Optimal Multiple Testing Procedure Under Linear Models
Thursday, January 17, 2013 - 3:30pm
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This paper considers the problem of optimal false discovery rate control under the linear model $Y = X \beta + \epsilon$ , where $\epsilon \sim N(0, sigma^2 I ) $. It is an extension of the normal mean model with arbitrary dependence. To solve the problem, we first propose an adjusted z-surrogate which simplifies the original data and captures the useful information. We show that many commonly used surrogates based on univariate associations are biased and inefficient. We then propose a testing procedure based on the adjusted z-surrogate so that the entire testing method is optimal, in the sense that, it can control mfdr level and minimize mfnr level asymptotically among all the methods based on the original data. Numerical results show that our method is able to control mfdr under linear models with correlated predictors while other methods fail to control the error rate. The proposed method also leads to a small mfnr. The procedure is further illustrated by an application to a genome-wide association study of blood pressure.