Melanie Wall

Compared to the literature surrounding linear factor analysis, the literature describing methods for nonlinear factor analysis models or their application is relatively sparse. This is no doubt due to the complexity of the distributions of the nonlinear factors and/or the distribution of the indicators of these factors which precludes the reliance soley on covariance matrices (as in the linear model) and makes difficult the direct use of maximum likelihood techniques. While McDonald (1962, Psychometrika) gives reference to a few earlier methods papers introducing the idea of nonlinear factors, a series of papers by McDonald throughout the 1960's appears to be the first attempt at providing a statistical framework to the nonlinear factor analysis model. The model considered was of the form Y = m + L F(x) + d, where the nonlinearity F( ) was in the factors x themselves, not in the coefficients L. It is this form of the model that has received the most attention since then. A recently growing series of papers on the topic of nonlinear latent variable models, in particular nonlinear (e.g. interaction) structural equation models, are all special cases of the nonlinear factor analysis model of the form above. Method of moments techniques, pseudo-maximum likelihood, the EM algorithm, and fully Bayesian frameworks have been considered for fitting this model. Recently Yalcin and Amemiya (2001, Statistical Science) introduced a more general nonlinear factor analysis model Y = F(L, x) + d, where the nonlinearity can be in both the factors and the coefficients. In this talk, a review of the nonlinear factor analysis literature is presented with discussion of the varying estimation strategies and a survey of their actual application in the literature.