Latent Curve Models A Structural Equation

This volume represents a comprehensive treatment of a model sometimes referred to as latent curve or growth curve models. Latent Curve Models analyzes LTMs from the perspective of structural equation modeling (SEM) with latent variables. Although the authors discuss simple regression-based procedures that are helpful in the early stages of LTM, most of the presentation will use SEMs as a driving tool throughout the text.

KENNETH A. BOLLEN, PhD, is Henry Rudolph Immerwahr Distinguished Professor of Sociology, Director of the Odum Institute for Research in Social Science, and an Adjunct Professor of Statistics at The University of North Carolina at Chapel Hill. He is the author of two books, including Structural Equations with Latent Variables (Wiley), and more than 100 scholarly papers.

PATRICK J. CURRAN, PhD, is Associate Professor of Psychology in the L. L. Thurstone Psychometric Laboratory at The University of North Carolina at Chapel Hill. He has made contributions to the development and application of new quantitative methodologies in the social sciences through his integrated program of research, writing, and teaching.

Preface

xi

Introduction

1

(15)

Conceptualization and Analysis of Trajectories

2

(6)

Trajectories of Crime Rates

3

(4)

Data Requirements

7

(1)

Summary

7

(1)

Three Initial Questions About Trajectories

8

(1)

Question 1: What Is the Trajectory for the Entire Group?

8

(1)

Question 2: Do We Need Distinct Trajectories for Each Case?

8

(1)

Question 3: If Distinct Trajectories Are Needed, Can We Identify Variables to Predict These Individual Trajectories?

9

(1)

Summary

9

(1)

Brief History of Latent Curve Models

9

(5)

Early Developments: The Nineteenth Century

10

(1)

Fitting Group Trajectories: 1900--1937

11

(1)

Fitting Individual and Group Trajectories: 1938--1950s

12

(1)

Trajectory Modeling with Latent Variables: 1950s--1984

12

(1)

Current Latent Curve Modeling: 1984--present

13

(1)

Summary

14

(1)

Organization of the Remainder of the Book

14

(2)

Unconditional Latent Curve Model

16

(42)

Repeated Measures

16

(3)

General Model and Assumptions

19

(2)

Identification

21

(4)

Case-By-Case Approach

25

(9)

Assessing Model Fit

29

(4)

Limitations of Case-by-Case Approach

33

(1)

Structural Equation Model Approach

34

(20)

Matrix Expression of the Latent Curve Model

36

(3)

Maximum Likelihood Estimation

39

(3)

Empirical Example

42

(2)

Assessing Model Fit

44

(4)

Components of Fit

48

(6)

Alternative Approaches to the SEM

54

(1)

Conclusions

55

(1)

Appendix 2A: Test Statistics, Nonnormality, and Statistical Power

55

(3)

Missing Data and Alternative Metrics of Time

58

(30)

Missing Data

58

(15)

Types of Missing Data

58

(4)

Treatment of Missing Data

62

(6)

Empirical Example

68

(4)

Summary

72

(1)

Missing Data and Alternative Metrics of Time

73

(14)

Numerical Measure of Time

73

(1)

When Wave of Assessment and Alternative Metrics of Time Are Equivalent

74

(3)

When Wave of Assessment and Alternative Metrics of Time Are Different

77

(2)

Reorganizing Data as a Function of Alternative Metrics of Time

79

(2)

Individually Varying Values of Time

81

(1)

Summary

82

(1)

Empirical Example: Reading Achievement

83

(4)

Conclusions

87

(1)

Nonlinear Trajectories and the Coding of Time

88

(38)

Modeling Nonlinear Functions of Time

89

(9)

Polynomial Trajectories: Quadratic Trajectory Model

89

(8)

Polynomial Trajectories: Cubic Trajectory Models

97

(1)

Summary

98

(1)

Nonlinear Curve Fitting: Estimated Factor Loadings

98

(5)

Selecting the Metric of Change

102

(1)

Piecewise Linear Trajectory Models

103

(3)

Identification

106

(1)

Interpretation

106

(1)

Alternative Parametric Functions

106

(7)

Exponential Trajectory

107

(2)

Parametric Functions with Cycles

109

(3)

Nonlinear Transformations of the Metric of Time

112

(1)

Nonlinear Transformations of the Repeated Measures

112

(1)

Linear Transformations of the Metric of Time

113

(7)

Logic of Recoding the Metric of Time

114

(1)

General Framework for Transforming Time

115

(5)

Summary

120

(1)

Conclusions

120

(1)

Appendix 4A: Identification of Quadratic and Piecewise Latent Curve Models

121

(5)

Quadratic LCM

121

(3)

Piecewise LCM

124

(2)

Conditional Latent Curve Models

126

(36)

Conditional Model and Assumptions

126

(2)

Identification

128

(4)

Structural Equation Modeling Approach

132

(7)

Implied Moment Matrices

133

(3)

Estimation

136

(1)

Model Fit

137

(2)

Interpretation of Conditional Model Estimates

139

(16)

Direct Effects of Exogenous Covariates on Latent Curve Factors

139

(2)

Indirect Effects of Covariates on Repeated Measures

141

(6)

Further Exploring Conditional Effects

147

(3)

Higher-Order Interactions

150

(5)

Empirical Example

155

(5)

yt on λt Regressions

157

(1)

Regions of Significance and Confidence Bands

158

(2)

Summary

160

(1)

Conclusions

160

(2)

The Analysis of Groups

162

(26)

Dummy Variable Approach

163

(7)

Various Level 2 Models

165

(1)

Identification, Estimation, and Model Fit

165

(1)

Empirical Example

166

(3)

Summary

169

(1)

Multiple-Group Analysis

170

(10)

Unconditional Latent Curve Model for Multiple Groups

170

(2)

Conditional Latent Curve Model for Multiple Groups

172

(8)

Unknown Group Membership

180

(4)

Empirical Example

182

(2)

Conclusions

184

(1)

Appendix 6A: Case-by-Case Approach to Analysis of Various Groups

184

(4)

Dummy Variable Method

185

(1)

Multiple-Group Analysis

185

(1)

Unknown Group Membership

186

(1)

Appendix Summary

187

(1)

Multivariate Latent Curve Models

188

(41)

Time-Invariant Covariates

188

(4)

Empirical Example: Conditional Model of Reading and Math Achievement

191

(1)

Time-Varying Covariates

192

(3)

Interpretation of Effects

194

(1)

Simultaneous Inclusion of Time-Invariant and Time-Varying Covariates

195

(3)

Interpretation of Conditional TVC Latent Curve Model

197

(1)

Empirical Example: Conditional TVC Model of Reading and Math Achievement

197

(1)

Multivariate Latent Curve Models

198

(9)

Multivariate Latent Curve Model with TICs

201

(3)

Multivariate Latent Curve Model with TICs and Factor Regressions

204

(1)

Empirical Example: Multivariate Conditional Latent Curve Model of Reading and Math Achievement

204

(3)

Summary

207

(1)

Autoregressive Latent Trajectory Model

207

(11)

Autoregressive (Simplex) Model

208

(2)

Unconditional Bivariate ALT Model

210

(4)

Conditional Bivariate ALT Model

214

(1)

Empirical Example: Income Data from the National Longitudinal Survey of Youth

215

(2)

Summary

217

(1)

General Equation for All Models

218

(7)

Examples of General Notation

220

(5)

Implied Moment Matrices

225

(3)

Conclusions

228

(1)

Extensions of Latent Curve Models

229

(34)

Dichotomous and Ordinal Repeated Measures

229

(16)

Assumptions Violated

230

(2)

Corrective Procedures

232

(8)

Empirical Example

240

(3)

Conditional Model

243

(1)

Summary

244

(1)

Repeated Latent Variables with Multiple Indicators

245

(14)

Single Repeated Measure versus Multiple Indicators

246

(1)

Multiple-Indicator Model

247

(7)

Invariance: Continuous Multiple Indicators

254

(3)

Dichotomous and Ordinal Multiple Indicators

257

(1)

Conditional Models

258

(1)

Summary

259

(1)

Latent Covariates

259

(3)

Conclusions

262

(1)

References

263

(12)

Author Index

275

(4)

Subject Index

279

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Latent Curve Models A Structural Equation Perspective

Summary

Author Biography

Table of Contents

Supplemental Materials