A five-year follow-up study was carried out to assess the relationship of diet and weight to the incidence of stomach cancer in 40- to 50-year-old males in a certain metropolitan area. Let K- denote the number of cases of stomach cancer found in the ith weight category of diet type/ as indicated in the following table:

Consider the following Poisson regression model for these data:
Model 1: E(Yij) = ijλij i = 1, 2, 3, 4 and j = l, 2
where ij = man-years at risk in the (i, j)th cell and λij = rate of development of stomach cancer in the (i, j)th cell, where λij, is modeled as follows:

a. Based on this model, give an expression for the rate ratio (RR) comparing low-cholesterol diet subjects to high-cholesterol diet subjects, adjusted for weight group.
b. Give a large-sample formula for the 95% confidence interval for the RR described in part (a).
c. How would one test whether there is a significant interaction between diet type and weight group?
d. Assuming that there is no interaction between weight group and diet type, consider the following model as an alternative to model 1:
Model 2: In λij, = a + γWi + βC
where Wi denotes the midpoint (in pounds) of weight group i.
Describe how to use a "difference between deviances" approach to evaluate whether model 2 fits the data significantly better than model 1. (State the null hypothesis, describe the test statistic in terms of deviances, and state the distribution of the test statistic, including degrees of freedom, under the null hypothesis.) How might one criticize the use of this approach?

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Low-cholesterol Diet (j 1) High-cholesterol Diet (j 2) Low weight Medlow weight (i# 2) Medhigh weight (i, 3) High weight G4) 12 Ya 41 V-11 .-10 ifweight group i otherwise In λ,-Σα.K + ßC In λί.-ΣαΜ + ßC where where if low-cholesterol diet (je 1) if high-cholesterol diet ( 2)