# How to test difference of proportion in the same group of patients tested at two time points?

I have to calculate a sample size for a study in which patients are treated for a set period of time with a drug. Subsequently, responders are followed up (but not treated) for an additional period of time to see if the symptoms return. The remaining patients (not followed up) are assumed to still be not-responders. So at the end of active treatment I expect to have 56% of responders and after end of the follow-up I assume that only 30% of original cohort can still be categorized as responders (observe that 44% of patients that did not present response at the end of treatment while not observed in the follow-up is categorized as non responders).

At first I thought about McNemar test, but the issue is with the fact that you cannot have non-responders at the conclusion of treatment become responders, so data for McNemar test look like this:

                          Follow-up
|yes|no |
End of treatment|yes |30 |26 |
|no  |0  |44 |


Observe that one cell has 0 (i.e. non responders cannot become responders). So what would be a correct way to test a difference in proportion between conclusion of treatment and follow-up treatment and as a consequence, how I should go about calculating sample size? I would be really grateful for any help!

You cannot test for a difference in proportion because one of the two proportions is zero (or one) by definition. One possible way out would be to estimate the proportion of responders who relapse during follow-up and estimate a confidence interval for that. If you can determine how wide you want your confidence interval to be then you can work out how many responders you would need and then multiply it up to get the number of people you would need to recruit bearing in mind your likely response rate.

• I am not sure if I fully understand. Proportion at the end of active treatment is expected to be 56% while in example above I assumed that after the follow-up it will be 30%. Nevertheless I thought to calculate lower bound of 95% CI (for assumed number of 100 patients) like this: binom.test(56,100,p=0.56) And then check at what number of patients this proportion will be statistically difference: binom.test(35,100,p=0.457) It is significant at 35. Then I thought to consult with the researcher if they consider such a proportion plausible. Is that correct? Thanks! – yarriofultramar Nov 30 '17 at 10:33