I am trying to do a priori sample size calculation based on published results. However, I am unable to obtain a reasonable estimate of the published effect size (which is not reported) as I am unable to obtain an estimate of the pooled standard deviation or the standard deviation of the difference.
The problem resides in the fact that it is a factorial experiment with two within-subjects factors ($2 \times 3$ levels). I only have the cell means and standard deviations (i.e., for the $2 \times 3$ table) but not the marginal means for the first factor (with 2 levels) which I need.
I know that the formula for the pooled standard deviation for independent samples (taken from wikipedia) is:
$$s_p = \sqrt{\frac{(n_1-1) s_1^2 + (n_2-1) s_2^2 + \cdots + (n_k-1) s_k^2}{n_1 + n_2 + \cdots + n_k - k}}.$$
But what is the formula for pooled standard deviation for dependent samples?
Means:
1A 1B
2a 3.24 3.01
2b 2.91 2.56
2c 3.01 3.05
Standard deviations:
1A 1B
2a 0.65 0.70
2b 0.68 0.60
2c 0.46 0.53
I want to obtain the effect size between 1A and 1B (so pool over levels of factor 2). Sample size is 27.