Chapter 2

Question 1

What are two key problems with studying mortality using idea of examining a set of lives over their lifetime?

Answer 1

(1) It takes too long: the experiment would take about 100 years to complete.
(2) In practice we would not be able to observe the deaths of all the lives in the sample – censoring. All we know about censored lives is that they died after a certain age.

Question 2

Where is censoring quite low - what type of data

Answer 2

Medical statistics - non-parametric estimation is very important here

Question 3

What are consequences of amending our observation experiment to be a shorter time frame and to have censoring

Answer 3

We no longer observe the
same cohort throughout their joint lifetimes so we will need to make some assumptions to ensure we are sampling from
the same distribution

Question 4

Explain censoring with examples

Answer 4

Censoring is present when we do not observe the exact length of a lifetime but observe only that its length
falls within some interval. ex: Emigrating and leaving study, withdrawing consent, entering study late etc

Question 5

Name some forms of censoring

Answer 5

Right censoring
Left censoring
Interval censoring
Random censoring
Non informative censoring
Type 1
Type 2

Question 6

Explain right censoring

Answer 6

Data are right censored if the censoring mechanism cuts short observations in progress. E.g.: the ending of a mortality investigation before all the lives being observed have died.

Question 7

Explain left censoring

Answer 7

Data is left censored if censoring mechanism prevent sus knowing when entry to the state we are observing happened ex: discovery of a medical condition,
patient fell ill with the disease at an unknown time t between appointments we just know the diagnosis date.

Question 8

Explain interval censoring

Answer 8

Data is interval censored if the observational plan only allows us to say an event of interest fell in some interval ex: we only know the year of death. Right and left censoring are a special kind of interval censoring

Question 9

Explain random censoring

Answer 9

Random censoring means time when observation of the ith lifetime is censored is a random variable Ci - Observation is censored if CI< Ti where Ti is the random lifetime of the ith life - special case of right censoring

Question 10

Explain non informative censoring

Answer 10

Cnesoring is non informative if it gives no info about the future lifetimes. If {Ti} and {Ci} are independent then censoring is non informatvive - meaning censoring happens for a reason completely unrelated to the study. usually we make this an assumption

Question 11

Explain Type 1 censoring

Answer 11

If censoring times are non in advance ie. variable Ci’s are constant, the mechanism is called Type 1 censoring. ex: You know the investiagtion might end on a certain date.

Question 12

Explain Type 2 censoring

Answer 12

If observation continued until a predetermined number of deaths occur this is type 2 censoring and number of deatsh is not random - this is uncommon in mortality studies but common in medical studies.

Question 13

Explain what the empirical distribution is and the Kaplan-Meier Estimator.

Answer 13

The empirical distribution of the survival function is known as the Kaplan Meier estimator of the distribution. The empirical distribution summarises all information in the data and is the best estimator of the distribution

Question 14

What is another name for the Kaplan Meier estimator

Answer 14

Product limit estimator

Question 15

What are the assumptions in the calculation of the kaplan meier estimator

Answer 15

1. Censoring is non infromative
2. Hazard of experiencing event is 0 at all durations other than where the event actually happens
3. Hazard of experiencing the event at particular tj where event takes place is Dj/ Nj
4. People censored are removed from stduy at duration which censroing takes place, or if at the time of an event, directly after the event

Question 16

How can we compare distributions graphically?

Answer 16

Confidence intervals - see how much they oevrlap

Question 17

What formula allows us to calculate the variance of our estimator - is it effective

Answer 17

Greenwoods formula - reasonable estimation over most t but tends to underestimate the variance in the tails of the distribution

Question 18

Explain the nelson aaeln estimator and what is estimates

Answer 18

Another non parametric way to calculate the empirical distribution function based on non infromative cenosring, it estimates the intergrated hazard

Question 19

What are three key assumptions when using maximum likelihood estimation

Answer 19

1. we know the mathematical form of the survival function
2. we assume the censoring is non informative
3. we assume deaths are independent of one another

Question 20

What is λj

Answer 20

the proportion of people dying at exact age tj

Question 21

What is the Fleming-Harrington estimator

Answer 21

Estimate of survival function using the Nelson aalen estimator - Fleming-Harrington 𝑆 𝑡 = exp −Λ 𝑡

Question 22

How can you use the Kaplan Meir estimator to determine if survival probabilities are the same as antoher experience

Answer 22

We know the variance of the Kaplan-Meier estimates, which is often estimated using
Greenwood’s formula. Hence we can draw confidence intervals about the estimate of the
survival function over the period for a group of policyholders. We know the
survival function of the typical policyholder, so if the former and its, say 95% confidence
interval falls outside the later curve then we
know that the two groups experience different mortality rates.

Or we could do simulation
Do random simulations of the numbers surviving the year based on the actual exposure
times/months. Estimate the prob of observing what we observed, assuming pop mortality, and then see if this is small (under 5%, say). If so, two experiences have different mortality at that confidence level