Odds Ratio and 95% Confidence Intervals for Binary Matched Outcome after Propensity Score Matching

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I am conducting a propensity score match analysis on the outcome of two different new cancer treatments where the outcome is binary (cancer-free or not cancer free). Following successful matching I get my paired 2x2 contingency table for my outcome between my matched pairs which looks like below;

                                                    **Treatment 1** 
                                           Not-Cancer Free     Cancer Free   
 **Treatment 2**        Not-Cancer Free           50               39
                        Cancer Free               53               60

I'd like to compare the outcomes to figure out if one treatment is better than the other by comparing odds ratios of being cancer free. I've been advised to conduct a McNemar's test due to the matched nature of the data which I do and get a p-value of 0.17 (non-significant). However, I've also been advised that instead of simply using the odds ratio normally used for such 2x2 tables (B/C --> 39/53 = 0.78 OR) that I should calculate the odds ratio and 95% confidence intervals using the methods shown in Agresti Alan, Min Yongyi. Effects and non‐effects of paired identical observations in comparing proportions with binary matched‐pairs data. Statistics in medicine. 2004 Jan 15;23(1):65-75. as it accounts for the matched nature of the data.

Unfortunately after reading this paper numerous times (especially it's odds ratio section) I can't figure out what the equations given for the odds ratio and 95% CI calculations are that they are referring to but know that they must be in there somewhere as other papers have cited this paper when referring to their odds ratios but don't share their methodology making it difficult to traceback.

If anyone has read this paper or has experience with odds ratios for matched binary data, can you please let me know how I can go about to get matched pair odds ratios. Thank you incredibly much in advance!

propensity-score-matching confidence-interval matching r
1 Answers

You can use McNemar exact test for the paired data. A point they are making in the paper and what the exact test uses are the off-diagonal elements (b,c) in the calculations. You can use exact2x2 package (https://cran.r-project.org/web/packages/exact2x2/exact2x2.pdf) to get the test results with 95%CI:

library(exact2x2)
# Set up your data as matrix
x<-matrix(c(50,53,39,60),2,2)
mcnemar.exact(x)

Gives:

    Exact McNemar test (with central confidence intervals)

data:  x
b = 39, c = 53, p-value = 0.175
alternative hypothesis: true odds ratio is not equal to 1
95 percent confidence interval:
 0.4738071 1.1339142
sample estimates:
odds ratio 
 0.7358491
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