Chapter 7

Question 1

Define a covariate

Answer 1

A covariate is any quantity recorded in respect of each life
such as age, sex, smoker or not, type of treatment, etc

Question 2

When do we use non parametric estimates

Answer 2

If the covariates partition the population into a small
number of homogeneous groups then non-parametric (e.g., Kaplan-Meier estimates) can be compared

Question 3

What is a more direct and transparent way to construct a model other than non parametric methods

Answer 3

A regression model

Question 4

Give example of how what covariates can be measured by

Answer 4

1. direct measurements
2. indicator variable
3. quantitative interpretation of a measurement ex: scale

Question 5

What does Zi denote

Answer 5

Vector of covariates

Question 6

What is the most widely used regression model

Answer 6

Proportional hazards (PH) model. This model is also known as the Cox model after its originator. It can
help us to identify the factors that influence the relative
levels of mortality between members of a population.

Question 7

How do covariates act on the baseline hazard

Answer 7

Multiplicatively

Question 8

Assuming the ith covariate takes only positive values then if the ith regression parameter is positive what is the effect on the hazard

Answer 8

If the ith regression parameter is positive the hazard rate
increases with the ith covariate. Greater the magnitude of the parameter the greater the effect

Question 9

Assuming the ith covariate takes only positive values then if the ith regression parameter is negative what is the effect on the hazard

Answer 9

If the ith regression parameter is negative the hazard rate
decreases with the ith covariate. Greater the magnitude of the parameter the greater the effect

Question 10

Under the cox models how does one compare hazards of different lives

Answer 10

Under the Cox model the hazards of different lives with
covariate vectors z1 and z2 are in the same proportion at all
times so the general shape of the hazard is determine by the baseline hazard while the exponential terms account for the differences between lives

Question 11

Why is the Cox model termed a semi-parametric approach

Answer 11

We are not primarily concerned with the precise form of the baseline hazard but with the effects of the covariates
we can ignore λ0(t) and estimate β from the data irrespective of the shape of the baseline hazard – this is
termed a “semi-parametric” approach

Question 12

what is S/e of Beta

Answer 12

Often s fitted Cox model will report the estimate of each β from the data and also the standard error. The standard error is the standard deviation of the sampling distribution of that parameter

Question 13

What is significant about the estimates of Beta

Answer 13

As the sample size on which the estimate of β is based tends to infinity the central limit theorem implies that the sampling distribution of the mean (estimate of Beta) is asymptotically
Normal

Question 14

How do we ensure beta is significant

Answer 14

Examine the confidence interval - if 0 is included int he interval it is not significant meaning that covariate has no effect on the hazard

Question 15

How does one estimate Beta

Answer 15

To estimate β it is usual to maximise the partial
likelihood. Note that the baseline hazard cancels out (hence partial) and the partial likelihood depends only on the order in which deaths are
observed

Question 16

Describe the partial likelihood in words

Answer 16

Force of mortality jth life/ force of mortality all lives : aka the partial likelihood gives the comparative risk of
a particular individual dying, given that a death occurs

Question 17

Whats the difference between the Kaplan Meiers method and the Cox method

Answer 17

Unlike the Kaplan-Meier method the partial likelihood
considers observed deaths only, not the times at which the
deaths occurred nor any censoring observed between deaths

Question 18

What are two ties that can be present int he data preventing the maximisation of the partial likelihood

Answer 18

(a) some dj > 1 or
(b) some observations are censored at an observed
death time

Question 19

How do we combat problem when some observations are censored at an observed death time

Answer 19

By Including the lives on whom
observation was censored at time tj in the risk set R(tj)
effectively assuming that censoring occurs just after the
deaths were observed.

Question 20

How do we combat problem where dj>1 and we can’t maximise the partial likelihood

Answer 20

Use Breslow’s approximation

Question 21

How do you maximise the likelihood

Answer 21

Find the log likelihood then differentiate this with respect to beta.
set equal to zero and solve for Beta estimate

Question 22

What is the likelihood ratio statistic a criteria for

Answer 22

Suppose we need to assess the effect of adding further covariates to the model. The effect these have on the model

Question 23

What are we assuming in calculation of the likelihood ratio statistic

Answer 23

In general, suppose we fit a model with p covariates and another model with p+q covariates. Each is fitted by maximising a likelihood. Let lp and lp+q be the maximised log-likelihoods. We assume the hypothesis that the extra q covariates have no effect in the presence of the original p covariates

Question 24

With the test statistic as the likelihood ratio what is our test

Answer 24

H0:βp+1 = βp+2 =…= βp+q = 0
Test statistic is likelihood ratio stat - H0 is rejected at the 5% significance level if the value of this is greater than the upper 5% point of Chi squared q degrees of freedom

Question 25

In a life assurance firm if a life does not meet the premium basis what are the options?

Answer 25

decline insuring them
Charge something reasonable above/below the standard premium (can sue this formula to calculate!)

Question 26

How is the likelihood ratio statistic the basis for various modelling building strategies

Answer 26

We start with null model (one with no covariates) and
add possible covariates one at a time or
We start with a full model which includes all possible
covariates and then try to eliminate those of no
significant effect.

Question 27

What else should we test for in relation to covariates in a model

Answer 27

Interactions - may have an additive effect as well as a multiplicative effect

Question 28

Why does a life insurance company wish to know how certain covariates affect mortality

Answer 28

So that it can charge
premiums that accurately reflect the risk for an individual

Question 29

State the proportional hazards model

Answer 29

A survival time T follows a Cox proportional hazards model if the hazard function
(force of mortality) for the i

th life with covariate zi = (zi1,zi2,….zip) can be written as:

…

Question 30

Why is the PH model so widely used?

Answer 30

This model is widely used when comparing the impact of different covariates on
mortality.
Shape of the hazard is determined by the baseline hazard while the exponential
term accounts for the differences between lives. If we know the β’s then we can
compare the lives.
The covariates act multiplicatively on the baseline hazard. [1]
If we are not primarily concerned with the precise form of the baseline hazard [1]
but with the effects of the covariates we can ignore λ0(t) and estimate β from the data
irrespective of the shape - a “semi-parametric”
approach
It is straightforward to estimate the (partial) likelihood from data as the baseline hazard cancels out and the partial likelihood depends only on the order in which deaths are
observed.