Power Analysis - Controlling Power
Power is the fourth element in this closed system - Given an effect size, and alpha, and sample size, power is known. As noted above, a "convention" exists that power should be set at 80% but this convention has no logical basis. The appropriate level of power should be decided on a case-by-case basis, taking into account the potential harm attendant on a Type I error, the determination of a clinically important effect, the potential sample size, as well as the importance of identifying an effect, should one exist.
Power analysis - ethical issues
Some studies involve putting patients at risk. At one extreme, the risk might involve a loss of time spent completing a questionnaire. At the other extreme, the risk might involve the use of an ineffective treatment for a potentially fatal disease. These issues are clearly beyond the scope of this discussion, but one point should be made here.
Ethical issues play a role in power analysis. If a study to test a new drug will have adequate power with a sample of 100 patients, then it would be inappropriate to use a sample of 200 patients since the second 100 are being put at risk unnecessarily. At the same time, if the study requires 200 patients to yield adequate power, it would be inappropriate to use only 100. These 100 patients may consent to take part in the study on the assumption that the study will yield useful results. If the study is under-powered, then the 100 patients have been put at risk for no reason.
Of course, the actual decision making process is complex. One can argue about whether "adequate" power for the study is 80%, or 90%, or 99%. One can argue about whether power should be set based on an improvement of 10 points, or 20 points, or 30 points. One can argue about the appropriate balance between alpha and beta. The point being made here is that these kinds of issues need to be addressed explicitly as part of the decision making process.
The null hypothesis vs. the nil hypothesis
Power analysis focuses on the study's potential for rejecting the null hypothesis. In most cases the null hypothesis is the null hypothesis of no effect (a.k.a. the nil hypothesis). For example, the researcher is testing a null hypothesis that the change score from time-1 to time-2 is zero. In some studies, however, the researcher might attempt to disprove the null hypothesis other than the nil. For example, "The intervention boosts the scores by 20 points or more". The impact of this is to change the effect size.
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