Contaminated Chi-Square Modeling and Large-Scale ANOVA Testing

Abstract

We propose a convenient moment-based procedure for testing the omnibus null hypothesis of no contamination of a central chi-square distribution by a non-central chi-square distribution. In sharp contrast with likelihood ratio tests for mixture models, there is no need for re-sampling or random field theory to obtain critical values. Rather, critical values are available from an asymptotic normal distribution, and there is excellent agreement between nominal and actual significance levels. This procedure may be used to model numerous chi-square statistics, obtained via monotonic transformations of F statistics, from large-scale ANOVA testing, such as that encountered in microarray data analysis. In that context, modeling chi-square statistics instead of p-values may improve detection of differential gene expression, as we demonstrate through simulation studies, while also reducing false declarations of the same, as we illustrate in a case study on aging and cognition. Our procedure may also be incorporated into a gene filtration process, which may reduce type II errors on genewise null hypotheses by justifying lighter controls for Type I errors. Let X i denote the rescaled test statistic (K-1) F i. With large (g 1 +g 2 +… +g K-K), X i is distributed approximately χ 2 K-1 (0) under the genewise null hypothesis, and approximately χ 2 K-1 (μ), under the genewise alternative hypothesis, for some μ. We explain this approximation in the Appendix. If g 1 , g 2 , …, g K are not large enough to warrant this approximation, then a more sophisticated approach may be employed to transform F statistics into chi-square statistics; one such approach is described in and used for our case study. Letting λ denote the proportion of genes for which mean expression Journa l o f B io m etrics & B io s ta tistics