# 辅导 MATH39512 Survival Analysis for Actuarial Science: example sheet 2讲解 Processing

MATH39512 Survival Analysis for Actuarial Science: example sheet 2

*=easy, **=intermediate, ***=difficult

* Exercise 2.1

The data in Table 1 relate to 12 patients who had an operation that was intended to correct a life-threatening condition, where week 0 is the week the investigation started and d=death and c=censored.

Patient number 1 2 3 4 5 6 7 8 9 10 11 12

Time of operation (in weeks) 0 0 0 4 5 10 20 44 50 63 70 80

Time observation ended (in weeks) 120 68 40 120 35 40 120 115 90 98 120 110

Reason observation ended c d d c c d c d d d d d

Table 1: Table corresponding to Exercise 2.1.

For this data set, compute the Kaplan-Meier estimate of the survival function of the survival time after operation.

* Exercise 2.2

Table 2 contains data on the (residual) lifetimes of a group of 10 people, where d=death and c=censored.

age observation started 60.0 62.3 63.5 64.8 65.5 66.0 72.0 74.1 74.5 75.6

age observation ended 73.2 69.7 74.9 72.2 69.7 79.6 82.0 79.6 83.8 77.8

reason observation ended d c c d d d c d c d

Table 2: Table corresponding to Exercise 2.2.

For this data set, determine the Kaplan-Meier, respectively, Nelson-Aalen estimate of the survival, respectively cumulative hazard function, of the residual lifetime at age 60.0.

** Exercise 2.3 (Sept 2005 CT4 exam IFoA)

A lecturer at a university gives a course on Survival Models consisting of 8 lectures. 50 students initially register for the course and all attend the first lecture, but as the course proceeds the numbers attending lectures gradually fall. Some students switch to another course. Others intend to sit the Survival Models examination but simply stop attending lectures because they are so boring. In this university, students who decide not to attend a lecture are not permitted to attend any subsequent lectures. The table below gives the number of students switching courses and stopping attending lectures after each of the first 7 lectures of the course.

Lecture number 1 2 3 4 5 6 7

Number of students switching courses 5 3 2 0 0 0 0

Number of students ceasing to attend lectures but 1 0 3 1 2 1 0

remaining registered for Survival Models

Table 3: Table corresponding to Exercise 2.3.

The university’s Teaching Quality Monitoring Service has devised an Index of Lecture Boringness. This index is defined as the Kaplan-Meier estimate of the proportion of students remaining registered for the course who attend the final lecture. In calculating the Index, students who switch courses are to be treated as censored after the last lecture they attend.

(a) Calculate the Index of Lecture Boringness for the Survival Models course.

(b) Explain whether the censoring in this example is likely to be independent.

** Exercise 2.4

The following data give the burn time of a number of candles of the same type. During the experiment some candles went out due to other reasons than the wax being fully usurped and hence there are censored values which are denoted by a +.

Time(hours): 5+, 6+, 6.5, 6.5, 6.5+, 7, 8+, 8.5, 9, 9, 9, 9.5.

(a) Use the data to calculate the Kaplan-Meier estimate of the survival function of the burn time and plot your estimate.

(b) Estimate, using the Kaplan-Meier estimate, the probablility that a candle which has burnt for more than 6.5 hours, will burn for at least two more hours.

(c) Estimate, using the Kaplan-Meier estimate, the expected burn time of a candle. (Hint: use Exercise 1.4.)

(d) Estimate, using the Kaplan-Meier estimate, the (conditional) expected burn time of a candle given that the candle has burnt less than 9 hours. (Hint: use Exercise 1.4.)

(e) Estimate, using the Kaplan-Meier estimate, the median burn time of a candle.

(f) How reliable/robust are each of the estimates found in (b)-(e)? What would happen if in the data 9.5 was replaced by 9.5+?