UNSW-MATH5916 Survival Analysis

Assignment 1

Please include this cover sheet with your individual submission.Submission date: Friday 20 April 2018, 5:00pm.

Name:

Student Number:

1

Assignment submission: Please submit your assignment by email (as an attachment) to a.pope@unsw.edu.au.

You may submit your assignment in hand-written form (after scanning or photographing), as an Rmarkdown notebook or some combination of these (see the Rmarkdown example in the folder Rmarkdown example), and you should include any R source code you use in a form that can be run independently of your particular computer set-up. Whichever method you use to submit your assignment, there should be included a pdf or jpeg file which represents the definitive hardcopy output.

If you wish to use other means of presenting your assignment (e.g. Matlab, Jupyter notebook) please send an email to a.pope@unsw.edu.au to discuss acceptability of your proposal.

Please provide explanations with your responses to questions, at a level where someone knowledgeable in statistics but unfamiliar with the details of the course can understand what you are doing. Just algebra, calculations or computer output is not enough. Be selective though!

1. Consider an experiment with time-to-event data subject to right censoring. Assume all times are measured from the same origin. For individual i in the study, let Ui denote the actual survival time, Ti denote the observed time, and Ci denote the censoring time. Let Di be an event indicator taking the value 1 if the event is observed to occur and 0 otherwise.

1. (a)   Explain why Ti = min{Ui, Ci}.

2. (b)   Explain why Di = I{Ti = Ui} = 1 − I{Ti = Ci}, where I is an indicator function (i.e. it takes value 1 when the statement inside the parentheses is true, 0 when it is false).

3. (c)   Assume type I censoring. Show that the contribution to the likelihood for indi- vidual i can be written

   h(Ti)Di exp{−H(Ti)}.
   Deduce that the likelihood depends on censoring times only for individuals actu-

   ally censored.

4. (d)   Now assume random independent censoring. Let g represent the density function of the censoring time random variables, and let G be their survival function. Show that in this case, the contribution to the likelihood for individual i is

   h(Ti)Di exp{−H(Ti)}g(Ti)1−DiG(Ti)Di.

5. (e)   Explain why the maximum likelihood estimates obtained from likelihoods con-

   structed using (c) and (d) are equivalent under independent censoring.

2

2. Example 4.3 of the Survival Analysis notes described a study of time to infection for burns patients, where the objective was to compare the effectiveness of two bathing care methods for preventing infection. An extract from the dataset is available from the course web page (see Ass1burn.csv in the Data directory). The four columns in the dataset are: (1) the observation number; (2) the treatment indicator (0=rou- tine method, 1=new method), (3) the censoring indicator (0=right censored due to discharge from hospital, 1=infection occurred), and (4) time to infection or censoring (days).

1. (a)   Fit Weibull models to the data, separately for each treatment group. Give pa- rameter estimates and 95% confidence intervals for the parameterization of the Weibull distribution used in the notes (where h(t) = λγtγ−1). Assume the pa- rameter estimates are normally distributed.

2. (b)   Obtain 95% confidence intervals for the median survival times in each group, based on the models in (a) and assuming a normal distribution for the log of the estimated median survival times.

3. (c)   Consider the Weibull model represented by the log-linear model log Ti = μ + αZi + σεi

   where εi has survival function Sε(u) = P(ε > u) = exp(−exp(u)) and Zi is the treatment group indicator: Zi = 0 if patient i receives the routine bathing care method and Zi = 1 if patient i receives the new bathing care method. Show that the survival function of Ti can be written as

   Si(t) = exp(−λβZitγ)
   where λ = exp(−μ/σ) and γ = 1/σ. Express β as a function of μ, α and σ. Give

   an interpretation of β in terms of hazard.

4. (d)   Fit the model in (c) to the burns data. Obtain maximum likelihood estimates for the parameters λ, β and γ. Also derive a 95% confidence interval for β, assuming normality, and interpret this interval.

5. (e)   On the same graph, plot the estimated hazard functions obtained from the model in (c) for the two treatment groups.

6. (f)   Test at the 5% level whether there is a difference in time to infection for the two bathing care methods, based on the model in (c).

7. (g)   Do you think the Weibull distribution produces a significantly better fit than the exponential distribution for these data? Provide an analysis to justify your answer.