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Latent transition analysis (LTA) for modeling discrete change in longitudinal data. Stephanie T. Lanza The Methodology Center The Pennsylvania State University UCLA May 1, 2006. The Methodology Center. A group of social scientists and statisticians working together - PowerPoint PPT Presentation
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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
The Methodology Center
http://methodology.psu.edu