This Monte Carlo study evaluated the performance of three estimation methods in fitting confirmatory factor analysis (CFA) models for ordered categorical data, with the focus on data with non-normal underlying distributions and small sample sizes. The three methods are: weighted least squares with mean and variance adjusted (WLSMV), Bayesian with non-informative priors (BN), and Bayesian with informative priors (BI). Design factors included sample sizes, factor structures, underlying continuous distributions, and categorical distributions. Results were evaluated based on the model-data fit, point estimates, and standard errors of point estimates. Results showed that Bayesian methods encountered less convergence problems than WLSMV. Bayesian methods were robust to the non-normality of underlying continuous distributions. WLSMV tended to perform equally well or slightly better than Bayesian methods except for some conditions with small sample sizes or highly non-normal underlying distributions.

This study addressed the argument that online courses rarely use discussion strategies that are specifically designed and constructed for soliciting learners' cognitive presence and higher-order thinking. Prompted by this argument, the authors conducted a meta-analysis of the empirical studies that examined the effectiveness of discussion strategies in online learning. More specifically, the meta-analysis was to reveal whether learners perform better in a strategic discussion session than when they engage in a conventional online discussion and under what conditions the learners' differences in performance are greater or smaller. Furthermore, the investigation examined the extent to which the online discussion strategies that included instructional elements and the pedagogical recommended by the literature make a difference in learners' performance.

Correct detection of measurement bias could help researchers revise models or refine psychological scales. Measurement bias detection can be viewed as a variable-selection problem, in which biased items are optimally selected from a set of items. This study investigated a number of regularization methods: ridge, lasso, elastic net (enet) and adaptive lasso (alasso), in comparison with maximum likelihood estimation (MLE) for detecting various forms of measurement bias in regard to a continuous violator using restricted factor analysis. Particularly, complex structural equation models with relatively small sample sizes were the study focus. Through a simulation study and an empirical example, results indicated that the enet outperformed other methods in small samples for identifying biased items. The alasso yielded low false positive rates for non-biased items outside of a high number of biased items. MLE performed well for the overall estimation of biased items.

Bayesian structural equation modeling (BSEM) is a flexible tool for the exploration and estimation of sparse factor loading structures; that is, most cross-loading entries are zero and only a few important cross-loadings are nonzero. The current investigation was focused on the BSEM with small-variance normal distribution priors (BSEM-N) for both variable selection and model estimation. The prior sensitivity in BSEM-N was explored in factor analysis models with sparse loading structures through a simulation study (Study 1) and an empirical example (Study 2). Study 1 examined the prior sensitivity in BSEM-N based on the model fit, population model recovery, true and false positive rates, and parameter estimation. Seven shrinkage priors on cross-loadings and five noninformative/vague priors on other model parameters were examined. Study 2 provided a real data example to illustrate the impact of various priors on model fit and parameter selection and estimation. Results indicated that when the 95% credible intervals of shrinkage priors barely covered the population cross-loading values, it resulted in the best balance between true and false positives. If the goal is to perform variable selection, a sparse cross-loading structure is required, preferably with a minimal number of nontrivial cross-loadings and relatively high primary loading values. To improve parameter estimates, a relatively large prior variance is preferred. When cross-loadings are relatively large, BSEM-N with zero-mean priors is not recommended for the estimation of cross-loadings and factor correlations.

This study compared the performance of (1) model selection methods based on a point estimate (MSM-P), (2) partially Bayesian methods, and (3) fully Bayesian methods, for testing factorial invariance. The MSM-P considered included the likelihood ratio test (LRT) and a number of information criteria (IC). The Bayesian methods tested included the deviance information criterion (DIC), widely available information criterion (WAIC), and leave-one-out cross-validation (LOO). We investigated true positive and false positive rates of each model selection method under conditions varying in sample size, proportion of non-invariant items, and pattern, location and magnitude of non-invariance. The impact of various prior choices on Bayesian model selection results was also examined. Results showed that DIC, WAIC, and LOO performed comparably well, had a good balance between true and false positive rates, were flexible given the prior choices, and may complement MSM-P in factorial invariance testing. Implications and recommendations based on the findings were discussed.