Partial omega-squared as an effect size in analysis of variance
Partial $$\omega^text{2}$$ has been suggested by Field (2013, p.473-4), 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 where MSE is the mean square of the error term.
(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. Baguley (2012, p.483) and Field (2013, p.473) give the above formula for $$\omega^text{2}$$ with Total SS + MSE in the denominator.
Olejnik and Algina further present and illustrate an alternative to the above 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 number of observations (scores).
$$\omega^text{2}$$ 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.
Field (2013, p.537-8) illustrates computing $$\omega^text{2}$$ for factorial between subjects ANOVAs which is somewhat more involved and uses variance components.
He also (pages 566-567) mentions that a different form of $$\omega^text{2}$$ to that discussed above is needed for a factor in a repeated measures analysis of variance since the former overestimates effect size if used in a repeated measures.
In particular for a factor in a one-way repeated measures ANOVA with k levels and n subjects
$$ \omega^text{2} = \frac{[(k-1)/nk] (MS(effect)-MSE)}{MSE + (MSB - MSE)/k + [(k-1)/nk] (MS(effect)-MSE)} $$
where MS(B) = (total variance(N-1)-SS(effect) - SSE)/(N-1). Field also suggests only using effect sizes for pairwise comparisons for repeated measures with more than one factor.
$$\omega^text{2}$$ is unaffected by small sample sizes unlike $$\eta^text{2}$$ which tends to overestimate the effect in smaller samples (Larson-Hall (2010), p.120; Field (2013), p.473) and, since it is a population based measure unlike $$\eta^text{2}$$, will consequently take a smaller value.
References
Baguley T (2012). Serious Stats. A guide to advanced statistics for the behavioral sciences. Palgrave Macmillan:New York.
Field AP (2005, p.417-419). Discovering statistics using SPSS. Sage:London. Formulae using ANOVA output to compute omega-squared for factorial designs.
Field A (2013) Discovering statistics using IBM SPSS statistics. Fourth Edition. Sage:London. On page 474 it is suggested using values for $$\omega^text{2}$$ of 0.01, 0.06, 0.14 to indicate small, medium and large effects respectively.
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.