Cohen d for 2x2 interaction

There is a nice answer to this question, but it assumes that you have the ANOVA table available.

My problem is different. Say I'm reading a paper describing a male vs female, disease vs control experiment, and I know the n's, means, and standard deviations for all four groups (healthy females, ... , diseased males), but I don't have the original data.

I estimate Cohen d for the sex:health interaction. Numerator: (male_disease-female_disease)-(male_healthy-female_healthy). Denominator: I pool the standard deviations.

In simulations with perfect data (normally distributed, etc), my estimated Cohen d does not match the Cohen d calculated from eta-squared: $$2\sqrt\frac{\eta^2}{1-\eta^2}$$

I am sure there must be a closed-form estimator for Cohen's d for this case, but having looked all over google and online stats books I can't seem to find the answer. Apologies if I've missed something obvious.

• Your conversion formula has an error. It should be $2\sqrt\frac{\eta^2}{1-\eta^2}$ – Jake Westfall Nov 6 '13 at 21:06
• Does using this corrected formula not fix the discrepancy? – Jake Westfall Nov 7 '13 at 15:48
• I don't have the original data. I only have summary data (means, standard deviations, and n's; see above). – user20281 Nov 7 '13 at 21:37
• I'm talking about the discrepancy in your simulations. Does this correction not fix that discrepancy? By the way, the pooled variance across all 4 conditions is the MSE. So you do have everything you need to compute Cohen's $d$ from your summary statistics according to the formula in the previous post. You don't need the raw data. – Jake Westfall Nov 7 '13 at 22:23
• Aha! Thanks for clarifying. I had made a typo here initially (apologies). In my simulations, my formula for pooled variance for the sex:disease interaction were within ~15% of MSE; is this good enough? I just want to answer the question: is the main disease severity effect common to both genders significantly bigger than the male-female difference in disease severity? This will help in interpreting our results when there are sex biases. Many thanks @JakeWestfall ! – user20281 Nov 8 '13 at 19:18