Post Hoc Power: A US Researcher's Guide

In the realm of statistical analysis, post hoc statistical power plays a crucial role in interpreting non-significant findings, particularly within disciplines heavily reliant on null hypothesis significance testing, such as psychology and medicine. Calculating post hoc statistical power, a practice debated among researchers at institutions like the National Institutes of Health (NIH), becomes essential after failing to reject the null hypothesis, where tools such as GPower aid in determining the probability of detecting an effect if one truly exists. The interpretation of post hoc statistical power* is especially pertinent for US researchers striving to draw meaningful conclusions from their data, despite the limitations and controversies surrounding its application, as highlighted by statisticians like Jacob Cohen.

Debunking the Myth of Post Hoc Power: Why It's Meaningless

In the realm of research design and statistical inference, statistical power plays a crucial role. It is the probability that a study will detect a statistically significant effect when a true effect exists. This a priori calculation is fundamental for planning studies with adequate sensitivity.

However, a controversial practice has emerged: the calculation of post hoc power, also known as observed or retrospective power. This involves estimating power after a study has been completed, typically based on the observed effect size and sample size.

The central argument of this piece is that post hoc power is not only uninformative but also fundamentally misleading. It provides no meaningful additional information beyond the already available p-value.

Instead, it perpetuates a flawed understanding of statistical inference.

Defining Statistical Power: A Crucial Concept

Statistical power is a critical component of sound research. It is the probability of rejecting the null hypothesis when it is, in fact, false.

In simpler terms, it is the ability of a study to find a real effect if that effect truly exists.

Power is directly influenced by several factors: the effect size (magnitude of the effect), the sample size (number of participants or observations), and the significance level (alpha, often set at 0.05).

A well-powered study minimizes the risk of a Type II error, or a false negative, where a real effect is missed.

Post Hoc Power: Definition and Misconceptions

Post hoc power is calculated after data collection. It uses the observed effect size from the study, along with the sample size and alpha level, to estimate the power of the test.

The intention behind post hoc power is often to justify non-significant results. For example, a researcher might argue that a non-significant finding is due to low power, rather than the absence of a true effect.

This interpretation is precisely where the problem lies.

The calculation of post hoc power is inherently tied to the p-value.

The Central Argument: Invalidity and Misleading Nature

Post hoc power offers no additional insight beyond what the p-value already conveys. It is mathematically derived from the p-value and sample size. Thus, it is simply a re-expression of existing information.

A high post hoc power associated with a non-significant result does not make the result any more meaningful. It also does not increase the likelihood that the null hypothesis is false.

The observed effect size is already reflected in the p-value and the confidence interval. These measures provide a more nuanced and informative assessment of the study's findings.

Focusing on post hoc power distracts from more critical considerations such as the practical significance of the observed effect. It also diverts attention from the limitations of the study design.

Structure of the Discussion

The following sections will delve deeper into the flaws of post hoc power. They will further clarify its origins, and offer superior alternatives for interpreting research results.

We will examine the historical context, explore why researchers use this problematic measure, and highlight the institutional perspectives that discourage its use.

Ultimately, this analysis aims to promote a more rigorous and insightful approach to statistical inference, one that moves beyond the misleading allure of post hoc power.

The Fatal Flaw: Why Post Hoc Power is Circular

Moving beyond the basic definition, we must confront the core issue that renders post hoc power fundamentally flawed: its inherent circularity. The calculation's dependency on the p-value and sample size transforms it into a redundant exercise that provides no fresh perspective on research findings. It merely re-expresses data already available in a different form.

Understanding Statistical Power: A Recap

Before delving into the specifics of circularity, it's crucial to reiterate the core components of statistical power. Statistical power is the probability of correctly rejecting a false null hypothesis. It is directly influenced by three key factors: the effect size (the magnitude of the observed effect), the sample size (the number of observations in the study), and the alpha level (the probability of making a Type I error, typically set at 0.05).

These elements are intricately linked, and understanding their relationship is essential for grasping the futility of post hoc power.

The Inherent Circularity of Post Hoc Power

The central problem with post hoc power lies in its derivation. Post hoc power is not an independent measure; it's directly calculated from the p-value and the sample size obtained in the study.

The P-Value and Sample Size Connection

Consider the formula typically used in calculating observed power; it implicitly incorporates the sample effect size and variability through its reliance on the p-value. A smaller p-value (stronger evidence against the null hypothesis) will mechanically result in a higher post hoc power.

Similarly, a larger sample size, all else being equal, will increase power. This interdependence reveals that post hoc power doesn't offer any novel insights beyond what the p-value and sample size already convey.

A Redundant Measure

The mathematical relationship that ties post hoc power to the p-value renders it a redundant measure. If you know the p-value and the sample size, you essentially know the post hoc power. Calculating it is akin to converting kilometers to miles, gaining no new fundamental information in the process. The circularity is the core reason that statisticians argue against its validity and use.

Misinterpreting High Post Hoc Power

A common misconception is that a high post hoc power somehow validates non-significant results. This is simply incorrect. A non-significant p-value remains non-significant, regardless of the calculated post hoc power. High post hoc power in a non-significant result merely indicates that the study was adequately powered to detect the observed effect size, which, unfortunately, was not large enough to achieve statistical significance given the sample size and variability.

It does not mean that a real effect exists that the study failed to detect. The reality is that the experiment failed to meet the criteria of statistical significance, whether that is due to noise within the data, a small sample size, or even a small effect size.

Effect Size Reflected in P-Values and Confidence Intervals

The observed effect size is already incorporated into the p-value and, more importantly, is explicitly quantified by the confidence interval. The confidence interval provides a range of plausible values for the true population effect size. Focusing on the confidence interval allows researchers to assess the precision of their estimate and to determine whether the range of plausible values is practically meaningful, regardless of the p-value.

Addressing Type II Errors

Type II errors (false negatives) occur when a study fails to reject a false null hypothesis. While understanding the risk of Type II errors is crucial, post hoc power does not mitigate this risk in future studies. Calculating post hoc power after a non-significant result doesn't change the fact that the study failed to detect an effect.

Future studies require proper a priori power analysis based on a realistic estimate of the effect size, not on the observed effect size from a failed study.

Historical Context: Cohen and the Origins of Power Analysis

Building on the understanding of post hoc power's limitations, a deeper exploration into the origins of power analysis clarifies its intended purpose. The concept, pioneered by Jacob Cohen, was designed as an a priori tool to guide research design—a far cry from its misapplication as a post hoc justification. Furthermore, critiques from figures like John Ioannidis regarding the over-reliance on p-values shed light on the broader context of statistical interpretation and its pitfalls.

Jacob Cohen and A Priori Power Analysis

Jacob Cohen's work laid the foundation for modern power analysis. His seminal texts provided researchers with a framework for determining the necessary sample size to detect a statistically significant effect, before conducting a study.

Cohen emphasized the critical interplay between sample size, effect size, alpha level (significance level), and statistical power. His approach was fundamentally prospective, aimed at optimizing study design and resource allocation.

He provided guidelines for estimating effect sizes based on prior research or theoretical expectations, enabling researchers to proactively plan for sufficient statistical power. This a priori approach is crucial for ensuring that studies are adequately powered to detect meaningful effects, minimizing the risk of Type II errors (false negatives).

Misinterpreting Cohen: Post Hoc Power is Not the Answer

Despite Cohen's clear focus on a priori planning, the concept of power analysis has been twisted into post hoc power calculations. This practice is a misinterpretation of Cohen's framework. Cohen himself never advocated for calculating power after a study is completed.

Post hoc power analysis attempts to retroactively justify non-significant results. It offers no meaningful insights into the study's design or the validity of its findings. Instead, it simply repackages the information already contained in the p-value and sample size.

The P-Value Problem: Broader Critiques of Statistical Significance

The misuse of post hoc power is symptomatic of a broader issue: the over-reliance on p-values as the sole criterion for evaluating research findings. John Ioannidis, among others, has highlighted the limitations and potential biases associated with p-value-centric research.

Ioannidis's work has exposed the alarming rate of false positives in scientific literature. He argues that factors such as publication bias, selective reporting, and flawed study designs contribute to an inflated number of statistically significant findings that may not be replicable or generalizable.

The obsession with achieving statistical significance can lead researchers to engage in questionable research practices, such as p-hacking or HARKing (Hypothesizing After the Results are Known), further undermining the integrity of scientific research.

Recognizing the flaws in relying solely on p-values and embracing comprehensive methods is critical to progressing from inadequate research practices.

Beyond Power: Better Alternatives for Interpretation

Having established the fundamental flaws of post hoc power, it's crucial to explore superior methodologies that offer a more insightful and accurate interpretation of research findings. These alternatives move beyond the limitations of simply recalculating power after the fact, providing richer context and more actionable information for researchers.

The Primacy of Confidence Intervals

One of the most valuable tools for interpreting effect sizes is the confidence interval. Unlike post hoc power, which offers little more than a restatement of the p-value, confidence intervals provide a range of plausible values for the true effect in the population.

Assessing Precision with Confidence Intervals

A narrow confidence interval indicates a precise estimate, suggesting that the sample statistic is likely close to the true population parameter. Conversely, a wide confidence interval implies greater uncertainty, reflecting a less precise estimate. Researchers should focus on the width of the confidence interval as a measure of the reliability of their findings.

Interpreting the Range of Plausible Values

The confidence interval allows researchers to assess the practical significance of their results. Even if a p-value is non-significant, a confidence interval can reveal whether the effect size, though not statistically significant, might still be meaningful in a real-world context. The lower and upper bounds of the interval provide a range of plausible effect sizes, enabling a more nuanced interpretation than a simple binary (significant/non-significant) decision.

Bayesian Statistics: Quantifying Evidence

Bayesian statistics offer a fundamentally different approach to statistical inference. Instead of focusing on p-values and null hypothesis significance testing, Bayesian methods provide a direct quantification of the evidence for or against a hypothesis. By incorporating prior beliefs and updating them with observed data, Bayesian analysis yields posterior probabilities that are often more intuitive and easier to interpret than frequentist p-values. While a full discussion is beyond our scope, understanding its availability is essential.

Equivalence Testing: Demonstrating Practical Insignificance

While traditional hypothesis testing focuses on rejecting the null hypothesis of no effect, equivalence testing aims to demonstrate that an effect is practically equivalent to zero. This approach is particularly useful when researchers want to show that a treatment or intervention has no meaningful impact. Equivalence testing defines a range of "equivalence" and assesses whether the observed effect falls within that range. If it does, the researcher can conclude that the effect is practically insignificant.

Benefits of Alternatives

Compared to post hoc power, confidence intervals, Bayesian statistics, and equivalence testing offer several significant advantages.

• They provide more informative measures of uncertainty and effect size.
• They facilitate a more nuanced interpretation of research findings.
• They encourage a shift away from the problematic reliance on p-values.

By embracing these alternative approaches, researchers can move towards a more rigorous and meaningful understanding of their data.

Addressing the Root Cause: Why Researchers Use Post Hoc Power

Having established the fundamental flaws of post hoc power, it's crucial to explore the underlying motivations driving its persistent use.

Understanding why researchers gravitate towards this problematic metric is essential for promoting better statistical practices and fostering more rigorous scientific inquiry.

The Allure of Post Hoc Power: Justification and Planning

Researchers often turn to post hoc power calculations for two primary reasons: justifying non-significant results and assessing the need for larger sample sizes in future studies.

The pressure to publish statistically significant findings can be intense, leading some researchers to seek ways to bolster seemingly weak results.

Post hoc power, despite its inherent limitations, can appear to offer a lifeline in these situations.

Furthermore, when a study fails to achieve statistical significance, researchers may wonder if a larger sample size would have yielded different results.

Post hoc power is sometimes used to estimate the sample size needed for a future study, but, as we have discussed, this is problematic.

Justifying Non-Significant Results to Reviewers and Editors

One of the most common motivations for using post hoc power is to appease reviewers or editors who question the validity of non-significant findings.

In the current academic climate, where publication metrics heavily influence career advancement, the pressure to produce statistically significant results can be overwhelming.

Researchers may feel compelled to demonstrate that their study had "sufficient power" to detect an effect, even if the observed effect was small or non-existent.

This can lead to the misinterpretation of post hoc power as a validation tool, when in reality, it provides no additional information beyond the p-value.

The temptation to use post hoc power as a shield against criticism highlights a deeper issue: the overemphasis on statistical significance at the expense of other important considerations, such as effect size and practical significance.

Assessing the Need for a Larger Sample Size

Another common reason for calculating post hoc power is to determine whether a future study with a larger sample size is warranted.

When a study yields non-significant results, researchers may question whether the lack of power contributed to the failure to detect a true effect.

However, using post hoc power to justify increasing the sample size in future studies is misguided.

Post hoc power is calculated using the observed effect size, which may be an unstable estimate of the true population effect.

Using this unstable estimate to plan future studies can lead to inaccurate power calculations and potentially wasteful allocation of resources.

Better Strategies: Focusing on Practical Significance

Instead of relying on post hoc power, researchers should focus on the practical significance of their findings.

Practical significance refers to the real-world importance or relevance of an effect, regardless of its statistical significance.

Even if a result is not statistically significant, it may still have practical implications.

For example, a small effect size may be meaningful in certain contexts, such as when the intervention is low-cost or easily implemented.

Researchers should carefully consider the potential implications of their findings, regardless of the p-value.

Designing Future Studies with A Priori Power Analysis

When planning future studies, researchers should conduct a well-designed a priori power analysis.

A priori power analysis involves estimating the sample size needed to detect an effect of a given size, based on prior research or theoretical considerations.

Unlike post hoc power, a priori power analysis is conducted before the study begins, allowing researchers to make informed decisions about sample size and other design parameters.

By focusing on a priori power analysis, researchers can increase the likelihood of detecting a true effect, if one exists, and avoid wasting resources on underpowered studies.

This approach requires careful consideration of the expected effect size, the desired level of statistical power, and the chosen significance level.

By addressing the underlying motivations for using post hoc power and providing better strategies, we can promote more rigorous and informative statistical practices in scientific research.

So, next time you're staring down a non-significant result, remember that post hoc statistical power isn't your magic bullet. It's more like a flashlight in a dark room – helpful for seeing what might have been, but not a substitute for thoughtful study design and a sufficiently large sample size in the first place. Good luck with your research!