In 2001, Hoenig and Heisey wrote an influential article, titled “The Abuse of Power: The Persuasive Fallacy of Power Calculations For Data Analysis.”  The article has been cited over 500 times and it is commonly cited as a reference to claim that it is a fallacy to use observed effect sizes to compute statistical power.

In this post, I provide a brief summary of Hoenig and Heisey’s argument. The summary shows that Hoenig and Heisey were concerned with the practice of assessing the statistical power of a single test based on the observed effect size for this effect. I agree that it is often not informative to do so (unless the result is power = .999). However, the article is often cited to suggest that the use of observed effect sizes in power calculations is fundamentally flawed. I show that this statement is false.

The abstract of the article makes it clear that Hoenig and Heisey focused on the estimation of power for a single statistical test. “There is also a large literature advocating that power calculations be made whenever one performs a statistical test of a hypothesis and one obtains a statistically nonsignificant result” (page 1). The abstract informs readers that this practice is fundamentally flawed. “This approach, which appears in various forms, is fundamentally flawed. We document that the problem is extensive and present arguments to demonstrate the flaw in the logic” (p. 1).

Given that method articles can be difficult to read, it is possible that the misinterpretation of Hoenig and Heisey is the result of relying on the term “fundamentally flawed” in the abstract. However, some passages in the article are also ambiguous. In the Introduction Hoenig and Heisey write “we describe the flaws in trying to use power calculations for data-analytic purposes” (p. 1). It is not clear what purposes are left for power calculations if they cannot be used for data-analytic purposes. Later on, they write more forcefully “A number of authors have noted that observed power may not be especially useful, but to our knowledge a fatal logical flaw has gone largely unnoticed.” (p. 2). So readers cannot be blamed entirely if they believed that calculations of observed power are fundamentally flawed. This conclusion is often implied in Hoenig and Heisey’s writing, which is influenced by their broader dislike of hypothesis testing  in general.

The main valid argument that Hoenig and Heisey make is that power analysis is based on the unknown population effect size and that effect sizes in a particular sample are contaminated with sampling error.  As p-values and power estimates depend on the observed effect size, they are also influenced by random sampling error.

In a special case, when true power is 50%, the p-value matches the significance criterion. If sampling error leads to an underestimation of the true effect size, the p-value will be non-significant and the power estimate will be less than 50%. When sampling error inflates the observed effect size, p-values will be significant and power will be above 50%.

It is therefore impossible to find scenarios where observed power is high (80%) and a result is not significant, p > .05, or where observed power is low (20%) and a result is significant, p < .05.  As a result, it is not possible to use observed power to decide whether a non-significant result was obtained because power was low or because power was high but the effect does not exist.

In fact, a simple mathematical formula can be used to transform p-values into observed power and vice versa (I actually got the idea of using p-values to estimate power from Hoenig and Heisey’s article).  Given this perfect dependence between the two statistics, observed power cannot add additional information to the interpretation of a p-value.

This central argument is valid and it does mean that it is inappropriate to use the observed effect size of a statistical test to draw inferences about the statistical power of a significance test for the same effect (N = 1). Similarly, one would not rely on a single data point to draw inferences about the mean of a population.

However, it is common practice to aggregate original data points or to aggregated effect sizes of multiple studies to obtain more precise estimates of the mean in a population or the mean effect size, respectively. Thus, the interesting question is whether Hoenig and Heisey’s (2001) article contains any arguments that would undermine the aggregation of power estimates to obtain an estimate of the typical power for a set of studies. The answer is no. Hoenig and Heisey do not consider a meta-analysis of observed power in their discussion and their discussion of observed power does not contain arguments that would undermine the validity of a meta-analysis of post-hoc power estimates.

A meta-analysis of observed power can be extremely useful to check whether researchers’ a priori power analysis provide reasonable estimates of the actual power of their studies.

Assume that researchers in a particular field have to demonstrate that their studies have 80% power to produce significant results when an important effect is present because conducting studies with less power would be a waste of resources (although some granting agencies require power analyses, these power analyses are rarely taken seriously, so I consider this a hypothetical example).

Assume that researchers comply and submit a priori power analysis with effect sizes that are considered to be sufficiently meaningful. For example, an effect of half-a-standard deviation (Cohen’s d = .50) might look reasonable large to be meaningful. Researchers submit their grant applications with a prior power analysis that produce 80% power with an effect size of d = .50. Based on the power analysis, researchers request funding for 128 participants. A researcher plans four studies and needs $50 for each participant. The total budget is $25,600.

When the research project is completed, all four studies produced non-significant results. The observed standardized effect sizes were 0, .20, .25, and .15. Is it really impossible to estimate the realized power in these studies based on the observed effect sizes? No. It is common practice to conduct a meta-analysis of observed effect sizes to get a better estimate of the (average) population effect size. In this example, the average effect size across the four studies is d = .15. It is also possible to show that the average effect size in these four studies is significantly different from the effect size that was used for the a priori power calculation (M1 = .15, M2 = .50, Mdiff = .35, SE = 1/sqrt(512) = .044, t = .35 / .044 = 7.92, p < 1e-13). Using the more realistic effect size estimate that is based on actual empirical data rather than wishful thinking, the post-hoc power analysis yields a power estimate of 13%. The probability of obtaining non-significant results in all four studies is 57%. Thus, it is not surprising that the studies produced non-significant results.  In this example, a post-hoc power analysis with observed effect sizes provides valuable information about the planning of future studies in this line of research. Either effect sizes of this magnitude are not important enough and research should be abandoned or effect sizes of this magnitude still have important practical implications and future studies should be planned on the basis of a priori power analysis with more realistic effect sizes.

Another valuable application of observed power analysis is the detection of publication bias and questionable research practices (Ioannidis and Trikalinos; 2007), Schimmack, 2012) and for estimating the replicability of statistical results published in scientific journals (Schimmack, 2015).

In conclusion, the article by Hoenig and Heisey is often used as a reference to argue that observed effect sizes should not be used for power analysis.  This post clarifies that this practice is not meaningful for a single statistical test, but that it can be done for larger samples of studies.