The following questions concern Poisson regression models fit to fictitious follow-up study data in which rates of disease are modeled as a function of age and smoking status. The SAS program codes and output for six models are provided on pages 775-780 following the last problem in this chapter. The dependent variable is called COUNT and the independent variables are AGE and SMOKE. There are three age groups (defined by age midpoint) and two levels of smoking status. The data are given in the following table:

AGE = 25 if 15 AGE = 45 if 35 AGE = 75 if 55 a. Using the problem code for the available computer output, note that the difference between model 1 and model 2 is that model 2 does not contain an offset term. Why does the lack of an offset term for model 2 suggest that smoking is preventive of the disease, while model 1 suggests that smoking is a risk factor?
b. Suppose that both model 1 and model 2 were fit to a different data set, and it was found that there was no difference in the estimated effects of smoking and age. What would that say about the structure of the data?
c. Use the Wald test to evaluate the statistical significance of the interaction term in model 3. In answering this question, state the model, state the null hypothesis, perform the statistical test, show the calculations, and draw conclusions.
d. What assumption is made for model 1 that is not made for model 5 concerning the effect of age on the rate of disease? Based on the output, should there be a preference for either of these two models? Explain.
e. Use deviance information to perform an LR test to assess the significance of the age variables in model 5 (controlling for smoking). Make sure to discuss appropri¬ate conclusions.
f. Use deviance information to perform an LR test to assess the significance of the interaction terms in model 6. Make sure to discuss appropriate conclusions.
g. Why is the deviance for model 6 equal to zero?
h. Determine the estimated rate ratio and 95% confidence interval for comparing the oldest age group (55 i. Determine the estimated rate ratio and 95% confidence interval for comparing the oldest age group (55 j. Are the results for part (h) different from the results for part (i)? Explain briefly.
k. Using the computer output, fill in the blanks in the following Poisson regression "ANOVA" table based on fitting models 1-6.

1. Based on answers to the previous questions and on any other considerations related to the computer output from fitting models 1-6, which of the six models seems to be the "best" model? Explain.

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SMOKE 0 SMOKE I AGE 25 AGE 45 AGE 75 | AGE-25 AGE-45 AGE-75 2,000 35 100 Person-time Count 1,000 15 3,500 36 1,500 28 1,000 32 Model number Number of Parameters DevianceDeviance d.f. Deviance/d.f 3 6.3636