| Abstract: | ![]()
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 Cagnone , S. , Mignani , S. ( 2007 ). Assessing the goodness-of-fit of a latent variable model for ordinal data . Metron LXV : 337 – 361 . [Google Scholar]; Maydeu-Olivares and Joe, 2005 Maydeu-Olivares , A. , Joe , H. ( 2005 ). Limited- and full-information estimation and goodness-of-fit testing in 2n contingency tables: A unified framework . J. Amer. Statist. Assoc. 100 ( 471 ): 1009 – 1020 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]; Reiser, 1996 Reiser , M. ( 1996 ). Analysis of residual for the multinomial item response model . Psychometrika 61 : 509 – 528 .[Crossref], [Web of Science ®] , [Google Scholar]). 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. |
