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Asymptotic Distribution of P Values in Composite Null Models

James M. Robins, Aad van der Vaart and Valerie Ventura
Journal of the American Statistical Association
Vol. 95, No. 452 (Dec., 2000), pp. 1143-1156
DOI: 10.2307/2669750
Stable URL: http://www.jstor.org/stable/2669750
Page Count: 14
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Asymptotic Distribution of P Values in Composite Null Models
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Abstract

We investigate the compatibility of a null model H0 with the data by calculating a p value; that is, the probability, under H0, that a given test statistic T exceeds its observed value. When the null model consists of a single distribution, the p value is readily obtained, and it has a uniform distribution under H0. On the other hand, when the null model depends on an unknown nuisance parameter θ, one must somehow get rid of θ, (e.g., by estimating it) to calculate a p value. Various proposals have been suggested to "remove" θ, each yielding a different candidate p value. But unlike the simple case, these p values typically are not uniformly distributed under the null model. In this article we investigate their asymptotic distribution under H0. We show that when the asymptotic mean of the test statistic T depends on θ, the posterior predictive p value of Guttman and Rubin, and the plug-in p value are conservative (i.e., their asymptotic distributions are more concentrated around 1/2 than a uniform), with the posterior predictive p value being the more conservative. In contrast, the partial posterior predictive and conditional predictive p values of Bayarri and Berger are asymptotically uniform. Furthermore, we show that the discrepancy p value of Meng and Gelman and colleagues can be conservative, even when the discrepancy measure has mean 0 under the null model. We also describe ways to modify the conservative p values to make their distributions asymptotically uniform.

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