of Voting Turnout in U.S. Presidential Elections

Table 4. Descriptive Statistics for ANES Voter Turnout Data, 1952 to 2004

Variables Description Mean SD Min Max

Dependent Variable

VOTE 1= Voted in U.S. presidential elections; 0 = Did not vote in U.S. presidential elections 0.76 0.43 0 1

Level-1 Variables MALE Respondent's sex: 1 = male; 0 = female 0.45 0.5 0 1 BLACK Respondent's race: 1 = black; 0 = nonblack 0.1 0.3 0 1 AGE Respondent's age at survey year 45.62 16.53 18 95 Centered around grand mean 0 16.53 -27.46 49.54 Religion Respondents' religious preference PROTESTANT 1 = Protestant; 0 = otherwise 0.66 0.47 0 1 CATHOLIC 1 = Catholic; 0 = otherwise 0.24 0.42 0 1 JEW 1 = Jew; 0 = otherwise 0.02 0.15 0 1

OTHER 1 = Other/None; 0 = otherwise 0.08 0.27 0 1

70

MARRIED Respondent's marital status: 1 = Currently married; 0 = otherwise 0.67 0.47 0 1

Occupational Class PROFESSIONAL 1 = Professional; 0 = otherwise 0.25 0.43 0 1 CLERICAL 1 = Clerical; 0 = otherwise 0.18 0.38 0 1 SKILLED 1 = Skilled; 0 = otherwise 0.31 0.46 0 1 LABORER 1 = Laborer; 0 = otherwise 0.03 0.17 0 1 FARMER 1 = Farmer; 0 = otherwise 0.04 0.19 0 1 NOT WORKING 1 = Not working; 0 = otherwise 0.2 0.4 0 1 Political affiliation Respondent's political affiliation DEMOCRATIC 1 = Democratic; 0 = otherwise 0.52 0.5 0 1 INDEPENDENT 1 = Independent; 0 = otherwise 0.1 0.3 0 1 REPUBLICAN 1 = Republican; 0 = otherwise 0.38 0.48 0 1 POLITICAL SOUTH¹ 1 = Political south; 0 = otherwise 0.27 0.45 0 1 Level-2 Variables N Min Max PERIOD Survey year 14 1952 2004 COHORT Five-year birth cohort 23 1859 1986

¹Includes the eleven session states: Alabama, Arkansas, Florida, Georgia, Louisiana, Mississippi, North Carolina, South Carolina, Tennessee, Texas, Virginia Note: N=19,766

71

To model the likelihood of voter turnout in U.S. Presidential Elections, we apply the HAPC-CCREM approach and specify the following model:

Level 1 or “Within-Cell” Model:Pr (VOTEijk = 1) = β0jk + β1AGEijk + β2AGE2ijk + β3MALEijk + β4BLACKijk + β5PROTESTANTijk + β6CATHOLICijk + β7JEWijk + β8PROFESSIONALijk + β9CLERICALijk + β10SKILLEDijk + β11FARMERijk + β12NOWORKijk + β13PSOUTHijk + β14CMARRIEDijk + β15DEMOCRATICijk + β16REPUBLICANijk

Level 2 or “Between-Cell” Model:β0jk = γ0 + u0j + ν0k , u0j ~ N(0, τu), ν0k ~ N(0, τv)

COMBINED MODEL: Pr (VOTEijk = 1) = β0jk + β1AGEijk + β2AGE2ijk + β3MALEijk + β4BLACKijk + β5PROTESTANTijk + β6CATHOLICijk + β7JEWijk + β8PROFESSIONALijk + β9CLERICALijk + β10SKILLEDijk + β11FARMERijk + β12NOWORKijk + β13PSOUTHijk + β14CMARRIEDijk + β15DEMOCRATICijk + β16REPUBLICANijk + u0j + ν0k + eijk (12)for i = 1, 2, …, njk individual within cohort j and period k;

j = 1, …23 birth cohorts;k = 1, …, 14 time periods (presidential elections).

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Table 5. HAPC Models of the ANES Voter Turnout Data, 1952-2004

Fixed Effects coefficient se t ratio p value

INTERCEPT -0.49 0.134 -3.66 0.003

AGE 0.025 0.001 19.22 <.000

AGE² -0.001 0.0001 -13.85 <.000

MALE 0.21 0.044 4.73 <.000

BLACK -0.02 0.058 -0.34 0.734

PROTESTANT 0.335 0.065 5.2 <.000

CATHOLIC 0.591 0.071 8.3 <.000

JEW 1.211 0.184 6.6 <.000

PROFESSIONAL 1.414 0.105 13.45 <.000

CLERICAL 1.037 0.107 9.73 <.000

SKILLED 0.285 0.098 2.9 0.004

FARMER 0.301 0.128 2.35 0.019

NOT WORKING 0.386 0.107 3.6 <.000

POLITICAL SOUTH -0.607 0.04 -15.23 <.000 CURRENTLY MARRIED 0.446 0.041 10.78 <.000

DEMOCRATIC 0.736 0.055 13.33 <.000

REPUBLICAN 0.963 0.059 16.4 <.000

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Random Effects

Cohort coefficient se t ratio p value

1859-1875 0.011 0.066 0.17 0.867

1876-1880 -0.01 0.064 -0.16 0.871 1881-1885 -0.022 0.063 -0.35 0.723

1886-1890 0.007 0.061 0.12 0.906

1891-1895 -0.026 0.059 -0.45 0.655

1896-1900 0.01 0.058 0.17 0.864

1901-1905 -0.028 0.055 -0.51 0.613

1906-1910 -0.047 0.053 -0.89 0.376

1911-1915 0.061 0.051 1.19 0.236

1916-1920 0.016 0.049 0.32 0.747

1921-1925 0.05 0.048 1.03 0.303

1926-1930 0.054 0.048 1.12 0.262

1931-1935 -0.003 0.049 -0.06 0.953

1936-1940 -0.033 0.05 -0.66 0.511

1941-1945 0.024 0.049 0.49 0.625

1946-1950 0.035 0.048 0.73 0.468

1951-1955 0.027 0.049 0.55 0.581

1956-1960 -0.106 0.05 -2.11 0.035

1961-1965 -0.032 0.053 -0.61 0.54

1966-1970 -0.014 0.057 -0.24 0.811

1971-1975 -0.003 0.06 -0.05 0.958

1976-1980 0.016 0.063 0.26 0.795

1981-1986 0.013 0.065 0.2 0.838

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Period

1952 -0.029 0.076 -0.39 0.7

1956 -0.102 0.068 -1.50 0.134

1960 0.274 0.081 3.4 0.001

1964 0.066 0.072 0.91 0.361

1968 0.02 0.073 0.28 0.783

1972 -0.066 0.063 -1.04 0.298

1976 -0.049 0.067 -0.74 0.459

1980 -0.055 0.073 -.75 0.453

1984 0.004 0.067 0.05 0.956

1988 -0.259 0.067 -3.85 <.000

1992 0.108 0.066 1.63 0.104

1996 -0.011 0.073 -0.16 0.876

2000 -0.074 0.075 -0.99 0.323

2004 0.174 0.087 2.01 0.045

Variance Components variance se z value p value

Cohort 0.004 0.003 1.33 0.16

Period 0.021 0.01 2.1 0.04

Deviance 94358.54

AIC 94305.54 Source: 1952-2004 American National Election Study (N = 19,766) #p<.1; *p<.05; **p<.01; ***p<.001

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Figure 2. Estimated Cohort and Period Effects and 95 Percent Confidence Bounds for NES Voter Turnout Model

Period Effect k0̂ Cohort Effect j0̂

0.5

0.55

0.6

0.65

0.7

0.75

0.8

Cohort

Pred

icte

d Pr

obab

ility

of V

otin

g

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Table 6. F-Tests for the Presence of Random Effects, ANES Data

Cohort Effects τu = 0 vs. τu > 0

Period Effects τv = 0 vs. τv > 0

Cohort and Period Effects τu = τv = 0 vs. τu or τv > 0

l0 9,695 9,695 9,695 lmax 9,679 9,662 9,646 r 39 30 53 m 16 16 16

NT 19,766 19,766 19,766 (lmax – l0)/(r – m) 0.6957 2.357 1.32 l0/( NT – r ) 0.4915 0.4912 0.4918 F 1.42 4.8 2.69

f0.95(r – m, NT – r) 1.53 1.69 1.7

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As in the case of trends in GSS verbal ability, this analysis of Presidential voting turnout finds:significant random variance components that reside

in all three levels of the APC data: individuals nested within cohorts and periods;

quadratic age effects that are not explained away by controlling for the effects of individual characteristics, and for period and cohort effects;

significant contextual effects of cohorts and periods on voting in Presidential elections;

but Presidential voting turnout is mainly a period story.

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Application 3: A HAPC-GLMM Analysis of GSS Data on Happiness, 1972-2004 (Yang 2008)

Research Questions: Who is happier? – Social stratification of subjective well-being Do people get happier with age and over time? How do social inequalities in happiness vary over the life course andby time? Born to be happy? Are there any birth cohort differences in happiness?

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Level 1 (Individual-Level) Model:

where yijk denotes the ordinal response happiness variable in the GSS data (very happy, pretty happy, not too happy) modeled with an ordinal logit HAPC-CCREM specification, and

Xp denotes a vector of other individual-level variables such as age by sex, age by race, and age by education interaction variables.

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Level 2 Model:


81

Some Findings:

82

Some Findings:

83

Some Findings:

84


As in the case of trends in GSS verbal ability and NES Presidential election voting probabilities, this analysis of the GSS happiness data finds:

significant random variance components that reside in all three levels of the APC data: individuals nested within cohorts and periods; quadratic age effects that are not explained away by controlling for the effects of individual characteristics, and for period and cohort effects; significant contextual effects of both cohorts and periods on voting in Presidential elections, i.e., interesting stories both for cohorts and periods.

Application 4: An Integration of the Hierarchical Age-Period-Cohort Model with Heteroscedastic Regression to Develop the HAPC-HR Model, Applied to Study Variations in Self-Reported Health Disparities in the U.S., 1984-2007 (Zheng, Yang, and Land 2011)

There are three standard approaches to the study of changes in health disparities:

(1) across the life course (e.g., House et al. 1994; Dannefer 2003),

(2) across cohorts (e.g., Lynch 2003; Warren and Hernandez 2007), and

(3) across time periods (e.g., Pappas et al. 1993; Goesling 2007).


All of these approaches have one thing in common:

They focus on changes in health disparities as estimated by conditional expectation functions (regressions) estimated on the basis of measured demographic and socioeconomic covariates.

This facilitates the estimation of between-group disparities, i.e., variations in health across groups or between-cell variation and temporal variations therein,

but

it ignores possible within-group disparities – variations in health inside groups or within-cell variation – and variations therein over time.

To examine Age-Period-Cohort variations in both health and health disparities, we:

intersect the HAPC model

with

a Heteroscedastic Regression (HR) model.

This allows us to both:

(1) disentangle age, period, and cohort effects, and

(2) separate within-group health disparities from between-group health disparities. The result is a Hierarchical-Age-Period-Cohort-Heteroscedastic-Regression Model (HAPC-HR) model.

Application to National Health Interview Survey (NHIS) data on self-reported health, 1984-2007: With individual-level demographic and socioeconomic that are established covariates of health used to define the cells in the Level-1 regression model:

sex (1 = male, 0 = female), race (1 = white, 0 = non-white), marital status (1 = married, 0 = unmarried), work status (1 = full/part time job and 0 = not

employed), education (years of formal education), and income (in 2007 dollars),

here are some results.

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Figure 1. Observed Means of Self-Rated Health, NHIS, 1984 to 2007.

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4

Year

Me

an

of

Se

lf-R

ate

d H

ea

lth

The whole sample

Men

Women

* The trends are adjusted for sample weights and smoothed by a three-point moving average.

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Figure 2. Observed Variances in Self-Rated Health, NHIS, 1984 to 2007.

1

1.1

1.2

1.3

1.4

1.5

Year

Va

ria

nce

in S

elf-

Ra

ted

He

alth

The whole sample

Men

Women

* The trends are adjusted for sample weights and smoothed by a three-point moving average

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Figure 5. Variations in Conditional Expected Values of Gender-Specific Self-Rated Health

across Age, Cohort and Period, with 95% Confidence Intervals.

3.1

3.3

3.5

3.7

3.9

4.1

4.3

4.5

4.7

AgeC

on

ditio

na

l E

xp

ecte

d V

alu

e o

f S

elf-R

ate

d H

ea

lth

Men

Women

3.45

3.5

3.55

3.6

3.65

3.7

3.75

3.8

3.85

Cohort

Con

ditio

nal E

xpec

ted

Valu

e of

Sel

f-Rat

ed H

ealth

Men

Women

3.45

3.5

3.55

3.6

3.65

3.7

3.75

3.8

3.85

Period

Con

ditio

nal E

xpec

ted

Val

ue o

f Sel

f-Rat

ed H

ealth

Men

Women

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Figure 6. Variations in Predicted Dispersion of Gender-Specific Self-Rated Health across

Age, Cohort and Period, with 95% Confidence Intervals.

0.55

0.65

0.75

0.85

0.95

1.05

1.15

1.25

Age

Pre

dic

ted

Dis

pe

rsio

n o

f S

elf-R

ate

d H

ea

lth

Men

Women

0.8

0.9

1

1.1

1.2

1.3

Cohort

Pred

icte

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ispe

rsio

n of

Sel

f-Rat

ed H

ealth

Men

Women

0.8

0.9

1

1.1

1.2

1.3

Period

Pre

dic

ted

Dis

pe

rsio

n o

f S

elf-

Ra

ted

He

alth

Men

Women

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Part IV: Third Research Design: Cohort Analysis of Accelerated Longitudinal Panels

References for Part IV:

Miyazaki, Yasuo and Stephen W. Raudenbush. 2000. "Tests for Linkage of Multiple Cohorts in an Accelerated Longitudinal Design." Psychological Methods 5:44-63.

Yang, Yang. 2007. “Is Old Age Depressing? Growth Trajectories and Cohort Variations in Late Life Depression.” Journal of Health and Social Behavior 48:16-32.

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Accelerated Longitudinal Panel Design

Definition: A longitudinal panel study of an initial sample of individuals from a broad array of ages (and thus birth cohorts) interviewed or monitored with three or more follow-up waves.

The design allows a more rapid accumulation of information on age and cohort effects than a single cohort follow-up study.

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Cohort

Age (Time)

Data Structure: Accelerated Longitudinal Panel Design

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For this research design, the HAPC Model becomes a Growth Curve Model of Individual Change with cohort interactions:Assess the intra-individual age changes and birth cohort

differences simultaneously;Assess differential cohort patterns in age changes: age-

by-cohort interaction effects;Period effects?

• The time period for an accelerated longitudinal panel study often is short (e.g., a decade or so), so the effects of period usually can be ignored;

• In growth curve models, age and time are the same variable, so the effects of period need not be estimated; and

• can be focused on the age-by-cohort interactions.• If period effects are of concern, estimate the HAPC-CCREM.

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Application: Cohort Variations in Age Trajectories of Depression in the Elderly (Yang 2007)Research Questions

• Does the age growth trajectory show an increase in depressive symptoms in late life?

• Is there cohort heterogeneity in levels of depressive symptoms and age growth trajectories of depressive symptoms?

• What social risk factors are associated with these effects?

Data• Established Populations for Epidemiologic Studies

of the Elderly (EPESE) in North Carolina: A four-wave panel study of older adults aged 65+ from 1986 to 1996

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Model SpecificationLevel-1 Repeated Observation Model

(11)

Yti = CES-D for person i at time t, for i =1, …, n and t = 1, …, Ti

Xpti = (marital status, economic status, health status,

stress and coping resources)

= expected CES-D for person i

= expected growth rate per year of age in CES-D for person i

= regression coefficient associated with Xpti

tip

ptipitiiiti eXAgeY 10

i0

i1

pi

),0(~ 2Neti

iid

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Model SpecificationLevel-2 Individual Model

(12)Zqi = (Female, Black, Education)

= expected CES-D for person i for the reference group (at median age in Cohort 1 at T1)

= main cohort effect coefficient: mean difference in CES-D between cohorts

= regression coefficient associated wit Zqi

= age effect coefficient: expected rate of change in CES-D

= age*cohort coefficient: mean difference in rate of change between cohorts

iq

qiqii rZCohort 0001000

iii rCohort 111101

110

010

1

0 ,0

0~

Nr

r

i

i iid

00

01

q0

10

11

100


Fixed Effect Model 1

(Total)

Model 7

(Net)

Intercept, 2.856*** 2.525***

Growth Rate: Age, 0.048*** -0.018

Cohort 0.244*** -0.213**

Age * Cohort -0.019# -0.040***

Random Effect Variance Component % Reduction

Level-1: Within person 36.987*** 35.109*** 5%

Level-2: In intercept 6.170*** 3.763*** 39%

In growth rate 0.057*** 0.051*** 11%

Goodness-of-fit

AIC (smaller is better) 51190.5 48167.4

BIC (smaller is better) 51215.6 48192.5

i0

i1

# p < .10; * p < .05; ** p < .01; *** p < .001.

00

01

10

11

2

0

1

Model Estimates

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Expected Growth Trajectories and Cohort Variations in Depression

a. Model 1-Gross Age and Cohort Effects

1

2

3

4

5

Age

CE

S-D

b. Model 7- Net Age and Cohort Effects

1

2

3

4

5

Age

CE

S-D

All cohort 1 cohort 2 cohort 3 cohort 4 cohort 5

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Summary of Findings:The gross age trajectory of depressive symptoms

during late life is positive and linear;There is substantial cohort heterogeneity in both

average levels of depressive symptoms and age growth trajectories of depressive symptoms;

The age growth trajectories of depressive symptoms are not significant after adjusting for cohort effects and risk factors associated with historical trends in education, life course stages, survival, health decline, stress and coping resources;

Net of all the factors considered, more recent birth cohorts have higher levels of depression.

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Conclusion

A Webpage has been developed that contains copies of our papers referenced in this presentation as well as others:

http://www.unc.edu/~yangy819/apc/index.html

Happy Hunting for Age, Period, and Cohort Effects!

1 Age-Period-Cohort Analysis: New Models, Methods, and Empirical Analyses Kenneth C. Land, Ph.D....

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