Measuring statistical evidence and multiple testing

Abstract

The measurement of statistical evidence is of considerable current interest in fields where statistical criteria are used to determine knowledge. The most commonly used approach to measuring such evidence is through the use of p-values, even though these are known to possess a number of properties that lead to doubts concerning their validity as measures of evidence. It is less well known that there are alternatives with the desired properties of a measure of statistical evidence. The measure of evidence given by the relative belief ratio is employed in this paper. A relative belief multiple testing algorithm was developed to control for false positives and false negatives through bounds on the evidence determined by measures of bias. The relative belief multiple testing algorithm was shown to be consistent and to possess an optimal property when considering the testing of a hypothesis randomly chosen from the collection of considered hypotheses. The relative belief multiple testing algorithm was applied to the problem of inducing sparsity. Priors were chosen via elicitation, and sparsity was induced only when justified by the evidence and there was no dependence on any particular form of a prior for this purpose.