In a factorial ANOVA there may be many effects estimated. Should I use a correction to control the Type I error rate?
This discussion assumes that the data were analyzed using effect coding (as opposed to dummy coding), and that the individual experimental conditions n’s are at least approximately equal across experimental conditions.
In our view, whether to use a correction to control the Type I error rate depends largely on whether you are operating from a conclusion-priority or decision-priority perspective (see Chapter 3 in Collins, 2018). If your purpose is classical null hypothesis testing for the purpose of drawing scientific conclusions about main effects and interactions, you are probably operating from a conclusion-priority perspective. In this case you may wish to use a correction.
Corrections to control the Type I error rate may be particularly important when effect estimates are correlated. It may be helpful to note that whereas dummy coding typically produces substantially correlated effect estimates, effect coding produces estimates that are uncorrelated in the case of equal n’s, or moderately correlated unless the n’s vary widely.
If your purpose is deciding on which components and component levels to select for inclusion in the optimized intervention, you are probably operating from a decision-priority perspective. Here we do not recommend a correction, for two reasons. First, we recommend either not using classical hypothesis-testing at all when working from a decision-priority perspective, or using it as a guiding framework rather than a strict set of rules. (In fact, a Bayesian approach would be more consistent with the decision-priority perspective; we are working on such an approach.) Second, corrections to control the Type I error rate can increase the Type II error rate to unacceptable levels.