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This article develops $p$-values for evaluating means of normal populationsthat make use of indirect or prior information. A $p$-value of this type isbased on a biased test statistic that is optimal on average with respect to aprobability distribution that encodes indirect information about the meanparameter, resulting in a smaller $p$-value if the indirect information isaccurate. In a variety of multiparameter settings, we show how to adaptivelyestimate the indirect information for each mean parameter while stillmaintaining uniformity of the $p$-values under their null hypotheses. This isdone using a linking model through which indirect information about the mean ofone population may be obtained from the data of other populations. Importantly,the linking model does not need to be correct to maintain the uniformity of the$p$-values under their null hypotheses. This methodology is illustrated inseveral data analysis scenarios, including small area inference, spatiallyarranged populations, interactions in linear regression, and generalized linearmodels.

Smaller $p$-values via indirect information