Linear functional relationships are intended to be symmetric and therefore cannot generally be accurately estimated using ordinary least squares regression equations. Orthogonal regression (OR) models allow for errors in both "Y" and "X" and therefore can provide symmetric estimates of these relationships. The most…

Ordinary least squares (OLS) regression provides optimal linear predictions of a dependent variable, y, given an independent variable, x, but OLS regressions are not symmetric or reversible. In order to get optimal linear predictions of x given y, a separate OLS regression in that direction would be needed. This report provides a least squares…

Although frequentist estimators can effectively fit ordinal confirmatory factor analysis (CFA) models, their assumptions are difficult to establish and estimation problems may prohibit their use at times. Consequently, researchers may want to also look to Bayesian analysis to fit their ordinal models. Bayesian methods offer researchers an…

Ordinary least squares (OLS) regression and orthogonal regression (OR) address different questions and make different assumptions about errors. The OLS regression of Y on X yields predictions of a dependent variable (Y) contingent on an independent variable (X) and minimizes the sum of squared errors of prediction. It assumes that the independent…

Prior studies investigating the effects of non-normality in structural equation modeling typically induced non-normality in the indicator variables. This procedure neglects the factor analytic structure of the data, which is defined as the sum of latent variables and errors, so it is unclear whether previous results hold if the source of…

We examined the effect of estimation methods, maximum likelihood (ML), unweighted least squares (ULS), and diagonally weighted least squares (DWLS), on three population SEM (structural equation modeling) fit indices: the root mean square error of approximation (RMSEA), the comparative fit index (CFI), and the standardized root mean square residual…

This Monte Carlo simulation study aimed to investigate confirmatory factor analysis (CFA) estimation methods under different conditions, such as sample size, distribution of indicators, test length, average factor loading, and factor structure. Binary data were generated to compare the performance of maximum likelihood (ML), mean and variance…

Weighted least squares (WLS), weighted least squares mean-and-variance-adjusted (WLSMV), unweighted least squares mean-and-variance-adjusted (ULSMV), maximum likelihood (ML), robust maximum likelihood (MLR) and Bayesian estimation methods were compared in mixed item response type data via Monte Carlo simulation. The percentage of polytomous items,…

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…

A simulation study was conducted to investigate the model size effect when confirmatory factor analysis (CFA) models include many ordinal items. CFA models including between 15 and 120 ordinal items were analyzed with mean- and variance-adjusted weighted least squares to determine how varying sample size, number of ordered categories, and…

Evaluations of measurement invariance provide essential construct validity evidence--a prerequisite for seeking meaning in psychological and educational research and ensuring fair testing procedures in high-stakes settings. However, the quality of such evidence is partly dependent on the validity of the resulting statistical conclusions. Type I or…

The infinitesimal jackknife, a nonparametric method for estimating standard errors, has been used to obtain standard error estimates in covariance structure analysis. In this article, we adapt it for obtaining standard errors for rotated factor loadings and factor correlations in exploratory factor analysis with sample correlation matrices. Both…

Estimation of multidimensional item response theory (MIRT) model parameters can be carried out using the normal ogive with unweighted least squares estimation with the normal-ogive harmonic analysis robust method (NOHARM) software. Previous simulation research has demonstrated that this approach does yield accurate and efficient estimates of item…

Exploratory factor analysis (EFA) is often conducted with ordinal data (e.g., items with 5-point responses) in the social and behavioral sciences. These ordinal variables are often treated as if they were continuous in practice. An alternative strategy is to assume that a normally distributed continuous variable underlies each ordinal variable.…

Ordinal variables are common in many empirical investigations in the social and behavioral sciences. Researchers often apply the maximum likelihood method to fit structural equation models to ordinal data. This assumes that the observed measures have normal distributions, which is not the case when the variables are ordinal. A better approach is…