The following set of questions relates to using Poisson regression methods to analyze data from an in vitro study of human chromosome damage. In this study, using Poisson regression is appropriate because we have "count" response data, where each count is the number of broken chromosomes in a sample of 100 cells taken from each of n = 40 individuals. (A Poisson distribution is appropriate for describing counts per unit of space€”e.g., 100 cells€”as well as for describing counts per unit of time.)
The design of the study described here involved randomly assigning each of the cell samples for the 40 individuals to one of four treatment groups consisting of 10 individuals each. The 10 sets of 100 cells in each treatment group were exposed to a particular combination of two drugs, A and B, as follows:

The exposed cells were then examined for chromosome breakage, and the number of broken chromosomes in the 100 cells for each individual was counted. The investigator wishes to assess the effects of drug use, and of drug interaction, on the rate of chromosome damage.
Let Yij denote the random variable representing the number of chromosome breaks counted for each individual i(i = 1, 2,..., 10) within treatment group j(j = 1, 2, 3, 4). Also, let ij = 100, where ijλij denotes the amount of "person-space" of observation. Assume that the drugs act in a multiplicative fashion on the rate of chromosome breakage (λij). The Poisson regression model under consideration can be written as

where Xijk denotes the value of the kth predictor for the ith individual in the jth treatment group
a. Based on the preceding scenario, write out the form of the Poisson regression model for In λij a function of the predictors of interest in this study. In doing so, make sure to explicitly define a set of categorical predictors to include in this model; these predictors should reflect the primary objective of the study.
b. Based on the model specified in part (a), describe two alternative ways one can test the null hypothesis of no interaction between the drugs. Include in the answer the null hypothesis (in terms of regression coefficients), the formula for the test statistic(s), the degrees of freedom for each test, and the distribution of the test statistic under the null hypothesis.
c. Fill in the blanks in the following Poisson regression ANOVA table, which is based on fitting various models to describe the rate of chromosome breakage:
Number of Deviance

d. According to the ANOVA table, which models provide good fit to the data? Explain briefly.
e. According to the ANOVA table, does drug A have a significant main effect? Explain briefly.
f. According to the ANOVA table, does drug B have a significant effect over and above the effect of drug A? Explain briefly.
g. According to the ANOVA table, do drugs A and B have a significant interaction effect? Explain briefly.
h. According to the ANOVA table (and in light of your answers to parts (d) through (g)), provide an appropriate expression for the rate ratio for the effect of drug A given drug B status relative to baseline and an expression for the rate ratio for the effect of drug B given drug A status relative to baseline.

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Treatment Groupj Number of Individuals n Drug Combination 10 10 10 10 neither A nor B A only Bonly both A and IB 40 Sp Number of Parameters Deviance d.f 39 Model for InA Deviance (D) Baseline (i.c., B (II) Main effect of drug A only II Main effect of drug B only (IV) Main effects of both drug A and drug B (V) All main effects and interaction 90 85 84 40 30