We consider a data set collected from a study on prostate cancer, available from http://www.biostat.au.dk/teaching/postreg/AllData.htm. One of study objectives is to see if information collected at a baseline exam of a patient with prostate cancer, could predict whether the tumor has penetrated the prostatic capsule. The data set contains 380 observations and 5 variables. A brief description is given in Table 8.6. Denote π = Pr {Y = 1|X1, X2, X3, X4} . Logistic regression is applied to model the relationship. To account for DPROS, three dummy variables {Z1, Z2, Z3} are introduced with the reference cell coding scheme.

(a) We first fit an additive logistic model Model I: logit(π) = β0 + β1Z1 + β2Z2 + β3Z3 + β4X3 + β5X4.

Provided in Table 8.7 are some of ML fitting results from SAS PROC LOGISTIC. Answer the following question(s) based on Model I. i. Suppose that there are two individuals, A and B, both having detection of capsular involvement (DCAPS, X3) but no nodule (DPROS, X2) found in the rectal exam. Patient A has a PSA value of 10 mg/ml higher than Patient B. Obtain a point estimate of the odds ratio for having tumor penetration that compares A versus B. Construct a Wald 95% confidence interval for this odds ratio. ii. Given an individual who has detection of capsular involvement (DCAPS, X3) but no nodule (DPROS, X2) found in the rectal exam and a PSA value of 20 mg/ml, apply Model I to predict the probability for this individual to have tumor penetration of prostatic capsule. Construct a Wald 95% confidence interval for this probability.

(b) Next, we consider an interaction model Model II: logit(π) = β0+β1Z1+β2Z2+β3Z3+β4X3+β5X4+β6X3·X4. Provided in Table 8.8 are some of ML fitting results from SAS PROC LOGISTIC. Answer the following question(s) based on Model II. i. Suppose that there are two individuals, A and B, both having detection of capsular involvement (DCAPS, X3) but no nodule