On the Post Hoc Power in Testing Mean Differences

Abstract

Retrospective or post hoc power analysis is recommended by reviewers and editors of many journals. Little literature has been found that gave a serious study of the post hoc power. When the sample size is large, the observed effect size is a good estimator of the true effect size. One would hope that the post hoc power is also a good estimator of the true power. This article studies whether such a power estimator provides valuable information about the true power. Using analytical, numerical, and Monte Carlo approaches, our results show that the estimated power does not provide useful information when the true power is small. It is almost always a biased estimator of the true power. The bias can be negative or positive. Large sample size alone does not guarantee the post hoc power to be a good estimator of the true power. Actually, when the population variance is known, the cumulative distribution function of the post hoc power is solely a function of the population power. This distribution is uniform when the true power equals 0.5 and highly skewed when the true power is near 0 or 1. When the population variance is unknown, the post hoc power behaves essentially the same as when the variance is known.