Simultaneous directional inference

TL;DR

This paper introduces a method for providing simultaneous post-hoc confidence bounds on the number of positive and negative parameters, improving inference accuracy in multiple hypothesis testing scenarios.

Contribution

It proposes a new approach for simultaneous inference on parameter signs using adjusted p-values, offering tighter bounds and computational efficiency compared to existing methods.

Findings

Bounds are often significantly tighter than existing methods.

Method is computationally feasible with polynomial time complexity.

Useful in evaluating treatment effects across multiple studies or subgroups.

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.