A note on goodness of fit test in latent variable models with categorical variables

Assessing the goodness-of-fit of latent variable models for categorical data becomes a problem in presence of sparse data since the classical goodness-of-fit statistics are badly approximated by the chi square distribution. A good solution to this problem is represented by statistical tests based on the residuals associated to marginal distributions of the manifest variables (Cagnone and Mignani, 2007; Maydeu-Olivares and Joe, 2005; Reiser, 1996). The quadratic form associated to the test involves the use of a generalized inverse of the covariance matrix of the sample proportions. In this article we prove that the rank of the Moore-Penrose generalized inverse is univocally determined and hence it can be used appropriately.