Nelson-Aalen, Cox, Caplan-Meier Models

Question 0

Derive expression for the survival function between ages 50 and 55 years and where hazard of death is given h(t) = α + bt

Answer 0

S₅₀(t) = P (T₅₀ >t) for 0 ≤ t ≤ 5
t
= exp ( ‑ ∫ h(u) du)
0

           t  = exp ( ‑ ∫ (α+ βu) du
           0

                                  t  = exp ( ‑ (αu + 1/2βu²) |
                                  0

= exp [ -αt+ 1/2βt² ]

Question 1

Define the hazard rate h(t) of random variable T lifetime

Answer 1

h(t) = [ lim(t) Ρ (Τ≤ t +dt | Τ>t) ] / dt

dt→0

Question 2

Right censoring?

Answer 2

If censoring mechanism cuts short observations in progress.

Example: ending an investigation on a fixed date

Question 3

Left censoring?

Answer 3

Censoring mechanism prevents us from knowing when entry in to the observation took place.
Example: Regular examination of patients reveals a condition of which onset fell between the examination.

Question 4

Interval censoring?

Answer 4

If observational plan allows us to say that event fell wining some interval of time.
Example: the calendar year of death

Question 5

Random censoring?

Answer 5

If the time C(i) at which observation of i-th lifetime is censored is a random variable. The observation will be censored if C(i) < T(i) where T(i) is a random lifetime of these i-th life.

Question 6

Non-informative censoring?

Answer 6

It gives no information about the lifetimes.
In case of random censoring, the independence of each pair T(i) and C(i) is sufficient to ensure that the censoring is non-informative.

Question 7

Type I censoring?

Answer 7

When censoring times are known in advance

Question 8

Type II censoring?

Answer 8

Observation is continued until a predetermined number of deaths has occurred. Simplify analysis because the number of events is non-random.

Question 9

Gompertz law of mortality

Answer 9

μₓ = B cˣ

As it is an exponential function and it is often a reasonable assumption for middle ages and older ages.

Question 10

Makenham’s Law?

Answer 10

μₓ = A + B cˣ

It incorporates a constant term which is sometimes interpreted as an allowance for accidental deaths, not depending on age.

Question 11

Definition of distribution function F(t)

Answer 11

F(t) = P [ T <= t ]

Question 12

Probability of death if deaths are uniformly distributed

Answer 12

₃q₃₇=3*q₃₇

Question 13

Assumptions underlying Poisson process

Answer 13

λ - continuous time
N(t) - integer valued (discrete process) with following conditions:
1) N(0) = 0
2) N(t) has independent increments
3) N(t) has Poisson distributed stationary increments.

Question 14

For death intervals

Age nearest birthday

Answer 14

(x -1/2 ; x+ 1/2)

Question 15

For death intervals

Age last birthday

Answer 15

(x , x + 1)

Question 16

For death interval

Age next birthday

Answer 16

(x -1, x)

Question 17

Census method for 2 year study

Answer 17

Eⁿ₊=∫₀²P*(t)dt _

Question 18

Estimate of mortality formula

Answer 18

μχ+f=θχ/Eχⁿ

It is mu hat in the formula..

Force of mortality is constant over the estimated period!

Question 19

Formula of probability of survival when μ is constant

Answer 19

tPx=exp(-μt)

Question 20

Three advantages of 2- state model over the binomial for the estimation of transition intensities where exact entries in and out of observation are known

Answer 20

1. 2 - state model can be easily extended to a multi-state model with increments as well as decrements but Binomial model is not easily extended.
2. 2 state model closely represents underlying process and takes into account the precise timing of each death. In the binomial model times of death are not used in the investigation only that each life died or survived. So Binomial is less precise.
3. Since exact dates are known in 2- state model it is easier to calculate the central exposed to risk and the MLE of mu. However we need extra assumptions (Balducci assumption). So for the binomial we use less information to estimate parameter q hence the estimator of a for the binomial model has a higher variance

Question 21

What assumption we use when using Census formula and using trapezium rule?

Answer 21

That the population P(t) varies linearly over the given period.

Question 22

Principle of correspondence

Answer 22

Correspondence between the deaths counted in numerator of an estimate of the force of mortality and the exposure counted in the denominator. A day will be included in exposure if the life died in that day is included in the count of death.

Question 23

Effect of estimation of using claims rather than deaths and using polices rather than lives.

Answer 23

Using claims is satisfactory when forecasting claims rather than deaths.
As for policies the assumption that the policies are not independent from one another unlike lives because of duplicates of policies hence it will give unbiased estimates but the standard errors of the estimate will be larger

Question 24

Outline differences between the two state model and the Poisson model when used it estimate transition rates

Answer 24

Poisson model is an approximation of the two state model. While the two state model can be specified as allow for increments this is not possible for the Poisson model.
The estimation of the transition rates in the two state model involves measurement of the two random variables - the observed number of decrements and the exposed to risk that gave rise to thee decrements.
The Poisson model assumes that the exposed to risk remains constant and it only measures the observed number of decrements.
The MLE estimators for both models are asymptotically unbiased and consistent. And formulae for the estimates are the same.

Question 25

Binomial model compared to with Poisson and two state models: strengths and weaknesses:

Answer 25

+ Estimate is easy to calculate using this actuarial estimator

• Model doesn’t use actual times of death - loss of information
• Balducci assumption might not be appropriate for older ages
• The initial expose to risk is not very intuitive measure
• The model doesn’t generalise easily where multiple decrements are required.

Question 26

Difference between Central exposed to risk and initial exposed to risk

Answer 26

Central exposed to risk is a measure of the number of lives alive during the year while initial exposed to risk is the number of lives alive at the beginning of the year.

Question 27

Compare advantages and disadvantages of the binomial and multi-state models in the situation when analysing mortality without distinguishing between causes of death

Answer 27

Binomial model is useful when only q functions are needed.

Both methods are equally straightforward when exposure is being calculated directly.

The multiple state approach can be slightly more complex if q is the only function of interest as mu has to be estimated first and then converted using q = 1 - exp (-mu).

On the other hand if census formula have been used the binomial model requires an additional approximation to calculate the initial exposed to risk where ,multiple state model does not require this adjustment even to estimate q.

In the absence if censoring within the year of age the binomial model is able to compute que x without any assumption regarding the pattern of mortality over the year of age. The multi- state model always requires the assumption that the force of mortality is constant between birthdays.

However in the more common situation in which censoring within the year of age occurs the binomial model requires additional assumption about the pattern of mortality over the year of age (usually Balducci assumption which is less realistic at most ages than assuming the force of mortality being constant).
Extract time of death is not necessary to estimate qx using binomial, and it is necessary to compute multi-state estimate even if only mu is required. When mortality is high multi-state model is more accurate. Variance of estimator for q is higher for the binomial model.

Question 28

Compare advantages and disadvantages of the binomial and multi-state models in the situation when analysing mortality when distinguishing between causes of death

Answer 28

Multi-state model can be easily extended from two states (alive and dead) to more states (alive, death cause 1, death cause 2 and etc…), estimators for the force of transition to death by each cause can be separately calculated, requiring similar data to before and having similar statistical properties (they are maximum likelihood estimates).

There is no extension of the binomial model to what is in effect multiple causes of decrement ( as rational of the binomial model requires only two outcomes dead or alive)

Question 29

List the data needed for the exact calculation of a central exposed to risk based on the age.

Answer 29

Date of birth
Date of entry into investigation
Date of exit from investigation

Question 30

Difference between the Central and Initial exposed to risk

Answer 30

Central exposed to risk is used as a denominator when estimating the force of mortality over an age range using the two- state model or the Poisson model. It is an actual time that the lives were observed within the rate interval during the study. An approximate value for the central exposed to risk can also be obtained using the census method.

Initial exposed to risk is used as a denominator when estimating mortality rate q over a a particular age range using the binomial model. An assumption Balducci assumption is required about the pattern of mortality over the age range. Calculation of the initial exposed to risk is the same as to the central exposed to risk except that lives who die during the rate interval continue to be counted up to the end of the interval. An approximate value for the initial exposed to risk can be obtained by adding half the number of deaths during the year to the central exposed to risk, which assumes that deaths occur uniformly over the rate interval.