Upper bounds for the number of true null hypotheses and novel
estimates for error rates in multiple testing
Nicolai Meinshausen and Peter Bühlmann
February 2004
Abstract
When testing multiple hypotheses simultaneously, a quantity of interest is the
number m0 of true null hypotheses.
We present a general framework for finding upper probabilistic bounds for
m0, that is estimates \mhat_0 with the property
P[\mhat_0 > m_0] > 1-alpha for any chosen level alpha. A conservative,
one-sided (1 - alpha) confidence interval for m0 is
then given by [0,\mhat_0]. Moreover, \mhat_0 can be used for novel
estimates of type I errors in multiple testing such as the false discovery rate.
Control of the family-wise error rate emerges as a special case in our
framework but suffers from vanishing power for a large number of tested
hypotheses. We present a different estimate such that the ability to detect
true non-null hypotheses increases with the number of tested hypotheses. A
detailed algorithm is provided. The method is valid under general and unknown
dependence between the test statistics.
We develop the method primarily for multiple testing of associations
between random variables. The method is illustrated with simulation
studies and applications to microarray data.
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