# Calculating Cohen's D for a contrast

I have run a t-test contrast by dummy coding three groups. Let's call them A, B, and C. The contrast is A and B against C. Now, I want to calculate an effect size for the resulting t-test. I am not sure of the best method. The formula that I have for Cohen's D indicates that the numerator is the difference between two means. I have the mean of C, but not sure what I should use for the mean of A and B. Similarly, the denominator is the pooled standard deviation, not sure what I should be pooling in this situation...Should I be calculating the mean and standard deviation of the combined observations in A and B?

I think you can use Cohen's $d'$ defined as $$d'= L_1/\sqrt{MS_{Error}},$$ where $L_1$ is the estimate of the contrast. In your case, this is simply $L_1=\bar{y}_A-\bar{y}_C$ (and $L_2=\bar{y}_B-\bar{y}_C$). This can be found, for example, in Equation 10.11 in Myers et al. (2010, p. 246).
There is probably also an $\eta^2$ for this, but I don't have a formula at hand right now.