First one should really avoid partial eta-squared. If you have a two-by-two design all the effects are differences in means so I think using the unstandardized differences is best. If you must use a standardized effect size metric use a d family statistic such as Hedges' g.

Thanks for your inputs, Thom. But, why do you say "one should really avoid partial eta-squared"? I have seen ANOVA effect size being reported through partial eta-squared in reputed journals, so I'm curious - what makes you say so? 

Hi Diptiman, the partial eta-squared is a sample effect size estimate and should not be used is because it is a positively biased estimate of the population effect size. The omega effect size is adjusted to provide better population estimates. I am not recommending the omega effect size; other effect size estimates could be also appropriate and I am just pointing out here that partial eta-squared is a biased estimate of the population effect size.

I thought I had replied, but somehow it didn't get through ...

1) It is a descriptive measure of variance explained - it describes variation accounted for in the sample. This relates to Patricia's points. I'm not sure this is is an indication of bias per se, because descriptive measures can be unbiased - it is just inappropriate to interpret them as estimates of population parameters. Other descriptive measures include R^2.

2) It is a standardised effect size metric. This means that the effect size is scaled in terms of standard deviations or variances. Typically one uses the sample estimate of the SD or variance and this is (in my view) deeply problematic because all sorts of factors influence the SD or variance that don't influence the magnitude of the effect. Generally I prefer unstandardised measures of effect size for this reason. These factors include the design of the study, the reliability of the measures and range restriction.

3) Partial eta squared is particularly bad because of the way it handles other factors or covariates in the design (including random factors such as the subjects effect in a repeated measures design). This tends to result in bigger effect size estimates than classical eta-squared (which is calculated as SSeffect / SStotal). Partial eta-squared is SSeffect / (SSeffect + SSerror). The denominator is only the same if there are no other effects or error terms in the analysis.

If you do use eta-squared don't interpret it as a population statistic, be aware of limitations (reliability, range restriction etc.) and calculate what is known as generalised eta-squared (Oleynik & Algina, 2003).

Generalised eta-squared attempts to equate the calculation of eta-squared as if each effect were from an independent measures design. I have simplified formulae for generalised eta-squared in my book.

Thank you, Patricia, and Thom for your replies! Thom, I appreciate your effort to break down your answer into minute details, and the way out you've suggested. I also found your article: Standardized or simple effect size: What should be reported?, 2009 - very detailed and helpful.  

Hello Diptiman, If you are interested in reporting the generalized eta-squared, we have a SAS paper to estimate this effect size for a variety of ANOVA designs. I hope it is helpful.

Thanks a lot, Patricia! Much appreciated! Will check your paper out.

Glad of being of help. Among the research we have done, we have research on effect sizes that you could also look at. These studies have been very through, thought-out simulations studies for providing evidence of the behavior of some of these effect sizes. I have uploaded the poster I presented for effect sizes used for dichotomized outcomes and I will update the full paper sometime today. Best, Patricia.

Dr. Lakens, that's wonderful of you to share the latest version of your spreadsheet. Thank you so much! Am sure it'll be a big help for me. Sincerely appreciated! 

Daniel, I add my sentiments to Diptiman's! Thank you so much for sharing your file!

Another resource to calculate generalized eta squared on repeated measures designs:

https://github.com/VeronicaMaki-Marttunen/Generalized-eta-square

Hi Veronica! Thank you so much for taking time to answer my question!

For calculation, please refer to this link:

https://real-statistics.com/analysis-of-covariance-ancova/effect-size-ancova/