Patrick J. Curran

Monte Carlo computer simulations were used to investigate the performance of three X 2 test statistics in confirmatory factor analysis (CFA). Normal theory maximum likelihood)~2 (ML), Browne's asymptotic distribution free X 2 (ADF), and the Satorra-Bentler rescaled X 2 (SB) were examined under varying conditions of sample size, model specification, and multivariate distribution. For properly specified models, ML and SB showed no evidence of bias under normal distributions across all sample sizes, whereas ADF was biased at all but the largest sample sizes. ML was increasingly overestimated with increasing nonnormality, but both SB (at all sample sizes) and ADF (only at large sample sizes) showed no evidence of bias. For misspecified models, ML was again inflated with increasing nonnormality, but both SB and ADF were underestimated with increasing nonnormality. It appears that the power of the SB and ADF test statistics to detect a model misspecification is attenuated given nonnormally distributed data. Confirmatory factor analysis (CFA) has become an increasingly popular method of investigating the structure of data sets in psychology. In contrast to traditional exploratory factor analysis that does not place strong a priori restrictions on the structure of the model being tested, CFA requires the investigator to specify both the number of factors

Simple slopes, regions of significance, and confidence bands are commonly used to evaluate interactions in multiple linear regression (MLR) models, and the use of these techniques has recently been extended to multilevel or hierarchical linear modeling (HLM) and latent curve analysis (LCA). However, conducting these tests and plotting the conditional relations is often a tedious and error-prone task. This article provides an overview of methods used to probe interaction effects and describes a unified collection of freely available online resources that researchers can use to obtain significance tests for simple slopes, compute regions of significance, and obtain confidence bands for simple slopes across the range of the moderator in the MLR, HLM, and LCA contexts. Plotting capabilities are also provided.

Confirmatory factor analysis (CFA) is widely used for examining hypothesized relations among ordinal variables (e.g., Likert-type items). A theoretically appropriate method fits the CFA model to polychoric correlations using either weighted least squares (WLS) or robust WLS. Importantly, this approach assumes that a continuous, normal latent process determines each observed variable. The extent to which violations of this assumption undermine CFA estimation is not well-known. In this article, the authors empirically study this issue using a computer simulation study. The results suggest that estimation of polychoric correlations is robust to modest violations of underlying normality. Further, WLS performed adequately only at the largest sample size but led to substantial estimation difficulties with smaller samples. Finally, robust WLS performed well across all conditions.

Longitudinal models are becoming increasingly prevalent in the behavioral sciences, with key advantages including increased power, more comprehensive measurement, and establishment of temporal precedence. One particularly salient strength offered by longitudinal data is the ability to disaggregate between-person and within-person effects in the regression of an outcome on a time-varying covariate. However, the ability to disaggregate these effects has not been fully capitalized upon in many social science research applications. Two likely reasons for this omission are the general lack of discussion of disaggregating effects in the substantive literature and the need to overcome several remaining analytic challenges that limit existing quantitative methods used to isolate these effects in practice. This review explores both substantive and quantitative issues related to the disaggregation of effects over time, with a particular emphasis placed on the multilevel model. Existing analytic methods are reviewed, a general approach to the problem is proposed, and both the existing and proposed methods are demonstrated using several artificial data sets. Potential limitations and directions for future research are discussed, and recommendations for the disaggregation of effects in practice are offered.

Preface. 1 Introduction. 1.1 Conceptualization and Analysis of Trajectories. 1.2 Three Initial Questions About Trajectories. 1.3 Brief History of Latent Curve Models. 1.4 Organization of the Remainder of the Book. 2 Unconditional Latent Curve Model. 2.1 Repeated Measures. 2.2 General Model and Assumptions. 2.3 Identification. 2.4 Case-By-Case Approach. 2.5 Structural Equation Model Approach. 2.6 Alternative Approaches to the SEM. 2.7 Conclusions. Appendix 2A: Test Statistics, Nonnormality, and Statistical Power. 3 Missing Data and Alternative Metrics of Time. 3.1 Missing Data. 3.2 Missing Data and Alternative Metrics of Time. 3.3 Conclusions. 4 Nonlinear Trajectories and the Coding of Time. 4.1 Modeling Nonlinear Functions of Time. 4.2 Nonlinear Curve Fitting: Estimated Factor Loadings. 4.3 Piecewise Linear Trajectory Models. 4.4 Alternative Parametric Functions. 4.5 Linear Transformations of the Metric of Time. 4.6 Conclusions. Appendix 4A: Identification of Quadratic and Piecewise Latent Curve Models. 4A.1 Quadratic LCM. 4A.2 Piecewise LCM. 5 Conditional Latent Curve Models. 5.1 Conditional Model and Assumptions. 5.2 Identification. 5.3 Structural Equation Modeling Approach. 5.4 Interpretation of Conditional Model Estimates. 5.5 Empirical Example. 5.6 Conclusions. 6 The Analysis of Groups. 6.1 Dummy Variable Approach. 6.2 Multiple-Group Analysis. 6.3 Unknown Group Membership. 6.4 Conclusions. Appendix 6A: Case-by-Case Approach to Analysis of Various Groups. 6A.1 Dummy Variable Method. 6A.2 Multiple-Group Analysis. 6A.3 Unknown Group Membership. 6A.4 Appendix Summary. 7 Multivariate Latent Curve Models. 7.1 Time-Invariant Covariates. 7.2 Time-Varying Covariates. 7.3 Simultaneous Inclusion of Time-Invariant and Time-Varying Covariates. 7.4 Multivariate Latent Curve Models. 7.5 Autoregressive Latent Trajectory Model. 7.6 General Equation for All Models. 7.7 Implied Moment Matrices. 7.8 Conclusions. 8 Extensions of Latent Curve Models. 8.1 Dichotomous and Ordinal Repeated Measures. 8.2 Repeated Latent Variables with Multiple Indicators. 8.3 Latent Covariates. 8.4 Conclusions. References. Author Index. Subject Index.