CONTENTS

List of Figures

List of Tables

Acknowledgments

Acronyms

PART I FUNDAMENTALS

CHAPTER 1 GENERAL INTRODUCTION

1.1 OVERVIEW

1.2 CONCEPTUAL FOUNDATION AND BRIEF HISTORY OF THE LATENT CLASS MODEL

1.3 WHY SELECT A CATEGORICAL LATENT VARIABLE APPROACH?

1.4 SCOPE OF THIS BOOK

1.5 EMPIRICAL EXAMPLE OF LCA: ADOLESCENT DELINQUENCY

1.6 EMPIRICAL EXAMPLE OF LTA: ADOLESCENT DELINQUENCY

1.7 ABOUT THIS BOOK

1.8 THE EXAMPLES IN THIS BOOK

1.9 SOFTWARE

1.10 ADDITIONAL RESOURCES: THE BOOK’S WEB SITE

1.11 SUGGESTED SUPPLEMENTAL READINGS

1.12 POINTS TO REMEMBER

1.13 WHAT’S NEXT

CHAPTER 2 THE LATENT CLASS MODEL

2.1 OVERVIEW

2.2 EMPIRICAL EXAMPLE: PUBERTAL DEVELOPMENT

2.3 THE ROLE OF ITEM-RESPONSE PROBABILITIES IN INTERPRETING LATENT CLASSES

2.4 EMPIRICAL EXAMPLE: HEALTH RISK BEHAVIORS

2.5 LCA: MODEL AND NOTATION

2.6 SUGGESTED SUPPLEMENTAL READINGS

2.7 POINTS TO REMEMBER

2.8 WHAT’S NEXT

CHAPTER 3 THE RELATION BETWEEN THE LATENT VARIABLE AND ITS INDICATORS

3.1 OVERVIEW

3.2 THE LATENT CLASS MEASUREMENT MODEL

3.3 HOMOGENEITY AND LATENT CLASS SEPARATION

3.4 THE PRECISION WITH WHICH THE OBSERVED VARIABLES MEASURE THE LATENT VARIABLE

3.5 EXPRESSING THE DEGREE OF UNCERTAINTY: MEAN POSTERIOR PROBABILITIES AND ENTROPY

3.6 POINTS TO REMEMBER

3.7 WHAT’S NEXT

CHAPTER 4 PARAMETER ESTIMATION AND MODEL SELECTION

4.1 OVERVIEW

4.2 MAXIMUM LIKELIHOOD ESTIMATION

4.3 MODEL FIT AND MODEL SELECTION

4.4 FINDING THE ML SOLUTION: MODEL IDENTIFICATION, STARTING VALUES, AND LABEL SWITCHING

4.5 EMPIRICAL EXAMPLE OF USING MANY STARTING VALUES: POSITIVE HEALTH BEHAVIORS

4.6 EMPIRICAL EXAMPLES OF SELECTING THE NUMBER OF LATENT CLASSES

4.7 MORE ABOUT PARAMETER RESTRICTIONS

4.8 STANDARD ERRORS

4.9 SUGGESTED SUPPLEMENTAL READINGS

4.10 POINTS TO REMEMBER

4.11 WHAT’S NEXT

PART II ADVANCED LCA

CHAPTER 5 MULTIPLE-GROUP LATENT CLASS ANALYSIS

5.1 OVERVIEW

5.2 INTRODUCTION

5.3 MULTIPLE-GROUP LCA: MODEL AND NOTATION

5.4 COMPUTING THE NUMBER OF PARAMETERS ESTIMATED

5.5 EXPRESSING GROUP DIFFERENCES IN THE LCA MODEL

5.6 MEASUREMENT INVARIANCE

5.7 ESTABLISHING WHETHER THE NUMBER OF LATENT CLASSES IS IDENTICAL ACROSS GROUPS

5.8 ESTABLISHING INVARIANCE OF ITEM-RESPONSE PROBABILITIES ACROSS GROUPS

5.9 INTERPRETATION WHEN MEASUREMENT INVARIANCE DOES NOT HOLD

5.10 STRATEGIES WHEN MEASUREMENT INVARIANCE DOES NOT HOLD

5.1 1 SIGNIFICANT DIFFERENCES AND IMPORTANT DIFFERENCES

5.12 TESTING EQUIVALENCE OF LATENTCLASSPREVALENCES ACROSS GROUPS

5.13 SUGGESTED SUPPLEMENTAL READINGS

5.14 POINTS TO REMEMBER

5.15 WHAT’S NEXT

CHAPTER 6 LATENT CLASS ANALYSIS WITH COVARIATES

6.1 OVERVIEW

6.2 EMPIRICAL EXAMPLE: POSITIVE HEALTH BEHAVIORS

6.3 PREPARING TO CONDUCT LCA WITH COVARIATES

6.5 HYPOTHESIS TESTING IN LCA WITH COVARIATES

6.6 INTERPRETATION OF THE INTERCEPTS AND REGRESSION COEFFICIENTS

6.8 EMPIRICAL EXAMPLE OF MULTIPLE COVARIATES AND INTERACTION TERMS: POSITIVE HEALTH BEHAVIORS

6.9 MULTIPLE-GROUP LCA WITH COVARIATES: MODEL AND NOTATION

6.10 GROUPING VARIABLE OR COVARIATE?

6.11 USE OF A BAYESIAN PRIOR TO STABILIZE ESTIMATION

6.12 COMBINING LATENT CLASSES TO APPLY BINOMIAL LOGISTIC REGRESSION IN LCA

6.13 SUGGESTED SUPPLEMENTAL READINGS

6.14 POINTS TO REMEMBER

6.15 WHAT’S NEXT

PART III LATENT CLASS MODELS FOR LONGITUDINAL DATA

CHAPTER 7 REPEATED-MEASURES LATENT CLASS ANALYSIS AND LATENT TRANSITION ANALYSIS

7.1 OVERVIEW

7.2 RMLCA

7.3 LTA

7.4 LTA MODEL PARAMETERS

7.5 LTA: MODEL AND NOTATION

7.6 DEGREES OF FREEDOM ASSOCIATED WITH LATENT TRANSITION MODELS

7.7 EMPIRICAL EXAMPLE: ADOLESCENT DEPRESSION

7.8 EMPIRICAL EXAMPLE: DATING AND SEXUAL RISK behavior

7.9 INTERPRETING WHAT A LATENT TRANSITION MODEL REVEALS ABOUT CHANGE

7.10 PARAMETER RESTRICTIONS IN LTA

7.11 TESTING THE HYPOTHESIS OF MEASUREMENT INVARIANCE ACROSS TIMES

7.12 TESTING HYPOTHESES ABOUT CHANGE BETWEEN TIMES

7.13 RELATION BETWEEN RMLCA AND LTA

7.14 INVARIANCE OF THE TRANSITION PROBABILITY MATRIX WHEN THERE ARE THREE OR MORE TIMES

7.15 SUGGESTED SUPPLEMENTAL READINGS

7.16 POINTS TO REMEMBER

7.17 WHAT’S NEXT

CHAPTER 8 MULTIPLE-GROUPS LATENT TRANSITION ANALYSIS AND LATENT TRANSITION ANALYSIS WITH COVARIATES

8.1 OVERVIEW

8.2 LTA WITH A GROUPING VARIABLE

8.3 MULTIPLE-GROUP LTA: MODEL AND NOTATION

8.5 HYPOTHESIS TESTS CONCERNING GROUP DIFFERENCES IN LATENT STATUS PREVALENCES AND TRANSITION PROBABILITIES: GENERAL CONSIDERATIONS

8.6 OVERALL HYPOTHESIS TESTS ABOUT GROUP DIFFERENCES IN LTA

8.7 TESTING THE HYPOTHESIS OF EQUALITY OF LATENT STATUS PREVALENCES ACROSS GROUPS AT TIME 1

8.8 TESTING THE HYPOTHESIS OF EQUALITY OF TRANSITION PROBABILITIES ACROSS GROUPS

8.9 INCORPORATING COVARIATES IN LTA

8.12 INCLUDING BOTH A GROUPING VARIABLE AND A COVARIATE IN LTA

8.13 COMBINING LATENT STATUSES TO APPLY BINOMIAL LOGISTIC REGRESSION IN LTA

8.14 THE RELATION BETWEEN MULTIPLE-GROUP LTA AND LTA WITH A COVARIATE

8.15 SUGGESTED SUPPLEMENTAL READINGS

8.16 POINTS TO REMEMBER

Topic Index

Author Index

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Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data:

Collins, Linda M.

Latent class and latent transition analysis / Linda M. Collins, Stephanie T. Lanza,

p. cm.

Includes bibliographical references and index.

ISBN 978-0-470-22839-5 (cloth)

1. Latent structure analysis. 2. Latent variables. 3. Statistical methods. I. Lanza,

Stephanie T., 1969– II. Title.

QA278.6.C65 2010

519.5′35—dc22  2009025970

10 9 8 7 6 5 4 3 2 1

For David and Kathy with admiration (LMC)

For Dad my first math teacher (STL)

List of Figures

1.1    Latent variable with three observed variables as indicators.

1.2    Adolescent delinquency latent class membership probabilities (Add Health public-use data, Wave I; N = 2,087). Note that the four probabilities sum to 1.

1.3    Probability of a “Yes” response to each delinquency item conditional on latent class membership (Add Health public-use data, Wave I; N = 2,087).

1.4    Probability of a “Yes” response to each delinquency item conditional on latent status membership (Add Health public-use data, Waves I and II; N = 2,087).

2.1    Latent class prevalences in pubertal development example (Add Health public-use data, Wave I; N = 469).

2.2    Item-response probabilities for measurement of Delayed Pubertal Onset latent class (Add Health public-use data, Wave I; N = 469). Response category labels appear in Table 2.5.

2.3    Item-response probabilities for measurement of Biologically Mature latent class (Add Health public-use data, Wave I; N = 469). Response category labels appear in Table 2.5.

2.4    Item-response probabilities for measurement of Visibly Mature latent class (Add Health public-use data, Wave I; N = 469). Response category labels appear in Table 2.5.

2.5    Item-response probabilities for measurement of Mature latent class (Add Health public-use data, Wave I; N = 469). Response category labels appear in Table 2.5.

2.6 Probability of endorsing alcohol and tobacco use items conditional on latent class membership (Youth Risk Behavior Survey, 2005; N = 13,840).

2.7 Probability of endorsing other drug use items conditional on latent class membership (Youth Risk Behavior Survey, 2005; N = 13,840).

2.8 Probability of endorsing sexual behavior items conditional on latent class membership (Youth Risk Behavior Survey, 2005; N= 13,840).

2.9 Prevalence of health risk behavior latent classes (Youth Risk Behavior Survey, 2005; N = 13,840).

2.10    Figure 1.1, repeated here for convenience. Latent variable with three observed variables as indicators. This figure illustrates local independence. There are arrows connecting observed variables X₁, X₂, and X₃ to the latent variable but no other arrows connecting any components of the observed variables to each other. This signifies that the three observed variables are related only through the latent variable.

2.11    Latent variable with three observed variables as indicators. This figure illustrates a violation of local independence. Observed variables X₂ and X₃ are related to each other not only through the latent variable, but also through their error components (e’s).

3.1 Probability of endorsing tobacco use behavior items conditional on latent class membership. Hypothetical data from Table 3.5 exhibit high homogeneity and high latent class separation.

3.2    Probability of endorsing tobacco use behavior items conditional on latent class membership. Hypothetical data from Table 3.6 exhibit high homogeneity overall and low separation between the Regular I and II latent classes.

3.3    Probability of endorsing tobacco use behavior items conditional on latent class membership. Hypothetical data from Table 3.7 exhibit low homogeneity and low latent class separation.

4.1    Unimodal likelihood function for a single parameter θ, indicative of good identification.

4.2    Multimodal likelihood function for a single parameter θ, indicative of underidentification.

4.3    Likelihood function for a single parameter θ. This function has a flat region, which suggests that the model being fit is unidentified.

4.4    Distribution of log-likelihood values for five-latent-class model of positive health behaviors based on 100 random sets of starting values (Monitoring the Future data, 2004; N = 2,065).

4.5    G², AIC, and BIC for models of positive health behaviors (Monitoring the Future data, 2004; N = 2,065).

4.6    G², AIC, and BIC for models of female pubertal development (Add Health public-use data, Wave I; N = 469).

4.7    G², AIC, and BIC for models of health risk behaviors (Youth Risk Behavior Survey, 2005; N = 13,840).

5.1    Prevalence of latent classes of positive health behaviors by gender (Monitoring the Future data, 2004; N = 2,065).

5.2    Prevalence of health risk behavior latent classes for each grade (Youth Risk Behavior Survey, 2005; N = 13,840).

6.1    Overall effect of one-standard-deviation increase in maternal education in positive health behavior example (Monitoring the Future data, 2004; N = 2,065). The Typical latent class is the reference group.

6.2    Effect of one-standard-deviation increase in maternal education for males and females in positive health behavior example (Monitoring the Future data, 2004; N = 2,065).

6.3    Effect of one-standard-deviation increase in maternal education on odds of membership in Healthy latent class as compared to membership in other latent classes combined, by gender (Monitoring the Future data, 2004; N = 2,065).

7.1    Patterns of heavy drinking across four developmental periods, corresponding to the latent classes in Table 7.2 (from Lanza and Collins, 2006). Note that more precise values for item-response probabilities appear in Table 7.1.

7.2    G², AIC, and BIC for latent transition models of adolescent delinquency (Add Health public-use data, Waves I and II; N = 2,087).

7.3    G², AIC, and BIC for latent transition models of adolescent depression (Add Health public-use data, Waves I and II; N = 2,061).203

7.4    Transition probabilities for latent transition model of adolescent depression (Add Health public-use data, Waves I and II; N = 2,061).206

8.1    Prevalences of latent statuses of adolescent depression at Time by gender (Add Health public-use data, Waves I and II; N = 2,061).237

8.2    Odds ratios associated with current cigarette use and lifetime marijuana use. Although not estimated, the odds ratio of 1 associated with the reference latent status, Not Depressed, is shown here for comparison purposes (Add Health public-use data, Waves I and II; N = 2,061).

List of Tables

1.1    Four Different Latent Variable Models

1.2    Proportion of Adolescents Responding “Yes” to Questions About Delinquent Behaviors (Add Health Public-Use Data, Wave I; N = 2,087)

1.3    Four-Latent-Class Model of Past-Year Delinquency (Add Health Public-Use Data, Wave I; N = 2,087)

1.4    Five-Latent-Status Model of Past-Year Delinquency (Add Health Public-Use Data, Waves I and II; N = 2,087)

2.1    Marginal Response Proportions for Female Pubertal Development Variables (Add Health Public-Use Data, Wave I; N = 469)

2.2    Response Patterns and Frequencies for Add Health Pubertal Development Data (Add Health Public-Use Data, Wave I; N = 469)

2.3    Item-Response Probabilities for a Hypothetical Two-Latent-Class Model

2.4    Item-Response Probabilities for a Hypothetical Three-Latent-Class Model

2.5    Four-Latent-Class Model of Female Pubertal Development in Seventh Grade (Add Health Public-Use Data, Wave I; N = 469)

2.6    Proportion of Students Reporting Each Health Risk Behavior (Youth Risk Behavior Survey, 2005; N = 13,840)

2.7    Five-Latent-Class Model of Health Risk Behaviors (Youth Risk Behavior Surveillance System Data; N = 13,840)

2.8    Hypothetical Example with Two Latent Classes and Two Observed Variables

2.9    Response Pattern Probabilities for Hypothetical Example in Table 2.8

3.1    Item-Response Probabilities from Four-Latent-Class Model of Female Pubertal Development (Add Health Public-Use-Data, Wave I; N = 469. From Table 2.5; repeated here for convenience)

3.2    Hypothetical Item-Response Probabilities Reflecting Independence of Observed Variables and Latent Variable

3.3    Item-Response Probabilities from Five-Latent-Class Model of Health Risk Behaviors (Youth Risk Behavior Survey, 2005; N = 13,840. From Table 2.7; repeated here for convenience)

3.4    Item-Response Probabilities for a Hypothetical Three-Latent-Class Model

3.5    Item-Response Probabilities for a Hypothetical Three-Latent-Class Model with High Homogeneity and High Latent Class Separation

3.6    Item-Response Probabilities for a Hypothetical Three-Latent-Class Model with High Homogeneity and Low Latent Class Separation

3.7    Item-Response Probabilities for a Hypothetical Three-Latent-Class Model with Low Homogeneity and Low Latent Class Separation

4.1    Marginal Response Proportions for Indicators of Positive Health Behavior (Monitoring the Future Data, 2004; N = 2,065)

4.2    Summary of Information for Selecting Number of Latent Classes of Positive Health Behaviors (Monitoring the Future Data, 2004; = 2,065)

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