step-stress models and associated inferencehome.iitk.ac.in/~kundu/chennai-aug-2014-1.pdf · 2015....

Step-Stress Models and Associated Inference

Debasis Kundu

Department of Mathematics & StatisticsIndian Institute of Technology Kanpur

August 19, 2014

Debasis Kundu Step-Stress Models and Associated Inference

Accelerated Life Test

Step Stress Test

Cumulative Exposure Model

Khamis and Higgins Model

Cumulative Risk Model

Stress Changes at Random Time

Different Other Issues and Open Problems

Outline

1 Accelerated Life Test

2 Step Stress Test

3 Cumulative Exposure Model

4 Khamis and Higgins Model

5 Cumulative Risk Model

6 Stress Changes at Random Time

7 Different Other Issues and Open Problems



Step Stress Test






Life Testing Experiments

Life testing experiments have gained popularity in recenttimes. The main aim of a life testing experiment is to measureone or more reliability characteristics of the product underconsideration. In a very classical form of the life testingexperiment certain numbers of identical items are placed onthe test under normal operating conditions, and the time tofailure of all the items are recorded.



Step Stress Test






Life Testing Experiments: Difficulties

However, due to substantial improvement of science andtechnology, the most of the items now-a-days are quitedurable, and hence one of the major difficulties of the lifetesting experiments is the time duration of the experiment. Toovercome this problem, the experimenters use accelerated lifetesting experiments, which ensure early failures of theexperimental units.



Step Stress Test






What is an Accelerated Life Test?

Accelerated life test is a popular experimental strategy to

obtain information on life distributions of highly reliable

product. The main idea is to submit materials to higher than

usual environmental conditions to ensure early failure. Data

obtained from such an experiment need to be extrapolated to

estimate lifetime distribution under normal conditions.



Step Stress Test







Stress Factors:

1 Single Stress2 Multiple Stress

Classical Stresses

1 Temperature2 Voltage3 Current4 Pressure5 Load



Step Stress Test







Advantages

1 If you increase the stress, reasonable number of failures areensured

2 Reduce the experimental time

Disadvantages

1 Need to know the exact relation between the stress and lifetime2 Model must take into account the effect of accumulation of

stress3 Model becomes more complex4 Even in simple case analysis becomes difficult



Step Stress Test






Acceleration Model

One of major concerns with the accelerated life testing is to findthe relationship between the lifetime under stress condition andlifetime under normal condition. A relationship between thelifetimes needs to be established.



Step Stress Test






Some Definitions

θ(s): Stress Function at the stress level s

X : The time to failure under stress

Y : The time to failure under normal condition

FX : The cumulative distribution function of X

FY : The cumulative distribution function of Y

φθ : The acceleration function

The acceleration function

φθ : [0,∞) → [0,∞).

andY = φθ(X )



Step Stress Test






Relations Between X and Y

The Distribution Functions

FY (y) = FX (φ−1θ (y))

The Density Functions

fY (y) = fX (φ−1θ (y))

∣∣∣∣d

dyφ−1θ (y))

∣∣∣∣

Hazard Functions

λY (y) =

∣∣∣∣d

dyφ−1θ (y))

∣∣∣∣λX (φ−1θ (y))



Step Stress Test






Different Forms of φθ

Linear acceleration function

φθ = θ(s)X ; θ(s) ≥ 1.

In this case

λY (y) =1

θ(s)λX (x).

This is a Proportional Hazard ModelPower Transform

Y = AX θ(s)

With such an acceleration function, a Weibull(α, β) is transformedto another Weibull(Aαθ, βθ ).



Step Stress Test






Outline


2 Step Stress Test








Step Stress Test






What is a Step-Stress Life Test?

A Step-Stress life test is a particular accelerated life-test. Youobserve the failure times of the objects at a particular stress level,then change the stress level to a different level. Observe the failuretimes in the new stress level, and then the change the stress levelagain and so on.



Step Stress Test






Step-Stress Life Test

Several Variations are possible depending on the need

1 The Stress can be changed at the pre specified timing. In thiscase number of failures is random

2 The Stress can be changed at the pre specified number offailures. In this case failure time is random.



Step Stress Test






Outline


2 Step Stress Test








Step Stress Test






Lifetime Distribution Under Step-Stress

Cumulative Exposure ModelIt is the most popular model assumption for analyzing step-stressdata. The main idea is to assume that the remaining lifetime ofthe specimens depends only on the current accumulated stress,regardless of how it has accumulated.



Step Stress Test







Let us consider simple step-stress model:Suppose:

F1 : The cumulative distribution function under stress s1

F2 : The cumulative distribution function under stress s2

G : The cumulative distribution function under a step-stresspattern

G can be obtained from F1 and F2 under the assumptions that thelifetime F1 (under stress s1) at the time t1 has an equivalent timeu2 of the distribution function F2 (under stress s2).



Step Stress Test






Graphical Representation

F 2

F1

τ u 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6



Step Stress Test






Simple Step-Stress: Fixed Time

[0, τ) → Stress is s1

[τ,∞) → Stress is s2



Step Stress Test






Data: Step-stress Life Tests

n : Number of items put on the test.

s1, s2 : Stress levels (Simple SSLT).

τ : Stress changing time (Pre-fixed).

n❥✻

0 τ

✛ s1 ✲ ✛ s2 ✲



Step Stress Test







n : Number of items put on the test.

s1, s2 : Stress levels (Simple SSLT).

τ : Stress changing time (Pre-fixed).

n❥✻

0 τ

✛ s1 ✲ ✛ s2 ✲

t1:n t2:n . . . tN1:n tN1+1:n . . . tn:n



Step Stress Test







Generalization

n : No of items placed on the test.s1, s2, s3, . . . , sm+1 : Stress levels.τ1 < τ2 < . . . < τm : Stress changing times (Pre-fixed).

n❥✻

0 τ1

✛ s1 ✲

. . . . . . . . . τm−1

✛ sm ✲

τm

✛ sm+1 ✲

t1:n . . . tN1:n tNm−1+1:n. . .tNm:n tNm+1:n. . .

tn:n



Step Stress Test






Model Formulation: Simple Step-Stress

The effect of change of stress from s1 tp s2 is to change thelifetime distribution at stress level s2 from F2(τ) to F2(τ − h),where the shifting time h is such that

F2(τ + h) = F1(τ).



Step Stress Test






Model Formulation: Simple Step-Stress

Based on the following assumptions:

1 θ1 and θ2 are the scale parameters associated with F1 and F2,respectively.

2 F1 and F2 belong to the same scale family.

3 F1 and F2 are absolute continuous

The shifting time h becomes

h =θ2θ1

τ − τ



Step Stress Test






CDF and PDF for CEM

The cumulative distribution function of the lifetime under thecumulative exposure model becomes

G (t) =

{G1(t) = F1(t) for 0 < t < τ

G2(t) = F2

(t + θ2θ1 τ − τ

)for τ < t < ∞

The corresponding probability density function becomes;

g(t) =

{g1(t) = f1(t) for 0 < t < τ

g2(t) = f2

(t + θ2θ1 τ − τ

)for τ < t < ∞



Step Stress Test






CDF and PDF for CEM

The cumulative distribution function of the lifetime under thecumulative exposure model becomes

G (t) =

F1(t) for 0 < t < τ1Fk (t + ak−1 − τk−1) for τk−1 < t < τk , k = 2, . . . ,mFm+1(t + am − τm) for τm ≤ t < ∞,

where

ak−1 = θk

k−1∑

i=1

(τi − τi−1

θi

)for k = 2, . . . ,m + 1,

with τ0 = 0 and θi is the scale parameter of Fi for i = 1, . . . ,m+1.



Step Stress Test






Exponential Distribution

If we assume that the lifetime distributions are exponential at allstress levels, in case of simple step-stress model the cumulativedistribution function takes the form:

G (t) =

{G1(t) = 1− e

−t/θ1 for 0 < t < τ

G2(t) = 1− e−

(t+

θ2θ1

τ−τ)/θ2 for τ < t < ∞

The corresponding probability density function becomes;

g(t) =

{g1(t) =

1θ1e−t/θ1 for 0 < t < τ

g2(t) =1θ2e−

(t+

θ2θ1

τ−τ)/θ2 for τ < t < ∞



Step Stress Test






Likelihood Function

Suppose N1 is the number of failures before time point τ , andn−N1 is the number of failures beyond time τ . Then based on theobservations: t1:n, . . . , tN1:n, . . . , tn:n, the log-likelihood functionwithout the additive constant becomes:



Step Stress Test






Likelihood Function

L(θ1, θ2) = −N1 ln θ1 −1

θ1

N1∑

i=1

ti :n − (n − N1) ln θ2 −1

θ2

n∑

i=N1+1

ti :n

−1

θ1(n − N1)τ +

1

θ2(n − N1)τ.



Step Stress Test






Maximum Likelihood Estimators

The maximum likelihood estimator of θ1 does not exist if N1 = 0,the MLE of θ2 does not exist if N1 = n, and the MLEs of θ1 andθ2 exist only when 1 ≤ N1 ≤ n − 1,



Step Stress Test







The maximum likelihood estimator of θ1 and θ2 when1 ≤ N1 ≤ n − 1, are

θ̂1 =

∑N1i=1 ti :n + (n − N1)τ

N1

θ̂2 =

∑ni=N1+1

(ti :n − τ)

n − N1

Clearly, these MLEs are conditional MLEs, conditional on1 ≤ N1 ≤ n − 1.



Step Stress Test






Properties of the Maximum Likelihood

Estimators

We would like to study the properties of θ̂1 and θ̂2. We obtain theconditional distributions of θ̂1 and θ̂2.One way to obtain is to consider the moment generating functionsof θ̂1 and θ̂2, namely:

Mθ̂1(t) = E (etθ̂1 |1 ≤ N1 ≤ n − 1)

Mθ̂2(t) = E (etθ̂2 |1 ≤ N1 ≤ n − 1)



Step Stress Test






Conditional Densities

The conditional densities of θ̂1 and θ̂2 are

fθ̂1(t) =

n−1∑

j=1

j∑

k=0

cj ,k fG

(t − τj ,k ; j ,

j

θ1

)

and

fθ̂2(t) =

n−1∑

j=1

wj fG

(t; j ,

j

θ2

)



Step Stress Test






Conditional Densities

The conditional densities of θ̂1 and θ̂2 can be obtained underdifferent censoring schemes:

1 Type-I censoring scheme

2 Type-II censoring scheme

3 Type-I Hybrid censoring scheme

4 Type-II Hybrid censoring scheme

5 Progressive Type-II censoring scheme



Step Stress Test






Confidence Intervals

Under the assumptions that the conditional probabilitiesPθ1(θ̂1 ≥ a) and Pθ2(θ̂2 ≥ a) are increasing functions of θ1 and θ2,respectively, the exact confidence intervals of θ1 and θ2 can beobtained by solving two non-linear equations.

Pθ̂1(θ̂1 ≥ θL) = 1−

α

2Pθ̂1(θ̂1 ≥ θU) =

α

2.

Pθ̂2(θ̂2 ≥ θL) = 1−

α

2Pθ̂2(θ̂2 ≥ θU) =

α

2.



Step Stress Test






Order Restricted MLEs

The basic idea of the step-stress experiment is to increase thestress so that early failure is observed. It means it is assumed thatthe expected lifetime at the stress level s2 should be smaller thanthe expected lifetime at the stress level s1. Therefore, the naturalrestriction on the parameter space becomes

θ1 ≥ θ2.



Step Stress Test






Likelihood Function

The inference on θ1 and θ2 should be performed under thatrestriction. The MLEs of θ1 and θ2 can be obtained by maximizingthe log-likelihood function under restriction i.e.

maxθ1≥θ2

L(θ1, θ2)

L(θ1, θ2) = −N1 ln θ1 −1

θ1

N1∑

i=1

ti :n − (n − N1) ln θ2 −1

θ2

n∑

i=N1+1

ti :n

−1

θ1(n − N1)τ +

1

θ2(n − N1)τ.



Step Stress Test







If θ̂1 ≥ θ̂2 thenθ̃1 = θ̂1 and θ̃2 = θ̂2.

Otherwise

θ̃1 = θ̃2 =1

n

n∑

i=1

ti :n.



Step Stress Test






Bayesian Approach

Bayesian approach seems to be a natural choice under this orderrestriction case namely θ1 ≥ θ2. Let us consider

α =θ2θ1

or θ1 =θ2α

0 < α < 1.

The following priors can be taken: π1(θ2) as inverted gamma andπ2(α) as beta.The Bayes estimates and the associated credible intervals also canbe obtained.



Step Stress Test






Other Lifetime Distributions

Different other lifetime models have been used for analyzingstep-stress data.

1 Weibull distribution

2 log-logistic distribution

3 generalized exponential distribution

4 two-parameter exponential distribution

5 gamma distribution



Step Stress Test






Weibull Distribution: CEM

Since for the CEM

G (t) =

{F1(t) if 0 < t < τ

F2(a(t)) if τ < t < ∞,

where a(t) = (θ2/θ1)τ + t − τ . Therefore, the PDF of Weibulldistribution becomes

g(t) =

{βθ1tβ1e−(t

β/θ1 if 0 < t < τβθ2(a(t))β−1e−(a(t))

β/θ2 if τ < t < ∞.



Step Stress Test






Outline


2 Step Stress Test








Step Stress Test







The CEM in case of Weibull distribution becomes quitecomplicated. Due to this reason Khamis and Higgins proposed thefollowing model. The hazard function of CEM in case ofexponential distribution takes the following form:

h(t) =

1θ1

if 0 < t < τ

1θ2

if τ < t < ∞



Step Stress Test






Khamis and Higgins Model: Weibull

Distribution

It is assumed that at the two different stress levels the lifetimedistributions follow Weibull distribution with the same shapeparameter, but different scale parameter. At the stress level si , thelifetime becomes;

f (t) =β

θitβ−1e−t

β/θi ; t > 0, β > 0, θi > 0



Step Stress Test






Khamis and Higgins Model: Weibull

Distribution

Khamis and Higgins proposed the following step-stress model forWeibull distribution:

h(t) =

βθ1tβ−1 if 0 < t < τ

βθ2tβ−1 if τ < t < ∞

The corresponding survival function becomes:

S(t) =

exp(− t

β

θ1

)if 0 < t < τ

exp(− t

β−τβ

θ2− τ

β

θ1

)if τ < t < ∞



Step Stress Test






Khamis and Higgins Model: MLEs

The MLEs cannot be obtained in explicit forms. Non-linearequations need to be solved to compute the MLEs.

Bootstrap or asymptotic distributions can be used forconstructing confidence intervals.

Order restricted inference is also possible, but computationallyit is quite demanding.



Step Stress Test






Khamis and Higgins Model: Bayesian

Inference

In case of order restricted inference i .e. when θ1 > θ2, Bayesianinference seems to be a natural choice in this case also. Similarprior assumptions can be made:The following priors can be taken: π1(θ2) as inverted gamma,π2(α) as beta, take π3(α) as a log-concave prior.The Bayes estimates and the associated credible intervals also canbe obtained.



Step Stress Test






Outline


2 Step Stress Test








Step Stress Test






Cumulative Risk Model: Exponential

Distribution

One of the problems for CEM model is that the hazard function isnot continuous. The cumulative risk model has been suggested toovercome this problem. In case of exponential distribution theCRM takes the following form:

h(t) =

θ1 if 0 < t < τ1a+ bt if τ1 ≤ t < τ2θ2 if t > τ2

The parameters a and b are such that h(t) becomes continuous.Therefore

a+ bτ1 = θ1 and a+ bτ2 = θ2.



Step Stress Test






Cumulative Risk Model: PDF

In case of exponential distribution, the PDF becomes

f (t) =

(a+ bτ1)e−(a+bτ1)t if 0 < t < τ1

(a+ bt)e−(at+b(t2+τ12)/2) if τ1 < t < τ2

(a + bτ2)e−(a+bτ2)t−b(τ

11−τ

22 )/2 if t > τ2



Step Stress Test






Cumulative Risk Model: CHF

The associated cumulative hazard function becomes

H(t) =

h1(t) if 0 < t < τ1h2(t) if τ1 < t < τ2h3(t) if t > τ2

whereh1(t) = (a+ bτ1)t

h2(t) = (a+ bτ1)τ1 + (t − τ1)2b/2

h3(t) = (a+ bτ1)τ1 + (τ2 − τ1)2b/2 + (t − τ2)(a+ bτ2)



Step Stress Test






Cumulative Risk Model: Estimation

The MLEs of the unknown parameters cannot be obtained inclosed form. But the least squares estimators which can beobtained by minimizing

n∑

i=1

(H(ti )− Ĥ(ti )

)2

with respect to the unknown parameters a and b are in explicitforms. Here

Ĥ(ti ) = − ln(Ŝ(ti )) = ln n − ln(n − i + 1)



Step Stress Test






Cumulative Risk Model: Weibull Distribution

In case of Weibull distribution the CRM takes the following form:

h(t) =

α1λ1tα1−1 if 0 < t < τ1

a+ bt if τ1 ≤ t < τ2α2λ2t

α2−1 if t > τ2

The parameters a and b are such that h(t) becomes continuous.Therefore

a+ bτ1 = α1λ1τα1−11 and a+ bτ2 = α2λ2τ

α2−12 .



Step Stress Test







Based on the hazard function h(t), the corresponding cumulativehazard function becomes;

H(t) =

H1(t) if 0 < t < τ1H2(t) if τ1 < t < τ2H3(t) if t ≥ τ2.



Step Stress Test







H1(t) = λ1tα1

H2(t) = λ1τα11 + β0(t − τ1) +

β12(t2 − τ21 )

H3(t) = λ1τα11 + β0(τ2 − τ1) +

β12(τ22 − τ

21 ) + λ2(t

α2 − τα22 )



Step Stress Test







The survival function S(t) is given by

S(t) = e−H(t) =

S1(t) if 0 < t < τ1S2(t) if τ1 < t < τ2S3(t) if t ≥ τ2.



Step Stress Test







The survival function S(t) is given by

S1(t) = e−λ1t

α1

S2(t) = e−λ1τ

α11 −β0(t−τ1)−

12β1(t

2−τ21 )

S3(t) = e−λ1τ

α11 −β0(τ2−τ1)−

12β1(τ

22−τ

21 )−λ2(t

α2−τα22 )



Step Stress Test







The corresponding PDF becomes;

f (t) = −d

dtS(t) =

f1(t) if 0 < t < τ1f2(t) if τ1 < t < τf3(t) if t ≥ τ2.



Step Stress Test







The corresponding PDF becomes;

f1(t) = α1λ1tα1−1e−λ1t

α1

f2(t) = (β0 + β1t)e−λ1τ

α11 −β0(t−τ1)−

12β1(t

2−τ21 )

f3(t) = α2λ2tα2−1e−λ2t

α2e−λ1τ

α11 −β0(τ2−τ1)−

12β1(τ

22−τ

21 )+λ2τ

α22



Step Stress Test






Outline


2 Step Stress Test








Step Stress Test






Step-Stress Model: Stress Changes at

Random Time

So far we have discussed the step-stress model where the stresschanges at a pre-specified time. In this set up the stress changes ata time where the pre-specified number of failures take place.Suppose it is assumed that the stress changes at the n1-th failure.Therefore we observe the failures at

t1:n < . . . < tn1:n < tn1+1:n < tn:n.



Step Stress Test






Stress Changes at Random Time: MLEs

Under the assumptions of cumulative exposure model if thelifetime distributions are exponential then

θ̂1 =T1

n1θ̂2 =

T2

n − n1,

where

T1 =

n1∑

i=1

ti :n + (n − n1)tn1:n

T2 =n∑

i=n1+1

(ti :n − tn1:n)



Step Stress Test






Outline


2 Step Stress Test








Step Stress Test






Optimum τ

How to choose τ so that some optimality criterion is satisfied?What can be a good optimality criterion?

1 Minimize the trace of the inverse of the Fisher informationmatrix.

2 Maximize the determinant of the Fisher information matrix.

3 Minimize the variance of the p-th percentile estimator of thedistribution function under normal stress.



Step Stress Test






Step-Stress Model with Cure Fraction

In a cure rate model it is assumed that the population consists oftwo types of items: susceptible and immune. Susceptibles arethose who are subject to failures and immunes are those who arenot subject to failures. In this case the survival function of thepopulation

S(t) = p + (1− p)S0(t)

In case of step-stress set up it is assumed that S0(t) has one ofthese step-stress models.



Step Stress Test






Step-Stress Model with Competing Causes of

Failures

In many life testing experiment one observes more than one causesof failures. When the time of failure and the associated cause offailure is also observed it is known that competing risk data. In thiscase it is assumed that the observed failure time T is as follows:

T = min{T1, . . . ,TM},

where T1, . . . ,TM are the failure times for different causes whichare observable. A competing risk data is as follows;

(T ,∆)

∆ denotes the cause of failure.Debasis Kundu Step-Stress Models and Associated Inference


Step Stress Test






Multiple Step-Stress Model

So far we have mainly discussed the simple step-stress model. Incase of multiple step-stress model the stress changes at

τ1 < τ2 < . . . < τk .

It is usually assumed that there is a link function of the differentparameters at the different stress levels. For example theassumption of the log-link function is very common, i .e.

ln θi = a+ bxi ; i = 1, . . . , k .

Here xi is some function of xi and it is known.



Step Stress Test






Open Problems: General

1 Goodness of fit tests

2 Comparison of the different models

3 Bayesian Prior choice

4 Optimal for simple step-stress and multiple step-stress.

5 Develop step-stress models with covariates



Step Stress Test






Open Problems: Specific

1 Develop order restricted inference for Weibull

2 Develop inference for three parameter Weibull

3 Bayesian inference for Weibull distribution

4 Develop step-stress models with covariates



Step Stress Test






Thank You


Main PartAccelerated Life TestStep Stress TestCumulative Exposure ModelKhamis and Higgins ModelCumulative Risk ModelStress Changes at Random TimeDifferent Other Issues and Open Problems

step-stress models and associated inferencehome.iitk.ac.in/~kundu/chennai-aug-2014-1.pdf · 2015....

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