Size and Power of Recent Tests for Homogeneity in Exponential Mixtures

The article investigates diagnostic procedures for finite mixture models. The problem is to decide whether given data stem from an exponential distribution or a finite mixture of such distributions. Recently, three new test approaches have been proposed, the modified likelihood ratio test (MLRT) by Chen et al. (2001 Chen , H. , Chen , J. , Kalbfleisch , J. D. ( 2001 ). A modified likelihood ratio test for homogeneity in finite mixture models . Journal of the Royal Statistical Society, B 63 : 19 – 29 .[Crossref] , [Google Scholar]), the ADDS test by Mosler and Seidel (2001 Mosler , K. , Seidel , W. ( 2001 ). Testing for homogeneity in an exponential mixture model . Australian and New Zealand Journal of Statistics 43 : 231 – 247 . [Google Scholar]), and the D-test by Charnigo and Sun (2004 Charnigo , R. , Sun , J. ( 2004 ). Testing homogeneity in a mixture distribution via the l ² distance between competing models . Journal of the American Statistical Society 99 : 488 – 498 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The size and power of these tests are determined by Monte Carlo simulation and their relative merits are evaluated. We conclude that the ADDS test shows always not much less and under some alternatives, in particular lower contaminations, considerably more power than its competitors. Also, new tables for the ADDS test are provided.     

Omnibus testing and gene filtration in microarray data analysis

When thousands of tests are performed simultaneously to detect differentially expressed genes in microarray analysis, the number of Type I errors can be immense if a multiplicity adjustment is not made. However, due to the large scale, traditional adjustment methods require very stringen significance levels for individual tests, which yield low power for detecting alterations. In this work, we describe how two omnibus tests can be used in conjunction with a gene filtration process to circumvent difficulties due to the large scale of testing. These two omnibus tests, the D-test and the modified likelihood ratio test (MLRT), can be used to investigate whether a collection of P-values has arisen from the Uniform(0,1) distribution or whether the Uniform(0,1) distribution contaminated by another Beta distribution is more appropriate. In the former case, attention can be directed to a smaller part of the genome; in the latter event, parameter estimates for the contamination model provide a frame of reference for multiple comparisons. Unlike the likelihood ratio test (LRT), both the D-test and MLRT enjoy simple limiting distributions under the null hypothesis of no contamination, so critical values can be obtained from standard tables. Simulation studies demonstrate that the D-test and MLRT are superior to the AIC, BIC, and Kolmogorov-Smirnov test. A case study illustrates omnibus testing and filtration.     

Testing homogeneity in discrete mixtures

This paper introduces W-tests for assessing homogeneity in mixtures of discrete probability distributions. A W-test statistic depends on the data solely through parameter estimators and, if a penalized maximum likelihood estimation framework is used, has a tractable asymptotic distribution under the null hypothesis of homogeneity. The large-sample critical values are quantiles of a chi-square distribution multiplied by an estimable constant for which we provide an explicit formula. In particular, the estimation of large-sample critical values does not involve simulation experiments or random field theory. We demonstrate that W-tests are generally competitive with a benchmark test in terms of power to detect heterogeneity. Moreover, in many situations, the large-sample critical values can be used even with small to moderate sample sizes. The main implementation issue (selection of an underlying measure) is thoroughly addressed, and we explain why W-tests are well-suited to problems involving large and online data sets. Application of a W-test is illustrated with an epidemiological data set.