The two posts below, both from the psych-postgrads jiscmail list, give thoughts about the choice of effect size to use in analysis of variance. The two most prevalent are generalized eta-squared and generalized omega-squared. These can be computed in R using four functions in the sjstats library on output from analyses of variance procedures, aov and anova, in R (see here.)
References
Baguley T (2012) Serious Stats: A guide to advanced statistics for the behavioral sciences. Palgrave: London.
Lakens D (2013) Calculating and reporting effect sizes to facilitate cumulative science: a practical primer for t-tests and ANOVAs. Front Psychol. https://doi.org/10.3389/fpsyg.2013.00863 and available on-line from here.
Posts
Thom Baguley suggests there is little difference in generalised eta squared or generalised omega squared (as below).
"I would suggest generalised eta squared or generalised omega squared (if you are using standardized effect sizes). This is briefly covered by Lakens (with simplified formula for generalised eta squared in my book Serious Stats; also output from the afex packages in R I think).
In most cases generalised omega squared is only slightly smaller than generalised eta squared unless you have pretty small samples (in which case the effect sizes are too imprecise to mean much of anything)".
From James Bartlett
"You could download the free stats package called 'JASP' and it provides a range of effect sizes for ANOVA including eta squared and omega squared. These are similar but omega squared is generally preferred as it provides a less biased estimate.
For more information on effect sizes and how to interpret them in ANOVAs, Daniël Lakens wrote a great paper: http://journal.frontiersin.org/article/10.3389/fpsyg.2013.00863/full" (see above link)