Impact of Sampling Variability When Estimating the Explained Common Variance

Assessing multidimensionality of a scale or test is a staple of educational and psychological measurement. One approach to evaluate approximate unidimensionality is to fit a bifactor model where the subfactors are determined by substantive theory and estimate the explained common variance (ECV) of the general factor. The ECV says to what extent the explained variance is dominated by the general factor over the specific factors, and has been used, together with other methods and statistics, to determine if a single factor model is sufficient for analyzing a scale or test (Rodriguez et al., 2016). In addition, the individual item-ECV (I-ECV) has been used to assess approximate unidimensionality of individual items (Carnovale et al., 2021; Stucky et al., 2013). However, the ECVand I-ECVare subject to random estimation error which previous studies have not considered. Not accounting for the error in estimation can lead to conclusions regarding the dimensionality of a scale or item that are inaccurate, especially when an estimate of ECVor I-ECV is compared to a pre-specified cut-off value to evaluate unidimensionality. The objective of the present study is to derive standard errors of the estimators of ECV and I-ECV with linear confirmatory factor analysis (CFA) models to enable the assessment of random estimation error and the computation of confidence intervals for the parameters. We use Monte-Carlo simulation to assess the accuracy of the derived standard errors and evaluate the impact of sampling variability on the estimation of the ECV and I-ECV. In a bifactor model for J items, denote Xj, j 1⁄4 1, ..., J , as the observed variable and let G denote the general factor. We define the S subfactors Fs, s2f1,..., Sg, and Js as the set of indicators for each subfactor. Each observed indicator Xj is then defined by the multiple factor model (McDonald, 2013)