# Multivariate Adjustments for Average Equivalence Testing

## TL;DR

This paper introduces a more powerful method for multivariate equivalence testing, addressing limitations of existing approaches when comparing multiple outcomes.

## Contribution

A finite-sample adjustment called multivariate α-TOST is proposed, improving power over conventional methods.

## Key findings

- The multivariate α-TOST is uniformly more powerful than the conventional multivariate TOST.
- The method accounts for arbitrary dependence between outcomes and adjusts the significance level accordingly.
- Simulation studies and a case study confirm the improved performance of the proposed method.

## Abstract

Multivariate (average) equivalence testing is widely used to assess whether the means of two conditions of interest are “equivalent” for different outcomes simultaneously. In pharmacological research for example, many regulatory agencies require the generic product and its brand‐name counterpart to have equivalent means both for the AUC and C
max pharmacokinetics parameters. The multivariate Two One‐Sided Tests (TOST) procedure is typically used in this context by checking if, outcome by outcome, the marginal 100(1−2α)% confidence intervals for the difference in means between the two conditions of interest lie within predefined lower and upper equivalence limits. This procedure, already known to be conservative in the univariate case, leads to a rapid power loss when the number of outcomes increases, especially when one or more outcome variances are relatively large. In this work, we propose a finite‐sample adjustment for this procedure, the multivariate α‐TOST, that consists in a correction of α, the significance level, taking the (arbitrary) dependence between the outcomes of interest into account and making it uniformly more powerful than the conventional multivariate TOST. We present an iterative algorithm allowing to efficiently define α*, the corrected significance level, a task that proves challenging in the multivariate setting due to the inter‐relationship between α* and the sets of values belonging to the null hypothesis space and defining the test size. We study the operating characteristics of the multivariate α‐TOST both theoretically and via an extensive simulation study considering cases relevant for real‐world analyses—that is, relatively small sample sizes, unknown and possibly heterogeneous variances as well as different correlation structures—and show the superior finite‐sample properties of the multivariate α‐TOST compared to its conventional counterpart. We finally re‐visit a case study on ticlopidine hydrochloride and compare both methods when simultaneously assessing bioequivalence for multiple pharmacokinetic parameters.