On a Generalized Model in Accelerated Life Testing

doi:10.4236/ojs.2012.22021

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Open Journal of Statistics, 2012, 2, 184-187

http://dx.doi.org/10.4236/ojs.2012.22021 Published Online April 2012 (http://www.SciRP.org/journal/ojs)

On a Generalized Model in Accelerated Life Testing

Eduardo L. Cruz1, Adolfo M. De Guzman2

1Department of Epidemiology and Biostatistics, University of the Philippines, Manila, Philippines

2School of Statistics, University of the Philippines, Quezon City, Philippines

Email: [email protected]

Received January 30, 2012; revised February 29, 2012; accepted March 14, 2012

ABSTRACT

The main objective of accelerated life tests in this setting is the recovery of the distribution of the random variable rep-

resenting lifetime which is difficult to observe at a certain level of a given stress factor. A general model for accelerated

life test is proposed that utilizes the inverse prob lem approach, that is, the variable is observ e at different level/s and the

transfer function is used to recover the elusive random variable (life time). The problem then is reduced to finding the

transfer function. We derive some properties of the proposed general model. The Lognormal distribution and the Ar-

rhenius model for lifetime are used as examples. Its relationship with the Cox proportional hazards model is also dis-

cussed.

Keywords: Accelerated Life Test; Arrhenius Model; Inverse Problem

1. Introduction

From the point of view of production and reliability en-

gineers, accelerated life testing is an important aspect of

product development, quality control and improvement.

Accelerated life testing is accomplished by applying in-

creased stress on the product or product component. It is

intended to produce data on strength and on lifetime of

material components and systems. For an excellent but

elementary exposition on this, the reader is referred to

Nelson [1].

Accelerated life testing is also very interesting from

the point of view of theory. Most lifetime data suffer

from the censoring problem of statistics. Lifetime obser-

vations usually exceed the interest time of the observer or

even his own lifetime. Accelerated life tests, however,

are performed to destruction thereby eliminating cen-

sored observations. The analysis of censored or incom-

plete data requires a modification of the usual statistical

methodology of independent and identically distributed

observations and their uncensored generalizations. Re-

cent developments point to the counting process approach

with the use of martingales as introduced by Aalen [2]

for analyzing survival or lifetime data. The audience is

referred to Anderson, Borgen, Gill and Keiding [3] and

Fleming and Harrington [4] for a comprehensive treat-

ment. The first volume covers extensively both theory

and applications of the counting process approach to sur-

vival analysis while the second volu me treats the analysis

of clinical data via the martingale approach. Both books

require some amount of mathematical as well as statisti-

cal sophistication. It takes more time for this approach to

be a practical statistical technology. An alternative to the

counting approach is provided by the Accelerated Life

Testing in some interesting cases. This is so because it

reduces the number of incomplete observations and even

eliminate them.

Section 2 of this paper describes a generalized model

for accelerated life testing. Its properties are stated and

discussed in Section 3 where some examples are also

provided. Section 4 gives some concluding remarks and

future work.

2. A Generalized Model for Accelerated Life

Test

Let X be a nonnegative random variable that represents

lifetime in the normal condition, that is, without any

stress applied to it, with unknown distribution F. Suppose

stress is applied on X and the observations say,

,,,

sns

XX

are made from a distribution, say Fs.

This setting is an example of the inverse problem ap-

proach in statistics where the stress S is a known operator.

That is,

X:S

SF F

,,,

or , (1)

in some space of random variables. Here the stress s may

have K levels, i.e. 12

ss.

An approach to accelerated life test model is given by

Cox and Oakes [5] and Barlow and Sheuer [6] and also

appears in Gnedenko, et al. [7]. This is described as fol-

lows. Denote a failure time distribution function under a

E. L. CRUZ ET AL. 185

normal stress condition by F0(·). The accelerated life

time transformation is given in terms of





 

and

F0(·) by the relat ionship



tzF tzA



 





,zA





,zA

 

, (2)

where is a positive function (or acceleration

function) connecting “time to failure” with a stress factor

z, and A is a vector of unknown parameters. For z = 0,

is assumed equal to 1. This relationship is a

scale transformation where a change in stress does not

result in a chang e in the sh ape of the distribution function

but changes its scale only. This relationship can be writ-

ten in terms of the acceleration function as



tzAt



. (3)

In other words, the relationship above is linear with a

time acceleration function



We now propose a model for accelerated life test gen-

eralizes the relationship in (3). This general model is

expressed in terms of random variables and thus is sim-

pler in structure. Let S be a nonempty set whose elements

we will refer to as stress space. For example, in the Ar-

rhenius model, S is the set of nonnegative real numbers

representing temperature. We define the proposed model



As X







, (4)

where X is the random variable under normal use,



is a

vector of parameters, s  S and



is called the

scaling or acceleration function which is a monotone

function continuous from the right. This model is a sto-

chastic process S





sS and will be called a general

lifetime model with stress space S. The model says the

random lifetime S

depends on the stress s, where stress

is a state or configuration of stress factors belonging to

some known stress space S. Information on the stress

space should be available in the underlying field such as

medicine for clinical trials, psychology for behavioral

studies, chemistry for phenomena depending on clinical

reactions, to name a few.

Two situations arise from this model: first, if stress s is

fixed or controlled, this model is interpreted as a general

accelerated life test model. If s is random, the model is

known as the survival model with covariates. In this pa-

per we consider only the first case where stress s is fixed.

We will refer to this model as the Generalized Model for

Accelerated Life Testing (GMALT). It will be shown

how this model relates to the well-known Arrhenius

model in reliability engineering. The Arrhenius model is

widely used to model product life as a function of tem-

perature. Applications include electrical insulations, solid

state and semiconductor devices, battery cells, and in-

candescent lamp filaments.

3. Elementary Properties of the GMALT

and Some Examples

As mentioned in the introduction, the main advantage of

the proposed model which is expressed as a scaling of

random variables is that its properties can be studied

more conveniently. In this section we present some ana-

lytic properties of the proposed model and state them as

propositions.

3.1. Mean and Covariance Function

Proposition 1. If S

is a gmalt with acceleration func-

tion







and stress space S, then the mean function

μs and the covariance function





are given by,

















sAsEX As







, S



 

and









,, ,

,,,

EAs XAsAt

As At

























SAs

where μX and X are the mean and variance of the

untransformed X, respectively. In particular, the variance

at any stress s  S is









. The proof of

these is straightforward and follows from the properties

of expectation. To keep things simple, we write A(s) in-

stead of







from now on.

The next property shows the effect of log transforma-

tion in the GMALT model. The proof follows directly

from applying the definition of logarithmic function and

mathematical expectation.

3.2. Effect of Log Transformation





glo S



Proposition 2. If

As X, then

has

constant variance



2Var logS



 and mean





0log

given by

 





, (6)





log

where 0











log

So the log transformation results in a constant variance

but a changing mean. In particular, since

and .



logEX













0, 1As



log 0As ,

the mean decreases since . An interesting

case is when





log

S has a Normal distribution.

Example 1. (Lognormal model) In the GMALT, S

Lognormal with mean S



2

and variance S



log if

is normal with mean, say



2

and variance





. In this

case the parameters of



exp 2







are and









22 22

expexp 1













expb







where



, and



2

and



 are as defined in Property 2 (See Hogg and

Craig [8] for pr operties of Lognormal distribution).

This example will enable us to make inferences about

X which is not observed in the GMALT model via the ac-

celeration function A(s), when it is known.

In the next example we show a situation where the

E. L. CRUZ ET AL.

186

model in Equation (4) fails. This example is also used to

verify the second property which states the effect of log

transforma tion on the GMALT.

Example 2. The data in Table 1 are hours to turn fail-

ure of a new Class-H insulation system tested in Motor-

ettes at 190, 220, 240, and 260 Temperature (˚C). The

original purpose of the experiment was to estimate the

median time to turn failure at the design of 180 Tem-

perature (˚C) [1].

To verify the second property in Section 3.2, we com-

pare the variances of the untransformed and the log-

transformed gmalt model. But we will see that Property 2

is satisfied on the three temperature levels only, as the

stress level 260 Temperature (˚C) seemed to have violat ed

an assumption of accelerated life model. In Tables 2 and

3 we see that the variances of both the untransformed and

the log-transformed lifetime at all temperature levels are

statistically different, but is not so for the first three lev-

els, where the variances of the log-transformed data are

not statistically different. Our observation, is that failure

Table 1. Hours to failure in an accelerated life test of class H

insulation in Motorettes.

Temperature (˚C) Levels

190˚C 220˚C 240˚C 260˚C

7228 1764 1175 600

7228 2436 1175 744

7228 2436 1521 744

8448 2436 1569 744

9167 2436 1617 912

9167 2436 1665 1128

9167 3108 1665 1320

9167 3108 1713 1464

10511 3108 1761 1608

10511 3108 1953 1896

Table 2. Test of homogenity of variance (all temperature

levels) of hours to failure and its log transformation Motor-

ettes data.

Life time of Motorette Levene Statistic Significance

Hours to failure 9.962 0.000

Log (Hours to failure) 8.373 0.000

Table 3. Test of homogenity of variance (190, 220 and 240

Temperature (˚C) levels only) of hours to failure and its log

transformation in class H insulation in Motorettes data.

Life time of Motorette Levene Statistic Significance

Hours to failure 7.708 0.001

Log (Hours to failure) 0.712 0.553

modes are different at these two sets of levels (the 190,

220 and 240 Temperature (˚C) degrees, and the 260

Temperature (˚C)), and that the stress level 260 Tem-

perature (˚C) may have destroyed the shape of the distri-

bution, thus violating the scaling assumption of the mo d e l ,

where a change in stress does not result in a change in

the shape of the distribution function but changes its

scale only. So, in this example we see that for the first

three temperature levels, the log transformation has re-

sulted in a constant variance but with still different (de-

creasing) mean, thus verifying Property 2. For this ex-

ample, therefore th e proposed model is applicable on the

first three stress levels only. The following will be useful

in interpreting the log mean of S and X.

Proposition 3. If Z is a log symmetric random variable

with log mean S







, i.e.





logEZ S



, then









exp median





, where S



 is as in Example 1.

Proof of Propositio n 3. From symmetry we have









log1 2





.

Since







log exp

PZ PZ





















exp median





then











, where F is the

distribution fun c tion of Z.

The result follows for absolutely continuous distribu-

tion functions F. We next relate the mean





2 and vari-

ance



 to the mean and variance of the untransformed

or “normal use” random variable X. To do that we need

the next result.

Proposition 4. Let X be a any nonnegative random

variable with distribution function F and finite mean











F and second moment . Then there

exists positive numbers



and



such that,





logEX and . In







Var logFFF



 











and

fact, Fx





logyx

where

. We may now state the next property.





Proposition 5. In the GMALT model with accelera-

tion function









0, 1As

and stress space S, the ex-

pectations and variance of the log transformation of

have the properties







log log

SFF

EX As





, (8)

and





Var logSFF







, (9)

where the random variable X with distribution F satisfies

the usual regularity conditions and



and



, are as

in Proposition 4.

The next result states the equivalence of our approach

to the classical Arrhenius model where the acceleration

function is given by

E. L. CRUZ ET AL.

187

 

exp







0,s

(11)

with is a function of temperature and 1



and 2



are constants characteristic of the product fail-

ure mechanism and test conditions [9 ].

3.3. Relationship to Cox Proportional Hazards

Model and Arrhenius Model

In the next result we show the equivalence of the

GMALT model and the Cox regression model or the pro-

portional hazards model. The reader is referred to Flem-

ing and Harrington [4] for an exposition. This result

shows that the dual of the GMALT model is the Cox

proportional hazards in the counting process approach to

survival analysis, the GMALT being the lifetime ap-

proach to survival analysis. The Arhennius life model is

also presented as special case.

Proposition 6. In the GMALT model with accelera-

tion function



with stress space S, the hazard func-

tion at any stress s  S is given by



sAs





0,x





 for all (13)



exp







where X is the covariate,

then 01

 

sx







which is the Cox

proportional hazards model with one covariate. In par-

ticular, if

s, 01





 and 12





 where s is

temperature, and 1



and 2



are constants, we get the

well-known Arhennius model [9].

Proof of Propositio n 6. Let . Then



PXy F



















PAsX yPX

FyAs







As y







where X

is the distribution of X. Also



 



yAs As



fyAs where fX is the

density of X. By definition the hazard function (at a given

stress level s) is

 



 



 

,11

sfyAs

FyAs







xs Fy

As x















by letting

yAs



3.4. Equivalence to Model of Cox and Oakes

Finally we show the equivalence of the proposed model

(4) to the one given by Cox and Oakes [5] and described

in Section 2.

Let X have distribution function F. Since

As X



 we have









, .

PXtPAsX tPX

FtAs

,tAs



















Taking









,1,











tsFs t



 gives











and the two are equivalent. Thus, the approach of Cox and

Oakes may be also taken as a special case of GMALT.

4. Concluding Remarks

In this paper, we considered a general model for acceler-

ated life testing and derive some of its properties. This

model is expressed in terms of the random variables

which is simpler, instead of distribution functions. An

important consideration is the choice of stress level,

where the GMALT model may fail. A threshold level

(stress) at which the scaling is no longer applicable must

be sought by the scientist. Future work may consider

where a test of hypothesis 0 applies. Rejec-

tion of this hypothesis means we can proceed to the test

or increase stress at a certain level. For the simple null

hypothesis



:1HAs





0:1HAs



, rejection in favor of the alter-

native





:1HAs



indicates that we can start doing

accelerated testing at stress level s where



1As



. For

products where





s is not known, the search for





s is the subject of many inquiries.

5. Acknowledgements

E. L. C. thanks the Office of the Vice President for

Academic Affairs of the University of the Philippines

where he was recipient of Doctoral Stude nt Grant.

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