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$$\phi = \sqrt{\frac{\mbox{abs(A-B)}}{\mbox{abs(C-D)}}}$$ |
then we can define $$\phi = \sqrt{\mbox{abs(A-B)} \mbox{abs(C-D)}}$$ |
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 |
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Cohen's d unpaired t, |
$$\frac{\mbox{16}}{\mbox{d}^text{2}}$$ + 2 |
0.2 |
0.5 |
0.8 |
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equal group sizes |
|
402 |
66 |
28 |
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Correlation |
8/$$r^text{2}$$ |
0.1 |
0.3 |
0.5 |
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Total sample size |
|
800 |
88 |
32 |
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$$\phi$$, 2x2 table |
8/$$\phi^text{2}$$ |
0.1 |
0.3 |
0.5 |
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Total sample size |
|
800 |
88 |
32 |
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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 $$\phi = \sqrt{\mbox{abs(A-B)} \mbox{abs(C-D)}}$$
Reference
Dunlap WP, Myers L. (1997) Approximating power for significance tests with one degree of freedom. Psychological methods 2(2) 186-191.