The utility of the maximum likelihood F-test was demonstrated as an alternative to the omnibus Chi-square test when evaluating model fit in confirmatory factor analysis with small samples, as it has been well documented that the likelihood ratio test (T[subscript ML]) with small samples is not Chi-square distributed.

A class of effect size indices are discussed that evaluate the degree to which two nested confirmatory factor analysis models differ from each other in terms of fit to a set of observed variables. These descriptive effect measures can be used to quantify the impact of parameter restrictions imposed in an initially considered model and are free…

There has been increasing interest in applying integrative data analysis (IDA) to analyze data across multiple studies to increase sample size and statistical power. Measures of a construct are frequently not consistent across studies. This article provides a tutorial on the complex decisions that occur when conducting harmonization of measures…

The use of multilevel modeling (MLM) to analyze nested data has grown in popularity over the years in the study of school psychology. However, with the increase in use, several statistical misconceptions about the technique have also proliferated. We discuss some commonly cited myths and golden rules related to the use of MLM, explain their…

American Educational Research Association (AERA) standards stipulate that researchers show evidence of the internal structure of instruments. Confirmatory factor analysis (CFA) is one structural equation modeling procedure designed to assess construct validity of assessments that has broad applicability for counselors interested in instrument…

Mixture factor analysis is examined as a means of flexibly estimating nonnormally distributed continuous latent factors in the presence of both continuous and dichotomous observed variables. A simulation study compares mixture factor analysis with normal maximum likelihood (ML) latent factor modeling. Different results emerge for continuous versus…

This article describes SIMREL, a software program designed for the simulation of alpha coefficients and the estimation of its confidence intervals. SIMREL runs on two alternatives. In the first one, if SIMREL is run for a single data file, it performs descriptive statistics, principal components analysis, and variance analysis of the item scores…

This paper discusses how a seldom-used statistical procedure, recursive regression (RR), can numerically and graphically illustrate data-driven nonlinear relationships and interaction of variables. This routine falls into the family of exploratory techniques, yet a few interesting features make it a valuable compliment to factor analysis and…

Measurement invariance is a necessary condition for the evaluation of factor mean differences over groups or time. This article considers the potential problems that can arise for tests of measurement invariance when the true factor-to-indicator relationship is nonlinear (quadratic) and invariant but the linear factor model is nevertheless…

To complement recent articles in this journal on structural equation modeling (SEM) practice and principles by Martens and by Quintana and Maxwell, respectively, the authors offer a consumer's guide to SEM. Using an example derived from theory and research on vocational psychology, the authors outline six steps in SEM: model specification,…

Exploratory factor analysis (EFA) and confirmatory factor analysis (CFA) have contributed to test development and validation in counseling psychology, but additional applications have not been fully realized. The author presents an overview of the goals, terminology, and procedures of factor analysis; reviews best practices for extracting,…

Structural equation modeling (SEM) refers to a family of statistical techniques that explores the relationships among a set of variables. Structural equation modeling provides an extremely versatile method to model very specific hypotheses involving systems of variables, both measured and unmeasured. Researchers can use SEM to study patterns of…