Growth Modeling Presentation Meadows

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Growth Models:Growth Models:A Practical GuideA Practical Guide

Sarah O. MeadowsSarah O. MeadowsCenter for Research on Child WellbeingCenter for Research on Child WellbeingPrinceton UniversityPrinceton University

October 15, 2007October 15, 2007

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Outline of Outline of PresentationPresentation What are growth models?What are growth models? Nuts and BoltsNuts and Bolts Hands-On ExampleHands-On Example Additional IssuesAdditional Issues

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Part I: Part I: What is a Growth What is a Growth Model?Model?

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What is a Growth What is a Growth Model?Model? A way to assess individual A way to assess individual

stability and change, both growth stability and change, both growth and decay, over time.and decay, over time.

A two-level, hierarchical model A two-level, hierarchical model that that models (1) within that that models (1) within individual change over time and individual change over time and (2) between individual (2) between individual differences in patterns of growth.differences in patterns of growth.

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A Rose by Any Other A Rose by Any Other Name . . .Name . . . Growth ModelsGrowth Models Trajectory ModelsTrajectory Models Growth Curve Growth Curve

ModelsModels Latent GMLatent GM Latent TMLatent TM Latent GCMLatent GCM Hierarchical Hierarchical

ModelsModels

Random Intercept Random Intercept ModelsModels

Random Random Coefficient ModelsCoefficient Models

Random Random Intercept/Random Intercept/Random Slope ModelsSlope Models

Variance Variance Component Component ModelsModels

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Why Latent?Why Latent?

Because we assume that whatever Because we assume that whatever process that is underlying the thing process that is underlying the thing we are modeling (or the behavior we we are modeling (or the behavior we observe) is actually unobserved, or observe) is actually unobserved, or latent. latent.

The characteristics we observe are a The characteristics we observe are a manifestation of this latent trajectory. manifestation of this latent trajectory.

This language grew out of structural This language grew out of structural equation modeling (SEM).equation modeling (SEM).

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Why use GM’s?Why use GM’s?

Everyone else is doing it!Everyone else is doing it!– EducationEducation– CriminologyCriminology– PsychologyPsychology– SociologySociology– Public HealthPublic Health

You have longitudinal data and are You have longitudinal data and are interested in change over time.interested in change over time.– You may want to explain those changes.You may want to explain those changes.– You may also believe that not everyone You may also believe that not everyone

follows the same path.follows the same path.

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How Have Others Used How Have Others Used GMs?GMs? ““Growth Trajectories of Sexual Risk Behavior in Growth Trajectories of Sexual Risk Behavior in

Adolescence and Young Adulthood.” Fergus, Adolescence and Young Adulthood.” Fergus, Zimmerman, & Caldwell. Zimmerman, & Caldwell. American Journal of Public American Journal of Public HealthHealth. 2007.. 2007.

““Individual Differences in the Onset of Tense Marking: A Individual Differences in the Onset of Tense Marking: A Growth Model Example.” Hadley & Colt. Growth Model Example.” Hadley & Colt. Journal of Journal of Speech, Language, and Hearing ResearchSpeech, Language, and Hearing Research. 2006.. 2006.

““Ten-Year Stability of Depressive Personality Disorder in Ten-Year Stability of Depressive Personality Disorder in Depressed Outpatients.” Laptook, Klein, & Dougherty. Depressed Outpatients.” Laptook, Klein, & Dougherty. The American Journal of PsychiatryThe American Journal of Psychiatry. 2006.. 2006.

““Verbal Learning and Everyday Functioning in Verbal Learning and Everyday Functioning in Dementia: An Application of Latent Variable Growth Dementia: An Application of Latent Variable Growth Curve Modeling.” Mast & Allaire. Curve Modeling.” Mast & Allaire. The Journals of The Journals of GerontologyGerontology. 2006.. 2006.

““You Make Me Sick: Marital Quality and Health Over the You Make Me Sick: Marital Quality and Health Over the Life Course.” Umberson, Williams, & Powers. Life Course.” Umberson, Williams, & Powers. Journal of Journal of Health and Social BehaviorHealth and Social Behavior. 2006. . 2006.

““Parental Divorce and Child Mental Health Trajectories.” Parental Divorce and Child Mental Health Trajectories.” Strohschein. 2005. Strohschein. 2005. Journal of Marriage and FamilyJournal of Marriage and Family..

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A Detailed ExampleA Detailed Example

““Stability and Change in Family Structure Stability and Change in Family Structure and Maternal Health Trajectories.” and Maternal Health Trajectories.” Meadows, McLanahan, & Brooks-Gunn. Meadows, McLanahan, & Brooks-Gunn. American Sociological ReviewAmerican Sociological Review. . Forthcoming.Forthcoming.

We wanted to know whether changes in We wanted to know whether changes in family structure, including transitions into family structure, including transitions into and out of coresidential relationships, had and out of coresidential relationships, had short-term impacts on health (i.e., crisis short-term impacts on health (i.e., crisis model) or long-term impacts on health model) or long-term impacts on health (i.e., resource model).(i.e., resource model).

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Example (cont.)Example (cont.)

Trajectories of maternal self-rated Trajectories of maternal self-rated health and mental health health and mental health problems from one year after problems from one year after birth to five years after birth.birth to five years after birth.

Two measures of family structure Two measures of family structure change:change:– Level 1: Time-VaryingLevel 1: Time-Varying– Level 2: Time-InvariantLevel 2: Time-Invariant

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Example (cont.)Example (cont.)

Results:Results:– Transitions, especially exits from Transitions, especially exits from

marriages, resulted in short-term marriages, resulted in short-term declines in physical health and short-declines in physical health and short-term increases in mental health term increases in mental health problems.problems.

– Little support for the resource model; Little support for the resource model; no growing gap in well-being between no growing gap in well-being between mothers who remained stably married mothers who remained stably married and those remained stably single, as and those remained stably single, as well as mothers who made transitions.well as mothers who made transitions.

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Figure 1. Mothers’ Mental Health TrajectoriesFigure 1. Mothers’ Mental Health Trajectories

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Figure 2. Mothers’ Household Income TrajectoriesFigure 2. Mothers’ Household Income Trajectories

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Figure 3. Fathers’ Mental Health TrajectoriesFigure 3. Fathers’ Mental Health Trajectories

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Figure 4. Fathers’ Earnings TrajectoriesFigure 4. Fathers’ Earnings Trajectories

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Part II: Nuts and Part II: Nuts and BoltsBolts

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Where Did GM’s Come Where Did GM’s Come From?From? Time Series Models Time Series Models

(Autoregressive)(Autoregressive) Repeated Measures ANOVA Repeated Measures ANOVA

– (Duncan & Duncan, 2004)(Duncan & Duncan, 2004) SEMSEM Multilevel Models (HLM)Multilevel Models (HLM)

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Hierarchical ModelsHierarchical Models

Traditional:Traditional:– Level 1: StudentsLevel 1: Students– Level 2: SchoolsLevel 2: Schools

Growth Models (a type of HM): Growth Models (a type of HM): – Level 1: Repeated ObservationsLevel 1: Repeated Observations– Level 2: IndividualsLevel 2: Individuals

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Unconditional ModelUnconditional Model

Level 1: Within IndividualLevel 1: Within Individual

Level 2: Between IndividualLevel 2: Between Individual

yit = αi + βit + εit

αi = α0 + ui

βi = β0 + vi

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A Latent TrajectoryA Latent Trajectory

α

Time

β

Latent Depression Trajectory

Dep

ress

ive

Sym

pto

ms

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Time-Invariant Time-Invariant CovariatesCovariates Level 1: Within IndividualLevel 1: Within Individual

Level 2: Between IndividualLevel 2: Between Individual

yit = αi + βit + εit

αi = α0 + α1xi1 + α2xi2 + . . . αkxik + ui

βi = β0 + β1xi1 + β2xi2 + . . . βkxik + vi

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Time-Varying VariablesTime-Varying Variables

Level 1: Within IndividualLevel 1: Within Individual

Level 2: Between IndividualLevel 2: Between Individual

yit = αi + βit + γt wit + εit Time-varying effect.

αi = α0 + α1xi1 + α2xi2 + . . . αkxik + ui

βi = β0 + β1xi1 + β2xi2 + . . . βkxik + vi

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Fixed vs. RandomFixed vs. Random

Fixed: Means of the latent Fixed: Means of the latent trajectory parameters (i.e., trajectory parameters (i.e., intercept and slope)intercept and slope)

Random: Variance of the latent Random: Variance of the latent trajectory parameters (i.e., trajectory parameters (i.e., indicates individual heterogeneity indicates individual heterogeneity around population means)around population means)

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Part III: An ExamplePart III: An Example

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SoftwareSoftware

MPlus – SEM basedMPlus – SEM based HLM – Hierarchical ModelingHLM – Hierarchical Modeling SAS – Proc TrajSAS – Proc Traj STATA STATA

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Data RequirementsData Requirements

Three observationsThree observations For polynomial curves you need For polynomial curves you need d d

+ 2 repeated measures, where + 2 repeated measures, where dd is is the degree of the polynomial.the degree of the polynomial.

Horizontal data file (i.e., one Horizontal data file (i.e., one person, one row).person, one row).

Convert data to .dat file.Convert data to .dat file.– Remember the order of the variables!!Remember the order of the variables!!

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Self-Rated HealthSelf-Rated Health

Mothers in FFCWSMothers in FFCWS ““In general, how is your health?”In general, how is your health?”

– Excellent (5)Excellent (5)– Very Good (4)Very Good (4)– Good (3)Good (3)– Fair (2)Fair (2)– Poor (1)Poor (1)

Repeated measures one, three, Repeated measures one, three, and five years after birth.and five years after birth.

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Setting the TrajectorySetting the Trajectory

Intercept Slope

SRH 1 SRH 3 SRH 5

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10

24

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ModelsModels

UnconditionalUnconditional– Model FitModel Fit

ConditionalConditional– Time-Invariant CovariatesTime-Invariant Covariates

MPlus GraphsMPlus Graphs Selection and CausationSelection and Causation

– Time-Varying CovariatesTime-Varying Covariates

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Model FitModel Fit

Chi-SquareChi-Square– Not Significant, but almost always is.Not Significant, but almost always is.

CFI (Comparative Fit Index)CFI (Comparative Fit Index)– Range: 0 – 1; 1 is best.Range: 0 – 1; 1 is best.

TLI (Tucker Lewis Index; or NNFI)TLI (Tucker Lewis Index; or NNFI)– Range: 0 – 1; 1 is best.Range: 0 – 1; 1 is best.

RMSEA (Root Mean Square Error RMSEA (Root Mean Square Error of Approximation)of Approximation)– Under .05 is good; above .10 is bad.Under .05 is good; above .10 is bad.

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Time-Invariant Time-Invariant CovariatesCovariates Age at BaselineAge at Baseline EducationEducation RaceRace Biological Parents Biological Parents

Mental Health Mental Health ProblemProblem

Lived with both Bio Lived with both Bio Parents at Age 15Parents at Age 15

Number of Previous Number of Previous RelationshipsRelationships

Baseline SRHBaseline SRH

Considered an Considered an AbortionAbortion

Positive Marriage Positive Marriage AttitudeAttitude

Prenatal Variables Prenatal Variables (medical care, drug (medical care, drug and alcohol use, and alcohol use, smoking)smoking)

Baseline Marital Baseline Marital StatusStatus

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Intercept Slope

SRH 1 SRH 3 SRH 5

11

10

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Time-Invariant Covariates

α β

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Figure 5. Mothers’ Self-Rated Health Trajectories.Figure 5. Mothers’ Self-Rated Health Trajectories.

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Selection IssuesSelection Issues

InterceptIntercept– Third factor is responsible for where Third factor is responsible for where

people start.people start. SlopeSlope

– Third factor is responsible for where Third factor is responsible for where people go.people go.

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Time-Varying Time-Varying CovariateCovariate Mental Health ProblemsMental Health Problems Range 0-3Range 0-3

– Includes CIDI Major depressive Includes CIDI Major depressive episode, binge drinking, and drug episode, binge drinking, and drug use.use.

– All occurred in the past 12-months.All occurred in the past 12-months.

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Intercept Slope

SRH 1 SRH 3 SRH 5

11

10

24

MH 1 MH 3 MH 5

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Part IV: Additional Part IV: Additional IssuesIssues

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Multi-ModelsMulti-Models

Multi-GroupMulti-Group– Growth process may vary for each Growth process may vary for each

group.group. Multi-ProcessMulti-Process

– Models more than one trajectory.Models more than one trajectory.

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MeasurementMeasurement

Latent Measures (Multiple Latent Measures (Multiple Indicators)Indicators)

Dichotomous/Categorical Dichotomous/Categorical VariablesVariables

Count VariablesCount Variables– ZIP ModelsZIP Models

SkewnessSkewness– Transform VariableTransform Variable– Semi-Continuous Growth ModelSemi-Continuous Growth Model

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Age-Based Growth Age-Based Growth ModelModel Synthetic cohortSynthetic cohort

– Sample members may contribute Sample members may contribute different amounts of information at different amounts of information at different times.different times.

Missing DataMissing Data– Drop Cases (default)Drop Cases (default)– Multiple ImputationMultiple Imputation– Full Information Maximum Likelihood Full Information Maximum Likelihood

(FIML)(FIML) Analysis: MISSINGAnalysis: MISSING

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Mixture ModelsMixture Models

Latent Class Models (LCM/LCA)Latent Class Models (LCM/LCA)– Group membership not known.Group membership not known.

Latent Class Growth Models Latent Class Growth Models (LCGM/LCGA)(LCGM/LCGA)– Group membership not known and is based Group membership not known and is based

on trajectory patterns.on trajectory patterns.– No variation is allowed within latent classes.No variation is allowed within latent classes.

Growth Mixture Models (GMM)Growth Mixture Models (GMM)– Group membership is not known and is based Group membership is not known and is based

on trajectory patterns.on trajectory patterns.– Allows for variation within latent classes.Allows for variation within latent classes.

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Contact InfoContact Info

[email protected]@princeton.edu