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MATH39512 Survival Analysis for Actuarial Science: example sheet 7
*=easy, **=intermediate, ***=difficult
* Exercise 7.1
Consider again Exercise 6.2.
(a) Use R to reproduce (i) the maximum likelihood estimate of the regression coefficient, (ii) the corresponding maximum (partial) likelihood value, (iii) the value of the test statistic for the score test and (iv) the value of the test statistic for the likelihood ratio test.
(b) Use R to obtain a 95% asymptotic confidence interval for the regression coefficient and the exponential of the regression coefficient.
** Exercise 7.2
Load the KMsurv package via library(KMsurv) that appeared earlier in Exercise 5.3. Load the data set kidtran by using the command data(kidtran). This gives a data frame called kidtran which consists of data on the time to death of 863 kidney transplant patients. Use help(kidtran) to get further information on this data set.
(a) In a Cox PH model with covariates gender, race and age, find the mle of the exponential of the regression coefficient of each covariate.
(b) Estimate the relative risk (or hazard ratio) of a male relative to a female.
(c) Estimate the hazard ratio of a black person relative to a white person.
(d) Estimate the hazard ratio of a black male relative to a white female.
(e) Use the likelihood ratio test to find out which of the three covariates have a significant, say at the 5% level, influence on the survival times of kidney transplant patients.
(f) In a Cox PH model with covariates gender, age and an interaction term gender ×age, find the mle of the regression coefficient of each covariate.
(g) How should the regression coefficient of the interaction term be interpreted? Is this interac- tion effect significant?
** Exercise 7.3
Consider again the data set channing from the KMsurv package considered in Exercise 5.3. (a) Work with time since entry into the retirement centre as the time scale.
(i) In a Cox PH model with the covariates age at entry and gender, find the mle of the regression coefficient of each covariate.
(ii) Perform the Wald test at the 5% significance level to test whether there is a difference in the survival time distribution between male and female residents and report your conclusions.
(iii) Use the Breslow estimate to estimate the probability that a female resident currently aged 75 years exactly and currently having been in the retirement centre for exactly 20 months is still alive exactly 5 years later.
(iv) Check if the proportional hazards assumption for the gender covariate is appropriate.
(b) Work with age as the time scale.
(i) In a Cox PH model with a single covariate gender, find the mle of the regression coeffi- cient of this covariate.
(ii) Perform the Wald test at the 5% significance level to test whether there is a difference in the survival time distribution between male and female residents and report your conclusions.
(iii) Use the Breslow estimate to estimate the probability that a female resident currently aged 75 years exactly and currently having been in the retirement centre for exactly 20 months is still alive exactly 5 years later.
(iv) Check if the proportional hazards assumption for the gender covariate is appropriate.
** Exercise 7.4
[This question is meant to be done (mostly) without the use of the survival package in R in order to see how estimation works differently in a stratified Cox PH model versus an unstratified Cox PH model.]
For a study into recidivism a number of convicted persons who spent time in jail were observed for some period of time. The data below displays for each individual how long it took to commit another offence carrying a prison sentence after he/she had been released from jail. Here a + next to the survival time indicates that the corresponding person did not carryout another major offence while being observed. Also recorded and displayed in the table are the amount of time each person spent in prison and whether he/she has a low or high income.
time to reoffence (in months) 1 4+ 6 12+ 15
amount of time jailed (in months) 5 2 9 10 10
income (h=high, l=low) l l h l h
(a) For modelling the rate of recidivism (i.e. the hazard rate of the time to recommit an offence after being released from prison) consider a Cox proportional hazards model with the amount of time jailed and income acting as covariates.
(i) Give the form of the hazard rate of an (arbitrary) individual in this Cox proportional hazards model.
(ii) Derive an explicit expression in terms of the two regression coefficients for the partial likelihood in this Cox proportional hazards model given the above data.
(iii) Estimate, within this Cox proportional hazards model, the probability that a convicted person who was jailed for 7 months and has a high income, will commit another offence within a year after release. Here you should use R to compute the mles of the regression coefficients.
(b) Consider now a stratified version of of the original Cox proportional hazards model where one stratifies over the income covariate.
(i) Give the form of the hazard rate of an (arbitrary) individual in this stratified Cox proportional hazards model.
(ii) Derive an explicit expression in terms of the single regression coefficient for the combined likelihood in this stratified Cox proportional hazards model given the above data.
(iii) Estimate the same probability as in (a)(iii) but now work within the stratified Cox proportional hazards model. Here you can use that the mle of the regression coefficient is −0.1142.