Model 1: Let's consider the logistic regression model, which we will refer to as Model 1, given by
log(pi / [1-pi]) = 0.25 + 0.32*X1 + 0.70*X2 + 0.50*X3 (M1),
Where X3 is an indicator variable with X3=0 if the observation is from Group A and X3=1 if the observation is from Group B. The likelihood value for this fitted model on 100 observations is 850.
(1) For X1=2 and X2=1 compute the log-odds ratio for each group, i.e. X3=0 and X3=1.
(2) For X1=2 and X2=1 compute the odds ratio for each group, i.e. X3=0 and X3=1.
(3) For X1=2 and X2=1 compute the probability of an event for each group, i.e. X3=0 and X3=1.
(4) Using the equation for M1, compute the relative odds associated with X3, i.e. the relative odds ratio of Group B compared to Group A.
(5) Use the odds ratios for each group to compute the relative odds of Group B to Group A. How does this number compare to the result in Question #4. Does this make sense?
Model 2: Now let's consider an alternate logistic regression model, which we will refer to as Model 2, given by
log(pi / [1-pi]) = 0.25 + 0.32*X1 + 0.70*X2 + 0.50*X3 + 0.1*X4 (M2),
Where X3 is an indicator variable with X3=0 if the observation is from Group A and X3=1 if the observation is from Group B. The likelihood value from fitting this model to the same 100 observations as M1 is 910.
(6) Use the G statistic to perform a likelihood ratio test of nested models for M1 and M2. State the hypothesis that is being tested, compute the test statistic, and test the statistical significance using a critical value for alpha=0.05 From these results should we prefer M1 or M2?