Problem - A clinical trial was conducted to determine whether a hormone treatment benefits women who were treated previously for breast cancer. Each subject entered the clinical trial when she had a recurrence. She was then treated by irradiation and assigned to either a hormone therapy group or a control group. The observation of interest is the time until a second recurrence, which may be assumed to follow an exponential distribution with parameter τθ (hormone therapy group) or θ (control group). Many of the women did not have a second recurrence before the clinical trial was concluded, so that their recurrence times are censored.
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TABLE - Breast cancer data.
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Hormone Treated
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Control
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Recurrence Times
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2
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4
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6
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9
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9
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9
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1
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4
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6
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7
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13
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24
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13
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14
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18
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23
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31
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32
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25
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35
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35
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39
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|
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33
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34
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43
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|
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|
|
|
|
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|
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Censoring Times
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10
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14
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14
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16
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17
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18
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I
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1
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3
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4
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5
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8
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18
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19
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20
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20
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21
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21
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10
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11
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13
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14
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14
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15
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23
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24
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29
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29
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30
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30
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17
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19
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20
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22
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24
|
24
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31
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31
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31
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33
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35
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37
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24
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25
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26
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26
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26
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28
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40
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41
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42
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42
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44
|
46
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29
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29
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32
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35
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38
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39
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48
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49
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51
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53
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54
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54
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40
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41
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44
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45
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47
|
47
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|
55
|
56
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|
|
|
|
47
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50
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50
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51
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In Table, a censoring time M means that a woman was observed for M months and did not have a recurrence during that time period, so that her recurrence time is known to exceed M months. For example, 15 women who received the hormone treatment suffered recurrences, and the total of their recurrence times is 280 months.
Let yHi = (xHi , δHi) be the data for the ith person in the hormone group, where xHi is the time and δHi equals 1 if xHi is a recurrence time and 0 if a censored time. The data for the control group can be written similarly.
The likelihood is then
L(θ, τ|y) ∝ θ(ΣδCi +ΣδHi)τ(ΣδHi) exp8 {-θ ΣxCi - τθΣxHi}.
You've been hired by the drug company to analyze their data. They want to know if the hormone treatment works, so the task is to find the marginal posterior distribution of τ using the Gibbs sampler. In a Bayesian analysis of these data, use the conjugate prior
f(θ, τ) ∝ θaτb exp{-cθ - dθτ}.
Physicians who have worked extensively with this hormone treatment have indicated that reasonable values for the hyperparameters are (a, b, c, d) = (3, 1, 60, 120).
a. Summarize and plot the data as appropriate.
b. Derive the conditional distributions necessary to implement the Gibbs sampler.
c. Program and run your Gibbs sampler. Use a suite of convergence diagnostics to evaluate the convergence and mixing of your sampler. Interpret the diagnostics.
d. Compute summary statistics of the estimated joint posterior distribution, including marginal means, standard deviations, and 95% probability intervals for each parameter. Make a table of these results.
e. Create a graph which shows the prior and estimated posterior distribution for τ superimposed on the same scale.
f. Interpret your results for the drug company. Specifically, what does your estimate of τ mean for the clinical trial? Are the recurrence times for the hormone group significantly different from those for the control group?
g. A common criticism of Bayesian analyses is that the results are highly dependent on the priors. Investigate this issue by repeating your Gibbs sampler for values of the hyperparameters that are half and double the original hyperparameter values. Provide a table of summary statistics to compare your results. This is called a sensitivity analysis. Based on your results, what recommendations do you have for the drug company regarding the sensitivity of your results to hyperparameter values?
I will need the equations and R CODES. R codes are VERY important. I am attaching the book information and the data file.
Attachment:- Assignment.rar