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 is the square root of (taken from wikipedia) .
But what is the formula for pooled standard deviation for dependent samples?
1A 1B 2a 3.24 3.01 2b 2.91 2.56 2c 3.01 3.05
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.