I am working on a meta-analysis and one of the studies has mentioned pre-operative data in mean and standard deviation and given mean difference and standard deviation along with the sample size. How do I calculate post-operative mean and standard deviation?
Hello Muhammad Ahmed. If you know the covariance between pre-op and post-op scores, you can do what the Stata code pasted below shows. If you do not know the covariance, the best you can do may be to plug in a range of estimates and carry out a kind of sensitivity analysis.
* Stata code
clear *
webuse fuel
generate diff = mpg1-mpg2
quietly summarize mpg1
generate n = r(N) in 1
generate m1 = r(mean) in 1
generate sd1 = r(sd) in 1
quietly summarize diff
generate mdiff = r(mean) in 1
generate sddiff = r(sd) in 1
quietly correlate mpg1 mpg2, covariance
generate cov12 = r(cov_12) in 1
* MeanDiff = m1 - m2
* MeanDiff + m2 = m1
* m2 = m1 - MeanDiff
generate m2 = m1 - mdiff in 1
* VarDiff = sd1^2 + sd2^2 - 2*cov12
* VarDiff - sd1^2 + 2*cov12 = sd2^2
generate sd2 = sqrt(sddiff^2 - sd1^2 + 2*cov12) in 1
list n m2 sd2 in 1
summarize mpg2
Here is the output from the last couple commands.
. list n m2 sd2 in 1, clean
n m2 sd2
1. 12 22.75 3.250874
. summarize mpg2
Variable | Obs Mean Std. dev. Min Max
-------------+---------------------------------------------------------
mpg2 | 12 22.75 3.250874 17 28
Here are the 10 pairs of data points, in case you want to tinker using your own software.
mpg1 mpg2
20 24
23 25
21 21
25 22
18 23
17 18
18 17
24 28
20 24
24 27
23 21
19 23
HTH.
The means are just subtracted to get the mean difference:
- meandiff = meanpost - meanpre
from this it follows that
- meanpost = meanpre + meandiff
The standard deviation of the difference can be obtained from the variances and the co-variance:
- vardiff = varpre + varpost - 2covpre,post
You are interested in the post-value:
- varpost = vardiff + 2covpre,post - varpre
actually you want the standard deviation, what is the square-root of the variance:
- sdpost = sqrt(varpost)
- = sqrt(vardiff + 2covpre,post - varpre)
- = sqrt(sddiff² + 2covpre,post - sdpre²)
If you don't know the covariance, you cannot get the standard deviation of the post values. However, you might give an upper and lower bound by plugging in some reasonable upper and lower values for the covariance (which you likely don't have).
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