A Power Primer: Tutorials in Quantitative Methods for Psychology – Cohen - 1992 - Article

## What is the problem?

On reason for why statistical power analysis in research is continuously ignored in behavioural science is the difficulty with the standard material. There has not been an increase in the probability of obtaining a significant result in the last 25 years. Why?

Everyone agrees on the importance of power analysis, and there are many ways to estimate sample sizes. But part of the reason for this could be the low level of consciousness about effect size; it is like the only worry about magnitude in a lot of psychological research is with regard to the statistical test result and its p value, not to the psychological phenomenon being studied. Some blame this on the precedence of Fisher’s null hypothesis testing; cut go-no-go decision over p=0.05. The author suggests that the neglect of power analysis represents the slow movement of methodological advance. Another suggestion is that researchers thinking the reference material for power analysis is too complicated.

## What are the components of power analysis?

Statistical power analysis uses the relationships between four variables involved in statistical inference:

1. The significance criterion α: the risk of falsely rejecting the null hypothesis (H0) and committing a type I error, α, represents a policy: the maximum risk of such a rejection.
2. Power: the statistical power of a significance test is the long-term probability, given ES, N, and α of rejecting H0. When ES does not equal zero, H0 is false, so failure to reject also causes an error (= type II error with probability of β). Power is equal to 1 – β (probability of rejecting a false H0. Taken with α = 0.05, power of 0.80 results in a β:α ratio of 4:1 (0:20 to 0.05) of the two risks.
3. Sample size (N): in planning the researchers needs to know the N needed to get the desired power for a specified alpha and hypothesized ES. N increases when:
1. Desired power increases
2. ES decreases
3. Α decreases.
4. Population effect size (ES): degree to which H0 is believed to be false. N or power can not be determined without the ES. The degree to which H0 is false is shown by the difference between H0 and H1 (ES). For all, H0 = ES is 0.

d – the ES index for the t-test of the difference between the independent means (difference in means divided by population standard deviation). The H0 is d = 0 (no difference between group means). The small, medium, and large ES’s (H1’s) are d – 0.20, 0.50, and 0.80.

Using Cohen’s table, we can find the necessary N’s for different powers and ES’s.

• To detect a medium difference between two independent sample means at α = 0.05 requires N = 64 in each group.
• For a significance test of a sample r at α = 0.01 when the population r is large, a sample size of 41 is required. At an α = 0.05 a sample size of 28.
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