Simultaneous Testing of Multiple Hypotheses Using Generalized pvalues KamWah Tsui and Shijie Tang University of Wisconsin, Madison Abstract
For problems of simultaneously testing many
hypotheses, Benjamini and Hochberg (1995) propose a procedure that
guarantees the False Discovery Rate (FDR) to be less than or equal to a
prefixed value. Here, FDR is the expected value of the ratio of the
number of incorrectly rejected hypotheses and the total number of
rejected hypotheses with the ratio defined to be zero if no hypothesis
is rejected. Lehmann and Romano (2005) give a comprehensive overview of
the development of the research in this area of multiple testing. To
our knowledge, the research in this area assumes that a usual pvalue
is available for each hypothesis. However, in circumstances when
nuisance parameters are present, usual pvalues that are free of
nuisance parameters may not exist. Tsui and Weerahandi (1989) introduce
the generalized pvalue method to handle testing problems where
nuisance parameters are present. In this paper, we describe a multiple
testing procedures using generalized pvalues with the FDR not
exceeding a prefixed value. Using an upper bound result from Tang and
Tsui (2007) for the probability of type I error using a generalized
pvalue for the BehrensFisher problem, we develop a procedure to
control FDR for the problem of simultaneously testing many
BehrensFisher problems. Finally, we use a data set from a microarray
experiment to illustrate our multiple testing procedure based on
generalized pvalues.
