This is something a colleague asked me about the other day. They needed to work out a sample size for future research comparing pre and post intervention outcomes, based on previous findings which stated the mean and SD pre-intervention, the mean and SD post-intervention, and the pre-post correlation. But they needed the SD of the differences between pre and post in order to do the sample size calculation. Here’s how you do it.
Let’s call the correlation (this only works for Pearson correlation by the way…) Cor(pre, post), and the mean of the pre-intervention outcomes Mean(pre). Similarly we can talk about Mean (post), SD(pre), SD(post). And we will use the variances, which are just the standard deviations squared: Var(pre)=SD(pre)*SD(pre); Var(post)=SD(post)*SD(post)
Now, there is a statistic we will use in passing called the covariance, Cov(pre, post). You don’t have to know about what it is because we are just using it as a stepping stone.
Work out Cov(pre, post) = Cor(pre, post) * SD(pre) * SD(post)
Then find Var(post-pre) = Var(pre) + Var(post) – (2 * Cov(pre, post))
and finally SD(post-pre) is the square root of Var(post-pre)
The mean difference is simply the difference of the means: Mean(post-pre)=Mean(post)-Mean(pre)