Latent Curve Models. A Structural Equation Perspective WILEY- INTERSCIENΠKENNETH A. BOLLEN

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1 Latent Curve Models A Structural Equation Perspective KENNETH A. BOLLEN University of North Carolina Department of Sociology Chapel Hill, North Carolina PATRICK J. CURRAN University of North Carolina Department of Psychology Chapel Hill, North Carolina WILEY- INTERSCIENΠA JOHN WILEY & SONS, INC., PUBLICATION

2 Contents Preface xi 1 Introduction 1 Conceptualization and Analysis of Trajectories, Trajectories of Crime Rates, Data Requirements, Summary, Three Initial Questions About Trajectories, Question 1: What Is the Trajectory for the Entire Group?, Question 2: Do We Need Distinct Trajectories for Each Case?, Question 3: If Distinct Trajectories Are Needed, Can We Identify Variables to Predict These Individual Trajectories?, Summary, Brief History of Latent Curve Models, Early Developments: The Nineteenth Century, Fitting Group Trajectories: , Fitting Individual and Group Trajectories: s, Trajectory Modeling with Latent Variables: 1950s-1984, Current Latent Curve Modeling: 1984-present, Summary, Organization of the Remainder of the Book, 14

3 CONTENTS 2 Unconditional Latent Curve Model Repeated Measures, General Model and Assumptions Identification, Case-By-Case Approach, Assessing Model Fit, Limitations of Case-by-Case Approach, Structural Equation Model Approach, Matrix Expression of the Latent Curve Model, Maximum Likelihood Estimation, Empirical Example, Assessing Model Fit, Components of Fit, Alternative Approaches to the SEM, Conclusions, 55 Appendix 2A: Test Statistics, Nonnormality, and Statistical Power, 55 3 Missing Data and Alternative Metrics of Time Missing Data, Types of Missing Data, Treatment of Missing Data, Empirical Example, Summary, Missing Data and Alternative Metrics of Time, Numerical Measure of Time, When Wave of Assessment and Alternative Metrics of Time Are Equivalent, When Wave of Assessment and Alternative Metrics of Time Are Different, Reorganizing Data as a Function of Alternative Metrics of Time, Individually Varying Values of Time, Summary, Empirical Example: Reading Achievement, Conclusions, 87 4 Nonlinear Trajectories and the Coding of Time Modeling Nonlinear Functions of Time, Polynomial Trajectories: Quadratic Trajectory Model, 89

4 CONTENTS Polynomial Trajectories: Cubic Trajectory Models, Summary, Nonlinear Curve Fitting: Estimated Factor Loadings, Selecting the Metric of Change, Piecewise Linear Trajectory Models, Identification, Interpretation, Alternative Parametric Functions, Exponential Trajectory, Parametric Functions with Cycles, Nonlinear Transformations of the Metric of Time, Nonlinear Transformations of the Repeated Measures, Linear Transformations of the Metric of Time, Logic of Recoding the Metric of Time, General Framework for Transforming Time, Summary, Conclusions, 120 Appendix 4A: Identification of Quadratic and Piecewise Latent Curve Models, 121 4A.1 Quadratic LCM, 121 4A.2 Piecewise LCM, Conditional Latent Curve Models Conditional Model and Assumptions, Identification, Structural Equation Modeling Approach, Implied Moment Matrices, Estimation, Model Fit, Interpretation of Conditional Model Estimates, Direct Effects of Exogenous Covariates on Latent Curve Factors, Indirect Effects of Covariates on Repeated Measures, Further Exploring Conditional Effects, Higher-Order Interactions, Î Empirical Example, ,1 Regressions, 157

5 viii CONTENTS Regions of Significance and Confidence Bands, Summary, Conclusions, The Analysis of Groups Dummy Variable Approach, Various Level 2 Models, Identification Estimation, and Model Fit, Empirical Example, Summary, Multiple-Group Analysis, Unconditional Latent Curve Model for Multiple Groups, Conditional Latent Curve Model for Multiple Groups, Unknown Group Membership, Empirical Example, Conclusions, 184 Appendix 6A: Case-by-Case Approach to Analysis of Various Groups, 184 6A.1 Dummy Variable Method, 185 6A.2 Multiple-Group Analysis, 185 6A.3 Unknown Group Membership, 186 6A.4 Appendix Summary, Multivariate Latent Curve Models Time-Invariant Covariates, Empirical Example: Conditional Model of Reading and Math Achievement, Time-Varying Covariates, Interpretation of Effects, Simultaneous Inclusion of Time-Invariant and Time-Varying Covariates, Interpretation of Conditional TVC Latent Curve Model, Empirical Example: Conditional TVC Model of Reading and Math Achievement, Multivariate Latent Curve Models, Multivariate Latent Curve Model with TICs, 201

6 CONTENTS Multivariate Latent Curve Model with TICs and Factor Regressions, Empirical Example: Multivariate Conditional Latent Curve Model of Reading and Math Achievement Summary, Autoregressive Latent Trajectory Model, Autoregressive (Simplex) Model, Unconditional Bivariate ALT Model, Conditional Bivariate ALT Model, Empirical Example: Income Data from the National Longitudinal Survey of Youth, Summary, General Equation for All Models, Examples of General Notation, Implied Moment Matrices, Conclusions, Extensions of Latent Curve Models Dichotomous and Ordinal Repeated Measures, Assumptions Violated, Corrective Procedures, Empirical Example, Conditional Model, Summary, Repeated Latent Variables with Multiple Indicators, Single Repeated Measure versus Multiple Indicators, Multiple-Indicator Model, Invariance: Continuous Multiple Indicators, Dichotomous and Ordinal Multiple Indicators, Conditional Models, Summary, Latent Covariates, Conclusions, 262 References 263 Author Index 275 Subject Index 279