Learning from a large number of chi-squared tests
George Mason University
Date: Friday, October 1, 2021
Location: JC Gold Room
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Efron (2011) investigated the merit and limitation of an empirical Bayes method to correct selection bias based on Tweedie's formula first reported in Robbins (1956). The exceptional virtue of Tweedie's formula for the normal distribution lies in its representation of selection bias as a simple function of the derivative of log marginal likelihood. Since the marginal likelihood and its derivative can be estimated from the data directly without specifying the prior distribution, bias correction can be carried out conveniently. We propose a Bayesian hierarchical model for chi-squared data such that the resulting Tweedie's formula has the same virtue as that of the normal distribution. Because the family of noncentral chi-squared distributions, the common alternative distributions for chi-squared tests, does not constitute an exponential family, our results cannot be obtained by extending existing results. Furthermore, the corresponding Tweedie's formula manifests new phenomena quite different from those of the normal data and suggests new ways to analyse chi-square data. Two real-data examples are discussed: gene expression difference among ethnic groups and higher-order interaction of gene expression in breast cancer metastasis. This is joint work with Lilun Du.