# Multivariate one-sided testing in matched observational studies as an adversarial game

## TL;DR

This paper introduces a multivariate one-sided sensitivity analysis method for matched observational studies, framing it as an adversarial game to optimize detection of causal effects across multiple outcomes with known directions.

## Contribution

It develops a convex optimization-based test statistic that maximizes design sensitivity and allows for familywise error control in one-sided sensitivity analyses.

## Key findings

- Test statistic converges to a chi-bar-squared distribution asymptotically.
- Achieves the largest possible design sensitivity among coherent tests.
- Enables one-sided sensitivity analysis with error control.

## Abstract

We present a multivariate one-sided sensitivity analysis for matched observational studies, appropriate when the researcher has specified that a given causal mechanism should manifest itself in effects on multiple outcome variables in a known direction. The test statistic can be thought of as the solution to an adversarial game, where the researcher determines the best linear combination of test statistics to combat nature's presentation of the worst-case pattern of hidden bias. The corresponding optimization problem is convex, and can be solved efficiently even for reasonably sized observational studies. Asymptotically the test statistic converges to a chi-bar-squared distribution under the null, a common distribution in order restricted statistical inference. The test attains the largest possible design sensitivity over a class of coherent test statistics, and facilitates one-sided sensitivity analyses for individual outcome variables while maintaining familywise error control through is incorporation into closed testing procedures.