Fitting of Mixtures with Unspecified Number of Components Using Cross Validation Distance Estimate

Estimation of the number of mixture components (k) is an unsolved problem. Available methods for estimation of k include bootstrapping the likelihood ratio test statistics and optimizing a variety of validity functionals such as AIC, BIC/MDL, and ICOMP. We investigate the minimization of distance between fitted mixture model and the true density as a method for estimating k. The distances considered are Kullback-Leibler (KL) and “L sub 2”. We estimate these distances using cross validation. A reliable estimate of k is obtained by voting of B estimates of k corresponding to B cross validation estimates of distance. This estimation methods with KL distance is very similar to Monte Carlo cross validated likelihood methods discussed by Smyth (2000). With focus on univariate normal mixtures, we present simulation studies that compare the cross validated distance method with AIC, BIC/MDL, and ICOMP. We also apply the cross validation estimate of distance approach along with AIC, BIC/MDL and ICOMP approach, to data from an osteoporosis drug trial in order to find groups that differentially respond to treatment.

A Diagnostic Test for the Mixing Distribution in a Generalised Linear Mixed Model

We introduce a diagnostic test for the mixing distribution in a generalised linear mixed model. The test is based on the difference between the marginal maximum likelihood and conditional maximum likelihood estimates of a subset of the fixed effects in the model. We derive the asymptotic variance of this difference, and propose a test statistic that has a limiting chi-square distribution under the null hypothesis that the mixing distribution is correctly specified. For the important special case of the logistic regression model with random intercepts, we evaluate via simulation the power of the test in finite samples under several alternative distributional forms for the mixing distribution. We illustrate the method by applying it to data from a clinical trial investigating the effects of hormonal contraceptives in women.