Empirical Null Distribution of -2log(lambda)

Abstract

Approximation of Non-central Chi-square distribution as an empirical distribution of log-likelihood ratio test statistics (-2logλ; abbreviated as LRT) has been a concern in the field of structural equation modeling. Under extremely severe misspecification (Chun & Shapiro, 2009) reported that non-central Chi-square is not a good choice. In this paper, we have used a bootstrap sampling procedure to investigate the empirical null distribution of LRT specifically in the context of a latent class model (LCM) via frequentist framework (that is, EM algorithm). We used two types of data sets. The first type includes those sets of data on which LCM had been carried out (published results; named as “training data”). The other type is that of those data sets which are not published earlier (i.e. “real” collected data; named as “test data”). Non-central χ² distribution with degrees of freedom equals to the expected value of bootstrap LRT and non-centrality parameter equals to inverse of the variance of bootstrapped LRT is found to be very well fitted empirical null distribution of LRT in case of LCM. These results will help in obtaining the significance value of LRT for deciding on the number of classes present in a latent variable.