survival 2
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Chapter 2 Hazard Models
In this Chapter, we will encounter some important concepts.The constant hazard model
The power hazard model
The exponential hazard model
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In chapter 1 we discovered that survival distributions may be equivalently specified by the probability density function, the cumulative distribution function, survival function and hazard function.
This means that if one of these representations is specified, then the others may be derived from it.
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In survival analysis, we always study the hazard function.
It is directly interpreted as imminent risk.
It may help identify the mechanism more effectively than the survival function.
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Researchers are likely to have first-hand knowledge of how imminent risk changes with time for the lifetimes being studied.
We would expect the shapes of the hazard functions for light-bulbs and people to be different.
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Section 2.1 The constant hazard model
A constant hazard model is usually proposed where imminent risk of failure does not change with time.
This characterises the exponential probability model.
.0for )( :model hazardConstant tth
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Conclusion
The lifetime variable T follows an exponential probability model with and we write
when the hazard is constant with .1)(
th
0
)exp(~ T
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The exponential model often fits survival data well.
Suppose we are modelling the lifetime of a system, Imagine events (such as toxic shocks to the system) occur according to a Poisson process with rate per unit time. Then , the number of shocks occurring in an interval of length t satisfies .
If the first such shock is fatal, then T, the time to system failure, satisfies
tX
)(~ tpoissonX t
tt
t eetXPtTPtS
!0
)()0()()(0
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Theorem (memoryless property)
If , then for any t>0 and s>0, it follows that
)exp(~ T
)()()( tStTPtTstTP
Remark:
An exponential lifetime is different from a human lifetime.
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Example 2.1
We simulate data from an exponential distribution with mean 100.
Two hundred data points are represented in the stem-and leaf plot: the size of each leaf unit is 10; the stems have increments of 20.
From this stem-and-leaf plot, we can compute hazard function based on the formula
)()()(
tTPtttTtP
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Section 2.2 The power hazard model
Suppose that and are constants. A power hazard model is usually proposed where imminent risk of failure is rapidly increasing with time
0 0
0for )( :model hazardPower 1 ttth
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The corresponding survival function is, for t>0,
The probability density function (pdf)
is
tduuduuh
eeetStt0
1
0)(
)(
)(')( tStf
.0 ,)( 1
tettft
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Definition:The lifetime random variable T follows a Weibull probability model with parameters and
, we write when T has a power hazard of the form
00
1)(
tth
),(~ WeibullT
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The parameter is very importantly a scale
parameter. It is easy to establish that the
following theorem holds.
Theorem:
a) If , then
b) If , then
)exp(~ T ).1exp(~T
),(~ WeibullT
)1,(~
WeibullT
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The mean and variance of the distribution are
somewhat more difficult to determine. For this we
will need the “gamma function”.
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Consider the random variable ,
the moments are given by
Now we have , then the
rth moment about 0 is given by
)1,(~ WeibullX
rdueudxexxXE u
rxuxrr 1)(
0
1
0
),(~ WeibullXT
rXEXETE rrrrrr 1)()()(
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Theorem:
If , then
and
),(~ WeibullT
11)(TE
2222 1121)()()(
TETETVar
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The Weibull distribution is a very flexible model in
a wide variety of situations: increasing hazards,
decreasing hazards, and constant hazards.
I. When , we have constant hazards.
II. When , we have increasing hazards.
III. When , we have decreasing hazards.
111
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Example 2.2
We simulate data from the Weibull distribution (2, √3).
Two hundred data points are represented in the stem-and leaf plot: the size of each leaf unit is 0.1; the stems have increments of 0.5.
From this stem-and-leaf plot, we can compute hazard function based on the formula
)()()(
tTPtttTtP
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a.Notice the shape of this stem-and-leaf plot; the right-skewness shows in a longer tail toward higher values.
b.The hazard estimates are 0.16, 0.43, 0.86, 0.95, 1.02,1.3. (Linear trend!)
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Section 2.3 The exponential hazard model
Suppose that and are constants. An exponential hazard model occurs frequently in actuarial science for modelling human lifetimes.
0b
0for exp1)( :model hazard lExponentia
tbut
bth
u
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The corresponding survival function is, for t>0,
Let and , it may be
expressed in the form
advocated as the Gompertz survival model.
bu
but
eetSt
bus
tdse
bduuh
expexpexp
)( 00
1)(
b1
bu
e
)1(exp)( tetS
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Definition:The lifetime random variable T follows a Gompertz probability model with parameters and
, we write when T has an exponential hazard of the form
0b u
0 ,exp1)(
tbut
bth
),(~ buGompertzT
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Section 2.4 Other hazards
Definition (lognormal distribution):
A positive valued survival variable T has a lognormal distribution and we write if . That is, X is normally distributed with mean and variance .
),(log~ normalT),(~log 2NTX e
2
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The survival function and the hazard function for the lognormal distribution do not have simple representations. If is the cumulative distributive function of the standard normal random variable, then has survival function
),(log~ normalT
t
tTPtTPtS
e
ee
log1
)log(log1)(1)(
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The density and hazard may be obtained by differentiation.
The major difficulty with the lognormal distribution is that the hazard rate is like a skewed mound in shapes: it initially increases, reaches a maximum and then decreases toward 0 as lifetimes become larger and larger.
This is hardly the stuff of lifetime modelling where hazards increase with old age!
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Definition (log-logistic distribution):
A positive valued survival variable T has a log-logistic distribution and we write
with and
if follows a logistic distribution with survival function
),(loglog~ isticT 0
TX elog
x
xSXexp1
11)(
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The survival function of the log-logistic distribution is therefore
where and .
t
ttS
tXPtTPtS
eeX
eT
11
logexp1
11)(log
)log()()(
01
exp
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The hazard function
ttthT
1)(
1
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Section 2.5 IFR (increasing failure rate)
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Definition:
The random variable T has increasing failure rate or IFR (respectively decreasing failure rate or DFR) if h(t), the hazard function of T, is an increasing (respectively decreasing) function of t.
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Figure 2.1 illustrates a bathtub-shaped hazard across support of the lifetime distribution.
Note that if , then T has IFR if
, DFR if and constant hazard if .
),(~ WeibullT
1 1 1
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The final example of this section concerns a new distribution which is clearly DFR as the hazard decreases for values beyond a fixed constant. (It is also hardly a model for human lifetimes!)
Definition
A positive-valued survival variable has a Pareto distribution and we write with and if T has hazard function
The support of this distribution is .
),(~ ParetoT 00
,
tt
th for )(
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The parameter acts as a threshold value beyond which data may be measured.
This applies particularly to certain types of astronomical data.
Where distant objects can not be detected if they are not of sufficient magnitude.
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