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,

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?