13_AS_4a_lec_a

The survival function then becomes s x z s0 x exp z t

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Unformatted text preview: sed on the 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 failure-time 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 10-year-old dog is in some way So a 10-year-old dog is in some alent to a 70-year-old human. In AFT term way equivalent probability of a we might say theto a 70-year-old 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 log-time and accelerated failure-time models 1 Linear log-time model 2 Accelerated failure-time model 3 Relationship between the log-time and accelerated failure-time models 4 Examples 6/15 Actuarial Statistics – Module 4a: Parametric models: Introduction to more advanced parametric models Relationship between the log-time and...
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