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Distributional Property of the Generalized p-value for the Behrens-Fisher Problem with Applications to Multiple Testing

The generalized p-value method introduced by Tsui and Weerahandi (1989) has been used to successfully provide finite sample solutions for many hypothesis testing problems when no solutions are available using the usual approach. Although simulation studies show that generalized p-values have similar distributional properties as ordinary p-values, no one has yet developed theoretical properties of generalized p-values. This paper provides some theoretical distributional properties of the generalized p-value for the Behrens-Fisher problem of testing the difference of two independent normal distribution means with possibly unequal distributional variances. We show why the probability that the generalized p-value for the Behrens-Fisher problem is less than a given value r in (0, 0.5), is approximately r. We use this theoretical result to obtain a multiple testing procedure based on generalized p-values, which controls the false discovery rate (FDR). In particular, we introduce a theoretically valid testing procedure that controls FDR in simultaneously testing several Behrens-Fisher problems. Our procedure to control FDR is an extension of the important work of Benjamini and Hochberg (1995), which is based on ordinary p-values. Our proof is constructive and concise and it shows clearly the essential step that makes the FDR result holds. It should be useful for development of further results. Data from a microarray experiment are used to illustrate our method.