Learning for Inference from Life-test Data: Testing for Confidence

Parametric Failure Data Analysis

5.5 INFORMATION FROM LIFE-TESTS: LEARNING FROM DATA

5.5.2 Learning for Inference from Life-test Data: Testing for Confidence

One attempt at avoiding the possible negativity of gT^∗ n is to introduce'gT^∗ n, the conditional value of sample information (cf. De Groot, 1984). Here, letd0denote that value of dwhich maximizes

UF d^def=

dFd

d₀is known as the Bayes’ decision with respect to the priorFd. We assume thatd₀ exists and is unique. Then

'gT^∗ n^def=max

d U FT^∗ n d−U FT^∗ n d₀

the difference between the expected utility from the Bayes’ decision using the dataT^∗ nand the expected utility usingd₀, the decision that would have been chosen had the data not been available. By definition, it is true that'gT^∗ n≥0 for everyT^∗, and that its expected value with respect toFT^∗, the marginal ofT^∗, is indeedgn.

INFORMATION FROM LIFE-TESTS: LEARNING FROM DATA 165

b²=c^*a³

b²=ca

α α⁺¹ α⁺² α⁺³

0

ψ Values of a

(shape parameter) Values of b

(scale parameter)

t₁ t₂

t₃

Figure 5.4 Values of the gamma scale and shape parameters for desired shannon information.

In Figure 5.4 testing will stop after Three failures for a specified value of C, where C is determined by a desired value ofI •. In the schemata discussed above, stopping between failures is not considered.

It is well known (cf. Lindley, 1956) thatI •is not invariant under transformations of, so that when interest centers around# the mean time to failure,I# •=I •+2Elog; (El-Sayyad, 1969). In this case the boundary in thea bdiagram isb²≈C^∗a³where the constant C^∗ depends onI# •, the amount of Shannon information that is desired. This boundary is illustrated by the convex dotted curve of Figure 5.4. A consequence is that for any fixed value ofand, the amount of testing needed to get the same amount of information about and

#will be different. Thus given a fixed amount of test time and a fixed number of test items, it matters whether a desired amount of information is needed about the failure rate, or about the mean#. More details pertaining to the implications of the concave and the convex boundaries of Figure 5.4 are in Singpurwalla (1996). Also discussed therein is an observation by Abel and Singpurwalla (1993), which shows that at any given point in timet, a failure is more informative than a survival, if interest centers around # the mean time to failure, but that the opposite is true when interest centers around the failure rate=#⁻¹. Specifically,I#; failure att > I#;

survival att) for allt, but thatI; failure att < I; survival attfor allt; (Figure 5.5).

Information Loss Due to Censoring

Recall, (section 5.4.1, that in practice life-tests are often censored to save test time and costs.

However, there is a price to be paid for this saving, namely, the loss of information. The purpose of this sub-section is to describe ways of assessing the said loss. I start with the case of Type I (or time truncated) censoring whereinnitems are put on a life test that is terminated at some pre-specified timet. Suppose thatk≥0 failures at times₁≤₂≤ · · · ≤_kare observed so that the available data isD_t=k _{1 2 k}. Following Brooks (1982), I focus attention on the exponential casePT > t=e^−t for >0 andt≥0, and compute the expected information inD_t about– via (5.48) – as

I_C=E_DtIDt n−I

Survival at t Failure at t

0

t Shannon information I(ξ; •)

Survival at t

Failure at t 0

t Shannon information I(λ; •)

Figure 5.5 A comparison of shannon information about failure rate and mean given failure or survival att.

where the expectation is taken with respect to the marginal distribution ofDt. Similarly, we compute I_C=E_DID n−I

whereDdenotes the case of no censoring; i.e. by settingt= . The difference betweenI_Cand I_Crepresents the loss of information due to censoring; we would expect thatI_C≥I_C. The detailed computations tend to be cumbersome, but if a gamma prior with shape (scale)is assumed, then according to Brooks (1982),IC−IC ≈ −¹₂Elog1−e^−t. An interesting follow on to the above is the work of Barlow and Hsiung (1983) who show that for a ‘time transformed exponential’, namely, the one whereinPT > s =e^−Rs, where Rsis a known function ofs I_C is concave and increasing in both Rs andn. The prior on is a gamma with shape (scale). Note that withRs=s, we have the exponential as a chance distribution, and that withRs =s, the Weibull distribution with a known shape parameter >0. The concavity ofIC enables us to assess the effects of increasing the test truncation timetandn, the number of items to be tested; it implies that there is a limit beyond which increasingnand/ortgives little added information. Figure 5.6 taken from Barlow and Hsiung (1983) illustrates this point;

here=279 and=478.

The concavity of expected information has also been noted by Ebrahimi and Soofi (1990), who consider the case of Type II censoring, namely subjectingnitems to a test that is stopped at ther-th failure so that the observed data isD_r=k _{1 2 r}. Furthermore, the authors remark that

I_C^∗=ED_rIDr n−I

increases inr but decreases in, the shape parameter of the gamma prior distribution of. 5.5.3 Life-testing for Decision Making: Acceptance Sampling

Whereas life-testing for enhanced appreciation of unknown parameters has merit in scientific contexts, in the industrial set-up life-testing is most likely done with the aim of making tangible decisions such as the acceptance of a batch of items. The best example of this application is the U.S. Department of Defense’s Military Standard 781 C plan of 1977. Here a sample ofn items from a batch ofN items is subjected to a life-test, and based on the results of the test, the untested items are accepted or rejected. We shall suppose that items which are surviving at the termination of the test are discarded and thatnis specified. The determination of an optimumn

INFORMATION FROM LIFE-TESTS: LEARNING FROM DATA 167 Shannon information

Test time t Shannon information

Increasing test time

2 3 4 5 6 7 8

1

Sample size

00

0 Sample size n

0

Figure 5.6 The effect of sample sizenand test timeton Shannon information (from Barlow and Hsiung, 1983).

is discussed later in section 5.6. In what follows, we describe a strategy to be used by a consumer who tests then≥0 items and then makes an accept/reject decision.

Sincenis assumes known,’s decision tree will have one decision node and three random nodes (section 1.4). This decision tree is shown in Figure 5.7.

Withngiven, tests thenitems and obtains as data – at the random node1– the quantity T^∗. OnceT^∗is obtainedtakes an action at the decision node1to either acceptAor to reject Rthe remainingN−nitems. Suppose that chooses to accept; then at the random node 2nature takes its course of action in that the value of the unknown parameter happens to be. For this sequence of events, working backwards from, experiences a utility A T^∗ n.

Similarly when chooses to reject and nature takes the action at 3, the utility to is R T^∗ n.

Suppose that’s prior on is Fd; then having obtainedT^∗, computes Fd T^∗, the posterior for. Furthermore, it is reasonable to suppose that

A T^∗ n= A and that R T^∗ n= R

1

3

1 2

T^* A=Accept

R=Reject n

(θ^{, A, T*, n)} θ^{(or T)}

θ^{(or T)} (θ^{, R, T*, n)} Figure 5.7 Decision tree for acceptance or rejection by.

this is becausen is chosen and the observed data T^∗ will not have any effect on ’s utility of accepting or rejecting, once is treated as being known. next invokes the principle of maximization of expected utility and accepts theN−nuntested items if:

AFd T^∗≥

RFd T^∗ (5.52)

otherwise will reject. As stated before, in section 5.5.1, if is a measure of the quality of the item, say the mean time to failure, then A will be an increasing function of. For example, A=log, suggesting that the utiles to for very large values ofget de facto saturated at some constant value.

There is a reinforcement to the argument which leads to (5.52) above. This is based on the premise that utilities are better understood and appreciated if they are defined in terms of observ- able entities, like lifelengths, instead of abstract parameters like. With that in mind, suppose then thatT₁ T₂ , the lifelengths of all theNitems in the batch are judged exchangeable, and suppose thatT denotes the generic lifelength of any one of the N−n untested items. Then givenT^∗, the predictive distribution ofT is

PT > t T^∗=

PT≥tFd T^∗ (5.53)

With the above in place, I now suppose that at nodes2and3, nature will yield a life-time T (instead of) so that’s terminal utilities are T A T^∗ nand T R T^∗ n. As before, I suppose that the above utilities do not depend onT^∗andn, so that’s criterion for accepting the untested items will be

t AFdt T^∗≥

t RFdt T^∗ (5.54)

where t Ais’s utility for acceptance whenT=t, andFdt T^∗is the probability density at t generated by (5.53), the predictive distribution of T; similarly t R. Here again it is reasonable to suppose that t Ais an increasing function oft. What about t R, the utility toof rejecting an item whose lifelengthT ist? One possibility is to let t R=0, assuming that there is no tangible regret in having rejected a very good item. Another possibility is to let t R=a3, where the constanta3encapsulates the utility of a lost opportunity.

Besides ease of interpreting utilities, there is another advantage to the formulation which leads us to (5.54). This is because the acceptance (or the rejection) criterion of (5.54) can come into play at any time during the life-test, not just at the time of a failure. The practical merit of this feature is timely decision making. To see how, suppose that the underlying chance distribution ofT is an exponential with#as the mean time to failure; i.e.PT > t#=exp−t/# t≥0 # >0.

Suppose that the prior on# is an inverted gamma distribution with shape (scale) parameter

; i.e. the prior distribution of=1/#is a gamma with shape (scale) parameter. Then Fd# =

exp

−

#

#⁻⁺¹ (5.55)

If n items are tested and k failures observed at time t, then the total time on test = k

1_i+n−kt, where_iis the time of thei-th failure; recall that₁≤₂≤ · · · ≤_k. It is evident

INFORMATION FROM LIFE-TESTS: LEARNING FROM DATA 169 (details left as an exercise for the reader) that the posterior distribution of#is also an inverted gamma with shape+kand scale+. That is

Fd# k=+^+k +k exp

−+

#

#^−+k+1 (5.56)

We now suppose that t A=a₁t−a₂and t R=a₃, wherea₁is a utile to for every unit of time that an item functions, and−a2 the utile to in accepting an item that does not function at all; a2 encompasses the cost to of purchasing and installing an item. With the above in place, an application of (5.54) would result in accepting the untested items as soon as

+

+k−1>a₂+a₃ a1

(5.57) the details which entail routine technical manipulations are left as an exercise for the interested reader. The essence of the intent of (5.57) can be graphically portrayed with the shape parameter on the horizontal axis and the scale parameter on the vertical axis. The boundary between An, the region of acceptance, andRn, the region rejection, is a line with slopea2+a3 /a1

(Figure 5.8).

It is instructive to note the parallels between Figure 5.8 and Figure 5.4. In Figure 5.8, the sample path of the +k + curve takes jumps of size nt₁ n−1₂−₁ n−kk+1−k 2n−1−n−2 n−n−1. Acceptance occurs at the point marked as aon the boundary, after the third failure, but prior to the fourth.

A more realistic version of t Awould be t A=a₁t^p−a₂forp≤1; this would make the utility function increasing but concave. When such is the case,’s criterion for acceptance will be

+^p+k−p

+k >a₂+a₃

a₁ (5.58)

α α⁺¹ α⁺² α⁺³

0 ψ

Shape parameter (n–1) (τ2–τ1)

(n–2) (τ3–τ2)

(n–3) (τ4–τ3) (a)

Scale parameter

1

Rejection region R(n) Acceptance region

A(n)

Boundary

nτ1

Figure 5.8 Parameters of the inverted gamma and’s acceptance–rejection region.

this suggests that the boundary between acceptance and rejection is a curve instead of the straight line of Figure 5.8.

The starting point in Figure 5.8 is shown to lie in the region of rejection. This means that based on prior knowledge alone,is unable to accept the lot of untested items. Had the point been in the region of acceptance,would accept the lot without any testing. There is an additional feature to the scheme illustrated by Figure 5.8. This pertains to the fact that had testing been continued after the pointahad been traversed by the sample path of+k +– say until all thenitems on test fail – then it is possible that the sample path would cross the boundary line again and re-enter the region of rejection. That is, acceptance could be premature. Assuming that nis optimally chosen (section 5.6), the implication of such a switch from acceptance to rejection would be that added evidence from the life-test suggests rejection as the prudent course of action. With this caveat in mind, why then should one stop testing as soon as the sample path of+k +reaches the pointa? One answer to this question is that since the Shannon information is concave in the test time (Figure 5.6) I would have gained a sufficient amount of knowledge about T by the time we accept so that a second crossing is unlikely. That is, our decision to accept is fairly robust. Furthermore, assuming that acceptance does not occur very early on in the test, a second crossing can happen if the sample path takes a small jump. But small jumps correspond to very small inter-failure times, and such inter-failure times could be seen as outliers, whose effects on the decision process should be tempered. To account for such outliers, it is important that the predictive distribution ofT have heavy tails. Finally, but more importantly, our set-up had not incorporated a disutility (i.e. a negative utility) associated with the time needed to conduct the test and the utility gained by making early decisions. Were such utilities to be incorporated for deigning ’s actions, then the cost of testing would offset the added information obtained by additional testing. Multiple crossings are the consequence of the sample path being close to the accept/reject boundary.

5.6 OPTIMAL TESTING: DESIGN OF LIFE-TESTING EXPERIMENTS The material of section 5.5 assumed thatn, the number of items tested, is specified. In actuality, nis also a decision variable – like accept or reject – and therefore, it should also be chosen by the principle of maximization of expected utility. Since the testing of each item entails costs, namely the cost of the item tested plus the cost of the actual conduct of the test, there is a disutility associated with the testing of an item. In what follows we assume that any tested item that does not fail during the test is worthless; i.e. it is discarded. As a start, we shall assume that the cost, in utiles, of procuring and testing an item to failure is a fixed constants, irrespective of how long it takes for the item to fail. Furthermore, the set-up costs associated with the conduct of the life-test are assumed to be negligible.

Figure 5.9 shows a decision tree that illustrates the problem of choosing an optimal sample sizenfor a life-testing experiment. We assume that the cost of sampling and testing is borne by a consumer, who chooses non the premise that all thenitems will be tested to failure.

The adversarial scenario wherein the sampling and testing costs are borne by a manufacturer

1 2 2

T^*

1 n θ (or T)

(θ, d, T^*, n) d(T ^*, n)

Figure 5.9 ’s Decision tree for sample size selection in a life-test.

OPTIMAL TESTING: DESIGN OF LIFE-TESTING EXPERIMENTS 171 will be considered in section 5.7. In Figure 5.9, the convention of Figures 1.1 and 5.7, and the notation of section 5.5 are used.

Figure 5.9 is an elaboration of the decision tree of Figures 5.7 in the sense that it has two decision nodes,1at whichselects ann, and2at whichtakes an actiondT^∗ nbased on the dataT^∗ revealed at the random node1. The decisiondT^∗ ncould be a tangible action, like the accept/reject choices of section 5.5.3, or it could be a probability distribution for an enhanced appreciation of(sections 5.5.1 and 5.5.2). At the random node2nature takes the action (or equivalently reveals aT) so that at the terminus of the tree experiences a utility d T^∗ n, whered=dT^∗ n. Following Lindley (1978) (and also the references therein), we assume that d T^∗ ndoes not depend onT^∗and that it is additive and linear inn. Thus d T^∗ n= d−ns (5.59) Working the tree backwards, will choose at the decision node2thatdfor which

dFdT^∗ n

is a maximum. At the decision node1, will choose thatnfor which

⎡

⎣E_T∗

⎧⎨

⎩max

d

dFdT^∗ n

⎫⎬

⎭−ns

⎤

⎦

is a maximum. Were to choosedandnas indicated above, then’s expected utility will be maxn

⎡

⎣E_T∗

⎧⎨

⎩max

d

dFdT^∗ n

⎫⎬

⎭−ns

⎤

⎦ (5.60)

Observe that (5.60) is an extension of the first part of (5.47), the expected utility due toT^∗ andn. In principle, the above pre-posterior analysis of’s actions provides a normative solution to the problem of choosing an optimal sample size.

If the goal of life-testing is to obtain an enhanced appreciation of, then d= p•= logp(section 5.5.1). In this case will choose thatnwhich yields

maxn

⎡

⎣E_T∗

⎧⎨

⎩

logFdT^∗ n

⎫⎬

⎭−ns

⎤

⎦ (5.61)

If the goal of life-testing is for to make a rational accept/reject decision, then it is best to formulate the problem in terms of the observableT(instead of the) as was done in section 5.5.3.

Once this is done,will choose thatnwhich results in maxn

E_T∗

maxAR t•FdtT^∗ n

−ns

(5.62)

where t•is t Aor t R, andFdtT^∗ nis the posterior predictive distribution ofT, were to take a sample of sizen and observe T^∗ as data. The details leading to (5.62) are relatively straightforward; they are left as an exercise for the reader. It is important to note that even though can make a decision to accept prior to the failure of all thenitems on test,’s

pre-posterior analysis for choosing an optimal should be based on the premise that testing will continue until allnitems fail.

An obstacle to using (5.61) and (5.62) is the difficulty of performing the underlying compu- tations. Even the simplest assumptions regarding the chance distributions and the priors on their parameters, lead to complex calculations. Computing expectations with respect to the distribution ofT^∗is certainly a hurdle. Thus efficient simulations and approximation schemes, suitably cod- ified for computer use, are highly desirable. The situation described here gets more complicated if we wish to incorporate the scenario in which the cost of life-testing is a function of boths, the cost of procuring an item, and also the duration of the life-test. If chooses an n, then expects the life-test to last until_n, the failure time of the last item to fail. Thus if incurs a disutility ofs^∗ utiles for each unit of time the test is in progress, then the expected disutility incurred by due to the conduct of the test isns+s^∗E_n E_nis the expected value of_n, the largest order statistic (out ofn) of the predictive distribution of thentimes to failure in the test. When this disutility is accounted for, (5.60) becomes

maxn

⎡

⎣E_T∗

⎧⎨

⎩max

d

dFdT^∗ n

⎫⎬

⎭−ns−s^∗E_n

⎤

⎦ (5.63)

SinceE_nwill entailn, the inclusion ofE_nin the term to be maximized overnis meaningful.

However, a question may arise as to whyE_nis not included in the term within the braces over which an expectation with respect to the distribution ofT^∗is taken. This is a subtle matter that deserves mention. It has to do with the fact that whenis contemplatingnat the decision node1, all that is available tois the predictive distribution of thentimes to failure based on

’s prior alone. However, at2whenis contemplating an action to take,has, in addition to the prior, theT^∗thathopes to observe. Thus the expectation with respect toT^∗of the quantity in braces of (5.63). The above points are best illustrated when we consider an evaluation of E_n. The predictive distribution of the times to failure of the items on test is based on the prior alone. Thus

PT > t=

PT > tFd Sincenis max1 2 n, its distribution function is

P_n≤t=1−PT≤tⁿ=

⎡

⎣1−

PT > tFd

⎤

⎦

n

LetFdndenote the probability density generated by the above distribution at the pointn. Then

En= 0

nFdn

Endepends onnsinceFdndepends onn– thus its inclusion in the term within brackets over whichnis maximized.

Thus to summarize, should wish to incorporate the cost of testing into a pre-posterior analysis, then will choose thatnwhich yields:

maxn

⎡

⎣E_T∗

⎧⎨

⎩max

d

dFdT^∗ n

⎫⎬

⎭−ns−s^∗ 0

_nFd_n

⎤

⎦ (5.64)