Testing Homogeneity in a Semiparametric Two-Sample Problem

Abstract

We study a two-sample homogeneity testing problem, in which one sample comes from a population with density f x and the other is from a mixture population with mixture density 1−λ f x λg x . This problem arises naturally from many statistical applications such as test for partial differential gene expression in microarray study or genetic studies for gene mutation. Under the semiparametric assumption g x f x e α βx , a penalized empirical likelihood ratio test could be constructed, but its implementation is hindered by the fact that there is neither feasible algorithm for computing the test statistic nor available research results on its theoretical properties. To circumvent these difficulties, we propose an EM test based on the penalized empirical likelihood. We prove that the EM test has a simple chi-square limiting distribution, and we also demonstrate its competitive testing performances by simulations. A real-data example is used to illustrate the proposed methodology.

PLEMT: A Novel Pseudolikelihood-Based EM Test for Homogeneity in Generalized Exponential Tilt Mixture Models

“…More specifically, the MELRT test has about 17% – 46% less power than that of the PLEMT test, the Kolmogorov-Smirnov test has about 40% – 90% less power than that of the PLEMT test, whereas the other four tests have essentially no power beyond Type I errors. This is consistent with the simulation results in Liu et al, (2012) that the MELRT test is more powerful than the EST test. When both mean and variance are different, the PLEMT test remains to be the most powerful one.…”

“…However, the choice of the value for the fixed λ is arbitrary, and the performance of the score test depends on the choice of λ . Alternatively, Liu et al, (2012) proposed an empirical likelihood function by adding penalty on λ . Rather than using the empirical likelihood, we propose the following penalized pseudolikelihood function to avoid the boundary and identifiability problems $${\mathbf{ℒ}}_{\mathit{\text{pp}}}false(\lambda ,\alpha ,{\beta }_1,{\beta }_{2}false)={\mathbf{ℒ}}_{p}false(\lambda ,\alpha ,{\beta }_1,{\beta }_{2}false)+C\text{ log}false(\lambda false),$$ where C is a positive number.…”

“…Recently, Qin and Liang (2011) derived a original score test under model (1) with a simple limiting chi-squared distribution. More recently, Liu et al, (2012) proposed a novel modified empirical likelihood ratio test under model (1) and developed an efficient and intuitive expectation–maximization (EM) algorithm for computing the test statistic. Despite the current success on the test of homogeneity in ETMM, the aforementioned methods only allow a scalar parameter β , which excludes important parametric distributions such as normal distributions with unequal means and unequal variances, gamma distributions with different shape and scale parameters, and beta distributions.…”

“…As recent studies have observed that cancer tissues and some complex disease cases can also be characterized by an increased variability in DNA methylation patterns (Hansen et al, 2011; Issa, 2011; Teschendorff et al, 2012; Xu et al, 2013), tests that ignore this feature may lead to a substantial loss of power. We therefore extended both the score test by Qin and Liang (2011) and the modified empirical likelihood ratio test by Liu et al, (2012) by generalizing x to k ( x ) = ( x, x 2 ) in equation (1) to account for differences in both means and variances. However, as the simulation results summarized in Section 2 of online supplementary materials suggested, both the extended score test and the extended modified empirical likelihood ratio test have inflated Type I errors.…”

“…Specifically, the right handside of equation (1) is extended to a general form of α + β T k ( x ), so that the multi-parameter ETMM includes many parametric models, such as the normal mixture model with unequal variances, the gamma mixture model, and the beta mixture model. Rather than estimating the baseline density function f (·) with the empirical likelihood procedure as in Qin and Liang (2011) and Liu et al, (2012), we construct a novel pseudolikelihood based on a conditioning procedure, which eliminates the baseline density function f (·) and avoids its estimation. To handle the non-regularity problems (i.e.…”