A cross-sectional study was carried out to assess the relationship of alcohol and smoking to blood pressure in 2,500 men ages 20 years or older in four North American population groups, each group utilizing a different clinic. The outcome variable was blood pressure status (BP): normotensive = 0, moderate hypertensive = Land severe hypertensive = 2. The primary exposure variables were alcohol status (ALC: (nondrinker = 0, light drinker = 1, and heavy drinker = 2, with 0 being the referent category) and smoking status (SMK: none = 0, less than 2 packs per day = 1, and 2 + packs per day = 2, with 0 being the referent category). Variables consider! I for control included AGE (under 40 = 0, at least 40 = 1), PA (physical activity: not regular = 0, regular = 1), and CLINIC (4 categories, with clinic 4 being the referent category).
a. State the logit form of a no-interaction ordinal (proportional odds) logistic mrxk that would describe the relationship of the outcome BP (treated ordinally) with the predictors ALC, SMK, AGE, PA, and CLINIC. In stating this model, make sure to treat all predictors as categorical variables.
(In questions to follow, we will refer to the exposure variables ALC and SMK as E variables; the control variables AGE, PA, and CLINIC will be referred to as K variables.)
b. Consider the following (crude) odds ratio estimates obtained from the crude 3 × 3 table relating ALC to BP:
1, = BP=2VS,0-1 (ALC = 2vs.O), 2 = BP=2VS,0–1,(ALC = 1V.,0)
3 = BP=1–2vs, 0(ALC = 2 vs. 0), 4 = BP=1 – 2 vs. 0(ALC = 1VS.0)
What relationships among the above odds ratios would one expect to see if the proportional odds model was appropriate for the variable ALC?
c. What does the score test for the proportional odds model allow one to evaluate, and how does it work (i.e., state the null hypothesis and how to proceed, de ing on whether you reject or do not reject the null hypothesis)?
d. For the model described in part (a), give an expression for the odds of being a moderate or severe hypertensive (i.e., BP S: 1) who is a heavy-drinking, 45-y old, 2+ pack-a-day smoker who does regular physical activity and comes from clinic 4.
e. For the model described in part (a), give an expression for the odds ratio for a moderate or severe hypertensive (i.e., BP > 1) that compares a heavy-drinking 2 + pack-a-day smoker to a light-drinking nonsmoker, controlling for AGE, and CLINIC.
f. For the model described in part (a), give an expression for the odds ratio for a normotensive (i.e., BP = 0) that compares a heavy-drinking, 2 + pack-a-day smoker to a light-drinking nonsmoker, controlling for AGE, PA, and CIMl
g. Suppose that the model in part (a) was extended to allow for two-way interactions (i.e., products involving two variables) between alcohol and age, alcohol and physical activity, alcohol and clinic, smoking and age, smoking and physical activity, smoking and clinic, and alcohol and smoking. Assuming that alcohol status and smoking status are exposure (E)variables of interest and that AGE, PA, and CLINIC ate control (V)variables, state which variables in the interaction model are EiVj variables and which are EiEj{variables. (Make sure to define the variables explicitly, keeping in mind that all the variables in the model are being treated as categorical variables.)
h. Starting with the model described in part (g), suppose that it was decided to determine a "best" model by first testing for significant interaction among the EjVj variables. Describe how one would simultaneously test for the significance of at least one of the EiVj product terms in this model. Make sure to state the null hypothesis in terms of model parameters, describe the formula for the test statistic, and give the distribution and degrees of freedom of the test statistic under the null hypothesis.
i. Assume that the only significant interaction terms among the EiVj variables were product terms involving alcohol status with age and smoking status with physical activity. State the model that remains for further assessment after this interaction assessment. Make sure to write the model in terms of the variables alcohol status, smoking status, age, and physical activity and the appropriate product terms.
j. Considering the answer to part (i), what is the formula for the odds ratio that compares heavy-drinking, 2 + pack-a-day smokers to light-drinking nonsmokers, controlling for AGE, PA, and CLINIC?
k. Using the odds ratio formula described in part (j), give a formula for a 95% confidence interval for the odds ratio for a 30-year-old man who does not do regular physical activity.
The general 95% CI formula for an odds ratio of the form exp[L], where = c1β1 +…..+ ckβk, is given by

Where

1. Suppose that one decides that there is no interaction between alcohol and smoking, that CLINIC is not a confounder but needs to be considered for reasons of precision, and that the only two significant interaction terms are between ALC and AGE and between SMK and PA. What is your "final" odds ratio expression for a moderate or severe hypertensive (i.e., BP > 1) that compares a heavy-drinking, 2 + pack-a-day smoker to a light-drinking nonsmoker, controlling for AGE, PA, and CLINIC?