Partial omega-squared as an effect size in analysis of variance

Partial $$\omega^{text{2}$$ has been suggested by Keppel (1991),pp 222-224) and Olejnik and Algina (2003) as an alternative to partial $$\eta}text{2}$$ when comparing the size of sources of variation across studies from analysis of variance.

(Partial) $$\omega^text{2} = \frac{\mbox{SS(effect)-df(effect) MSE(effect)}}{\mbox{SS(effect)+(N-df(effect))MSE}}$$

In the special case of a between subjects ANOVA with b groups and a total of N subjects the denominator in the above may be rewritten as below.

SS(effect)+(N-df(effect))MSE = SS(group) + (N-df(group))MSE

= SS(group) + (N-(b-1))MSE = SS(group) + (N-b+1)MSE = SS(group) + (N-b)MSE + MSE

= Total SS + MSE = SS(group) + SSE + MSE = SS(group) +((df(error term)+1) x MSE)

since in a one-way ANOVA between subjects design the total SS = SS(group) + (N-b)MSE. The latter denominator term (SS(group) +((df(error term)+1) x MSE)) will also be correct for any main effect in a factorial ANOVA since the df(effect) takes the same form as above ie equal to the number of groups - 1 provided the remaining factors are non-random ie chosen by the experimenter.

Olejnik and Algina further present and illustrate for ANOVAs with upto three factors a more general formula for partial $$\omega^text{2}$$ to take into account any non-manipulated factors in the ANOVA.

(Partial) $$\omega^text{2} = \frac{\mbox{SS(Effect) - df(effect)MSE}}{\mbox{d(SS(Effect) - df(effect)MSE)} + \sum_{M} [\mbox{SS(M) - df(M)(MSE of M)}] + \mbox{N MSE}}$$

where d=1 if the effect of interest is non-random and 0 otherwise, M are any sources of variation which include at least one random factor and N is the total sample size.

$$\omega^text{2}$$ may take values between $$\pm$$ 1 with a value of zero indicating no effect. A negative value will result if the observed F is less than one.

The numerator in $$\omega^text{2}$$ compares the mean square of the effect (=SS(effect)/df(effect)) to its mean square error which theoretically should be equal (giving a difference, as measured in the numerator, of zero) because the ratio of the mean squares may be approximated by a F distribution which takes a minimum value of one indicating no statistical evidence of the effect.

$$\omega^{text{2}$$ is unaffected by small sample sizes unlike $$\eta}text{2}$$ which tends to overestimate the effect in smaller samples (Larson-Hall (2010), p.120) and, since it is a population based measure unlike $$\eta^text{2}$$, will consequently take a smaller value.

References

Olejnik S and Algina J (2003). Generalized Eta and Omega Squared Statistics: Measures of effect size for some common research designs. Psychological Methods 8(4) 434-447.

Keppel G (1991). Design and analysis:A researcher's handbook. Prentice-Hall:Englewood Cliffs, NJ

Larson-Hall J (2010). A Guide to Doing Statistics in Second Language Research Using SPSS. Routledge:New York.