A fast score test for generalized mixture models

Abstract

In biomedical studies, testing for homogeneity between two groups, where one group is modeled by mixture models, is often of great interest. This paper considers the semiparametric exponential family mixture model proposed by Hong et al. (2017) and studies the score test for homogeneity under this model. The score test is nonregular in the sense that nuisance parameters disappear under the null hypothesis. To address this difficulty, we propose a modification of the score test, so that the resulting test enjoys the Wilks phenomenon. In finite samples, we show that with fixed nuisance parameters the score test is locally most powerful. In large samples, we establish the asymptotic power functions under two types of local alternative hypotheses. Our simulation studies illustrate that the proposed score test is powerful and computationally fast. We apply the proposed score test to an UK ovarian cancer DNA methylation data for identification of differentially methylated CpG sites.

Keywords: DNA methylation; asymptotics; conditional likelihood; nonregular problem; semiparametric mixture model.

Figure 1.

Power curves of the proposed score test, PLEMT test by Hong et al. (2017), MPRT proposed by Ning and Chen (2015), PELT proposed by Liu et al. (2012), EST proposed by Qin and Liang (2011), t-test, and Wilcoxon test. In the normal model, we took (μ₁, σ₁) = (0, 1) and (μ₂, σ₂) = (1, 1.5) (M&V scenario),(μ₁, σ₁) = (0, 1) and (μ₂, σ₂) = (1, 1) (M only scenario), and (μ₁, σ₁) = (0, 1) and (μ₂, σ₂) = (0, 2.5) (V only scenario). In the Beta model, we took (α₁, β₁) = (3, 3) and (α₂, β₂) = (2, 3) (M & V scenario), (α₁, β₁) = (2, 2) and (α₂, β₂) = (1, 2.256) (M only scenario), and (α₁, β₁) = (3, 3) and (α₂, β₂) = (1, 1) (V only scenario).

Figure 2.

Receiver operating characteristics (ROC) curves for the proposed test, the PLEMT test and t-test using logistic regression as the classification method.