Parametric Failure Data Analysis
5.8 ACCELERATED LIFE-TESTING AND DOSE–RESPONSE EXPERIMENTS
Accelerated life-testing was alluded to in section 5.4.1 as a way of economizing on test time by elevating the environmental conditions under which a life test is performed.Nominaloruse conditions stressmeans the environmental conditions under which an item is designed to be used. The elevated conditions under which identical copies of the item are tested are known asaccelerated stresses. Anaccelerated life-testis a life-test in which items are tested under accelerated stresses to gain information about its lifelength under nominal stress. This type of testing is done when it is not physically possible to conduct life-tests under use conditions stress, or when testing under use conditions entails an excessive amount of test time to yield failures.
Accelerated life-testing came into being in the space age; however, to the best of our knowledge, it was pioneered at the Bell Telephone Laboratories in the context of assessing the reliability of underwater telephone cables.
The viability of accelerated life-testing as a way to learn about lifetimes under use conditions stress depends on an ability to relate the information gained at one stress level to that which would be gained at lower stress levels. Thus a knowledge of the underlying physics of failure plays a key role here. Such knowledge is encapsulated in what is known as thetime transformation function.
If the time transformation function is ill chosen, then the resulting inferences can be misleading.
In principle, there could be several time transformation functions, each appropriate for a particular stress regime. Subsequent to the initial papers that laid out a statistical framework to model this problem (Singpurwalla 1971a, b) much has been written on the topic of accelerated life testing.
A recent treatise on this topic is by Bagdonavicius and Nikulin (2001). In section 5.8.1, I start with a formal introduction to the accelerated life-testing problem by presenting two commonly occur- ring versions; one based on time to failure data and the other based on the proportion of observed failures. Section 5.8.2 is an overview of the technology of Kalman filtering, my proposed method for dealing with inferential issues underlying the accelerated life-testing problem. Section 5.8.3 pertains to an illustration of the application of the Kalman filter algorithm for inference and extrapolation in accelerated tests; the material here is expository with details kept to a minimum.
Section 5.8 ends with the important issue of the design of experiments for accelerated testing; the material of section 5.8.4 draws on the notions of entropy and information discussed in section 5.5.1.
5.8.1 Formulating Accelerated Life-testing Problems
Suppose that Tj denotes the time to failure of an item under stress Sj,j=1 m, where S1< S2<· · ·< Sm; ‘Sj< Sk’ denotes the fact thatSj is less severe thanSk. Severity of stress
manifests itself in the fact thatSj< Sk⇒Tj
≥stTk; i.e. for allt≥0 PTj≥t≥PTk≥t. LetSube the use conditions stress and suppose thatSu< S1; thenS1 S2 Sm are accelerated stresses.
We assume that it is possible to conduct life tests at all themaccelerated stresses, but that it may or may not be possible to do any testing at use conditions stressSuand any stressSu< Su. To keep our discussion simple, suppose thatnitems are tested underSj and that the test is stopped at the time of occurrence of thek-th failure; forj=1 m; i.e. we have a Type II censored test at all the accelerated stress levels. In actuality, it is most likely that a large number of items will be tested at low stress levels so thatn1≥n2≥ · · · ≥nm, wherenjdenotes the number of items tested under stressSj; our assumption thatnj=nfor alljsimplifies the notation without a compromise in generality. Similarly, with the assumption of a commonkover all stress levels.
LetTjl l=12 kdenote the time to failure of thel-th item when it experiences a stress Sj j=1 mLetTjldenote thel-th order statistic of theTjls; i.e.Tj1≤Tj2≤ · · · ≤Tjk. Given that thetjl’s,j=1 m l=1 k, our aim is to make statements of uncertainty aboutTu;tjlis a realization ofTjl.
In order to do the above, we need to specify a relationship between Tj and Tu for each j j=1 m. It is important that there be some physical or practical justification for such relationships. These relationships are the time transformation functions mentioned before, and in principle are under the preview of subject matter specialists. Since specifying several rela- tionships, one for eachTj, can be cumbersome, a simplifying strategy is to assume a common general form over all theTjs with the only variable being theSjs. An example is the famous Power Law Modelin which forj=1 m, the random quantitiesFjTjl,l=1 k, are related to each other via the unknown parameters1 2>0 as:
FjTjl=exp−1Sj2Tjl (5.68)
here,Fjt=PTj≥t. Implicit to the above relationship is the assumption that the underlying chance distribution is exponential.
Traditionally, the Power Law and other such time transformation functions such as the Arrhenius and Eyring laws (Mann, Schafer and Singpurwalla, 1974), have been stated in terms of the unknown parameters of the underlying chance distributions. The specification of (5.68) is due to Blackwell and Singpurwalla (1988). It is a generalization of a tradi- tional power law model for accelerated life-testing, namely, PTj> tj=exp−t/j, with j=1Sj2. Thus in FjTjl=exp−1Sj2t, we replace t by Tjl to produce the model of (5.68). There are two advantages to our proposed model. One is that it is a relation- ship between the observables like the Tjls instead of parameters like the j’s. The second is that the model provides a framework for invoking a computationally efficient filtering algorithm. A similar strategy may be adopted when the underlying chance distribution is a Weibull. However, the filtering mechanism will entail computational difficulties due to non- linearities.
The data from an accelerated life-test tjl, j=1 m l=1 k, is used to make inferences about1 and2, and these in turn are used to a obtain predictive distribution for Tu, namely PTu≥tu. The Kalman filter model, also known as the Dynamic Linear Model or a State Space Model, provides a nice mechanism for accomplishing the said inference.
The Kalman filter model is a general methodology with applications to other problems in reliability. It is overviewed in section 5.8.2, following which I will continue with the problem posed above. But first I give below another version of the accelerated life-testing problem, a version that is prompted by scenarios in dose–response studies in the biomedical context, and in damage assessment experiments in the engineering reliability context. This second version of the accelerated life-testing problem is also amenable to analysis via a Kalman filter model.
ACCELERATED LIFE-TESTING AND DOSE–RESPONSE EXPERIMENTS 177 Dose–Response and Damage Assessment Studies
In dose–response studies, several subjects are treated to a dose, and the proportion of subjects that respond to the dose are noted. The dose is a proxy for the stress in accelerated life-testing, and the response could be a subject’s failure or survival. Thus in a dose–response experiment the response is a binary variable, whereas in conventional accelerated life-testing the response is the time to failure. The testing is done at several levels of the dose, and generally, these levels are higher than the nominal dose. The aim is to make an inference about the subject’s response at the nominal dose. In damage assessment studies, the situation is slightly different in the sense that often it is not possible to test an item at any desired stress that is pre-specified; the consequence that only one item can be tested at any level of the stress. The response in damage assessment studies is a number between zero and one (both inclusive) indicating the extent of damage done; often, the response is a subjective judgment. Here again, items are tested at the several levels of the accelerated stresses, the aim being to assess damage at a nominal stress.
Both the dose–response and the damage assessment scenario can be formulated in a unified way as described below. In the interest of clarity, the notation used below is different from that used in the context of conventional accelerated testing.
Quantified values of the dose (or the damage inflicting stress) will be denoted byX, where X takes valuesx0≤x <. The response of a subject to a dosex will be denoted byYx, where 0≤Yx≤1, for anyx. In drug testing, whereinxdenotes the dose that is administered to a patient, the relationship between Yxandx is known as thepotency curveor the dose–
response function. In the context of damage assessment,Yx=0 implies the total demolition of an item under study, andYx=1 denotes a total resistance to the damage causing agent. It is reasonable to suppose thatYxis non-increasing inx, withY0=1 andY=0. Whereas one can propose several plausible models for describing this behavior ofYx, the model I consider here is of the form
EYx =exp−x (5.69)
where >0 are unknown parameters. Alternate strategies involve taking a non-parametric approach wherein no parametric function relatingYxtoxis the sole basis of an analysis. More about this is said later in section 9.3.
Observe that the right-hand side of equation (5.69) is the survival function of a Wiebull distri- bution. Arguments that support this choice for a model are given by Meinhold and Singpurwalla (1987), who illustrate its generality for describing the several possible shapes thatEYx , the expected dose–response function, can take. Figure 5.11, taken from the above reference, illustrates this feature.
The data from a dose–response (or a damage assessment) experiment involving testing at doses x1 xm will consist of the responses Yx1 Yxm. The pairxj Yxj j=1 m, will be used in a Kalman filter model for making inferences aboutand, and these in turn will enable us to assessYxu, the response at nominal dosexu. The specifics on how to proceed with the above can be best appreciated once the Kalman filter model and the filtering algorithm are introduced. I therefore digress from the problems at hand and return to them in section 5.8.3, following an overview of Kalman filtering.
5.8.2 The Kalman Filter Model for Prediction and Smoothing
The Kalman filter model (KF) is based on two equations, an ‘observation equation’, and a
‘system or state equation’, with each equation containing an error term having a Gaussian distribution with parameters assumed known. The observation equation, given below, says that
1.0
0.8
0.6
0.4
0.2
0.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
x y
α=1.0 β=5.0
α=0.5 β=1.5
α=1.0 β=2.5 α= 2.0
β= 1.5 α=4.0
β=2.5
Figure 5.11 A Plot ofEYx=exp−Xfor different choices of values ofand.
for some index, sayxj, the responseYxjis related to an unknown state of naturexj, via the relationship:
Yxj=Fxjxj+vxj (5.70)
whereFxjis a known scalar. The quantityvxjis an error term, assumed Gaussian with mean 0 and varianceVxj, also assumed known; this is denoted as ‘vxj∼N0 Vxj. The state of naturexjis assumed to be related to its predecessorxj−1via thesystem equation:
xj=Gxjxj−1+wxj (5.71)
wherewxj∼N0 Wxj with Wxj assumed known. Also, the vxj and the wxj j= 12 , are serially and contemporaneously uncorrelated; furthermore, the Gxj is also assumed known. For equation (5.71) to be meaningful, the xj’s need to be ordered, as say x1≥x2≥ · · · ≥xj≥ · · ·; this will ensure that thexj’s will also be ordered. Equations (5.70) and (5.71) are given in scalar form; in principle, they could also be in vector form. Furthermore, some, but not all, of the error theory assumptions can be relaxed to encompass inter-dependence and non-Gaussianity; see, for example, Meinhold and Singpurwalla, (1989).
Traditionally, the KF model, which is comprised of the observation and the system equations, is introduced and discussed with time as an index. My choice of thexj’s as indices is to facilitate linkage with the accelerated life-testing scenario of section 5.8.1, in particular the dose–response and the damage assessment models. Thus in what follows, it makes sense to replacexjbyj, so thatYxjis simplyYjandxjisj; similarly the other quantities. With the above notation in place, suppose now thatj−1∼N,j−1 $j−1, forj=12 ; forj=1, we assume that,0and
$0are user specified. Then, the KF mechanism yields the recursive result that on observingYj,
j∼N,j $j (5.72)
ACCELERATED LIFE-TESTING AND DOSE–RESPONSE EXPERIMENTS 179
where,jand$jare given by theforward recursive equationsas:
,j=Gj,j−1+KjYj−FjGj,j−1
$j=1−KjFjRj Rj=G2j$j−1+Wj and
Kj=RjFjFj2Rj+Vj−1 (5.73)
the details, which entail a Bayesian prior to posterior updating, are in Meinhold and Singpurwalla (1983).
The recursive nature of (5.72) makes it suitable for the situation wherein given any indexj, the focus is on predictingYj+1via inference aboutj, the latter being based onYj Yj−1 Y1. As an illustration, suppose that Fj=Gj=1 for all j; then from (5.70) and (5.71) we see that Yj+1=j+1+vj+1=j+wj+1+vj+1. But j∼N,j $j, where,j and$j are given by (5.73). Consequently, givenYj Yj−1 Y1, we note that the predictive distribution ofYj+1is a Gaussian with mean,jand variance$j+Wj+1+Vj+1. Thus the KF model provides a recursive mechanism for predicting the future values of an observable given its past values.
The above predictive scheme is appropriate in the context of a time series, that is, when the indexj represents time. A time series is a collection of observations that are obtained over time; for example, the hourly temperature reading at a certain location. Once Yj+1 becomes available, attention will shift toj+1 andYj+2; the assessment ofj, which has been based on Y1 Y2 Yj, will not be revised. On the other hand, for those situations wherein all theYjs, saymin all, are simultaneously presented, it is meaningful to base inference about anyjon all the availableYjs – not justY1 Y2 Yj. Such an approach is calledKalman filter smoothing, and is appropriate in the context of dose–response and damage assessment studies wherein all theYjs are simultaneously available.
To describe the smoothing formulae, we need to slightly expand our notation. Let,jm and$jmdenote the mean and the variance of the distribution ofjbased onmobservations Y1 Y2 Ym. Then, it can be shown that thebackwards recursion equations(Meinhold and Singpurwalla, 1987) yield:
,jm=,jj+Jj+1,j+1m−Gj+1,jj and (5.74)
$jm=$jj−Jj+1$j+1m−Rj+1Jj+1
whereJj=$j−1j−1GjRj−1The details are in Appendix A of the paper mentioned above.
Note that,jjand$jjare the mean and variance, respectively, of the distribution ofjbased onY1 Y2 Yjalone, and are obtained via (5.73). The predictive distribution ofYm+1will be based on inference aboutm, namely, thatm∼N,mm $mm, with,mmand$mmgiven by (5.74). This completes our review of the KF model.
5.8.3 Inference from Accelerated Tests Using the Kalman Filter
The time transformation functions of section 5.8.1 are natural candidates for prescribing the observation equations of the Kalman filter model. Once this connection is made, and appropriate change of variables introduced to ensure that the error theory assumptions of the underlying KF models are satisfied, inference from accelerated life-tests proceeds, at least in principle, in a straightforward manner; the details tend to get messy and are therefore kept to a minimum. To see this, let us first consider the dose–response (damage assessment) scenario of section 5.8.1.
Filtering and Smoothing Dose–Response (Damage Assessment) Data
Recall the time transformation function of (5.69), namely EYx =exp−x, where x∈0is the dose. To facilitate an application of the KF algorithm, we need to introduce a transformation ofYxthat has a Gaussian distribution. For this, we define a random variable Y∗x=log−logYx, and require thatY∗xhave a Gaussian distribution with meanx and variance2x. The merits of this transformation will become clear in the sequel, but for now it suffices to say that withY∗x∼Nx 2x Yxhas what Meinhold and Singpurwalla (1987) call adouble lognormal distribution; its probability density atyxis of the form:
√ 1
2xyx−logyxexp
− 1
22xlog−logyx−x2
for 0≤yx≤1 Figure 5.12 taken from Meinhold and Singpurwalla (1987) shows plots of this density for different values ofx=, andx=. These plots show the versatility of this distribution to represent one’s subjective opinions about the damage phenomena in the01interval.
Properties of the double lognormal distribution are given in Meinhold and Singpurwalla (1987). The one of immediate interest to us here is its meanEYxx 2x, which for small values ofxis approximately exp−expx. However,EYx is also equal to exp−x, and soEY∗xx=x≈log+logx. This relationship forms the basis
μ=0 σ=1 μ=–1
σ=1 μ= 0.25
σ=0.25
0.0 0.2 0.4 0.6 0.8 1.0
y
μ=–2 σ=1
μ=0.5 σ=0.5
0 1 2 3 4 5
f (y)
Figure 5.12 Plots of the double lognormal density for different combinations of values ofand.