Simultaneous directional inference
Abstract
We consider the problem of inference on the signs of $n>1$ parameters. We aim to provide $1\alpha$ posthoc confidence bounds on the number of positive and negative (or nonpositive) parameters. The guarantee is simultaneous, for all subsets of parameters. Our suggestion is as follows: start by using the data to select the direction of the hypothesis test for each parameter; then, adjust the $p$values of the onesided hypotheses for the selection, and use the adjusted $p$values for simultaneous inference on the selected $n$ onesided hypotheses. The adjustment is straightforward assuming that the $p$values of onesided hypotheses have densities with monotone likelihood ratio, and are mutually independent. We show that the bounds we provide are tighter (often by a great margin) than existing alternatives, and that they can be obtained by at most a polynomial time. We demonstrate the usefulness of our simultaneous posthoc bounds in the evaluation of treatment effects across studies or subgroups. Specifically, we provide a tight lower bound on the number of studies which are beneficial, as well as on the number of studies which are harmful (or nonbeneficial), and in addition conclude on the effect direction of individual studies, while guaranteeing that the probability of at least one wrong inference is at most 0.05.
 Publication:

arXiv eprints
 Pub Date:
 January 2023
 DOI:
 10.48550/arXiv.2301.01653
 arXiv:
 arXiv:2301.01653
 Bibcode:
 2023arXiv230101653H
 Keywords:

 Statistics  Methodology
 EPrint:
 59 pages, 11 figures, 7 tables