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covariates. The survival function then becomes
S (x ; z ) = S0 x exp θZ T , where θ = (θ1 , · · · , θp ) is a vector of regression coeﬃcients.
Note:
a change in covariate values changes the time scale from the
baseline time scale x
exp θz T is called an acceleration factor 4/15 Assumption the effect of covariates is multiplicative (p
tional) with respect to survival time, w
for PH models the underlying assumption
the effect of covariates is multiplicative w
spect to the hazard. Actuarial Statistics – Module 4a: Parametric models: Introduction to more advanced parametric models
Accelerated failuretime model AFT—Multiplicative effect with
survival time
Illustration
PH—Multiplicative effect with
hazard 5/15 It is often the that dogs grow
To illustrate saididea underlying the AFT a
tion, considertimes faster of dogs. It is oft
older seven the lifespan than
that dogs grow older seven times faster th
humans.
mans. So a 10yearold dog is in some way
So a 10yearold dog is in some
alent to a 70yearold human. In AFT term
way equivalent probability of a
we might say theto a 70yearold dog su
human.
past 10 years equals the probability of a
SD(t)
SH(7t)
=
In this framework dogs can be
surviving past 70 years. Similarly, we might
probability of a dog surviving past 6 years
viewed, on average, as
the probabilitythrough lifesurviving past 4
accelerating of a human 7 times
Survival Function
Survival Function
because 42 equals 6 times 7. More generally
For Dogs
For Humans
faster than humans.
say SD (t) = SH (7t), where SD (t) and SH (t)
survival functions for dogs and humans,
AFT models:
In AFT terminology, the probability of a In this framework dogs can be view
tively. dog surviving past 10
Describe “stretching out” or of a human surviving past 70through life 7 time
years equals the probability
average, as accelerating years.
contraction of
than humans. Or from the other perspect
More generally survival timeSD (t ) = SH (7t ).
we can say
lifespan of humans, on average, is stretche Actuarial Statistics – Module 4a: Parametric models: Introduction to more advanced parametric models
Relationship between the logtime and accelerated failuretime models 1 Linear logtime model 2 Accelerated failuretime model 3 Relationship between the logtime and accelerated failuretime models 4 Examples 6/15 Actuarial Statistics – Module 4a: Parametric models: Introduction to more advanced parametric models
Relationship between the logtime and...
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