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arXiv · 2301.01653

Simultaneous directional inference

Ruth Heller·Aldo Solari
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Abstract

We consider the problem of inference on the signs of $n>1$ parameters. We aim to provide $1-\alpha$ post-hoc confidence bounds on the number of positive and negative (or non-positive) 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 one-sided hypotheses for the selection, and use the adjusted $p$-values for simultaneous inference on the selected $n$ one-sided hypotheses. The adjustment is straightforward assuming that the $p$-values of one-sided 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 post-hoc 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 non-beneficial), 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.

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Ruth Heller, Aldo Solari. 2023-01-04. Simultaneous directional inference. https://doi.org/10.1093/jrsssb%2Fqkad137

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