Testing a Point Null Hypothesis: The Irreconcilability of P Values and Evidence

The problem of testing a point null hypothesis (or a "small interval" null hypothesis) isconsidered. Of interest is the relationship between the P value (or observed significance level) and conditional nd Bayesian measures of evidence against he null hypothesis. Although one might presume that a small P value indicates the presence of strong evidence against he null, such is not necessarily the case. Expanding on earlier work [especiafly Edwards, Lindman, and Savage (1963) and Dickey (1977)], it is shown that actual evidence against a null (as measured, say, by posterior probability or comparative likelihood) can differ by an order of magnitude from the P value. For instance, data that yield a P value of .05, when testing a normal mean, result in a posterior probability of the null of at least .30 for any objective prior distribution. ("Objective" here means that equal prior weight is given the two hypotheses and that the prior is symmetric and nonincreasing away from the null; other definitions of"objective" will be seen to yield qualitatively similar esults.) The overall conclusion is that P values can be highly misleading measures of the evidence provided by the data against he null hypothesis.