I have a 3 way mixed ANOVA, where I have 2 significant main effects, one significant two way interaction and a significant three way interaction.

I've been told to not describe the lower order effects if the three way interaction is significant so I haven't (but have put inferentials in table).

After getting the significant 3 way I ran a post-hoc test to compare the means, using the EMMEANS command ( as per the handout I got). As its ANOVA it gives partial $\eta^2$, I know how to change the main effects and interaction into eta n2 (using the within effects table and between effects table obviously depending on what sort of factor it is)

But with the pairwise comparisons which I got from doing the EMMEANS command:

 EMMEANS = TABLES (type*wggroup*match) COMPARE (match)
/EMMEANS = TABLES (wggroup*type) COMPARE (type)

How do I change the partial $\eta^2$ in the multivariate table to $\eta^2$?

As I now have the inferentials in my text for the pairwise comparisons but am unable to convert the effect size to $\eta^2$ from the multivariate table to interpret it correctly.

The only tables I am given from EMMEANS is the pairwise comparisons (which doesn't have the inferentials) and then below that is the multivariate table (with inferentials).


First up, I don't think it's the case that you should not describe the lower order effects if the three-way ANOVA is significant. They are orthogonal effects and each tells you something different. The higher order things qualify the lower order ones (e.g. the significant ABC three-way interaction tells you that what's happening in your A*B two-way interaction differs depending on the level of C), but all can give you useful information. Why ignore it?

Why do you want to use eta-squared over partial eta-squared? Eta-squared is problematic - as more factors go into the model, eta-squared drops because it's based on SStotal (i.e. it underestimates the effect size). It is calculated by taking your SS for the effect of interest and dividing it by the relevant SStotal.

Partial eta-squared takes this into account. It's calculated by taking the SS for the effect of interest and dividing it by the sum of that SSeffect and SSerror (http://www.theanalysisfactor.com/calculate-effect-size/). That is, the effects that have been accounted for by other factors are not taken into account.

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    $\begingroup$ The lower order effects are only testing, eg, the difference between means at the reference level of the factors. That is, it is a particular simple effect, but it isn't clear that it is the simple effect that you necessarily care about. $\endgroup$ – gung Jan 13 '15 at 23:45

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