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ST227 Survival Analysis
生存分析代写 Instructions to candidates This paper contains four questions. Answer ALL FOUR. Question 1: 25 marks Question 2: 30 marks Question 3: 25 marks
Instructions to candidates
This paper contains four questions. Answer ALL FOUR.
Question 1: 25 marks
Question 2: 30 marks
Question 3: 25 marks
Question 4: 20 marks
The marks in brackets reflect marks for each part of a question.
Time allowed Reading Time:
(b) Define in R the density function for T20. This definition may involve a numerical integral. [ 5 marks]
(c) Calculate the expected remaining lifetime for a 20-year-old individual. [ 5 marks]
(d) Define in R the cumulative distribution function of T20. This definition may involve a numerical integral. [ 5 marks]
(e) Discuss how one can numerically find the median of this distribution. Outline the approach only, you are not required to solve for the median. [ 5 marks]
[Total 25 marks]
This questions is divided into two parts. Both parts use the same data set of fully observed lifetimes given below:
80 75 38 45 62 65 77 92 65 60
55 67 72 46 64 35 68 52 45 94
i. Derive algebraically the probability density function for lifetime and write down the joint-likelihood of the given sample. [5 marks]
ii. Using optim and the initial values λ0 = 67 and γ0 = 0.2233, numerically obtain the maximum likelihood estimators of the model parameters. [8 marks]
iii. For the purpose of lifetime modelling, what range of values for γ would yield a sensible model? [2 marks]
[Total 30 marks]
Cancer patients who are in remission are observed and the number of days until the symptoms reappear is recorded. Some records have been right-censored. The data set is provided in a spreadsheet named cancer.xlsx and the columns therein are:
• time: the time until reappearance of symptoms in number of days.
• event: an indicator variable taking value 0 if the record has been right-censored and 1 if fully observed.
• fullyObserved: logical variable indicating whether the record has been fully observed.
• sex: categorical variable with value 0 for male (the reference group) and 1 for female.
(a) Calculate the Kaplan-Meier estimate for survival probabilities. [15 marks]
In this question, we will fit a Cox Proportional Hazard model on the same data set in Question 3, with time as the response variable and sex as the categorical covariate.
(a) By using the survival package, calculate the MLE for the Cox Proportional Hazard Model. [10 marks]
(b) Based on the output you have generated, perform the z-test, Score test, and Likelihood Ratio test on the following hypotheses:
H0 : β = 0, vs H1 : β ≠ 0.
(c) It is hypothesised that cases with remission time greater than 125 belong to a different class of cancer. Create a data frame containing this subset of cases. [3 marks]
[Total 20 marks]