Cohen's f Statistic

Calculation

Cohen's f is calculated as

where σm is the standard deviation (SD) of population means (mi) represented by the samples and σ is the common within-population SD; σ= MSE½. MSE is the mean square of error (within groups) from the overall ANOVA F test. It is based on the deviation of the population means from the mean of the combined populations or the mean of the means (M).

for equal sample sizes and

for unequal sample sizes.

Examples

Example 1

Table 1 provides descriptive statistics for a study with four groups and equal sample sizes. ANOVA results are shown. The calculations below result in an estimated f effect size of .53, which is considered large by Cohen standards. An appropriate interpretation is that about 50% of the variance in the dependent variable (physical health) is explained by the independent variable (presence or absence of mental or physical illnesses at age 16).

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Example 2

Table 2 provides descriptive statistics for a study very similar to the one described in Example 1. There are four groups, but the sample sizes are unequal. ANOVA results are shown. The calculations for these samples result in an estimated f effect size of .27, which is considered medium by Cohen standards. For this study, an appropriate interpretation is that about 25% of the variance in the dependent variable (physical health) is explained by the independent variable (presence or absence of mental or physical illnesses at age 16).

Cohen's f and d

Cohen's f is an extension of Cohen's d, which is the appropriate measure of effect size to use for a t test. Cohen's d is the difference between two group means divided by the pooled SD for the two groups. The relationship between f and d when one is comparing two means (equal sample sizes) is d = 2f. If Cohen's f=0.1, the SD of k(k≥2) population means is one tenth as large as the SD of the observations within the populations. For k = two populations, this effect size indicates a small difference between the two populations: d = 2f= 2 ∗ 0.10 = 0.2.

Cohen's f in Equation 1 is positively biased because the sample means in Equation 2 or 3 are likely to vary more than do the population means. One can use the following equation from Scott Maxwell and Harold Delaney to calculate an adjusted Cohen's f:

Applying Equation 4 to the data in Table 1

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