Testing and modelling non-normality within the one-factor model

Maximum likelihood estimation in the one-factor model is based on the assumption of multivariate normality for the observed data. This general distributional assumption implies three specific assumptions for the parameters in the one-factor model: the common factor has a normal distribution; the residuals are homoscedastic; and the factor loadings do not vary across the common factor scale. When any of these assumptions is violated, non-normality arises in the observed data. In this paper, a model is presented based on marginal maximum likelihood to enable explicit tests of these assumptions. In addition, the model is suitable to incorporate the detected violations, to enable statistical modelling of these effects. Two simulation studies are reported in which the viability of the model is investigated. Finally, the model is applied to IQ data to demonstrate its practical utility as a means to investigate ability differentiation.