The comparability of linear and nonlinear latent variable analyses of skewed dichotomous items

Abstract

Factor-analytic methods founded on the multivariate linear model are often employed to scale behavior rating instruments. When items are scored as dichotomies and endorsements are skewed, the data violate the assumption of a multivariate normal distribution (MND). While the mathematical ramifications of such have been widely characterized, an empirical evaluation of MND- and non MND-based latent variable solution concordance has not been reported. This research investigated whether MND-based latent variable models employing previously-reported latent dimension extraction and item retention restrictions yielded a solution which was supported by two methods which do not assume a MND: (a) bi-factor analysis, and (b) latent class analysis. Standardization data of the Bristol Social Adjustment Guides, an instrument which evaluates social adjustment in school, were randomly divided into scaling (N = 1,505) and confirmatory (N = 1,000) samples. Scaling methods included: (a) first-order principal component analysis (PCA) and common factor analysis (CFA), (b) second-order CFA, and (c) confirmatory, oblique, principal-component cluster analysis. Bi-factor and latent class models evaluated the goodness-of-fit of confirmatory sample data to the item structure and behavior styles indicated via MND-based linear scaling. One solution comprising four first-order and two second-order latent dimensions, overactivity (OVER) and underactivity (UNDER), was derived and supported by both confirmatory methods. The bi-factor model was superior to simple structure for OVER and UNDER items, X\sp2 difference: OVER (39, N = 1,000) = 67, $p \u3c .005$; and UNDER (33, N = 1,000) = 100, $p \u3c .005$. A three-dimensional (over-, under-, and normoactivity) confirmatory latent class model was fitted, X\sp2 (4, N = 1,000) = 1.29, $p \u3e .80$, while alternative one- and two-class models were rejected at the $\u3c$.01 alpha level. PCA and CFA yield concordant solutions with dichotomous, skewed, item data under accepted extraction and retention criteria. Latent structures identified through MND-based linear factoring are, under these circumstances, supported by models which do not assume a MND