Module 2: NP Models

Question 1

Type 1 Right Censoring definition

Answer 1

Event (e.g. failure such as death) is observed only if it occurs
prior to some prescribed time C_R (right censoring time).

Question 2

Type 1 Right Censoring example

Answer 2

Life Insurance PH surrenders their policy.

Question 3

Type 2 Right censoring

Answer 3

Observations continues until a predetermined number (r) of events have occured.

Question 4

Left sensoring

Answer 4

Death/Event occurs before the observation starts.

So we only know that T is less than the left censor point C_L

Question 5

Interval Censoring

Answer 5

T is only known to fall within an interval.

Question 6

Truncation

Answer 6

Events within a certain period are observed.

Otherwise no infomation available.

Question 7

Example of truncation

Answer 7

Insurance claim where deductible is not reached.

Question 8

Random bounds is when

Answer 8

The censoring/truncation point is subject to randomness.

Question 9

C_i is the

Answer 9

Time when the i^th observation is censored.

Its a RV.

Question 10

Non-informative censoring is when

Answer 10

it gives no information about lifetimes

Question 11

Example of non-informative censoring

Answer 11

The end of an investigation period.

Question 12

Condition for non-informative censoring

Answer 12

Independance of all T’s and C’s

Question 13

Example of informative censoring

Answer 13

Those in better health are more likely to withdraw from life insurance.

Question 14

Why do truncation likelihoods have a quotient

Answer 14

They are conditional on actually being able to observe the occurance in the first place.

Question 15

N represents

Answer 15

population of N lives.

Question 16

m represents

Answer 16

the amount of deaths we observe

(N-m deaths are right censored)

Question 17

t₁2k

represent…

Answer 17

The ordered times of death

(Multiple can occur at the same time)

Question 18

d_j represents

Answer 18

Number of deaths that occured at time t_j

Question 19

c_j represents

Answer 19

Number of lives that are right censored between [t_j , t_j+1)

Question 20

t_j1 , t_j2 etc are the…

Answer 20

Times that obersvations are censored in the period [t_j,t_j+1)

Question 21

n_j represents

Answer 21

The number of lives alive at risk time t_i^- (just before time t_i)

Question 22

Discrete hazard function eqn

Answer 22

λ_j = Pr [T = t_j |T ≥ t_j ]

Question 23

Discrete hazard function implies what about the survival function?

Answer 23

S(t) = 1 − F(t) = Π _j:tj≤t (1 − λj)

Question 24

Non-informative sensoring means

Answer 24

Time to sensoring is indepedent to time of death. Ie censoring time tells us nothing.

Question 25

λ is

Answer 25

d/n for a certain j

Question 26

S(t) for the KM estimator is

Answer 26

Π (1- d/n)

(product from j=1 to t)

Question 27

Where is S(t) KM poorly defined?

Answer 27

beyond t_k

we could either set; they all die, or all survive.

In practice find a middle ground.

Question 28

Variance of KM

Answer 28

(1-F(t))^{2 *} Σ d / n(n-d)

summed from j=1 to t

Question 29

How to find CI of S(t) KM

Answer 29

S(t) +- Zquantile * sqrt(Var [S(t)])

Question 30

NA estimator S(t) =

Answer 30

exp( -Σd/n)

sum from j=1 to t

Question 31

NA estimator and KM estimator related via….

Also which is bigger ?

Answer 31

Taylor series approximation. See notes.

KM < NA for all t

Question 32

How do we compare survival functions…

Answer 32

use H₀ : h₁(t)=h₂(t)

then see if we can reject it.

Ie looking for a test statistic to be small enough to accept. <0.05

Question 33

Trick with data when comparing survial funcs;

Answer 33

must amalgamate the data essentially then compare the sample against the group.