Troubleshooting
Problem
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.
REFERENCE: [Cohen].
Cohen, Jacob. Statistical Power Analysis for the Behavioral Sciences, 2nd ed., New Jersey: Lawrence Erlbaum Associates, Inc., 1988.
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16 April 2020
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