Matched pairs design: individual confidence intervals for pre and post I have a matched pairs design. I have a 2x2 table showing success/failure in pre and post:





Post (yes)
Post (no)




Pre (yes)
0
10


Pre (no)
20
30




My colleague would like a confidence interval on the proportion of post (yes). How can I obtain one?
My start: The observations are conditionally independent given the "pre" status, right?
So can I just use the standard Wald-type CI: $\hat{p} \pm z_{\alpha/2}\sqrt{\hat{p}(1-\hat{p})/n}$, where $\hat{p} = (0+20)/(0+10+20+30)$, $z_{\alpha/2}=1.96$ for a 95% confidence interval and $n=0+10+20+30 = 60$?
 A: If I am understanding your question correctly, yes you can.  You could consider incorporating a log or logit link function, i.e.
$$\text{exp}\Big( \text{log}\{\hat{p}\}\pm z_{\alpha/2}\Big[\sqrt{\hat{p}(1-\hat{p})/n}\Big]/\hat{p}\Big)$$
$$\text{or}$$
$$\text{logit}^{-1}\Big( \text{log}\Big\{\frac{\hat{p}}{1-\hat{p}}\Big\}\pm z_{\alpha/2}\Big[\sqrt{\hat{p}(1-\hat{p})/n}\Big]/[\hat{p}(1-\hat{p})]\Big).$$
The log link function is useful if your estimates are near 0 and your sample size is small.  The logit link function is useful if your estimates are near 0 or near 1 and your sample size is small.  Your estimate of $20/60=0.33$ is not close to zero by most standards, but the link functions may still help to improve the coverage of the confidence interval.  For your estimate of $0.33$ they will shorten the lower confidence limit and lengthen the upper limit relative to a Wald interval using an identity link.  Here log refers to the natural log with base $e$.










Wald with identity link
(0.21, 0.45)


Wald with log link
(0.23, 0.48)


Wald with logit link
(0.23, 0.46)




With your sample size all of the intervals look similar.
If, for example, you had witnessed 3 events in a sample of size 30 then the confidence limits would be










Wald with identity link
(-0.01, 0.21)


Wald with log link
(0.03, 0.29)


Wald with logit link
(0.03, 0.27)




Of course we would never report a negative proportion so the lower limit using the identity link would be truncated to 0 (not inclusive).
