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Unformatted text preview: erence due to covariates
and NOT the baseline hazard  we can ignore λo (t ) and
concentrate on the function g (Z ) only (β terms in Cox).
10/45 Actuarial Statistics – Week 4: The Cox Regression Model
On the proportionality of hazard rates Example
The following data for each patient have been recorded:
0 for females
1 if patient attended Hospital B
Z1 =
Z2 =
1 for males
0 otherwise
Z3 = 1
0 if patient attended Hospital C
otherwise
T Suppose the force of mortality at time t is modelled by λ0 (t )e β Z
ˆ
and the parameter values have been estimated to be β1 = 0.031,
ˆ2 = −0.025 and β3 = 0.011.
ˆ
β
Compare the force of mortality for a female patient who attended
Hospital A with that of:
1
2
11/45 a female patient who attended Hospital B
a male patient who attended Hospital C Actuarial Statistics – Week 4: The Cox Regression Model
On the proportionality of hazard rates Solution
1 The hazard rate at time t for a female who attended Hospital
A is
λfemale ,A (t ) = λ0 (t )
and the hazard rate at time t for a female who attended
Hospital B is
λfemale ,B (t ) = λ0 (t )e −0.025
The ratio is
λfemale ,A (t )
= e 0.025 = 1.0253
λfemale ,B (t ) 12/45 2 So we estimate that the hazard rate for a female who
attended Hospital A is 2.53% higher than that of a female
who attended Hospital B.
ratio: e −0.042 = 0.9589 Actuarial Statistics – Week 4: The Cox Regression Model
Estimation of the regression parameters β 1 Introduction
2 Main assumptions
3 On the proportionality of hazard rates
4 Estimation of the regression parameters β
5 Hypothesis tests on the β ’s
6 Estimation of the full survival function
7 Diagnostics for the Cox regression model 12/45 Actuarial Statistics – Week 4: The Cox Regression Model
Estimation of the regression parameters β Partial likelihood estimation of β
To estimate β = (β1 , β2 , . . . βp ), we will select the ones that
maximize the partial likelihood under the following assumptions:
noninformative censoring
lives are independent
Notation:
Ordered times of observed death are t1 < t2 < · · · < tk
dj : number of deaths occuring at time tj (1 ≤ j ≤ k )
We will label all the lives as 1, · · · , n.
R (tj ) is the set of the label of all lives who are still under study
at a time just prior to tj . (Note R (tj ) and nj are not identical)
Use (j ) to denote the label of the life who dies at tj
13/45 Actuarial Statistics – Week 4: The Cox Regression Model
Estimation of the regression parameters β Two cases We will distinguish two cases:
1 assume that only one death occurs at each tj
(dj = 1 for 1 ≤ j ≤ k ) 2 Relax that assumption.
In practice there might be ties in the data, that is
1
2 some dj > 1; or
some observations are censored at an observed lifetime. This can signiﬁcantly complicate the partial likelihood as one
needs to include the lives censored at time tj in the risk set
R (tj ) and all permutations of simultaneous events. 14/45 Actuarial Statistics – Week 4: The Cox Regression Model
Estimation of the regression parameters β Case 1  one death at each observed lifetime
The partial likelihood for the death at tj is the ﬁrst death of the lives Pr the individual (j ) dies at tj in R (tj ) occurs at tj
and only one death at tj
λ tj ; Z(j ) exp − tj
0 tj
0 λ(s , Zi )ds i =(j ) = λ (tj ; Zi ) exp − =
i λ tj ; Z(j )
=
λ (tj ; Zi )
R (tj ) tj
0 λ (s ; Zi ) ds i R (tj ) i R (tj ) 15/45 exp − λ(s , Z(j ) )ds λ0 (tj ) exp β Z(T)
j
λ0 (tj )...
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