A Quasi-Likelihood Approach to Assess Model Fit in Quadratic and Interaction SEM.

For the assessment of model fit in linear structural equation modeling (SEM), several fit measures have been developed that use an unconstrained mean and covariance structure, but cannot be readily applied to SEM with quadratic and interaction effects. In this article, we propose the novel quasi-likelihood ratio test (Q-LRT) to evaluate global fit of nonlinear SEM models. The Q-LRT is based on a simplification of the quasi-maximum likelihood method for the estimation of model parameters. An empirical application of the Q-LRT is demonstrated for data in a study about aging in men. Results from a Monte Carlo study show that the Q-LRT performs reliably when sample size is sufficiently large. Also, simulations suggest robustness of Q-LRT for moderately skewed latent exogenous variables.

Disentangling the Common Variance of Perfectionistic Strivings and Perfectionistic Concerns: A Bifactor Model of Perfectionism.

Perfectionism nowadays is frequently understood as a multidimensional personality trait with two higher-order dimensions of perfectionistic strivings and perfectionistic concerns. While perfectionistic concerns are robustly found to correlate with negative outcomes and psychological malfunctioning, findings concerning the outcomes of perfectionistic strivings are inconsistent. There is evidence that perfectionistic strivings relate to psychological maladjustment on the one hand but to positive outcomes on the other hand as well. Moreover, perfectionistic strivings and perfectionistic concerns frequently showed substantial overlap. These inconsistencies of differential relations and the substantial overlap of perfectionistic strivings and perfectionistic concerns raise questions concerning the factorial structure of perfectionism and the meaning of its dimensions. In this study, several bifactor models were applied to disentangle the common variance of perfectionistic strivings and perfectionistic concerns at the item level using Hill et al.'s (2004) Perfectionism Inventory (PI). The PI measures a broad range of perfectionism dimensions by four perfectionistic strivings and four perfectionistic concerns subscales. The bifactor-(S - 1) model with one general factor defined by concern over mistakes as the reference facet, four specific perfectionistic strivings factors, and three specific perfectionistic concerns factors showed acceptable fit. The results revealed a clear separation between perfectionistic strivings and perfectionistic concerns, as the general factor represented concern over mistakes, while the perfectionistic strivings factors each explained a substantial amount of reliable variance independent of the general factor. As a result, factor scores of the specific perfectionistic strivings factors and the general factor had differential relationships with achievement motivation, neuroticism, conscientiousness, and self-efficacy that met with theoretical expectations, while results for manifest subscale scores were ambiguous. Our results question the existence of reliable sub-constructs of perfectionistic concerns independent of the general factor when defined by concern over mistakes.

A Heterogeneous Growth Curve Model for Nonnormal Data.

The heterogeneous growth curve model (HGM; Klein & Muthén, 2006 ) is a method for modeling heterogeneity of growth rates with a heteroscedastic residual structure for the slope factor. It has been developed as an extension of a conventional growth curve model and a complementary tool to growth curve mixture models. In this article, a robust version of the heterogeneous growth curve model (HGM-R) is presented that extends the original HGM with a mixture model to allow for an unbiased parameter estimation under the condition of nonnormal data. In two simulation studies, the performance of the method is examined under the condition of nonnormality and a misspecified heteroscedastic residual structure. The results of the simulation studies suggest an unbiased estimation of the heterogeneity by the HGM-R when sample size was large enough and a good approximation of the heteroscedastic residual structure even when the functional form of the heteroscedasticity was misspecified. The practical application of the approach is demonstrated for a data set from HIV-infected patients.

Evaluation of model fit in nonlinear multilevel structural equation modeling.

Evaluating model fit in nonlinear multilevel structural equation models (MSEM) presents a challenge as no adequate test statistic is available. Nevertheless, using a product indicator approach a likelihood ratio test for linear models is provided which may also be useful for nonlinear MSEM. The main problem with nonlinear models is that product variables are non-normally distributed. Although robust test statistics have been developed for linear SEM to ensure valid results under the condition of non-normality, they have not yet been investigated for nonlinear MSEM. In a Monte Carlo study, the performance of the robust likelihood ratio test was investigated for models with single-level latent interaction effects using the unconstrained product indicator approach. As overall model fit evaluation has a potential limitation in detecting the lack of fit at a single level even for linear models, level-specific model fit evaluation was also investigated using partially saturated models. Four population models were considered: a model with interaction effects at both levels, an interaction effect at the within-group level, an interaction effect at the between-group level, and a model with no interaction effects at both levels. For these models the number of groups, predictor correlation, and model misspecification was varied. The results indicate that the robust test statistic performed sufficiently well. Advantages of level-specific model fit evaluation for the detection of model misfit are demonstrated.