Analyzing unreplicated two-level factorial designs by combining multiple tests

There are several objective tests for analyzing unreplicated two-level factorial designs. However, there is no single test that can detect all patterns of possible active effects. Tests are sensitive to the number and/or the magnitude of active effects. Therefore, it is reasonable to combine recommended tests into a single test to provide researchers with a testing approach that leverages many existing methods to detect different patterns of active effects. The problem is how to combine multiple dependent tests into a single test. In this article, we review four methods for combining dependent tests and present four combined tests. In addition, we review four recommended object tests for detecting active effects. We also propose a new test procedure that can detect active effects when the number of active effects is large. We finally evaluate these nine tests (five original tests and four combined tests) in terms of controlling the type I error rate and the power performance via a simulation study. Simulation results show that the combined test that is based on the Jacobi polynomial expansion can be recommended as a test procedure to detect active effects.