Statistical Power in Evaluations That Investigate Effects on Multiple Outcomes
A Guide for Researchers
In education research and in many other fields, researchers are often interested in testing the effectiveness of an intervention on multiple outcomes, for multiple subgroups, at multiple points in time, or across multiple treatment groups. The resulting multiplicity of statistical hypothesis tests can lead to spurious findings of effects. Multiple testing procedures (MTPs) are statistical procedures that counteract this problem by adjusting p-values for effect estimates upward. While MTPs are increasingly used in impact evaluations in education and other areas, an important consequence of their use is a change in statistical power that can be substantial. Unfortunately, researchers frequently ignore the power implications of MTPs when designing studies. Consequently, in some cases, sample sizes may be too small, and studies may be underpowered to detect effects as small as a desired size. In other cases, sample sizes may be larger than needed, or studies may be powered to detect smaller effects than anticipated.
Researchers typically worry that moving from one to multiple hypothesis tests and thus employing MTPs results in a loss of power. However, that need not always be the case. Power is indeed lost if one focuses on individual power: the probability of detecting an effect of a particular size or larger for each particular hypothesis test, given that the effect truly exists. However, in studies with multiplicity, alternative definitions of power exist that in some cases may be more appropriate. For example, when testing for effects on multiple outcomes, one might consider 1-minimal power: the probability of detecting effects of at least a particular size on at least one outcome. Similarly, one might consider ½-minimal power: the probability of detecting effects of at least a particular size on at least ½ of the outcomes. Also, one might consider complete power: the power to detect effects of at least a particular size on all outcomes. The choice of definition of power depends on the objectives of the study and on how the success of the intervention is defined. The choice of definition also affects the overall extent of power.
This paper presents methods for estimating statistical power, for multiple definitions of statistical power, when applying any of five common MTPs — Bonferroni, Holm, single-step and step-down versions of Westfall-Young, and Benjamini-Hochberg. The paper also presents empirical findings on how power is affected by the use of MTPs. To contain its scope, the paper focuses on multiplicity that results from estimating effects on multiple outcomes. The paper also focuses on the simplest research design and analysis plan that education studies typically use in practice: a multisite, randomized controlled trial (RCT) with the blocked randomization of individuals, in which effects are estimated using a model with block-specific intercepts and with the assumption of constant effects across blocks. However, the power estimation methods presented can easily be extended to other modeling assumptions and other study designs.