I'm testing the independence of two variables, A and B, stratified by C. A and B are binary variables and C is categorical (5 values). Running Fisher's exact test for A and B (all strata combined), I get:
## (B) ## (A) FALSE TRUE ## FALSE 1841 85 ## TRUE 915 74 OR: 1.75 (1.25 -- 2.44), p = 0.0007 *
where OR is the odds ratio (estimate and 95% confidence interval), and
* means that p < 0.05.
Running the same test for each stratum (C), I get:
C=1, OR: 2.31 (0.78 -- 6.13), p = 0.0815 C=2, OR: 2.75 (1.21 -- 6.15), p = 0.0088 * C=3, OR: 0.94 (0.50 -- 1.74), p = 0.8839 C=4, OR: 1.48 (0.77 -- 2.89), p = 0.2196 C=5, OR: 3.38 (0.62 -- 34.11), p = 0.1731
Finally, running Cochran-Mantel-Haenszel (CMH) test, using A, B, and C, I get:
OR: 1.56 (1.12 -- 2.18), p = 0.0089 *
The result from the CMH test suggests that A and B are not independent at each stratum (p < 0.05); however, most of the within stratum tests were non-significant, which would suggest that we don't have enough evidence to discard that A and B are independent at each stratum.
So, what conclusion is right? How to report the conclusion given those results? Can C be considered a confounding variable or not?
EDIT: I performed the Breslow-Day test for the null hypothesis that the odds ratio is the same across strata, and the p-value was 0.1424.