Unformatted text preview: j ) 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 – Module 3: Semiparametric methods: 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 – Module 3: Semiparametric methods: 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 – Module 3: Semiparametric methods: Cox Regression Model...
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