A Method for Testing Additivity in Unreplicated Two‐Way Layouts Based on Combining Multiple Interaction Tests

There are several non‐functional‐form‐based interaction tests for testing additivity in unreplicated two‐way layouts. The distinctness of the various tests in this setting is somewhat unfortunate, as no single test can detect all patterns of possible interaction, and the tests are sensitive to a particular pattern of interaction. Therefore, it is reasonable to combine multiple interaction tests for testing additivity. The main problem is how we can combine multiple dependent tests into a single test such that it has an acceptable performance in terms of controlling the type I error rate and maximizing the power. In this paper, we review six methods for combining interaction tests which utilize six different non‐functional‐form‐based interaction tests. We also propose two novel combination methods based on the Jacobi polynomial expansion and the Gaussian copula. We first compare these eight combination methods in terms of controlling the type I error rate via a simulation study. We then compare the six interaction tests and four combination methods in terms of power. The simulation results show that our proposed combination method that is based on the Jacobi polynomial expansion has the best performance in terms of controlling the type I error rate and maximizing the power. Therefore, we recommend this combination method as a test procedure for testing additivity. The major contribution of this work is to provide researchers with a testing approach which leverages many existing methods to detect disparate forms of non‐additivity. This alleviates the need for researchers to naively choose among highly specialized methods by instead allowing for a general approach that captures a number of interaction effects with reasonable power.