MLE, equivalence of hypotheses, odds ratio

4 Problem 4: Equivalence of hypotheses - 15 points Consider a binary random variable X and Y, and let army : P[X : :12, Y : 3;] denote the probability of each outcome, i.e. (X: Y) = {(05 U) (05 1)3 (13 0)1 (13 1)} Let nmy denote the counts of each of these outcomes in a contingency table. 1. Explain how testing Whether the probability of Y = 1 differs when X = 0 versus X = 1 i.e. P[Y 1|X 0] P[Y 1|X 1] is the same as testing whether the odds ratio for the association between (X ,Y) is 1 (but may use different test statistics). 2. Explain why the MLE for the expected counts E(n:1:g) under the null that X ,Y are indepndent is the row totals times column totals divided by the total sample size. (hint... What is the MLE for army under the null?) You do not need to do any derivations here1 and can get full credit through a careful explanation discussing various things in class