Survival Models: Week 3

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 Estimation - Parametric

• Often we will assume that the random survival times of a population follow a
particular parametric distribution defined by f (t). For example, last week we
saw an example where f (t) = λ exp(−λt).
• Once we have assumed a particular distribution we need a means of
estimating the parameters that define the distribution.
• For the moment we will ignore censoring, that is, our data are complete.

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 Method of Moments (MOM)

Method of moments is a relatively simple form of estimation that involves

equating sample moments with population moments. The population moments
for a random variable X are E(X j ) and the corresponding sample moments are
−1
Pn j
n i=1 i . The method is best illustrated via an example.
x

The future lifetimes of a population follow an exponential distribution

(f (t) = λ exp(−λt)). Based on a sample of survival times x1 , x2 , ..., xn , compute
the method of moments estimator of λ.

For the exponential distribution we know that E(X) = λ1 and that the first
−1
Pn
sample moment is n i=1 xi . Equating the sample and population moments
1 −1
Pn −1
Pn −1
gives λ̂ = n i xi and λ̂ = (n i=1 xi ) .
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 Maximum Likelihood Estimation (MLE)

Maximum likelihood estimation involves maximizing the likelihood of the

observed data, viewed as a function of the unknown parameters.

The future lifetimes of a population follow an exponential distribution. Based on

a sample of survival times x1 , x2 , ..., xn , compute the maximum likelihood
estimate of λ.
Qn
1. Write the likelihood: L(λ|x1 , ..., xn ) = i=1 λ exp(−λxi ).
2. Typically easier to work with log-likelihood
Pn
l(λ|x1 , ..., xn ) = logL(λ|x1 , ..., xn ) = nlog(λ) − i=1 λxi .
dl(λ) n
Pn
3. Maximize the likelihood w.r.t λ. dλ = λ − i=1 xi .
−1
Pn −1
4. Setting the derivative to zero and solving gives λ̂ = (n i=1 x i ) .
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 Maximum Likelihood Estimation (MLE)

Asymptotically (as n → ∞) maximum likelhood estimators have the following

properties:
1. Consistent: θ̂ → θ.
1
2. Normally distributed with variance I(θ) , where I(θ) is the Fisher Information.
d2
3. I(θ) = −E( dθ2 log L(θ)).

Note: The above results mean that for large enough n, our MLE θ̂ is
1
approximately N (θ, I(θ) ).

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 Maximum Likelihood Estimation (MLE)

The number of deaths in a particular town (each week) follows a Poisson

θ k exp(−θ)
distribution (f (X, θ) = P (X = k) = k! ).Over a period of 11 weeks the
following death counts were observed 6, 6, 0, 3, 5, 4, 9, 5, 3, 7, 6. Compute the MLE
of θ and provide a variance for your estimate.

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 R Example

poisson.L<-function(mu,x) {

n<-length(x)
logL<-sum(x)*log(mu)-n*mu
return(-logL)

optim(par=1,fn=poisson.L,method="BFGS",x=c(6,6,0,3,5,4,9,5,3,7,6))

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 R Example

#another example: normal distribution

normal.L<-function(theta,x) {
mu<-theta[1]
sigma2<-theta[2]
n<-length(x)
logL<- -.5*n*log(2*pi) -.5*n*log(sigma2) -
(1/(2*sigma2))*sum((x-mu)**2)
return(-logL)
}
xdata<-rnorm(20,2,sd=2)
optim(par=c(1,1),fn=normal.L,method="BFGS",x=xdata)
mean(xdata)
var(xdata) #not exactly the mle of sigma2

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 R Example

n=5

Frequency

0 200
1 2 3 4 5 6

mle5

n=10,000
Frequency

0 150

2.94 2.96 2.98 3.00 3.02 3.04 3.06

mle10000

Figure 1: Histograms of MLE for n=5 and n=10,000

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 R Example

#sampling distribution of MLE for n= 5 nd n=1000

#based on a Poisson distribution with lambda=3
mle5<-rep(0,1000)
mle10000<-rep(0,1000)
for(i in 1:1000) {
mle5[i]<-mean(rpois(5,3))
mle10000[i]<-mean(rpois(10000,3))
}
par(mfrow=c(2,1))
hist(mle5,main="n=5")
hist(mle10000,main="n=10,000")
Note: Normality assumption is more appropriate when n goes large.

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 A Non-parametric Approach

If we observe N lifetimes a non-parametric estimate of the cumulative

distribution function at time t is:

d(t)
F̂ (t) = ,
N
where d(t) is the number of the N observed lifetimes that are ≤ t.
• An alternative to specifying a particular parametric distribution is to use a
“distribution free” or non-parametric approach.
• One advantage of a non-parametric approach is that we do not need to
specify a particular distribution - can be more robust.
• One disadvantage of a non-parametric approach is that it can be less
“efficient” than a suitable parametric approach.
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 A Non-parametric Approach

d(t)
F̂ (t) =
N
where d(t) is the number of the N observed lifetimes that are ≤ t.
• d(t) is binomial(N, F (t)).
• E(F̂ (t)) = E( d(t)
N ) = N −1
× N × F (t) = F (t). (unbiased).
1
• V (F̂ (t)) = N F (t)(1 − F (t))

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 R Example

#example of estimating F(t) using parametric

#and non-parametric approach.

observed<-rexp(20,0.25)
lammle<-1/mean(observed)
Fparam<-1-exp(-lammle*seq(0,max(observed),by=0.01))
Fnonparam<-rep(0,length(seq(0,max(observed),by=0.01)))
j<-1
for(i in seq(0,max(observed),by=0.01)) {
Fnonparam[j]<-sum(observed<=i)/length(observed)
j<-j+1
}
plot(seq(0,max(observed),by=0.01),Fparam,type="l",main=
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"Estimation of CDF",xlab="Time",ylab="Probability")
lines(seq(0,max(observed),by=0.01),Fnonparam,col="red")

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 R Example

Estimation of CDF

0.8
0.6
Probability

0.4
0.2
0.0

0 2 4 6 8 10

Time

Figure 2: Comparison of parametric and non-parametric estimation

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 Censoring

• Censoring refers to the situation where we only know that an observation

(survival time) falls in a particular interval - we do not know the exact value
of the observation.
• Right censoring - only know that survival time is equal to or larger than a
particular value. Example: subject moves interstate or investigation ends.
• Left censoring - do not know when condition of interest started. Example:
survival once contract a particular disease.
• Interval censoring - only know survival time falls in a particular interval.
Both left and right censoring are forms of interval censoring.

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 Censoring

• Random censoring - let Ci represent the time of censoring of the ith life and
Ti the random lifetime of the ith life. Life is censored if Ci < Ti .
1. Definition 1: Censoring is random if Ci is a random variable.
2. Definition 2: Censoring is random if Ci and Ti are independent random
variables. (This is the most common definition)
• Informative/non-informative censoring - Censoring is non-informative if it
provides no information about the future lifetime (Ti ). Definition 2 of
random censoring implies non-informative censoring, definition 1 does not.
The methods we will speak about in this course will rely on the fact that the
censoring is non-informative.

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