Unformatted text preview: j zij j =1 where:
λo (t ) is the baseline hazard
β1 , β2 , . . . βp are the regression parameters
zi 1 , zi 2 , . . . zip are the covariates for the i th subject
Note:
In this formulation only λo (t ) depends on time but is
independent of the covariates
p
conversely, exp
j =1 βj zij is independent of t but
dependent on the covariates
6/45 Actuarial Statistics – Week 4: The Cox Regression Model
Main assumptions Interpretation  sign of β
If βj is positive, the hazard rate increases with the j th
covariate, ie there is a positive correlation between hazard rate
and the j th covariate
If βj is negative, the hazard rate decreases with the j th
covariate, ie there is a negative correlation between hazard
rate and the j th covariate 7/45 Discussions:
If obese individuals are more likely to suﬀer from major heart
disease, what’s the sign of the regression covariate associated
with the covariate representing weight?
If individuals who drink a high volume of nonalcoholic liquids
are less likely to suﬀer from liver disease, what’s the sigh sigh
of the regression parameter associated with the covariate
representing liquid intake? Actuarial Statistics – Week 4: The Cox Regression Model
Main assumptions Interpretation  magnitude of β the sheer magnitude of the β does not say much (as this
depends on how the covariates have been deﬁned)
so need of hypothesis test to check that β = 0 at a signiﬁcant
level
if β is estimated with standard techniques then it is easy to
check their level of signiﬁcance
(more in the model building section) 8/45 Actuarial Statistics – Week 4: The Cox Regression Model
On the proportionality of hazard rates 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 8/45 Actuarial Statistics – Week 4: The Cox Regression Model
On the proportionality of hazard rates Relative risk The ratio of hazard rates of two diﬀerent lives x and y , p
p
βj zxj
exp
j =1
λ (t ; Zx )
=
= exp βj {zxj − zyj }
p
λ (t ; Zy )
exp
βz
j =1 j yj j =1 is constant at all times, which explains the qualiﬁcation of
proportional hazards model.
This ratio is also called the relative risk of an individual with
risk factor Zx as compared to an individual with risk factor Zy . 9/45 Actuarial Statistics – Week 4: The Cox Regression Model
On the proportionality of hazard rates Advantages of proportionality
Under the Cox model, diﬀerences of hazard rates of diﬀerent
groups (diﬀerent covariates) are accounted for via the
exponential term (linear function inside the exponential),
which leads to a simple expression for the relative risk
The Cox model is not the only model with proportional
hazards; one can generalise Cox to (see Exercise 3.2)
λ(t ; Zi ) = λ0 (t )g (Zi )
where g (Z ) is any function of Z , but not t . What additional
properties should g (Z ) have?
If we are only interested in the di...
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