Evaluation of Structural Equation Mixture Models: Parameter Estimates and Correct Class Assignment

Abstract

Structural equation mixture models (SEMMs) are latent class models that permit the estimation of a structural equation model within each class. Fitting SEMMs is illustrated using data from 1 wave of the Notre Dame Longitudinal Study of Aging. Based on the model used in the illustration, SEMM parameter estimation and correct class assignment are investigated in a large-scale simulation study. Design factors of the simulation study are (im)balanced class proportions, (im)balanced factor variances, sample size, and class separation. We compare the fit of models with correct and misspecified within-class structural relations. In addition, we investigate the potential to fit SEMMs with binary indicators. The structure of within-class distributions can be recovered under a wide variety of conditions, indicating the general potential and flexibility of SEMMs to test complex within-class models. Correct class assignment is limited.

Notes

¹SEMM performance should first be evaluated at a single time point before combining time points in a more complex model fitted to longitudinal data. Extension to structural equation growth mixture modeling is currently being evaluated.

²Item selection procedures in the context of mixture models are currently being investigated by the authors.

³Researchers intending to fit SEMMs must set aside the necessary computational resources, as models with categorical indicators could take considerable computation time.

⁴Entropy has been suggested as a measure of classification accuracy when true class membership is unknown. However, entropy can perform poorly in the context of FMMs (Lubke & Muthén, 2007).

⁵The multivariate Mahalanobis distance between two classes is given as MD = Anderson and Bahadur (1962) provide a distance measure which is more accurate when class covariance matrices are unequal. In the current work, the MD are approximately equal to the Anderson and Bahadur measure in most conditions, and results for those conditions in which this is not true are marked by empty cells in the results table.

⁶In this study, error variance was held constant across all conditions, with the side effect that not only factor variances but also item reliabilities differ when comparing balanced and unbalanced factor variance conditions. However, Lubke and Muthén (2007) found that comparable differences in item reliabilities only had minor effects on FMM performance.

⁷Whereas different class proportions do not influence class separation, factor variances do influence class separation. The mean of Factor 1 in Class 1 E(F _1C1) is adjusted across (im)balanced factor variance conditions such that class-invariant and class-specific factor variances are not confounded with class separation as measured by the multivariate Mahalanobis distance.

⁸Formulas for the BIC and saBIC are given in the Appendix.

⁹For example, in unbalanced factor variance conditions, the empirical standard errors are much larger than average standard errors, which could indicate an insufficient number of data sets within condition.

¹⁰An insufficient number of starting values can also lead to nonconvergence. In empirical analyses, the number of starting values can be increased as needed. However, increasing the number of starting values will not compensate for model misspecification or model nonidentification.

¹¹When reporting on parameter estimates in a mixture simulation study, the issue of class switching must be addressed. Because class labeling is arbitrary, there can be fitted models in which Class 1 subjects were labeled Class 2 subjects and vice versa. This creates problems when averaging a parameter across data sets within a condition. When the within-class factor models are the same, switching can be corrected by relabeling the class-specific parameters prior to summarizing. In this study, this was not possible for the SEMM123 model but is feasible for the unstructured FMM. For the conditions examined herein, switching was not problematic except in conditions where convergence rates were low enough to preclude serious interpretation of parameter estimates as discussed at the end of the section on convergence rates.