hybrid censoring scheme: an introductionhome.iitk.ac.in/~kundu/chennai-aug-2014-2.pdf · lifetime...

Hybrid Censoring Scheme: An Introduction

Debasis Kundu

Department of Mathematics & StatisticsIndian Institute of Technology Kanpur

August 19, 2014

Debasis Kundu Hybrid Censoring Scheme: An Introduction

Lifetime Data Analysis

Different Censoring Schemes

Type-I Hybrid Censoring

Type-II HCS

Generalized Hybrid Censoring Scheme

Outline

1 Lifetime Data Analysis

2 Different Censoring Schemes

3 Type-I Hybrid Censoring

4 Type-II HCS

5 Generalized Hybrid Censoring Scheme





Type-II HCS


What is Lifetime Data Analysis?

Lifetime data analysis is used to analyze data in which thetime until the event is of interest.





Type-II HCS


Example

1 Time until tumor recurrence.

2 Time until cardiovascular death after some treatmentintervention.

3 Time until AIDS for HIV patients.

4 Time until a machine part fails.





Type-II HCS


Type of Data

1 Time to event data is usually restricted to be positive and hasa skewed distribution.

2 The data are censored.





Type-II HCS


What is censoring?

Censoring is present when we have some information about asubject’s event time, but we do not know the exact event time.





Type-II HCS


Cause

There are generally three reasons why censoring might occur in astudy.

1 A subject/ an item does not experience the event before thestudy ends.

2 A person is lost to follow up during the study period.

3 An item has been withdrawn from the study.

These are examples of right censoring.





Type-II HCS


Outline




4 Type-II HCS






Type-II HCS


Type-I Censoring Scheme

Type-I censoring occurs when a study is designed to end after afixed time period T , determined before starting the experiment.





Type-II HCS


Type-I Censoring

A total of n units is placed on a life testing experiment. Letthe ordered lifetimes of these items be denoted byX1:n, · · · ,Xn:n respectively. The test is terminated when apre-determined time, T , on test has been reached. It is alsousually assumed that the failed items are not replaced.Suppose k items fail before time point T , the failure time ofthe k items are observed.





Type-II HCS


Type-II Censoring

A total of n units is placed on a life testing experiment. Thetest is terminated when exactly r failures occur. The failuretime of the r items are observed.





Type-II HCS


Type-I Censoring: Advantage and

Disadvantage

Advantage: Experimental time is fixed.

Disadvantage: Number of failures can be very small.





Type-II HCS


Type-II Censoring: Advantage and

Disadvantage

Advantage: Number of failures is fixed.

Disadvantage: Experimental time can be very large.





Type-II HCS


Outline




4 Type-II HCS






Type-II HCS


Type-I Hybrid Censoring Scheme

The mixture of Type-I and Type-II censoring schemes is known asthe hybrid censoring scheme.





Type-II HCS


Type-I HCS

A total of n units is placed on a life testing experiment. Thelifetimes of the sample units are independent and identically(i .i .d .) random variables . Let the ordered lifetimes of theseitems be denoted by X1:n, · · · ,Xn:n respectively. The test isterminated when a pre-chosen number, r < n, out of n itemsare failed, or when a pre-determined time, T , on test has beenreached, i .e. the test is terminated at a random timeT∗ = min{Xr :n,T}. It is also usually assumed that the faileditems are not replaced.





Type-II HCS


Available Data

When the data are Type-I hybrid censored, we have one of thefollowing two types of observations;

Case I: {x1:n < · · · < xr :n} if xr :n ≤ T

Case II: {x1:n < · · · < xd :n} if xr :n > T ,

here d denotes the number of observed failures that occurbefore time point T .





Type-II HCS


Associated Problems

Under parametric assumption of the underlying distribution,estimation of the parameter(s)

Finding the distribution of the parameter(s)

Finding the confidence interval(s)

Finding the optimum censoring scheme (Choosing r and T ).





Type-II HCS


Exponential Distribution

It is assumed that the lifetime distribution is a one parameterexponential distribution with mean θ, i .e., the PDF is

f (x ; θ) =1

θe−

xθ ; x > 0, θ > 0





Type-II HCS



The likelihood function based on the Type-I Hybrid Censoredsample becomes:

l(θ) =

−d ln θ − 1θ

[∑di=1 xi :n + (n − d)T

]if xr :n ≤ T

−r ln θ − 1θ [∑r

i=1 xi :n + (n − r)xr :n] if xr :n > T .





Type-II HCS



Based on the observed sample, it can be easily seen that the MLEof θ does not exist if d = 0, and if d > 0, it is given by

θ =

1d

∑di=1 xi :n + (n − d)T if xr :n ≤ T

1r

∑di=1 xi :n + (n − r)xr :n if xr :n > T .





Type-II HCS


Conditional Distribution of the MLE

How to obtain the conditional distribution of θ?Compute the conditonal moment generating function of θ, i.e.

Mθ|D>0

(t) = E (etθ|D > 0)





Type-II HCS



Using the uniqueness property of the MGF, it can be shown thatconditional distribution of θ can be written as a generalizedmixture of gamma and shifted gamma distributions.





Type-II HCS



Now using the moment generating functions of gamma and shiftedgamma distribution, one can obtain the PDF of θ for 0 < x < nT

as

fθ(x) = (1− qn)−1

[r−1∑

d=1

d∑

k=0

Ck,d g

(x − Tk,d ;

d

θ, d

)+ g

(x ;

r

θ, r)

+r

(n

r

) r∑

k=1

(−1)kqn−r+k

n − r + k

(r − 1

k − 1

)g(x − Tk,r ;

r

θ, r)]

,

where q = e−T/θ, Ck,d = (−1)k(n

d

)(d

k

)qn−d+k ,

Tk,d = (n − d + k)T/d , and g(.) is a gamma PDF.





Type-II HCS



It is clear that the PDF of θ, is a generalized mixture ofgamma and shifted gamma PDFs, when the mixingcoefficients may be negative also. The cumulative distribution

function (CDF) and the survival function (SF) of θ can beobtained in terms of incomplete gamma function.





Type-II HCS


Observations

Because of the explicit expression of the PDF of the MLE,different moments can be easily obtained. It is observed thatthe MLE of θ is a biased estimate of θ.

It can also be observed that as T → ∞,

fθ(x) = g

(x ;

r

θ, r).

It is the well known result that2r θ

θhas a chi-square

distribution with 2r degrees of freedom.

Substitution r = n, we get the PDF of the MLE of θ for theusual Type-I censored case.





Type-II HCS


Exact Confidence Intervals

A set of two sided 100(1-α)% confidence intervals for θ:[

2Sχ22,α/2

,∞]

if d = 0

[2S

χ22d+2,α/2

, 2Sχ22d,1−α/2

]if 1 ≤ d ≤ r − 1

[2S

χ22r,α/2

, 2Sχ22r,1−α/2

]if d = r ,

,

here S is the total time on test, i .e.

S =

∑di=1 xi :n + (n − d)T if xr :n ≤ T

∑di=1 xi :n + (n − r)xr :n if xr :n > T ,





Type-II HCS


Comments

The authors provided a formal proof in the ‘with replacement’case, and mentioned that the proof can be extended for thewithout replacement’ case also. It is not very clear that how theproof will be in the ‘without replacement’ case, as one of the majorassumption in their proof (‘with replacement’ case) is that for0 ≤ j ≤ r − 1,

P{j items fail at the decision time} =e−nT/θ (nt/θ)j

j!.

Clearly, the above assumption is no longer valid for the ‘withoutreplacement’ case and their proof very much depend on thatassumption.





Type-II HCS


Exact Confidence Interval

Based on the exact distribution of θ, and based on the fact thatPθ(θ > b) is an increasing function of θ for fixed b, the exact twosided 100(1-α)% symmetric confidence interval of θ, then θL andθU can be obtained by solving the following two non-linearequations;

α

2= FθL(θ), 1− α

2= FθU (θ).

HereFθ(x) = P(θ ≤ x).

θL and θU need to be computed numerically by solving the abovetwo non-linear equations.





Type-II HCS


Bayes Estimates

Bayesian solution seems to be a very natural choice in this case,and fortunately it turns out to be quite simple.

Prior:

Draper and Guttman assumed that θ has a inverted gamma priorwith the following PDF;

π(θ) =λβ

Γ(β)θ−(β+1)e−λ/θ; θ > 0,

here β > 0, and λ > 0 are the hyper parameters.





Type-II HCS


Based on the above prior, the Bayes estimate with respect tosquared error loss function is

θBayes =(Sd + λ)

(d + β − 1),

if (d + β) > 1, where

Sd =

nT if d = 0

∑di=1 xi :n + (n − d)T if 1 ≤ d ≤ r − 1

∑ri=1 xi :n + (n − r)xr :n if d = r .

Under the noninformative prior β = λ = 0, the Bayes estimatematches with the MLE.





Type-II HCS


Credible Interval

Interestingly, a 100(1-α)% credible interval of θ can be easilyobtained as

(2(Sd + λ)

χ22(d+β),α/2

,2(Sd + λ)

χ22(d+β),1−α/2

),

if d + β > 0, and here χ2m,α denotes the point of the central χ2

distribution with m degrees of freedom that leaves an area α in theupper tail.





Type-II HCS


Two-Parameter Exponential Distribution

Recently some work has been done on two-parameter exponentialdistribution, i .e. when both the location and scale parameters arepresent, i .e. the individual item has the PDF

f (x ; θ, µ) =1

θe−

1θ(x−µ); θ > 0, x > µ.

It is observed that for n ≥ 2, the MLEs of both θ and µ exist, andthey can be seen easily seen as

µ = x1:n,

and the MLE of θ is same as before and it can be obtained byreplacing xi :n with xi :n − x1:n.





Type-II HCS


Difficult Part

Although, the MLEs of θ and µ can be obtained in explicit form, itis not easy to obtain the joint PDF of µ and θ when they exist.The joint moment generating function of θ to µ is possible toobtain, but from their the inversion has not yet been possible.

The PDFs of θ and µ can be obtained along the same line. Fromthe marginal PDFs, the corresponding confidence intervals can beobtained.





Type-II HCS


Bayesian Inference

Prior:In this case it is assumed that λ = 1/θ has a Gamma distributionand µ has a non-informative prior.

Based on these priors, the posterior distribution of λ given µ is alsoa gamma distribution. The posterior distribution of µ is

l(µ|Data) ∝ 1

(A0 − µ)n+d; µ < x1:n.

Therefore, it is possible to generate samples from the jointposterior distribution function of λ and µ, and that can be used tocompute Bayes estimate and also to construct the credibleintervals.





Type-II HCS


Weibull Distribution

Estimation and the associated inferential procedures are quitedifferent when the lifetime distribution of the individual item is aWeibull random variables. Let us assume that the shape and scaleparameters as α and λ respectively, i .e. it has the following PDF;

f (x ;α, λ) =α

λ

(xλ

)α−1e−(x/λ)α , x > 0,

where α > 0 and λ > 0 are the natural parameter space.





Type-II HCS


Available Data

Remember we have one of the following two types ofobservations;

Case I: {x1:n < · · · < xr :n} if xr :n ≤ T

Case II: {x1:n < · · · < xd :n} if xr :n > T ,

here d denotes the number of observed failures that occurbefore time point T .





Type-II HCS


Likelihood

Based on the available sample as described in the beginning of thissection, the likelihood function for Case I and Case II are

l(α, λ) =(αλ

)r r∏

i=1

(xi :nλ

)α−1e−{

∑ri=1(xi :n/λ)

α+(n−r)(xr :n/λ)α}

and

l(α, λ) =

(αλ

)d∏di=1

(xi :nλ

)α−1e−{

∑di=1(xi :n/λ)

α+(n−r)(T/λ)α} if d > 0

e−n(T/λ)α if d = 0

respectively. The MLEs can be obtained by maximizing thelog-likelihood function with respect to the unknown parameters αand λ.





Type-II HCS


MLE of the Unknown Parameters

As expected the MLEs of α and λ cannot be obtained in explicitform. It has been observed that the MLE of α can be obtained byfinding the unique solution of a fixed point type equation

h(α) = α,

where for Case I

h(α) = −r

(r∑

i=1

ln xi :n +1

u(α)

[r∑

i=1

xαi :n ln xi :n + (n − r)xαr :n ln xr :n

])−1

and for Case II

h(α) = −d

(d∑

i=1

ln xi :n +1

u(α)

[d∑

i=1

xαi :n ln xi :n + (n − d)Tα lnT

])−1





Type-II HCS


MLE: Contd.

u(α) =

r (∑r

i=1 xαi :n + (n − r)xαr :n)

−1 for Case I

d(∑d

i=1 xαi :n + (n − d)Tα

)−1for Case II

Simple iterative scheme can be used to solve for α, the MLE of α,from the above equation. Start with some initial guess of α, sayα(0), obtain α(1) = h(α(0)), and proceeding in this manner we canobtain α(k+1) = h(α(k)). Stop the iteration procedure, when|α(k+1) − α(k)| < ǫ, some pre-assigned tolerance level. Once α isobtained then λ, the MLE of λ, can be obtained as λ = u(α)1/α.





Type-II HCS


Approximate MLEs

To avoid the numerical computation, approximate maximumlikelihood estimates have been proposed, which have explicitexpressions. It has been used in many cases particularly, when thefamily is a location scale family.

Although, Weibull family is not a location scale family, but logWeibull family is a location scale family..

Suppose a random variable X has Weibull distribution with PDF asgiven before, then Y = lnX has the extreme value distributionwith PDF

fY (y ;µ, σ) =1

σe(y−µ)/σ−e(y−µ)/σ

; −∞ < y < ∞,

here µ = lnλ, and σ = 1/α.Debasis Kundu Hybrid Censoring Scheme: An Introduction




Type-II HCS


AMLEs

The main idea is to work with Y = lnX , when X has aWeibull distribution. Make a Taylor series approximation tothe two normal equations upto first order terms, and obtainthe estimates in explicit forms.





Type-II HCS


AMLEs

Let us denote yi :n = ln xi :n, and T = T . The likelihood functionbecomes:

L(µ, σ) ∝ 1

σr

r∏

i=1

g(zi :n){G (zr :n)

}n−rCase I

L(µ, σ) ∝ 1

σd

d∏

i=1

g(zi :n){G (V )

}n−dCase II

where

g(z) = ez−ez , G (z) = e−ez , zi :n =yi :n − µ

σ,

V =T − µ

σ, µ = lnλ, σ =

1

α.





Type-II HCS


AMLEs

The log-likelihood functions become Let us denote yi :n = ln xi :n,and T = T . The likelihood function becomes:

l(µ, σ) = −r lnσ +r∑

i=1

ln g(zi :n) + (n − r) ln G (zr :n) Case I

l(µ, σ) = −d lnσ +d∑

i=1

ln g(zi :n) + (n − d) ln G (V ) Case II





Type-II HCS


Normal Equations:

Case I

−r∑

i=1

g ′(zi :n)

g(zi :n)+ (n − r)

g(zr :n)

G (zr :n)= 0

−r −r∑

i=1

g ′(zi :n)zi :ng(zi :n)

+ (n − r)g(zr :n)zr :n

G (zr :n)= 0





Type-II HCS


Normal Equations:

Case II

−d∑

i=1

g ′(zi :n)

g(zi :n)+ (n − d)

g(V )

G (V )= 0

−d −d∑

i=1

g ′(zi :n)zi :ng(zi :n)

+ (n − d)g(V )zr :n

G (V )= 0





Type-II HCS


Estimates:

µ = A− Bσ and σ =−D +

√D2 + 4rE

2r





Type-II HCS


Bayesian Inference

This problem has a nice Bayesian solution also. For the Bayesianinference the following Weibull PDF has been considered:

f (x ;α, λ) = αλxα−1e−λxα ; x > 0.

Prior:It is assumed that λ has a gamma prior, and α has a PDF which islog-concave, and they are independent.

Posterior:In this case the posterior distribution of λ given α is gamma, andthe posterior distribution α has a log-concave PDF.





Type-II HCS


Bayes Estimates and Credible Intervals

It is possible to generate samples from the joint posteriordensity function of α and λ. Using the generated samples, theBayes estimates and the associated credible intervals can beobtained.





Type-II HCS


Log-Normal Distribution

The PDF of a log-normal distribution is for −∞ < µ < ∞, σ > 0;

f (x ;µ, σ) =1√2πσx

e−

(ln x−µ)2

2σ2 x > 0.





Type-II HCS


Likelihod Function

L(µ, σ) ∝(1

σ

)R

exp

{−(ln xi :n − µ)2

2σ2

}{1− Φ

(c − µ

σ

)}n−R

where

R =

{r for Case Id for Case II

and c =

{ln xr :n for Case IlnT for Case II





Type-II HCS


Log-Likelihod Function

l(µ, σ) = −R lnσ− 1

2σ2

R∑

i=1

(yi :n−µ)2+(n−R) ln

{1− Φ

(c − µ

σ

)}





Type-II HCS


Normal Equations

∂l

∂µ=

1

σ2

R∑

i=1

(yi :n − µ) + (n − R)φ(z∗)

σΦ(−z∗)= 0

∂l

∂σ= −R

σ+

1

σ3

R∑

i=1

(yi :n − µ) + (n − R)z∗φ(z∗)

σΦ(−z∗)= 0,

where

z∗ =c − µ

σ





Type-II HCS


Expectation-Maximization Algorithm

1 Treat this as a missing value problem:

2 Complete observation: The whole sample. (not observable)

3 Missing observations: The censored observations.

4 Try to estimate the missing observations based on theobserved samples.

5 X = (x1:n, . . . , xR:n) be the observed data

6 U = (u1, . . . , un−R) be the censored data.

7 (W,U) is the complete data.





Type-II HCS


Complete log-likelihood function

lc(µ, σ) = −n lnσ − 1

2σ2

R∑

i=1

(ln xi :n − µ)2 − 1

2σ2

n−R∑

i=1

(ln ui − µ)2

In this case

µ =1

n

[R∑

i=1

ln xi :n +n−R∑

i=1

ui

]

σ =1

n

[R∑

i=1

(ln xi :n − µ)2 +n−R∑

i=1

(ln ui − µ)2

]





Type-II HCS


E-Step

ls(µ, σ) = −n lnσ− 1

2σ2

R∑

i=1

(ln xi :n−µ)2− 1

2σ2

n−R∑

i=1

E((lnUi − µ)2|Ui > c

)





Type-II HCS


Outline




4 Type-II HCS






Type-II HCS


Type-II HCS

Like conventional Type-I censoring, the main disadvantage ofType-I HCS is that most of the inference results are obtained underthe condition that the number of observed failures is at least one,and more over there may be very few failures occurring up to thepre-fixed time T . In that case the efficiency of the estimator(s)may be very low.

Because of this alternative hybrid censoring scheme that wouldterminate the experiment at the random time T ∗ = max{Xr ,n,T}has been proposed. It is called Type-II HCS.





Type-II HCS


Type-II HCS

It has the advantage of guaranteeing that at least r failures areobserved at the end of the experiment. If the r failure occursbefore time T , the experiment continues up to time point T . Onthe other hand, if the r -th failure does not occur before time T ,then the experiment continues until r -th failure takes place.





Type-II HCS


Comparison

Type-I HCS: In this case the termination time is pre-fixed, whichis clearly an advantage. However, if the mean lifetime of theexperimental item is not small compared to the pre-fixedtermination time T , then with high probability, far fewer than r

failures may be observed before the termination T . This willdefinitely has an adverse effect on the efficiency of the inferentialprocedure based on Type-I HCS.





Type-II HCS


Comparison

Type-II HCS: In this case, the termination time is a randomvariable, which is clearly a disadvantage. On the other hand forType-II HCS, more than r failures may be observed at thetermination time, and this will result in efficient estimationprocedure in this case.





Type-II HCS


Outline




4 Type-II HCS






Type-II HCS


Generalized Type-I HCS

Suppose n identical items are put on a life test at the time point 0.Fix r , k ∈ {1, 2, · · · , n} and T ∈ (0,∞), such that k < r < n.

If the k -th failure occurs before time T , terminate the experimentat min{Xr :n,T}. If the k-th failure occurs after time T , terminatethe experiment at Xk:n.

It is clear that this HCS modifies the Type-I HCS by allowing theexperiment to continue beyond time T if very few failures hadbeen observed up to time point T . Under this censoring scheme,the experimenter would like to observe r failures, but is willing toaccept a bare minimum of k failures.





Type-II HCS


Generalized Type-II HCS

Consider a life-testing experiment in which n items are put on atest. Fix r ∈ {1, 2, · · · , n}, and T1,T2 ∈ (0,∞), where T1 < T2.

If the r -th failure occurs before the time point T1, terminate theexperiment at T1. If the r -th failure occurs between T1 and T2,terminate the experiment at Xr :n. Otherwise, terminate theexperiment at T2.

This hybrid censoring scheme modifies the Type-II HCS byguaranteeing that the experiment will be completed by time T2.Therefore, T2 represents the absolute longest that theexperimenter allows the experiment to continue.





Type-II HCS


Thank You


hybrid censoring scheme: an introductionhome.iitk.ac.in/~kundu/chennai-aug-2014-2.pdf · lifetime...

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