Cohen’s d, Cohen’s f, and  PDF

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Cohen’s d, Cohen’s f, and ...

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Cohen’s d, Cohen’s f, and 2 Cohen’s d , the parameter, is the difference between two population means divided by their common standard deviation. Consider the Group 1 scores in dfr.sav. Their mean is 3. The sum of the squared deviations about the mean is 9.0000. Since there are nine scores, the population variance is 9/9 = 1, and the standard deviation is 1. Group 2 has the same number of scores, sum of squared deviations about the mean, variance and standard deviation, but a mean of 4. Cohen’s d is (4-3)/1 = 1. By Cohen’s benchmarks for d, this is a large effect. Cohen’s f, the parameter, is the standard deviation of the population means divided by their (3  3.5) 2  (4  3.5)2  0.25, yielding a common standard deviation. The variance of 3 and 4 is 2 standard deviation of .5. Cohen’s f is .5/1 = .5. For the two population case, a d of 1 is equivalent to an f of .5. By Cohen benchmarks for f, .5 is a large effect. If you hate arithmetic, you can use G*Power to calculate the value of f. Select F, ANOVA, fixed effects, omnibus, one way, a priori, enter the means and standard deviations, Calculate:

Cohen_d_f_r

2 If you conduct a simple linear regression relating the scores in dfr.sav to group membership, you will obtain  = .447, a close to large effect by Cohen’s benchmarks for rho. The proportion of variance in the scores explained by group membership is .4472 = .20. This is a squared point-biserial correlation coefficient, but is more commonly referred to as eta-squared. With equal population sizes, the relationship between f and 2 is f  populations, that is

2 . For our 1  2

.20 = .5. 1  .20

Suppose we have three populations, as in drf3.sav. The means are 3, 4, and 5, and each (3 4) 2  (4 4) 2  (5  4) 2 within-population variance is 1. The variance of 3, 4, 5 is  2/3, yielding a 3 standard deviation of .8165. Cohen’s f is .8165/1 = .8165.

The among groups sum of squares for these data is 18 and the total sum of squares is 45, for an 2 of 18/45 = .40. If you use multiple regression to predict the scores from the two dummy variables representing group membership, you obtain R2 = .40. Calculating f from the 2 , .40  .8165. G*Power will convert 2 to f for you. Click Determine, Effect Size from Variance. 1 .40 Enter 2 and (1- 2). Click Calculate. f 

3

The magnitude of Cohen’s f is affected by way in which the cases are allocated to the populations. As shown above, when we had equal sample sizes f = .8165. If we changed the sample sizes to 50 in each group, the f would remain the same (but power would be increased). Now look what happens if we deviate from equal population sizes: With most of the scores in the groups that differ most from each other, f increases. With most of the scores in the group which differs least from the others, f drops.

Notes Despite d, f, and his other effect size parameters being parameters, Cohen did not represent them with Greek letters. Instead, he used bold-faced Roman characters. Upon my request, Jacob Cohen (Jake) sent me several reprints of his articles. Every time the reprint had a handwritten salutation and signature. I regret that I never met him in person. My friend, and Institutional Research statistician, Chuck Rich, retired recently and passed on to me Cohen’s book on power analysis. I always wanted that book, but always was too cheap to buy it (I did buy his book on multiple regression). I have been enjoying reading Cohen’s own words. Thank you, Chuck. Karl L. Wuensch, January, 2016...