@Rick_SAS: Thank you for your reply and sorry for a late response from me. I agree with you that I have a small sample size as the contingency table is matrix(data=c(17, 31, 5, 1), nrow=2, ncol=2, byrow=FALSE). However, the numbers in my first post are the exact number rather than asymptotic approximation. As you see that I take p-value from the first table (last row) and the exact confidence limits from the last two rows of the last table (see attachment please). They are all exact as I understand correctly from the output of SAS. Please see here an article talking about this problem. https://academic.oup.com/biostatistics/article/11/2/373/267728
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