Population and sample simulation approaches were used to compare the performance of parallel analysis using principal component analysis (PA-PCA) and parallel analysis using principal axis factoring (PA-PAF) to identify the number of underlying factors. Additionally, the accuracies of the mean eigenvalue and the 95th percentile eigenvalue criteria were examined. The 95th percentile criterion was preferable for assessing the first eigenvalue using either extraction method. In assessing subsequent eigenvalues, PA-PCA tended to perform as well as or better than PA-PAF for models with one factor or multiple minimally correlated factors; the relative performance of the mean eigenvalue and the 95th percentile eigenvalue criteria depended on the number of variables per factor. PA-PAF using the mean eigenvalue criterion generally performed best if factors were more than minimally correlated or if one or more strong general factors as well as group factors were present.

ICAP is a theory of active learning that differentiates students' engagement based on their behaviors. ICAP postulates that Interactive engagement, demonstrated by co-generative collaborative behaviors, is superior for learning to Constructive engagement, indicated by generative behaviors. Both kinds of engagement exceed the benefits of Active or Passive engagement, marked by manipulative and attentive behaviors, respectively. This paper discusses a 5-year project that attempted to translate ICAP into a theory of instruction using five successive measures: (a) teachers' understanding of ICAP after completing an online module, (b) their success at designing lesson plans using different ICAP modes, (c) fidelity of teachers' classroom implementation, (d) modes of students' enacted behaviors, and (e) students' learning outcomes. Although teachers had minimal success in designing Constructive and Interactive activities, students nevertheless learned significantly more in the context of Constructive than Active activities. We discuss reasons for teachers' overall difficulty in designing and eliciting Interactive engagement.

If data exhibit multidimensionality, key conditional independence assumptions of unidimensional models do not hold. The current work pursues posterior predictive model checking, a flexible family of model-checking procedures, as a tool for criticizing models due to unaccounted for dimensions in the context of item response theory. Factors hypothesized to influence dimensionality and dimensionality assessment are couched in conditional covariance theory and conveyed via geometric representations of multidimensionality. A simulation study investigates the performance of the model-checking tools for dichotomous observables. Key findings include support for the hypothesized effects of the manipulated factors with regard to their influence on dimensionality assessment and the superiority of certain discrepancy measures for conducting posterior predictive model checking for dimensionality assessment.

The results support the efficacy of this ERGT and the durability of treatment gains.

A number of psychometricians have argued for the use of parallel analysis to determine the number of factors. However, parallel analysis must be viewed at best as a heuristic approach rather than a mathematically rigorous one. The authors suggest a revision to parallel analysis that could improve its accuracy. A Monte Carlo study is conducted to compare revised and traditional parallel analysis approaches. Five dimensions are manipulated in the study: number of observations, number of factors, number of measured variables, size of the factor loadings, and degree of correlation between factors. Based on the results, the revised parallel analysis method, using principal axis factoring and the 95th percentile eigenvalue rule, offers promise.