A power primer - summary of an article by Cohen (1992)

A power primer. 
Cohen (1992)
Psychological Bulletin

The tables of this article are missing

Abstract

Effect-size indexes and conventional values for these are given for operationally defined small, medium, and large effects.

Method

Statistical power analysis exploits the relationships among the four variables involved in statistical inference.

• Sample size (N)
• Significance certerion (α)
• Population effect size (ES)
• Statistical power

Each is a function of the other three. It is most useful to determine the N necessary to have a specified power for given α and ES.

The significance criterion α

α represents the maximum risk of mistakenly rejecting the null hypothesis (committing a Type I error). This is usually .05. α risk may be defined as one or two sided.

Power

The statistical power of a significance test is the long-term probability, given the population ES, α, and N of rejection the H₀. When the ES is nit equal to zero, H₀ is false, so failure to reject it also incurs an error (Type II error). For any given ES, α, and N, its probability of occurring is β. Power is 1 – β, the probability of rejecting a false H₀.

Taken the conventional α = .05, power of .80, there is a α:β ratio of 4:1 of the two kinds of risks.

Sample size

In research planning, the investigator needs to know the N necessary to attain the desired power for the specified α and hypothesized ES. N increases with an increase in the power desired, a decrease in the ES and in α.

For statistical tests involving two or more groups, N is the necessary size for each group.

Effect size

The effect size (ES) is the degree to which the H₀ is believed to be false.

In the Neyman-Pearson method of statistical inference, an alternative hypothesis H₁ is counterpoised against H₀. The degree to which H₀ is false is indexed by the discrepancy between H₀ and H₁ and is called the ES. Each statistical test has its own ES index.   All the indexes are scale free and continuous, ranging upward from zero. For all, the H₀ is that ES = 0.

To convey the meaning of any given ES index, it is necessary to have some idea of its scale.

The ES index for the t test of the difference between independent means is d, the difference expressed in units of the within-population standard deviation.

Statistical tests

The most common test in psychological research

• The t-test for the difference between two independent means
• The t test for significance of a product-moment correlation coefficient r
• The test for the difference between two independent rs
  Accomplished as a normal curve test through the Fisher z transformation of r
• The binominal distribution or, for large samples, the normal curve test that a population proportion.
• The normal curve test for the difference between two independent proportions
• The chi-square
  Goodness of fit or association in two-way contingency tables
• One-way analysis of variance
• Multiple and multiple partial correlation

Because all tests of population parameters that be either positive or negative are two-sided, their ES indexes are absolute values.

The ES posited by the investigator is what (s)he believes holds for the population. The sample size that is found is conditional on the ES.