Modified Distribution-Free Goodness of Fit Test Statistic

Abstract

Covariance structure analysis and its structural equation modeling extensions have become one of the most widely used methodologies in social sciences such as psychology, education, and economics. An important issue in such analysis is to assess the goodness-of-fit of a model under analysis. One of the most popular test statistics is the asymptotically distribution free (ADF) test statistic introduced by Browne in 1984. The ADF statistic can be used to test models without any specific distribution assumption (e.g., multivariate normal distribution) of the observed data. Despite its advantage, it has been shown in various empirical studies that unless sample sizes are extremely large, this ADF statistic can perform very poorly in practice. In this paper, we provide a theoretical explanation for this phenomenon and further propose a modified test statistic that improves the performance in samples of realistic size. The proposed statistic deals with the possible ill conditioning of the involved large scale covariance matrices.

Keywords: covariance structures, distribution-free test statistic, asymptotics, chisquare distribution, ill-conditioned problem