Inconsistent multiple testing corrections: The fallacy of using family-based error rates to make inferences about individual hypotheses

During multiple testing, researchers often adjust their alpha level to control the familywise error rate for a statistical inference about a joint union alternative hypothesis (e.g., “H_1,1 or H_1,2”). However, in some cases, they do not make this inference. Instead, they make separate inferences about each of the individual hypotheses that comprise the joint hypothesis (e.g., H_1,1 and H_1,2). For example, a researcher might use a Bonferroni correction to adjust their alpha level from the conventional level of 0.050 to 0.025 when testing H_1,1 and H_1,2, find a significant result for H_1,1 (p < 0.025) and not for H_1,2 (p > 0.025), and so claim support for H_1,1 and not for H_1,2. However, these separate individual inferences do not require an alpha adjustment. Only a statistical inference about the union alternative hypothesis “H_1,1 or H_1,2” requires an alpha adjustment because it is based on “at least one” significant result among the two tests, and so it refers to the familywise error rate. Hence, an inconsistent correction occurs when a researcher corrects their alpha level during multiple testing but does not make an inference about a union alternative hypothesis. In the present article, I discuss this inconsistent correction problem, including its reduction in statistical power for tests of individual hypotheses and its potential causes vis-à-vis error rate confusions and the alpha adjustment ritual. I also provide three illustrations of inconsistent corrections from recent psychology studies. I conclude that inconsistent corrections represent a symptom of statisticism, and I call for a more nuanced inference-based approach to multiple testing corrections.