The ranking probability approach and its usage in design and analysis of large-scale studies

Abstract

In experiments with many statistical tests there is need to balance type I and type II error rates while taking multiplicity into account. In the traditional approach, the nominal α-level such as 0.05 is adjusted by the number of tests, K , i.e., as 0.05/K. Assuming that some proportion of tests represent "true signals", that is, originate from a scenario where the null hypothesis is false, power depends on the number of true signals and the respective distribution of effect sizes. One way to define power is for it to be the probability of making at least one correct rejection at the assumed α-level. We advocate an alternative way of establishing how "well-powered" a study is. In our approach, useful for studies with multiple tests, the ranking probability Px,u is controlled, defined as the probability of making at least x correct rejections while rejecting hypotheses with u smallest P-values. The two approaches are statistically related. Probability that the smallest P-value is a true signal (i.e., P 1,1 ) is equal to the power at the level 0:5=K , to an excellent approximation. Ranking probabilities are also related to the false discovery rate and to the Bayesian posterior probability of the null hypothesis. We study properties of our approach when the effect size distribution is replaced for convenience by a single "typical" value taken to be the mean of the underlying distribution. We conclude that its performance is often satisfactory under this simplification; however, substantial imprecision is to be expected when K is very large and u is small. Precision is largely restored when three values with the respective abundances are used instead of a single typical effect size value.