# What is the Bayesian counterpart to the McNemar test (of marginal homogeneity)?

In analogy to a similar post on continuous data, I ask the following. In paired discrete data, we test the frequentist hypothesis of equivalent marginal distributions of two discrete variables $X$ and $Y$ with the McNemar test. For example when both variables are dichotomous with cell probabilities

$$P(X=i,Y=j) = \pi_{ij} \quad; \quad i,j \in \{0,1\}$$

and marginal probabilities $\pi_{+j}$ and $\pi_{i+}$, marginal homogeneity is tested one-sided as

$$H_0: \pi_{1+} \ge \pi_{+1},$$

which is equivalent to

$$H_0: \pi_{10} \ge \pi_{01},$$

known as symmetry.

What is the adequate Bayesian analysis for this contingency table?

• In a pithy response, a subjective Bayesian would place zero prior probability on your null hypothesis (in the vast majority of scenarios) and therefore there is no reason for such a test. – jaradniemi Feb 15 '17 at 20:14
• Have you searched the literature? A quick search turned up this article which seems relevant. journals.math.tku.edu.tw/index.php/TKJM/article/viewFile/605/… – jaradniemi Feb 15 '17 at 20:15
• @jaradniemi Thanks. To make it more applicable to Bayesian thinking I turned it into a one-sided hypothesis. I searched the literature without success. Will check out the link you sent. – tomka Feb 15 '17 at 20:29
• With the new version, I would just treat it as a parameter estimation problem and calculate the area under the posterior consistent with the hypothesis. – jaradniemi Feb 16 '17 at 15:37