1. Derive the score test for log(ψ) = 0 (i.e., ψ(OR)=1) for 1:1 matched data with dichotomous exposure. Compare your result to McNemar's test; why are they different?
Hint: Use the likelihood presented in class (for 1:1 matched, dichotomous exposure) to obtain the score function (first derivative of log likelihood with respect toψ) and the information (minus the expectation of the second derivative of log likelihood with respect toψ, where the expected values of n10 and n01 are those expected under the null hypothesis). Then the score test is the ratio of (score2/information), evaluated at the null hypothesis.
2. Show that the Mantel-Haenszel estimate for 1:1 matching with dichotomous exposure gives the maximum likelihood estimate for the odds ratio (i.e. the ratio of discordant pairs).