In this question, we change the response in Problem9.23 to a three-level response by grouping the counts of UTIs into three levels, 0,1 , and \(\geq 2\). Fit the following models and test whether there is a significant group difference.

(a) Model the ordinal three-level UTI count outcome with the following cumulative model with random intercept

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

\(j=1\) and 2 , where \(\gamma_{i 1}\) and \(\gamma_{i 2}\) are the probabilities of having no UTI and having at most one UTI, respectively, and \(\alpha_{i} \sim N\left(0, \sigma^{2}\right)\) is the random intercept.

(b) Change the link function in (a) to probit and cloglog.

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

\[\operatorname{logit}\left(\gamma_{i j}\right)=\beta_{0}+\beta_{1} \text { Gender }_{i}+\beta_{2} \text { Age }_{i}+\beta_{3} \text { Education }_{i}+\beta_{4} \text { Group }_{i},\]\(j=1\) and 2 , with the independent working correlation.

(d) Change the link function in

(e) to probit and cloglog.

Problem  9.23 is:

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9.23 Use the longitudinal Catheter Self Management Study data for this question ("in- take.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 logit (Tij) = ai + Bo+ BGender; + B2 Age; + B3 Education; + B4Groupi, where Tij is the probability of having UTIs during the last two months for the ith subject at the jth assessment, Gender is the indicator variable for female, and a ~N(0, 2) 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 logit (i) Bo+ BGender; + B2 Age, + B3Education + B4Groupi, with the independent working correlation. (d) Change the link function in (c) to probit and clog-log.