The true effect size remains unknown even after the mean difference between the two groups is observed. Yet, the study provides some useful information about the effect size. Whereas the a priori power analysis relied exclusively on guess-work, observed power uses the effect size that was observed in a reasonably large sample of participants.
Everything else being equal, effect size estimates based on participants are more likely to match the true effect size than those based on 0 participants. The observed effect size can be entered into a power analysis to compute observed power. One question examined by Yuan and Maxwell is whether it can be useful to compute observed power after a study produced a significant result.
The other question is whether it can be useful to compute observed power when a study produced a non-significant result. The actual mean difference that is observed after the study happens to match the true effect size exactly. A non-significant result is difficult to interpret. On the one hand, the means trend in the right direction. On the other hand, the mean difference is not statistically significant. The p-value suggests that a mean difference of this magnitude would occur in every second study by chance alone even if music intervention had no effect on IQ at all i.
Statistically, the correct conclusion is that the study provided insufficient information regarding the influence of music lessons on IQ. Computing observed power merely provides some quantitative information to reinforce this correct conclusion.
This suggests that the study had insufficient power to produce a significant result, if the effect size in the sample matches the true effect size. Yuan and Maxwell discuss false interpretations of observed power. One false interpretation is that a significant result implies that a study had sufficient power.
Power is a function of the true effect size and observed power relies on effect sizes in a sample. It is therefore possible that observed power is considerably higher than the actual power of a study. Another false interpretation is that low power in a study with a non-significant result means that the hypothesis is correct, but that the study had insufficient power to demonstrate it. The problem with this interpretation is that there are two potential reasons for a non-significant result.
One of them, is that a study had insufficient power to show a significant result when an effect is actually present this is called the type-II error. The second possible explanation is that the null-hypothesis is actually true there is no effect. A non-significant result cannot distinguish between these two explanations.
Yet, it remains true that the study had insufficient power to test these hypotheses against each other. Yuan and Maxwell focus on a design in which a sample mean is compared against a population mean and the standard deviation is known. To modify the original example, a researcher could recruit a random sample of children, do a music lesson intervention and test the IQ after the intervention against the population mean of with the population standard deviation of 15, rather than relying on the standard deviation in a sample as an estimate of the standard deviation.
This scenario has some advantageous for mathematical treatments because it uses the standard normal distribution. However, all conclusions can be generalized to more complex designs. Thus, although Yuan and Maxwell focus on an unusual design, their conclusions hold for more typical designs such as the comparison of two groups that use sample variances standard deviations to estimate the variance in a population i.
Yuan and Maxwell also focus on one-tailed tests, although the default criterion in actual studies is a two-tailed test. This means that an observed z-score has to exceed a value of 1. To illustrate this with an example, assume that the IQ of children after a music intervention is The test-statistic z is the ratio of the effect size and sampling error. Based on this result, a researcher would be justified to reject the null-hypothesis there is no effect of the intervention and to claim support for the hypothesis that music lessons lead to an increase in IQ.
Importantly, this hypothesis makes no claim about the true effect size. It merely states that the effect is greater than zero. The significance test merely rejects the possibility that the effect size is 0 or less i. The red curve shows the standard normal distribution for the null-hypothesis.
The blue curve shows the non-central distribution. The distribution shows how z-scores would be distributed for a set of exact replication studies, where exact replication studies are defined as studies with the same true effect size and sampling error.
The figure also illustrates power by showing the critical z-score of 1. On the left side are studies where sampling error reduced the observed effect size so much that the z-score was below 1.
On the right side are studies with significant results. The area under the curve on the left side is called type-II error or beta-error. The area under the curve on the right side is called power 1 — type-II error. In sum, the formula. Power is a function of the non-centrality parameter and not just the effect size. Thus I would have included sqrt n also on the left side of the formula ].
Because the formula relies on the true effect size, it specifies true power given the unknown population effect size.
To use it for observed power, power has to be estimated based on the observed effect size in a sample. The important novel contribution of Yuan and Maxwell was to develop a mathematical formula that relates observed power to true power and to find a mathematical formula for the bias in observed power. The formula implies that the amount of bias is a function of the unknown population effect size.
Yuan and Maxwell make several additional observations about bias. The second important observation is that systematic bias is never greater than 9 percentage points.
The last observation has important implications for the interpretation of observed power. This finding suggests that the study was underpowered and that a more powerful study might produce a significant result. As a result, bias does not undermine the conclusion. In this case, observed power might be used to argue that a study had sufficient power because it did produce a significant result.
This would indicate that there was a relatively high chance to end up with a non-significant result. However, systematic bias implies that observed power is more likely to underestimate true power than to overestimate it. Thus, true power is likely to be higher. Again, observed power is conservative when it comes to the interpretation of power for studies with significant results.
This would suggest that systematic bias is not a serious problem for the use of observed power. Moreover, the systematic bias is never more than 9 percentage-points. In sum, Yuan and Maxwell provided a valuable analysis of observed power and demonstrated analytically the properties of observed power. Based on their analyses, Yuan and Maxwell draw the following conclusions in the abstract of their article. 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.
Once can then compare this noncentral F-distribution to your critical value in order to get the "observed power.
Again, while the formulas and rejection region will differ based on your statistical test, the idea of how to calculate observed power are universal. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. How is SPSS calculating 'observed power'? Ask Question.
Asked 3 years, 5 months ago. Active 3 years, 5 months ago. Viewed 4k times. Does anyone know or can point me to some documentation? Improve this question. I was hoping that there was some documentation out there in general. Add a comment. How could I do this? Is effect size even necessary in studies that are not experimental? Post-hoc power analysis of negative result usually produces very low power when sample size is modest note that I did nor say small.
In our genetics case-control study we found negative result with a sample of per group. Result may not change even if we repeat it with say cases. But how do we establish this statistically. In another project, we found negative result with first 30 samples but positive result after analyzing samples. Interesting points!
I think there are some forms of post-hoc power analyses that are appropriate. I agree wholeheartedly with everything you have said. I understand that this method would closely align with confidence intervals. However, I think it will demonstrate under-powered studies with more impact. Particularly in non-inferiority trials that claim non-inferiority when that are massively under-powered. Yes, you can always calculate the effect size you could detect with a certain level of power.
But, there is never information that goes beyond the p-value. So, knowing how sensitive your design was is always good info to have, but it is difficult to use it as a way to draw inferences from data. I know that editorials are mass-produced, including similar criticisms. However, I don't know anyone who showed "how can I calculate post-hoc power" in the case of Welch's t-test, especially when the sample sizes of the two groups are different.
I'm asking to show it in a calculation code or mathematical formula in following community but, Our reviewer's comment is this: The authors should provide a back-to-the envelope assessment of what is the power of the tests given the sample size a classic reference to look at would be Andrews, Donald W. Are you familiar with this approach?
It is talking about an inverse power function. Thank you. Thanks for this informative blog post. I fully agree on the nonsense of an observed power analysis with the effect size estimated from the current data.
However, as you state later, a post-hoc power analysis based on an effect size of theoretical interest can be very useful. In the case of cognitive modeling, the data points going into estimation of conditional parameters further back in the model crucially depend on the level of performance achieved in earlier processes and thus cannot be estimated a priori. Here, a post-hoc power analysis based on the level of performance achieved in prior parameters and then asking whether meaningful, theoretically predicted differences in further back parameters would have been observed with sufficient power is the only informative power analysis that can be conducted.
I would greatly appreciate if you could clarify that post-hoc power analysis is more than observed power analyses and that this rightful critique only concerns a specific subset of post-hoc power analyses referred to as observed power analysis. Friday, December 19, Observed power, and what to do if your editor asks for post-hoc power analyses. This blog post is now included in the paper "Sample size justification" available at PsyArXiv.
Observed power or post-hoc power is the statistical power of the test you have performed, based on the effect size estimate from your data. Observed power differs from the true power of your test, because the true power depends on the true effect size you are examining. However, the true effect size is typically unknown, and therefore it is tempting to treat post-hoc power as if it is similar to the true power of your study.
In this blog, I will explain why you should never calculate the observed power except for blogs about why you should not use observed power. Observed or post-hoc power and p -values are directly related.
It looks like a curve, but the graph is basically a scatter plot of a large number of single observations that fall on a curve expressing the relation between observed power and p -values. Yes, that is exactly the same curve these observations fall on.
The only difference is how often we actually observe high p -values or have low observed power. You can see there are only a few observations with high p -values if we have high power compared to medium power , but the curve stays exactly the same. I hope these two figures drive home the point of what it means that p -values and observed power are directly related: it means that you can directly convert your p -value to the observed power, regardless of your sample size or effect size.
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