Question 1

Population censuses in a certain country are taken each year on the President’s birthday, provided that the President’s astrological advisor deems the taking of a census favourable. Censuses record the age of every inhabitant in completed years (that is, curtate age). Deaths in this country are registered as they happen, and classified according to age nearest birthday at the time of death.

Below are some data from the three most recent censuses.

Age in completed years | Population 2006 (thousands) | Population 2009 (thousands) | Population 2010 (thousands) |

64 | 300 | 320 | 350 |

65 | 290 | 310 | 330 |

66 | 280 | 300 | 320 |

Between the censuses of 2006 and 2009 there a total of 3,000 deaths to inhabitants aged 65 nearest birthday, and between the censuses of 2009 and 2010 there were a total of 1,000 deaths to inhabitants aged 65 nearest birthday.

- Estimate, stating any assumptions you make, the death rate at age 65 years for each of the following periods:
- The period between the 2006 and 2009 censuses
- The period between the 2009 and 2010 censuses (8 marks)
- Explain the exact age to which your estimates apply. (2 marks)

Question 2

- State the form of the hazard function for the Cox Regression Model, defining all the terms used. (2 marks)
- State two advantages of the Cox Regression Model. (2 marks)

Susanna is studying for an on-line test. She has collected data on past attempts at the test and has fitted a Cox Regression Model to the success rate using three covariates:

Employment Z_{1} = 0 if an employee, and 1 if self-employed.

Attempt Z_{2} = 0 if first attempt, and 1 if subsequent attempt.

Study time Z_{3} = 0 if no study time taken, and 1 if study time taken.

Having analysed the data Susanna estimates the parameters as:

- Employment 0.4
- Attempt -0.2
- Study time 1.15

Bill is an employee. He has taken study time and is attempting the test for the second time. Ben is self-employed and is attempting the test for the first time without taking study time.

- Calculate how much more or less likely Ben is to pass, compared with Bill. (4 marks)

Susanna subsequently discovers that the effect of the number of attempts is different for employees and the self-employed.

- Explain how the model could be adjusted to take this into account. (2 marks)

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