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# A quick guide to choice of sample sizes for Cohen's effect sizes

Dunlap and Myers (1997) suggest rules of thumb for sample sizes using Cohen's effect size rules of thumb and equations that they have derived in some simple cases. These are given in the table. The sample sizes guarantee between 80% and 90% power to detect the given effect sizes.

 Effect Size Formula Small Medium Large Cohen's d unpaired t, $$\frac{\mbox{16}}{\mbox{d}^text{2}}$$ + 2 0.2 0.5 0.8 equal group sizes 402 66 28 Correlation 8/$$r^text{2}$$ 0.1 0.3 0.5 Total sample size 800 88 32 $$\phi$$, 2x2 table 8/$$\phi^text{2}$$ 0.1 0.3 0.5 Total sample size 800 88 32

Dunlap and Myers (1997) show in their appendix that for a 2x2 table of proportions of form

 Column 1 Column 2 Row 1 $$p_text{11}$$ $$p_text{12}$$ Row 2 $$p_text{21}$$ $$p_text{22}$$

with

A = $$\frac{p_text{11}}{p_text{11}+p_text{12}}$$, B = $$\frac{p_text{21}}{p_text{21}+p_text{22}}$$

C = $$\frac{p_text{11}}{p_text{11}+p_text{21}}$$, D = $$\frac{p_text{12}}{p_text{12}+p_text{22}}$$

then we can define $$\phi = \sqrt{\mbox{abs(A-B)} \mbox{abs(C-D)}}$$

In addition Maxwell (2000) mentions rules of thumb for power to detect the $$R^text{2}$$ of a single predictor with outcome in a multiple regression with a total of p predictors and a type I error of 0.05. These are give below:

 Regression on p predictors Small:$$R^text{2}$$=0.02 Medium:$$R^text{2}$$=0.13 Large:$$R^text{2}$$=0.26 Total sample size 80% Power 392+p 52+p 22+p Total sample size 90% Power 526+p 70+p 30+p

Reference

Dunlap WP, Myers L. (1997) Approximating power for significance tests with one degree of freedom. Psychological methods 2(2) 186-191.

Maxwell SE (2000) Sample size and multiple regression analysis, Psychological methods 5(4) 434-458.

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