Latent transition analysis (LTA)Latent transition analysis (LTA)for modeling discrete change in for modeling discrete change in

longitudinal datalongitudinal data

Stephanie T. LanzaThe Methodology Center

The Pennsylvania State University

UCLA

May 1, 2006

The Methodology CenterThe Methodology Center

• A group of social scientists and statisticians working together

• Work on statistical methods and applications with direct relevance to important scientific questions

• Primarily motivated to work on methods relevant to study of substance use and abuse

The Methodology CenterThe Methodology Center

• Examples of research topics:– Latent class, latent transition analysis– Missing data, theory and applications– Adaptive interventions– Optimal design of behavioral interventions– Analysis of data from intensive data

collection methods– Economic cost-effectiveness analysis– Risk assessment

Bethany BrayHwan Chung

Linda M. CollinsDavid LemmonTammy Root

Joseph L. Schafer

Recent collaborators on LTARecent collaborators on LTA

OutlineOutline

• Overview of what LCA and LTA can do

• LTA

• Advanced topics and future directions

• Some research questions you might address using LTA

OutlineOutline

• Overview of what LCA and LTA can do

• LTA

• Advanced topics and future directions

• Some research questions you might address using LTA

Ideas underlying LCAIdeas underlying LCA

• Individuals can be divided into subgroups, or latent classes, based on unobserved construct

• Subgroups are mutually exclusive and exhaustive

• True class membership in unknown

Ideas underlying LCAIdeas underlying LCA

• Measurement of that construct typically based on several categorical indicators

• There may be error associated with the measurement of the latent classes

• Like confirmatory factor analysis (specify number of classes), but latent variable is categorical

• Latent class membership probabilities– e.g. probability of membership in Advanced

Substance Use latent class

• Item-response probabilities– e.g. probability of reporting marijuana use

given membership in Advanced Substance Use latent class

Parameters estimated in LCAParameters estimated in LCA

Example of LCA:Example of LCA:Depression in adolescenceDepression in adolescence

Sad– Couldn’t shake blues– Felt depressed– Felt lonely– Felt sad

Lanza, S. T., Flaherty, B. P., & Collins, L. M. (2003). Latent class and latent transition analysis. J. A. Schinka, & W. F. Velicer (Eds.), Handbook of Psychology: Vol. 2. Research Methods in Psychology (pp. 663-685). Hoboken, NJ: Wiley.

Eight indicators of adolescent depression:

Disliked– People unfriendly– Disliked by people

Failure– Life was failure– Life not worth living

Example of LCA:Example of LCA:Depression in adolescenceDepression in adolescence

Five latent classes of depression:

Male Female

No depression 46% 41%

Sad 12% 24%

Disliked 22% 10%

Sad and disliked 12% 14%

Full depression 8% 10%

Ideas underlying LTAIdeas underlying LTA

• LTA is a longitudinal extension of latent class models

• Some development can be represented as movement through discrete categories or stages

• There may be error associated with the measurement of the discrete categories

• Different people may take different paths• This heterogeneity may be unobserved (latent)

Ideas underlying LTAIdeas underlying LTA

1

2

3

4

Compare this approach to growth curve approach

Ideas underlying LTAIdeas underlying LTA

• LTA provides a way of fitting models with these characteristics:

– Change is stage sequential– Longitudinal– Measurement error– Developmental heterogeneity

Example 1:Example 1:Depression in adolescenceDepression in adolescence

Five stages of depression, two times:

Grade 12:

Grade 11:

No

Dep Sad

Dis-

liked

Sad+

Disliked

De-

pression

No depression .77 .12 .09 .02 .01

Sad .33 .49 .07 .07 .05

Disliked .24 .08 .46 .16 .06

Sad+Disliked .01 .22 .28 .40 .10

Depression .07 .22 .00 .13 .58

Example 2: Substance use over Example 2: Substance use over time, effect of pubertal timingtime, effect of pubertal timing

Eight stages of substance use, two times:

No Use

Alcohol

Cigarettes

Alcohol+Cigarettes

Alcohol+Cigarettes+Drunk

Alc+Cig+Drunk+Marijuana

Cigarettes+Marijuana

Alcohol+Cigarettes+Marijuana

Example 3: Substance use and Example 3: Substance use and delinquency over timedelinquency over time

OutlineOutline

• Overview of what LCA and LTA can do

• LTA

• Advanced topics and future directions

• Some research questions you might address using LTA

Latent transition analysis (LTA)Latent transition analysis (LTA)

• An extension of latent class theory to longitudinal data

• Provides a way of estimating and testing models of stage-sequential development in longitudinal data

• In LCA, latent classes are static• In LTA, latent statuses (stages) are dynamic

LTALTA

• A multiple‑indicator latent Markov model

• Estimates prevalence of stages and incidence of transitions between stages adjusted for measurement error

We will use LTA to:We will use LTA to:

• Fit a stage-sequential model of substance use onset in seventh-grade females

• Include pubertal timing as grouping variable• Examine the following:

– Proportion of girls with early timing– Group differences in substance use at Grade 7– Group differences in advancement in substance

use from Grade 7 to Grade 8

From Lanza & Collins (2002) Prevention Science

A model of substance use onsetA model of substance use onset

No Use

Alcohol

Cigarettes

Alcohol+Cigarettes

Alcohol+Cigarettes+Drunk

Alc+Cig+Drunk+Marijuana

Cigarettes+Marijuana

Alcohol+Cigarettes+Marijuana

Study participantsStudy participants

• From Waves I and II of The National Longitudinal Study of Adolescent Health, known as Add Health (Resnick et al., 1997)

• Used only females in Grade 7 at Wave 1

• N = 966

Indicators of substance useIndicators of substance use

ALCOHOL Have you had a drink of beer, wine or liquor – not just a sip or taste of someone else’s drink – more than 2 or 3 times in your life?– 1=no, 2=yes

CIGARETTES Have you ever tried cigarette smoking, even just 1 or 2 puffs?– 1=no, 2=yes

5+ DRINKS Over past 12 months, on how many days did you drink five or more drinks in a row?– 1=never, 2=one or two days in past 12 months, or more

DRUNK Over past 12 months, on how many days have you gotten drunk or “very, very high” on alcohol?– 1=never, 2=one or two days in past 12 months, or more

MARIJUANA How old were you when you tried marijuana for the first time?– 1=never tried, 2=all other ages

Indicators of pubertal timingIndicators of pubertal timing

• Breast size relative to grade school1 = On-time/late timing (same size/little bigger)

2 = Early timing (a lot bigger)

• Body becomes curvy1 = On-time/late timing (as curvy, somewhat curvy)

2 = Early timing (a lot more curvy)

LTA notationLTA notation

• Y represents an array of cells of the contingency table– Cells are formed by crosstabulating:

• Indicators of the dynamic latent variable measured at two or more times

• Indicator(s) of grouping variable

• S refers to number of latent statuses (stages)• a = 1, … S at Time 1, b = 1, …S at Time 2

Parameters in LTA modelsParameters in LTA models

• = probability of being in group c(e.g. the probability of being in the early pubertal timing group)

• = probability of being in latent status a at Time 1 given membership in group c(e.g. the probability of being in the No Use latent status at Time 1 given membership in the early pubertal timing group)

ca|

c

Parameters in LTA modelsParameters in LTA models

• = probability of membership in latent status b at Time t+1 given membership in latent status a at Time t and membership in group c

(e.g. the probability of being in the Advanced Substance Use latent status at Time 2, given membership in the No Use latent status at Time 1 and membership in the early pubertal timing group)

cab ,|

Parameters in LTA modelsParameters in LTA models

Time 2

Time 1

3|33|13|1

2|32|22|1

1|31|21|1

• parameters arranged in transition probability matrix

Parameters in LTA modelsParameters in LTA models

• The parameters are item-response probabilities

e.g. the probability of a particular response to an item (such as reporting drunkenness) given

- time

- latent status membership

- group membership• These parameters allow you to name the latent

statuses, test measurement invariance

The LTA modelThe LTA model

C

c

S

a

S

bcabcacyYP

1 1 1,||)(

For two times:

(one term for each manifest item)

EstimationEstimation

• LTA models can be estimated using WinLTA

• This program is available free of charge on our web site

http://methodology.psu.edu/http://methodology.psu.edu/

Overview of the LTA procedureOverview of the LTA procedure

• LTA is a confirmatory procedure– You tell the program some things about the model:

• number of groups• number of latent statuses• number of times• number of manifest items• number of response categories per item

– And some instructions about estimation

• The program then estimates the parameters

,,,

Overview of the LTA procedureOverview of the LTA procedure

• WinLTA uses the EM algorithm

• Handles missing data, makes MAR assumption

Overview of the LTA procedureOverview of the LTA procedure

• LTA computes expected response pattern proportions according to the model and estimated parameters

• These expected response pattern proportions are compared to the observed response pattern proportions. – This comparison is expressed in the likelihood

ratio statistic G2

Parameter restrictionsParameter restrictions

• The LTA user has three options for estimation of EACH parameter:

– Free estimation

– Constraining a group of parameters to be equal

– Fixing the parameter to a pre-specified value (such as 0)

Reasons for choosing parameter Reasons for choosing parameter restrictionsrestrictions

• To help improve identification by reducing the number of parameters to be estimated

• To express features of the model you wish to test

Examples of parameter Examples of parameter restrictionsrestrictions

• Constraining parameters equal across times– Measurement invariance across time– This assures that the latent statuses can be interpreted the

same way across times

• Constraining parameters equal across groups– Measurement invariance across groups

• Fixing elements of the transition probability matrix – This expresses a model of development

Examples of parameter Examples of parameter restrictionsrestrictions

• A model of no backsliding

Time 2

Time 1

3|3

2|32|2

1|31|21|1

00

0

Examples of parameter Examples of parameter restrictionsrestrictions

• A model of no change

Time 2

Time 1

100

010

001

Response probabilities conditional Response probabilities conditional on latent status membership ( )on latent status membership ( )

Stage of Substance Use Alcohol Cig

5+

Drinks Drunk Mar

No Use .01 .01 .01 .01 .01

Alcohol .97 .01 .01 .01 .01

Cigarettes .01 .94 .01 .01 .01

Alcohol + Cigarettes .97 .94 .01 .01 .01

Cigarettes + Marijuana .01 .94 .01 .01 .91

Alcohol + Cigarettes + Drunk .97 .94 .76 .81 .01

Alc + Cigarettes + Marijuana .97 .94 .01 .01 .91

Alc + Cig + Drunk + Marijuana

.97 .94 .76 .81 .91

Based on these parameters, what would YOU name the stages?

Prevalence of substance use Prevalence of substance use stages given pubertal timing ( )stages given pubertal timing ( )

Stage of Substance Use

Early

Timing

(19.7%)

On-time/Late

Timing

(80.3%) p < .05

No Use .29 .54 *

Alcohol .09 .08

Cigarettes .18 .19

Alcohol + Cigarettes .18 .09 *

Cigarettes + Marijuana .02 .02

Alcohol + Cigarettes + Drunk .08 .04

Alc + Cigarettes + Marijuana .05 .02

Alc + Cig + Drunk + Marijuana .11 .04 *

,

Advancement in substance use Advancement in substance use from Grade 7 to Grade 8 ( )from Grade 7 to Grade 8 ( )

No Use A C AC CM ACD ACM ACDM

No Use .78/.53* .06/.07 .09/.14 .01/.10 .02/.02 .02/.05 .00/.02 .02/.07

A --- .72/.61 --- .06/.05 --- .09/.24 .09/.04 .05/.05

C --- --- .66/.44 .11/.31 .08/.02 .09/.11 .01/.02 .05/.10

AC --- --- --- .66/.61 --- .10/.20 .10/.03 .15/.16

CM --- --- --- --- .26/.41 --- .20/.19 .54/.40

ACD --- --- --- .37/.38 --- .23/.27 .07/.04 .33/.32

ACM --- --- --- --- --- --- .70/.16 .30/.84

ACDM --- --- --- --- --- --- .10/.45 .90/.55

• Note: Early timing probabilities in bold• Early-timing group more likely to advance from No Use

Advancement in substance use Advancement in substance use from Grade 7 to Grade 8from Grade 7 to Grade 8

• Summing across certain cells:– 40% early-developing females increase in use– 26% of on-time or late-developing females increase in

use

• Early-maturing females are 1.5 times more likely to advance in substance use regardless of their level of use in seventh grade (p < .05)

Drawbacks and limitations of LTADrawbacks and limitations of LTA

• Not suitable for small samples

• Hypothesis testing flexible but not easy

• No consensus on best approach for model selection

OutlineOutline

• Overview of what LCA and LTA can do

• LTA

• Advanced topics and future directions

• Some research questions you might address using LTA

Advanced topics andAdvanced topics andfuture directionsfuture directions

• LCA for repeated measures• Data augmentation (DA)• Associative LTA (ALTA)• LCA with covariates

LCA for repeated measuresLCA for repeated measures

• LTA examines pairs of times• Here, each latent class represents trajectory

through stage sequence across 3 or more times

• Advantage over growth curve models: no smooth function of time necessary; discontinuous development modeled

• Example: Lanza & Collins (in press). Journal of Studies on Alcohol.

A mixture model of discontinuous development A mixture model of discontinuous development in heavy drinking from ages 18 to 30:in heavy drinking from ages 18 to 30:

The role of college enrollmentThe role of college enrollment

Lifetime Heavy Drinking and No Heavy Drinking

0

1

High School College YoungAdult

Adult

Hea

vy D

rink

ing

.

53.7%

16.9%

Increasing Trend

0

1

High School College YoungAdult

Adult

Hea

vy D

rink

ing

.

3.7%

8.7%

Short-term Heavy Drinking

0

1

High School College Young Adult Adult

Hea

vy D

rink

ing

.

3.7%

2.6%

Decreasing Trend

0

1

High School College Young Adult Adult

Hea

vy D

rink

ing

.

4.4%

6.3%

Heavy Drinking Pathway

College

Enrollment

No College

Enrollment p-valueNo heavy drinking 55.9% 51.3%

Young adulthood only 2.3% 4.5%

Young adult + adult 1.6% 5.4% +

College age only 8.1% 0.0% **

College + YA + adult 5.3% 9.1%

High school + college age 2.7% 5.4%

High School + college + YA 8.8% 6.0%

Lifetime heavy drinking 15.3% 18.3%

A mixture model of discontinuous development A mixture model of discontinuous development in heavy drinking from ages 18 to 30:in heavy drinking from ages 18 to 30:

The role of college enrollmentThe role of college enrollment

Data Augmentation (DA)Data Augmentation (DA)

• Data augmentation provides:

– Standard errors for LTA parameters– Bayesian approach to estimation of

parameters and standard errors– A very flexible approach to hypotheses

testing in LCA and LTA– A framework for model assessment

(posterior predictive check distribution)

Associative LTA (ALTA)Associative LTA (ALTA)

• A model where one stage sequence predicts another

• Example: development in substance use and concurrent development in risky sexual behavior

• Using this approach, you can test hypotheses about different degrees of relationship between two stage sequences– e.g. do transitions in one sequence predict

transitions in the other?

SAS PROC LCASAS PROC LCA

• New SAS procedure available at http://methodology.psu.edu/lca

• LCA with covariates involves regressing latent class variable on set of predictors

• PROC LCA is for– LCA with covariates (latent class regression analysis)– Multiple-groups LCA– Test measurement invariance with keyword– Save posterior probabilities to SAS data file

OutlineOutline

• Overview of what LCA and LTA can do

• LTA

• Advanced topics and future directions

• Some research questions you might address using LTA

Some research questions you Some research questions you might address using LTAmight address using LTA

I. Stage-sequential development in…• Tobacco, alcohol, inhalants, marijuana

– Onset model or current use model

• Comorbid behaviors– Substance use and problem behavior– Substance use and risky sexual behavior

• Intentions Experimentation Regular or Problem Use Dependence

Some research questions you Some research questions you might address using LTAmight address using LTA

I. Example research questions:• What stages of substance use (involving alcohol

use, tobacco use, inhalant use, and marijuana use) are necessary to characterize the onset process during early adolescence?

• What is the prevalence of the stages at Time 1?• What is the probability of transitioning from one

stage at Time 1 to another stage at Time 2?

Some research questions you Some research questions you might address using LTAmight address using LTA

II. Possible grouping variables…• Gender• Cohort• Poverty below threshold• Pubertal status

– Early v. On-time / Late

Some research questions you Some research questions you might address using LTAmight address using LTA

II. Example research questions:• Are children in households with low income

more likely to be using multiple substances at Time 1?

• Are males more likely to advance in substance use than females?

• Are females with early pubertal timing more likely to transition to stages of substance use characterized by regular smoking?

Some research questions you Some research questions you might address using LTAmight address using LTA

III. Possible covariates…• Temperament• Parental monitoring• Income-to-needs• Age• Academic achievement• Depression

Some research questions you Some research questions you might address using LTAmight address using LTA

III. Example research questions:• Does child temperament predict membership

in stages of substance use at Time 1?• What is the increase in odds of membership in

the ‘Advanced Substance Use’ stage (relative to the No Use stage) for a one-unit change in academic achievement?

• How does age relate to the probability of transitioning from ‘Alcohol’ to ‘Alcohol + Drunkenness’

Thank You!Thank You!

Stephanie Lanza

SLanza@PSU.EDU

The Methodology Center

http://methodology.psu.edu