In treatment studies it is common to report multiple outcome measures from the same subjects. The treatment effects on these outcomes are typically correlated so this should be taken into account when modeling the data. However, papers only publish means and standard deviations before and after the intervention. So given for example a report of three tests:
data.frame(mean.pre=c(1,5,100), sd.pre=c(0.2, 1, 10), mean.post=c(3, 2, 80), sd.post=c(0.3, 0.9, 11), rownames=c('test1','test2','test3')) mean.pre sd.pre mean.post sd.post test1 1 0.2 3 0.3 test2 5 1.0 2 0.9 test3 100 10.0 80 11.0
... would it be possible to calculate the 3 x 3 covariance matrix of the standardized mean differences
d = (mean.post - mean.pre)/sd.pre between
Bonus info: sometimes the standard deviation of the change
sd(mean.post-mean.pre) can be obtained through t-statistics and p-values and sometimes the correlation
cor(mean.pre, mean.post) is also known. Does that add anything?
If the above is not sufficient to calculate covariances, can we at least put a bound on the possible covariances given this data?