Survival and hazard functions ST3242: Introduction to Survival Analysis Alex Cook
August 2008
ST3242 : Survival and hazard functions
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Reminder
Last time in Surval Analysis. . . Last time we introduced one of two main differences between survival analysis and the statistics you have learned until now: Censoring & truncation Survival & hazard functions
ST3242 : Survival and hazard functions
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Simulated example
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Let x ∼ Ga(α, β) be simulated survival times we know come from a Gamma distribution.
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ST3242 : Survival and hazard functions
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Simulated example
Mean excluding censored individuals: 1.9 Mean pretending censored individuals have lives of length 2.5: 2.2 True simulated mean: 2.5 Result of censoring Standard descriptive methods are not very relevant to survival data.
ST3242 : Survival and hazard functions
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Simulated example
Mean excluding censored individuals: 1.9 Mean pretending censored individuals have lives of length 2.5: 2.2 True simulated mean: 2.5 Result of censoring Standard descriptive methods are not very relevant to survival data.
ST3242 : Survival and hazard functions
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Simulated example
Mean excluding censored individuals: 1.9 Mean pretending censored individuals have lives of length 2.5: 2.2 True simulated mean: 2.5 Result of censoring Standard descriptive methods are not very relevant to survival data.
ST3242 : Survival and hazard functions
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Aim of lecture Introduce two alternative summaries of the distribution of survival times: Survival function S(t) Hazard function h(t) Lecture plan survival function hazard function parametric survival and hazard functions example
ST3242 : Survival and hazard functions
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Aim of lecture Introduce two alternative summaries of the distribution of survival times: Survival function S(t) Hazard function h(t) Lecture plan survival function hazard function parametric survival and hazard functions example
ST3242 : Survival and hazard functions
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Survival function
Recall standard notation f (t) is the density function for T Rt F (t) = 0 f (τ ) dτ is the (cumulative) distribution function F (t) = p(T < t) is the probability failure occurs by time t
ST3242 : Survival and hazard functions
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Survival function
Definition If T is the time of failure, then S(t) is the survival function. S(t) = p(T > t), i.e. the probability of surviving at least to time t. It is defined on the domain t ∈ [0, ∞). As a probability, it has range S(t) ∈ [0, 1].
ST3242 : Survival and hazard functions
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Survival function
Properties S(0) = 1 i.e. no one starts off dead limt→∞ S(t) = 0 i.e. everyone dies eventually S(ta ) ≥ S(tb ) ⇐⇒ ta ≤ tb i.e. S(t) declines monotonically R∞ S(t) = 1 − F (t) = t f (τ ) dτ
ST3242 : Survival and hazard functions
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Survival function
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Typically, the population survival function is smooth, though estimates of it are not.
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ST3242 : Survival and hazard functions
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Note
It is implicitly assumed that all individuals undergo the event if they have enough time:
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all individuals die if event is death all individuals get lung cancer if don’t die first all individuals start smoking marijuana if don’t die first
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We can get around this using mixture or cure models.
ST3242 : Survival and hazard functions
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Note
It is implicitly assumed that all individuals undergo the event if they have enough time:
u
u
all individuals die if event is death all individuals get lung cancer if don’t die first all individuals start smoking marijuana if don’t die first
d
We can get around this using mixture or cure models.
ST3242 : Survival and hazard functions
10/1
Note
It is implicitly assumed that all individuals undergo the event if they have enough time:
u
u
all individuals die if event is death all individuals get lung cancer if don’t die first all individuals start smoking marijuana if don’t die first
d
We can get around this using mixture or cure models.
ST3242 : Survival and hazard functions
10/1
Note
It is implicitly assumed that all individuals undergo the event if they have enough time:
u
u
all individuals die if event is death all individuals get lung cancer if don’t die first all individuals start smoking marijuana if don’t die first
d
We can get around this using mixture or cure models.
ST3242 : Survival and hazard functions
10/1
Note
It is implicitly assumed that all individuals undergo the event if they have enough time:
u
u
all individuals die if event is death all individuals get lung cancer if don’t die first all individuals start smoking marijuana if don’t die first
d
We can get around this using mixture or cure models.
ST3242 : Survival and hazard functions
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Why survival function?
There is a very popular non-parametric method to estimate S(t) even with censoring.
ST3242 : Survival and hazard functions
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Estimating the survival function
Parametrically If you specify a parametric model for T with parameters ˆ ˆ θ, S(t) follows from θ. Use MLE to estimate θ, or take a Bayesian approach to ˆ incorporate uncertainty in S(t) more flexibly.
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Estimating the survival function
Non-parametrically If you cannot justify a particular distribution for T use e.g. the Kaplan–Meier estimate of S(t) (see next lecture).
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Example
If T ∼ We(κ, λ) with f (t) = λκt κ−1 exp{−λt κ }, what is S(t) =?
ST3242 : Survival and hazard functions
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Hazard function h(t) is the hazard function at time t. Definition h(t) = lim∆t→0
