Unformatted text preview: exp β ZiT
i R (tj ) = exp β Z(T)
j
exp β ZiT
i R (tj ) Actuarial Statistics – Week 4: The Cox Regression Model
Estimation of the regression parameters β Multiplying the likelihood over all deaths gives the partial
likelihood, whose maximum is our estimate for β : T
k exp β Z(j )
ˆ = β L (β ) =
is maximal
β exp β ZiT j =1 i R (tj ) 16/45 This is a partial likelihood since it considers only likelihood of
the deaths (censored observations contribute to the
denominator) and does not depend on the times of death (just
the order) (this is diﬀerent to the Kaplan Meier approach).
The numerator depends on information for the individual who
experiences death; The denominator utilises information about
all information about all lives who have not yet experienced
death.
ˆ
β are asymptotically normal Actuarial Statistics – Week 4: The Cox Regression Model
Estimation of the regression parameters β Example
A group of seven lives was observed over a period of time as part
of a mortality investigation. Each of the lives was under
observation at all ages from age 45 until death or policy expiry:
Life
1
2
3
4
5
6
7 Sex
F
M
F
M
M
M
F Age at Exit
47
50
52
55
64
65
65 Reason
L
D
D
L
D
L
L where “L” means “Lapse” and “D” denotes “Death”.
17/45 Actuarial Statistics – Week 4: The Cox Regression Model
Estimation of the regression parameters β The following proportional hazards model has been proposed:
λ(t ; Z ) = λ0 (t ) e β Z
where Z = 1 for males and 0 for females. Write down the partial
likelihood. 18/45 Actuarial Statistics – Week 4: The Cox Regression Model
Estimation of the regression parameters β Case 2: partial likelihood in present of ties in the data
We will consider Breslow’s approximation:
exp β sjT k L (β ) =
j =1 dj
exp β ZiT i R (tj ) where sj is the sum of the covariate vectors Z of the dj lives
observed to die at time tj .
This approximation works well when the number of ties are
relatively small
There are other methods like Efron’s method (see K&M,
Section 8.4)  Breslow is an option in R (coxph) and SAS
(proc phreg)
19/45 Actuarial Statistics – Week 4: The Cox Regression Model
Estimation of the regression parameters β Example
An investigation was carried out into the survival times (measured
in months) of patients in hospital following liver transplants. For
patient i , the covariates are zi 1 = 0 for placebo, 1 for treatment X,
and zi 2 = weight of the patient (measured in kg).
The observed lifetimes (with weights in brackets) were as follows:
Placebo Treatment X
3 (83)
6∗ (58)
9 (68)
11 (73)
14 (75) 14 (68)
16 (86) 14∗ (49)
Observations with an asterisk represent censored observations.
Using Breslow’s approximation, what contribution to the partial
likelihood is made by deaths at time 14?
20/45 Actuarial Statistics – Week 4: The Cox Regression Model
Estimation of the regression parameters β Solution Denote β = (β1 , β2 ).
Just prior to time 14, 4 lives are at risk and their covariates are
(0, 75), (1, 68), (1, 49) and (0, 86).
At time 14, there are two deaths with covariates (0, 75), (1, 68).
Hence, the contribution to the partial likelihood made by deaths at
time 14 is approximated by
exp β (0 + 1, 75 + 68)T
2 [exp (β (0, 75...
View
Full
Document
This document was uploaded on 04/03/2014.
 Three '14

Click to edit the document details