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Effect Size: Relationship between partial Eta-squared, Cohen's f, and Cohen's d



When I request "Display: Estimates of Effect Size" in SPSS GLM (after clicking the Options... button), I find that SPSS reports the partial Eta-Squared statistic. I would prefer another index of effect size, such as Cohen's f or Cohen's d (the standardized range of population means). Can I use SPSS to calculate these?

Resolving The Problem

SPSS cannot calculate Cohen's f or d directly, but they may be obtained from partial Eta-squared. Cohen discusses the relationship between partial eta-squared and Cohen's f :

eta^2 = f^2 / ( 1 + f^2 )

f^2 = eta^2 / ( 1 - eta^2 )

where f^2 is the square of the effect size, and eta^2 is the partial eta-squared calculated by SPSS. (cf. [Cohen], pg. 281.) Therefore,

f = sqr( eta^2 / ( 1 - eta^2 ) ).

If the model is a Univariate ANOVA with two groups, and the number of observations in each group is equal, then the standardized range of population means, Cohen's d, is given by

d = 2*f

([Cohen], pg. 276.)

When there are more than two means, Cohen considers three patterns of dispersion:

Pattern 1: Minimum variability
Pattern 2: Intermediate variability
Pattern 3: Maximum variability

Henceforth we will take there to be k means, where k > 2.

For Pattern 1, the dispersion is minimized when the intermediate means are all at the midpoint of the range, and then:

d = f * sqr(2*k)

([Cohen], pg. 277.)

For Pattern 2, it is assumed that the k means are equally spaced through the range. Then:

d = 2 * f * sqr(3*(k-1)/(k+1))

([Cohen], pg. 279.)

For Pattern 3, maximum dispersion for an even number of means occurs with half at one extreme and the other half at the other:

d = 2 * f (k even)

while for an odd number of means, there will be one additional mean at one extreme:

d = 2 * f * k / sqr(k^2 - 1) (k odd)

([Cohen], pp. 279-280).

Discussion of the power associated with these effects is beyond the scope of this note. Please consult [Cohen] or another reference.

Cohen, Jacob. Statistical Power Analysis for the Behavioral Sciences, 2nd ed., New Jersey: Lawrence Erlbaum Associates, Inc., 1988.

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Modified date:
16 April 2020