Fundamentals of Survival Time Analysis
Functions of Survival Time
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Probability density function: f(t)
The probability of the failure time
occurring at exactly time t (out of the
whole range of possible t’s).
unconditional failure rate
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Survival function: 1-F(t)
The goal of survival analysis is to estimate and compare
survival experiences of different groups.
Survival experience is described by the cumulative survival
function:
Example: If t=100 years, S(t=100) = probability of
surviving beyond 100 years.
7
Cumulative survival
Same hypothetical data,
plotted
as cumulative distribution rather
than density:
8
Cumulative survival
9
Hazard Function: new concept
Hazard rate is an instantaneous
incidence rate.
10
Hazard function
Hazard function
The hazard function is also known as the
Instantaneous failure rate, force of
mortality, conditional mortality rate, and
age-specific failure rate.
11
Hazard Curve
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Example 2.1
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Curves: PDF, Survival and Hazard
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15
Hazard vs. density
This is subtle, but the idea is:
When you are born, you have a certain
probability of dying at any age; that’s the
probability density (think: marginal
probability)
Example: a woman born today has, say, a 1%
chance of dying at 80 years.
However, as you survive for awhile, your
probabilities keep changing (think: conditional
probability)
Example, a woman who is 79 today has, say, a
5% chance of dying at 80 years.
16
A possible set of probability density, failure, survival,
and hazard functions.
F(t)=cumulative failure
S(t)=cumulative survival
h(t)=hazard function
f(t)=density function
17
A probability density we all
know: the normal distribution
What do you think the hazard looks like
for a normal distribution?
Think of a concrete example. Suppose
that times to complete the midterm
exam follow a normal curve.
What’s your probability of finishing at
any given time given that you’re still
working on it?
18
f(t), F(t), S(t), and h(t) for different normal
distributions:
19
Examples: common functions
to describe survival
Exponential (hazard is constant over
time, simplest!)
Weibull (hazard function is increasing or
decreasing over time)
20
f(t), F(t), S(t), and h(t) for different exponential
distributions:
21
Parameters of
the Weibull
distribution
f(t), F(t), S(t), and h(t) for different Weibull
distributions:
22
Exponential
23
With numbers…
Probability of
developing
disease
at
year 10.
Incidence rate (constant).
Probability of
surviving
past year 10.
(cumulative risk through year 10 is 9.5%)
Why isn’t the cumulative
probability of survival just
90% (rate of .01 for 10
years = 10% loss)?
24
Example…
Recall this graphic.
Does it look Normal, Weibull,
exponential?
25
Example…
One way to describe the survival
distribution here is:
P(T>76)=.01
P(T>36) = .16
P(T>20)=.20, etc.
26
Example…
Or, more compactly, try to describe this as an
exponential probability function—since that is how
it is drawn!
Recall the exponential probability distribution:
If T ~ exp (
h
), then
P(T=t) =
h
e
-
h
t
Where
h
is a constant rate.
Here:
Event time, T ~ exp (Rate)
27
Example…
To get from the instantaneous probability
(density), P(T=t) =
h
e
-
ht
, to a cumulative
probability of death, integrate:
Area to the left
Area to the right
28
Example…
Solve for
h
:
29
Example…
This is a “parametric” survivor function,
since we’ve estimated the parameter
h
.
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Hazard rates could also change over
time…
Example: Hazard rate
increases linearly with time.
31
32
Relating these functions
(a little calculus just for fun…):
33
Getting density from hazard…
Example: Hazard rate
increases linearly with time.
34
Getting survival from hazard…
Understanding survival time analysis involves concepts like probability density functions, cumulative survival functions, the hazard function, and more. This analysis helps estimate and compare survival experiences in different groups, providing valuable insights into predicting outcomes in various scenarios.
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Presentation Transcript
Probability density function: f(t) The probability of the failure time occurring at exactly time t (out of the whole range of possible t s). unconditional failure rate + ( ) P t T t t = ( ) lim f t t 0 t 2
PDF 3
Survival Function or Curve Let T denote the survival time S(t) = P(surviving longer than time t ) = P(T > t) The function S(t) is also known as the cumulative survival function. 0 S( t ) 1 (t)=number of patients surviving longer than t total number of patients in the study
Survival function: 1-F(t) The goal of survival analysis is to estimate and compare survival experiences of different groups. Survival experience is described by the cumulative survival function: F(t) is the CDF of f(t), and is more interesting than f(t). = = ( ) 1 ( ) 1 ( ) S t P T t F t Example: If t=100 years, S(t=100) = probability of surviving beyond 100 years. 6
Cumulative survival Same hypothetical data, plotted as cumulative distribution rather than density: Recall pdf: 7
Cumulative survival P(T>20) P(T>80) 8
Hazard Function: new concept AGES Hazard rate is an instantaneous incidence rate. 9
Hazard function + ( / ) P t T t t T t = ( ) lim h t t 0 t ( ) f t = Hazard from density survival and : (t) h ( ) S t Derivation (Bayes rule): + + ( & ) ( ) ( ) t P t T t dt T t P t T t dt f t dt = + = = = ( ) ( / ) h t dt P t T t dt T t ( ) ( ) ( ) P T t P T t S 10
Hazard function The hazard function is also known as the Instantaneous failure mortality, conditional mortality rate, and age-specific failure rate. rate, force of 11
Hazard Curve 12
Example 2.1 13
Hazard vs. density This is subtle, but the idea is: When you are born, you have a certain probability of dying at any age; that s the probability density (think: marginal probability) Example: a woman born today has, say, a 1% chance of dying at 80 years. However, as you survive for awhile, your probabilities keep changing (think: conditional probability) Example, a woman who is 79 today has, say, a 5% chance of dying at 80 years. 15
A possible set of probability density, failure, survival, and hazard functions. f(t)=density function F(t)=cumulative failure h(t)=hazard function S(t)=cumulative survival 16
A probability density we all know: the normal distribution What do you think the hazard looks like for a normal distribution? Think of a concrete example. Suppose that times to complete the midterm exam follow a normal curve. What s your probability of finishing at any given time given that you re still working on it? 17
f(t), F(t), S(t), and h(t) for different normal distributions: 18
Examples: common functions to describe survival Exponential (hazard is constant over time, simplest!) Weibull (hazard function is increasing or decreasing over time) 19
f(t), F(t), S(t), and h(t) for different exponential distributions: 20
f(t), F(t), S(t), and h(t) for different Weibull distributions: Parameters of the Weibull distribution 21
Exponential = ( ) h t h Constant hazard function: ht = = = ( ) ( ) P T t f t he Exponential density function: Survival function: hu hu ht ht = = = = = ( ) ( ) 0 P T t S t he du e e e t 22 t
Why isnt the cumulative probability of survival just 90% (rate of .01 for 10 years = 10% loss)? With numbers cases/pers 1 0 . = ( ) on year h t Incidence rate (constant). Probability of developing disease at year 10. 01 . 1 . 10 ( ) = = = = ( 10 ) 01 . 01 . 0.009 P t e e Probability of surviving past year 10. 01 . t = = ( ) 90 5 . % S t e (cumulative risk through year 10 is 9.5%) 23
Example Recall this graphic. Does it look Normal, Weibull, exponential? 24
Example One way to describe the survival distribution here is: P(T>76)=.01 P(T>36) = .16 P(T>20)=.20, etc. 25
Example Or, more compactly, try to describe this as an exponential probability function since that is how it is drawn! Recall the exponential probability distribution: If T ~ exp (h), then P(T=t) = he-ht Where h is a constant rate. Here: Event time, T ~ exp (Rate) 26
Example To get from the instantaneous probability (density), P(T=t) = he-ht, to a cumulative probability of death, integrate: ( ) h t = = P(T t) he t ( ) ( ) h t h t = = P(T ) t 1 he e Area to the left 0 ( ) ( ) h t h t = = ( ) 1 1 ( ) P T t e e Area to the right 27
Example ( ) h age = ( ) P T age e Solve for h: 36 h( ) = 0 16 . e = ln 16 (. 36 ) h ln 16 (. ) = h 36 05 . h 28
Example . 05 ( ) age = ( ) P T age e This is a parametric survivor function, since we ve estimated the parameter h. 29
Hazard rates could also change over time = ( ) 0 . = 1 * t h t Example: Hazard rate increases linearly with time. h(5) .05 = h(10) .1 30
Relating these functions (a little calculus just for fun ): ( ) f t = Hazard from density survival and : (t) h ( ) S t = Survival from density S(t) : ( ) f u du t ( ) dS t = Density from survival ( f : ) t dt t 0 ( ( ) ) h u du = Density from hazard ( f : ) ( ) t h t e t 0 ( ( ) ) h u du = Survival from hazard S(t) : e d = Hazard from survival : (t) - ln ( ) h S t dt 32
Getting density from hazard = ( ) 0 . = 1 * t h t Example: Hazard rate increases linearly with time. h(5) .05 = h(10) .1 t ( ( ) ) h u du = Density from hazard ( f : ) ( ) t h t e 0 t t ( 01 . ) . 01 tdu udu 2 005 . t = = = ( f ) . 01 * . 01 ( ) . 01 ( ) t te t e t e 0 0 . 005 ( 25 ) . 125 = = = = ( ) 5 . 01 ) 5 ( 05 = . . 044 f t e e 005 . 5 . 100 ( ) = ( 1 . = = ( 10 ) 10 ) 1 . . 06 f t e e 33
Getting survival from hazard = ( ) 0 . = 1 * t h t h(10) .1 = h(5) .05 t ( ( ) ) h u du = Survival from hazard : S(t) e 0 t ( 01 . ) udu 2 . 005 t = = S(t) e e 0 005 . 100 ( ) = = 10 ( ) = . 60 S e . 005 ( 25 ) = ) 5 ( S . 88 e 34
