The Use of Multiplicity Corrections, Order Statistics and Generalized Family-Wise Statistics with Application to Genome-Wide Studies

The most important decision faced by large-scale studies, such as those presently encountered in human genetics, is to distinguish between those tests that are true positives from those that are not. In the context of genetics, this entails the determination of genetic markers that actually underlie medically-relevant phenotypes from a vast number of makers typically interrogated in genome-wide studies. A critical part of these decisions relies on the appropriate statistical assessment of data obtained from tests across numerous markers. Several methods have been developed to aid with such analyses, with family-wise approaches, such as the Bonferroni and Dunn-Šidàk corrections, being popular. Conditions that motivate the use of family-wise corrections are explored. Although simple to implement, one major limitation of these approaches is that they assume that p-values are i.i.d. uniformly distributed under the null hypothesis. However, several factors may violate this assumption in genome-wide studies including effects from confounding by population stratification, the presence of related individuals, the correlational structure among genetic markers, and the use of limiting distributions for test statistics. Even after adjustment for such effects, the distribution of p-values can substantially depart from a uniform distribution under the null hypothesis. In this work, I present a decision theory for the use of family-wise corrections for multiplicity and a generalization of the Dunn-Šidàk correction that relaxes the assumption of uniformly-distributed null p-values. The independence assumption is also relaxed and handled through calculating the effective number of independent tests. I also explicitly show the relationship between order statistics and family-wise correction procedures. This generalization may be applicable to multiplicity problems outside of genomics.