Introduction
We consider effect-size measurements for two-way ANOVA based on the correlation coefficient. These are similar to the effect sizes described in More ANOVA Effect Sizes. These measures also carry over to ANOVA with more than two factors.
Partial Eta-squared
For two-way ANOVA there are three types of effects: Row, Column, and Interaction (actually four types if you include the Error effect). The eta-squared effect size for each of these is computed as in the one-factor case, namely
Note that the sum of these effect sizes is 1. If these effect sizes are respectively .10, .15, .25, .50, then you can conclude, for example, that the interaction effect is more important than each of the main effects. These sorts of judgments are particularly relevant when none of the effects are significant, especially in a pilot study with a small sample.
In the case of factorial ANOVA, another measure of effect size is the partial eta-squared, which is defined as follows where effect = Row, Col, Int or W (error),
Note that in the case of one-way ANOVA eta-squared and partial eta-squared produce the same value.
Partial Omega-squared
For two-way ANOVA, we have the following version of omega-squared for each effect:
More importantly, the partial omega-squared value is
Examples
For Example 2 of ANOVA using Regression, for the interaction effect
Partial η2 = 5960.067/(5960,067+10808) = .3554
which is the same value reported by Real Statistics’ Two Factor ANOVA data analysis tool. Also
Partial ω2 = (5960.067 – 2 ⋅ 450.333) / (5960.067 + (30-24)450.333) = .2725
Examples Workbook
Click here to download the Excel workbook with the examples described on this webpage.
References
MRC CBU Wiki (2009) Partial and generalized omega-squared as effect sizes in analysis of variance
https://imaging.mrc-cbu.cam.ac.uk/statswiki/FAQ/os2
Olejnik, S., Algina, J. (2003) Generalized eta and omega squared statistics: measures of effect size for some common research designs
https://coshima.davidrjfikis.com/EPRS8540/uploadf07/olejnik_PsychMeth2003.pdf