Use the intake data for the Catheter Study to study the association between urinary tract infection (UTI) and demographic characteristics including age, gender, and marital status (ms). There are seven levels for the \(\mathrm{ms}\) in the original data, and we combine levels 2 to 6 to make it a categorical variable with three levels. Fit a log-linear Poisson regression model for the UTI counts with age, gender, and the three-level marital status as predictors (main effects only, and no interactions)

(a) Write down the null hypothesis that there no differences in the UTI outcome among the three marital statuses in terms of the model parameters.

(b) Fit the model using the MLE and test the null hypothesis in part (a).

(c) Fit the model using EE and test the null hypothesis in part (a).

(d) What is the predicted count of UTIs of a 50-year-old unmarried (original \(\mathrm{ms}=\) 1) female?