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### Question for Chegg:A clinical trial was conducted to determine if hormone treatment benefits women who were previously treated for breast cancer. Each subject entered

### Question for Chegg:A clinical trial was conducted to determine if hormone treatment benefits women who were previously treated for breast cancer. Each subject entered the clinical trial when they had a recurrence. They were then treated with irradiation and assigned to a hormone treatment group or a control group. The observation of interest is the time until a second recurrence, which can be assumed to follow an exponential distribution with parameter (hormone therapy group) or (control group).In the table, a censoring time \( Q \) means that a woman was observed for \( Q \) months and did not have a recurrence during that time, so the recurrence time exceeds \( Q \) months. For example, 15 women who received hormone treatment had recurrences, and the total recurrence time was 280 months.Let \( y_i^H =(x_i^H,\delta_i^H)\) be the data for the \( i \)-th person in the hormone group, where \( x_i^H \) is the time \( y_i^H \) if it is a recurrence time and 0 if it is a censoring time. The data for the control group can be written similarly.The likelihood is given by:\[ p(y|\theta,\tau)\propto (\theta \tau)^{\sum \delta_i^H +\sum \delta_i^C}\exp \left\{-\theta \left(\sum x_i^C +\tau \sum x_i^H \right)\right\}\]You have been hired by the pharmaceutical company to analyze their data. They want to know if hormone treatment works. Your task is to find the marginal posterior distribution of \(\tau \) using Gibbs sampling.The conjugate prior is given by:\[ p(\theta,\tau)\propto \theta^{a-1}\tau^{b-1}\exp \{-c\theta - d\tau \theta \}\]Doctors who have worked extensively with this treatment have indicated that reasonable values for the hyperparameters are \((a, b, c, d)=(3,1,60,120)\).
1. Derive the necessary conditional distributions to implement Gibbs sampling.
2. Program and run your Gibbs sampler. Use a set of convergence diagnostics to evaluate the convergence and mixing of your algorithm. Interpret the diagnostics.
3. Calculate summary statistics of the estimated joint posterior distribution, including marginal means, standard deviations, and a 95% credible interval for each parameter. Create a table with these results.

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