Use the longitudinal Catheter Self Management Study data for this question ("intake.csv" contains demographic and baseline information, and "catheter.csv" contains follow-up measurements). We model the binary response for having any UTIs during the last two months with gender, age, education, and treatment as predictors. Fit the following models and test whether there is a significant group difference.

(a) Model the binary UTI severity outcome with the following logistic model with random intercept

\[\operatorname{logit}\left(\pi_{i j}\right)=\alpha_{i}+\beta_{0}+\beta_{1} \text { Gender }_{i}+\beta_{2} \text { Age }_{i}+\beta_{3} \text { Education }_{i}+\beta_{4} \text { Group }_{i}\]

where \(\pi_{i j}\) is the probability of having UTIs during the last two months for the \(i\) th subject at the \(j\) th assessment, Gender is the indicator variable for female, and \(\alpha_{i} \sim N\left(0, \sigma^{2}\right)\) is the random intercept.

(b) Change the link function in (a) to probit and clog-log.

(c) Change the GLMM in (a) to GEE for the marginal model

\[\operatorname{logit}\left(\pi_{i j}\right)=\beta_{0}+\beta_{1} \text { Gender }_{i}+\beta_{2} \text { Age }_{i}+\beta_{3} \text { Education }_{i}+\beta_{4} \text { Group }_{i},\]

with the independent working correlation.

(d) Change the link function in

(e) to probit and clog-log.