# Analysis 2x2x2 contingency table

I want to analyze whether the association between Variable A (expected value of two different options: high vs. low) and Variable B (choice proportions for each option) differs between the two levels of Variable C (manufacturer of the options: X vs. Y).

I have only one measurement point per participant, i.e. each participant made only one choice (high EV option or low EV option), and whether the manufacturer was X or Y was manipulated between-ps.

During my online research I stumbbled across the Cochran Mantel Haenszel Test and logistic regression. I ran a logistic regression including A, C, and AxC as predictors and B as the outcome; the interaction was not significant. However, a CMH (A as rows, B as columns, C as levels) indicated that the association between A and B was different for the two levels of C.

Now I wonder what approach is most appropriate for my research question or how to integrate these diverging findings, respectively. Any help is highly appreciated.

• It is not super clear to me just what LR model you fitted or how you set up the MH test. Can you edit in the output? If the data-set is confidential can you provide a made-up version which illustrates the issue? – mdewey Aug 26 at 13:43
• @Penguin_Knight This is exactly what I did, thank you. However, the p-value of the interaction corresponds only to the Test of Homogeneity of the Odds Ratio, and not to the Test of Conditional Independence (which is crucial for my question, if I understood it correctly). – Lafayote Aug 26 at 14:15
• ^ I see. But from your 1st paragraph, the whole statement points to testing of homogeneity ("... differs between the two levels...") And it's unclear to me that if the two odds ratios differ (aka the association between A&B changes at different levels of C), how is that not conditional dependence. Hopefully someone more knowledgeable in this matter can help. – Penguin_Knight Aug 26 at 14:23
• @Penguin_Knight I just had an Aha effect, thank you so much! I think I could successfully sort my thoughts and now understand what is going on in my data. Again, many thanks for stimulating this. – Lafayote Aug 26 at 14:32
• You might also look into Breslow-Day test and Woolf test. – Sal Mangiafico Aug 27 at 14:47

The logistic regression and MH analysis should come up with the same conclusion, here is the logistic regression on top (with B as dependent, A, C, and AC interaction as independent) and Mantel-Haenszel test at the bottom. The two p-values are highlighted for your reference.

While both of them test homogeneity of odds ratio, I believe it can inform if conditional dependence is likely or not. As in linear regression the a significant interaction term would imply i) the association between A and B depends on C, and also ii) the association between B and C depends on A.

Here is the Stata code I used in case anyone would like to replicate:

clear
input a b c freq
1 1 1 77
1 0 1 92
1 1 0 76
1 0 0 63
0 1 1 68
0 0 1 53
0 1 0 57
0 0 0 81
end

expand freq

logit b a##c

cc b a, by(c)

• The Mantel-Haenszel test for odds ratio isn't the same test as the Cochran-Mantel-Haenszel test the OP asked about. It is the case that MH test will return the same result as the logistic regression described (p value for the interaction). It sounds like this is what the OP is looking for. The CMH test is similar to running a logistic regression for A only, and ignoring any effect of C or A*C. (Not exactly the same.) – Sal Mangiafico Aug 27 at 14:47