The Proportional Hazards Model in Reliability

Thomas A. Mazzuchi; Shell Research Labs; Amsterdam

Refik Soyer; The George Washington University; Washington

Raymond V. Spring; Army Research, Development, and Engineering Center; Natick

Key Words: Covariate Analysis, Machine Tool Reliability, Software Reliability, Accelerated Life Testing, Reliability Growth.

_ _ _ _ _ IheJ"Kr_i!,-,,- l!19!ieLf9J"S!uvival analysis introduced by Cox (Ref. 4) was developed with applications for "industrial reliability studies and medical studies" in mind. While this model has had a significant imPact on the biomedical field, it has received little attention in the reliability literature. In fact, most references which address the use of this model are in proceedings or European reliability journals and are difficult to obtain. In addition, many of these attempts have been criticized for misuse of the model. In this paper we present an overview of the Cox model (often referred to as the proportional hazards model) and discuss the use of this model in several typical and important reliability applications. We further discuss possible extensions to the theory that arise out of these reliability applications.

1. - Introduction and Overview

The regression model for survival analysis introduced in (Ref. 4) WllII developed with applications for "industrial reliability studies and medical studies" in mind. The Cox model is referred to in the literature as the Proportional Hazards Model (PHM). While this model has had a significant impact on the biomedical field, it has received little attention in the reliability literature. Only recently has the model been used for the analysis of hardware reliability (Ref. 2), software reliability (Refs. 3, S and 11) and repairable systems (Ref. 1).

In the traditional life testing model, it is assumed that a certain number of identical items are tested under identical conditions. This is due to statistical rather than practical considerations as under these assumptions the times to failure of these items are independent and identically distributed and thus are more easily analyzed. The problem with this approach is that often actual field data may violate the underlying assumptions in that (Ref. 1):

a. Items tested may not be totally indistinguishable;

b. The test conditions may vary from item to item;

c. A specific parametric model for the underlying density or hazard function cannot be specified.

Furthermore, in many cases in practice the testing environment changes and therefore the failure behavior of the units is not identical. The PHM enables us to describe such changes in the failure behaVior by using concomitant variables-or covariates- to model the environment. The attractive feature of the PHM is that it accounts for the effect of both the aging and the environment on the failure behavior of the items.

In view of the above, we believe that the PHM can be used to model the dynamic nature of the failure rate in many instances in reliability in which failure is a function of operating environment as well as time. In this paper we consider five different scenarios :

• Obtaining inference from reliability field data which is typically nonhomogeneous,

• Monitoring the reliability of devices such as machine tools whose failure behavior is a function of the operatina environment,

• Determining the reliability of software as a function of programming I testing environment,

• Making inference from accelerated life tests with multiple stresses and stress levels,

• Modeling reliability growth,

and show how each of the above fits into the framework of the PHM. We also discuss procedures for inference from the above scenarios.

A synopsis of our paper is as follows: In Section 2 we present an overview of the PHM and associated estimation procedures. In Section 3, we introduce the above scenarios and discuss how they fit into the PHM framework. In Section 4 we give some comments on extensions to the PHM theory that arise out of reliability applications.

2. Overview of the Proportional Hazards Model

To introduce some notation, let T denote the lifetime of an item under consideration. We assume that T has some density function f(t) and corresponding reliability function R(t). Most often the failure characteristics of the test item are best modeled via the failure rate (or hazard) function

f(t) )"(t) - R(t) ,

(2.1)

where Mt)dt is approximately the conditional probability of failure in (t,t+dtj given survival to time t. Choices for the form or restrictions on Mt) are usually done with the physics of failure, aging characteristics, or simple statistical convenience in mind.

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The PHM is developed on the notion that the railure rate or an item is not only a function of age, but is also arrected by concomitant variables. That is, if It. is the failure rate function of an item, then

Mt I�) = It.o<t) g(fI. � ). (2.2)

where �t) is the baseline failure rate which is a function of time alone. � is a vector of concomitant variables, /! is a vector of unknown coefficients and g is a positive function of /! and�. The coefficients /! are a measure of the importance or weight of each covariate. As such, if for BOme i fJ· = O. then the variable X; is has no effect on the failure �te and should not be considered as part of the failure environment. The PHM gets its name from the fact that due to Eq. (2.2). the ratio of any two individual hazard functions is time invariant (i .... any two hazards are proportional).

The usual form considered ror g(§ • �) is

due to its nonnegativity property but other forms have also been used. We specify some alt.rnativ. forms in Section 3.

(2.3)

The PHM can be used both in a parametric and nonparametric analysis. A specific form for /I.o(t) may be 8811Umed for a complete parametric analysis. In the nonparametric analysis the baseline failure rate function (or baseline r.llabllity fUlICtion) is esti_ted .mpirically. Below.

__ lIfe_liv8_aD-01l8l"view_ol _both o.ylMrian and n{1l!!-&!yeaien estimation procedures.

2.1 Non-Bayesian Esti_tion Procedures

The nonparam.tric approach proposed by Cox (Ref. 4) uses the weU known reliability relationship

t -I R(t) = e 0 Mu) du • (2.4)

to obtain an alternate form for the PUM as

R(t I/!.�) = Ro(t)&< /!.�) • (2.5)

wh.re RoCt) is the baseline reliability function. Given a random _mple of n test items, yieldina r distinct failure times tl1l < �2l < ... < tCrl' and n-r censored failure times, fet N(il �"note the itfdex set of items at riak of failure just prior to ti_ ttl}' for i-I . .... r. Cox suggested the use of the PiD'tial lftelthood 'tmeHon for estimatinl !!.

(2.6)

wh.r. X . is the covariate vector associated with the item ramnl .;rVime trt). We note that Eq. (2.6) is not a true likelihood functl�n in that it cannot be derived as the probability of som. observable outcome for a given model. Instead. it is derived on heuristic grounds. Thoup the above is not a true likelihood. it may be treated as one and maximum likelihood estimates of fl. _y il. are obtained. These are often referred to as maximum-partial likelihood estimates. Appropriate tests and interval estimates for components of fl may then be obtained usinl asymptotic nonael theory Or the use of the likelihood ratio techniques.

By a88umina fl = iJ and forminl the full likelihood function for the baBiiline reliability • the maximum likelihood estimate of RoCt) is obtained as

and thus the esti_ted reliability function for an item with covariate � is &iven by

Tied failure times and/or lrouped failure data can lead to some difficulties but alternative approaches have been suaested for dealina with these problems. Esti_tion for fully para_tric forms of the PUM _y be obtained by simply formulatina the complete likelihood function and usina _ximum likelihood procedures. Numerical difficulties _y arise as a result of the increased number of parameters.

4.2 Bayesian Esti_tion Procedures

Even thouah the use of the PHM is weU docu_ted in the statistical literature, there is a dearth of papers dealina with a proper Bayesian analysis of the model. Th. Bayesian esti_tion procedure centers around obtainina a prior distribution for the baseline cumulativ. hazard rat.

"o(t) = I: ho<u) duo (2.8)

Th. cumulativ. hazard rate _y be written in terms of the sum of random variables (Ref. 6),

whera

i A_rt.) = La. ''0' I j-I J

i = L -Lo&{1 -q .l.

j-l .r

(2.9)

(2.10)

(assumina R(tj) > 0) is the hazard coatribution in the ,*" interval. AaUminl prior distributions ror the variables a, (or qi) indirectly defines the prior distribution of "fI.t). For _the_tiel convenience it is assulled that these variables are independ.nt.

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The gamma distribution is choosen as the prior distribution for al. Letting X - G(a'p) denote the fact that X ia distributed according to a gamma density of the form

tP a-I -fjx f(x) = rea) x e , (2.11)

for a, fj, x > 0, then ai - G(c(Aij(tl) -Aij(t'_I )� c). The prior parameters AiI(t) and c are chosen as lbe prior best guess of the baseline cumulative hazard function and the measure of strength of conviction in the prior best guess. This can be demonstrated by the fact that

AO·[t.) -AO·(t. 1) VARlaj) = 1 c 1-

The above constitutes a gamma process as the prior distribution for A(t).

(2.128)

(2.12b)

To date the Bayesian analysis of the PHM proceeds by estimating the coefficients fj1.' ... .PM using the partial likelihood of Eq. (2.6) and obtaiOlng the posterior estimates of the cU1Dmulative hazard function conditioned on these estimates. The posterior distribution of A is very complicated and, difficult to analyze.

This is clearly not a fully Bayesian procedure. For a fully Bayesian procedure we would proceed by specifying a parametric forlll for �(t) and assuming that the elements of 8 are random Quantities independent of the D8rameters

------ ---of .... Apttr.-Then- pOsterior-estimates are obtained via usual Bayeslan procedures. Difficulty will arise in that closed form posterior expressions cannot be obtsined. This problem can be handled through the use of posterior approximation techniques as sugested in Mazzuchi and Soyer (Ref. 9).

3. Overview of Application Areas for the PHM

3.1 Inference from Field Test Dats

Mazzuchi and Spriog (Ref. 10) consider the analysis of field data from a test designed to assess the ability of a certain permeable (uniform) material to repel contaminating agents under various scenarios. A scenario consists of a specific dose level, contaminant type, and exposure type. Several external factors (i.e .. those not related to the contaminant) are also considered. These include such variables as prior wear time of uniforms, duties performed with uniforms, and sample swatch location.

Sample swatches of the material are tested against two contaminating agents at four different dose levels usin. either sinale or multiple challeoge test. After the initial agent challenge, the state of the material sample is observed over a fixed period of time at fixed time intervals. Interest centers around the "life length" of the material after the initial agent challenge (i. e, contamination) where a "failure" of the material sample is given when a certain portion of the agent penetrates the material. The statistical goal is to determine the reliability of the material after contamination. The determination of the effects (if any) of agent type, dose level, and external factors on reliability are also an important considerations. These types of tests are usually marked by participation but not by control on the part of the experimenter. As a result of the above testing scenario, nonhomogeneity of the failure data is inevitable.

A PHM is assumed for describing the life test results. The baseline hazard function is determined for each agent type while dose level, exposure type, and external factors are treated as co variates affecting the life length of the material. For each agent type, the effect of the covariates is defined through Eq. (2.3) with covariates indicating the dose level. the exposure type, the prior wear time, swatch location, troop unit, and uniform size. Both the estimate of R(t) and the tests for fj. = O. i = I, . . . • 6, are considered.

1

3.2 Inference from Accelerated Life Testing

In reliability studies it is a common practice to subject items to an environment which is more severe than the normal operating enviroment so that failures can be induced in a short amount of test time. A more severe environment can be created by increasing one or more of the stress levels which constitute the environment to values which: are greater than their usual levels. Such tests are called il(;celerated IWe tests. The main problem with inference from accelerated life tests is that uncertainty statements about the failure behavior of the items at usual stress conditions have to be made using life length data from the more severe stress conditions.

Assume that testing is done using k accelerated stress levels denoted by S .. i = 1 • . . .• k. We note here that while it is usual to denote the stress environment by a single value S, this value may be a function of several environmental factors such as voltage. vibration level. and temperature. At each stress level S .. n, items are tested for a predetermined and fixed time length or.

In Mazzuchi and Soyer (Ref. 8) it is assumed that the failure rate function for items tested under stress S, is constant and denoted by II,. Thus, given II, the failure distribution under stress S, is described by an exponential density. Furthermore, given II .. failure times for items tested under stress S, are judged to be independent. Given the data, the goal is to make inferences about the failure behavior of an item operating at use stress (normal) conditions Su. In so doing, it is most common to assume a functional relationship between the failure rate and the applied stress level. Such relationship is known as an il(;ce/er.1On or time transformation function. Commonly used models for describing such relationship are the ArrlMnlus l..a.N, the Eyring l..a.N. and the PCYNer law (Ref. 7. p. 421). In (Ref. 8) tbe Power Law

(3.1)

is used wbere a and fl are unknown coefficients which describe the stress effect on failure rate. The above model

. constitutes a parametric PHM where Si is the siogle

covartate and a represents the baseline hazard. By explicitly defining Si in terms of environmental factors, Eq. (3.1) becomes a PIIM with multiple covariates. Tbus functional forms for this relationship may be evaluated as well as the importance of the specific environmental factors. Mazzuchi and Soyer (Ref. 8) consider inference for the Bingle covariate case where a and fj are changing from one stress level to another.

3.3 Assessment oC Machine Tool Reliability

In an automated machining environment, it is important to allBe8ll the life of a cutting tool during which the quality of the workpiece is acceptable. The ability to effectively anticipate the end of useful life of a tool would result in increaaed productivity via diminished inventory costs, optimal replacement policies, improved planning, and reduced waste.

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Because of the lack of a universally acceptable physical theory of tool failure, an empirical relationship called Taylor Life Equation- henceforth TLE- has been in use for quite some time. The TLE describes the relationship between tool life and the machining variables. Specifically, if T denotes the tool life, Xl the cutting speed, X2 the feed rate, and X3 the depth of cut, then the TLE specifies that

tJo T-X tJ. X tJ2 X tJ3

' I 2 3

(3.2)

where tJO' tJ1, tJ2 and tJ3 are unknown c�fficients which change with the tool-workp,ece comb,natton. The TLE has also been used to select cutting conditions such as speed, feed rate and depth of cut to obtain a desired tool life. The Taylor Equation accounts for the effect of cutting variables but fails to account for the aging and wearout characteristics of the tool.

In view of the above, Mazzuchi and Sayer (Ref. 9) propose a PHM to describe the stochastic behavior of machine tool life. To model the stochastic nature of tool life, they consider the Weibull hazard model:

(3.3)

as a baseline failure rate to account for the aging and wearout characteristics of the tool. The authors select" vector X to include cutting variables such as feed rate , depth or cut , and cutting speed as well as variables which describe a specific cutting environment with specific workpiece-tool combination, cutting directions, etc. On the basis of the relationship between tool life and cutting variables implied by the TLE, they specify

M tJ. OXi " i_I

where M is the number of covariates used in the modeJ.

(3.4)

Mazzuchi and Sayer (Ref. 9) consider a fully Bayesian approach for inference for the above model by aBSuming a., and tJ·, i = I, ... M, are independent normal random variables �nd tJO is a lognormal random �ariable. An approximation by Tierney and Kadane (Ref. 12) 1S used for obtaining posterior distributions of the parameters involvnd. In considerina a Bayesian approach, the physical meaning of these parameters and existence of empirical and engineering information enables the specification of prior distributions for the parameters.

3.4 ABSetI9ment of Software Reliability

As noted in Wightman and Bendell (Ref. 3) the mainstream software reliability models neglect many important program features such as length, type of input, number and type of branching alternatives, number of loops, and number oi calis oi prOlrams. in reaiity, periormance is also related to factors external to software, such as intensity of testing, manpower availability in test, familiarity of software in development, skill level, level of integration and test phase. The conventional software prediction mociuis-ao-noi. aUow iot expiicit modeliul of these factors which affect the failure process. Factors such as these can be naturally incorporated and there effects naturally esti_ted by PHM. In view of the above, Wightman and Benden (Ref. 3) analyze some data seta using a non-Bayesian, nonparametric approach with Eq. (2.3)

reflecting the effect of such covariates as time in days since start that failure occured, number of previous failures, and time in days since last failure. Font (Ref. 5) considers parametric PHM in which the baseline hazard is modeled by a standard software reliability model.

3.5 Modeling Reliability Growth

A complex, newly developed system undergoes several stages of testing before it is put into operation. After each stage of testing, corrections and modifications are made to the system with the hope of increasing ita reliability. This procedure is termed reliability growth. It is important to recotlnize that a particular modification or series of modifications can lead to a deterioration in performance of the system. The important statistical issue is how to model and describe the changes in the performance of the system as a result of the modifications. Most of the current approaches consider only the effect of the testing phase on the failure behavior without explicitly modeling the testing environment for each phase. With a PHM model one can easily incorporate factors such as number and complexity of modifications, duration of testing stage, and types of failure observed and test for their effect on the system performance. Such a model also enables an analyst to assess whether reliability growth or decay has taken place as a result of modifications. This approach is currently under investigation.

4. Closing Commenta

Due to the inherent differences between biometry and reliability various theoretical extension to existing PHM machinery will need to be developed. The _in issues at present are the treatment of time varying covariates, the design of experiments using covariate information, and the control problem. Each of these extensions will carry a new set of problems which will need to be resolved to fully develop the theory. Thus the PHM provides the reliability theoretician and practitioner with new areas of research.

REFERENCES

1. Ascher, H. (1983). Regression analysis of repairable systems reliability, Electronic Systems Effectiveness and Life Cycle CORtina, J. K. Skwirzynski (ed.), Sprinaer-Verlag, Berlin, pp. 119-133.

2. Bendell, A. et al (1985). Proportional hazard modeling in reliability analysis - an application to break discs on high speed trains, to appear in Quality and Relfabflfty International.

3. 'Bendell, A. and Wight_n, D. W. (1986). The practical application of proportional hazard modeling, Proceedinas of the 5th National Reliability Conference, Birmingham, Enaland pp. 2B7311-16.

4. Cox, D. R. (1972). Regression models and life tables (with diBcwmion), Journal 0/ tM Royal Statistical Socfdy. Serie8 B, 34, pp. 187 -220.

S. Font, V. (1985). Une approche de fiabiUti des logiciels; modeles classiques et ..adele lineaire generalise, Tlulsi8 L 'Umveraite Paul SabaUer de Toulouse, France.

6. Kalbfleisch, J. D. (1978). Nonparametric Bayesian analysis of survival time data, Journal 0/ tM Royal Statistical Society, Series B 40, pp. 212-220.

7. Mann, N. Roo Schafer, R. E. and Sinapurwa1le, N. D. (1974). Methods for Statistical Analyais of Reliability and Life Da� John Wiley.

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8. Mazzuchi, T. and Soyer, R. (1987). A dynamic general linear model for inference from accelerated life teeta. Submitted for publication.

9. Mazzuchi, T. and Soyer, R. (1988). A_ment of machine tool reliability usina a proportional hazard model. Submitted for publication.

10. Mazzuchi, T. and Spring, R. (1988). AnalyBia of field test reliability data uaing a proportional hazard model. Submitted for publication.

11. Nagel, P. M. and Skrivan, J. A. (1985). Software reliability; repetitive run experimentation and modeling, Technical Report BCS-40336, Boeing Computer Company.

12. Tierney, L. and Kadane, J. B. (1986). Accurate approximations for posterior moments and marginal densities. Journal 01 the American Statistical Association, Vol. 81, pp. 82-86.

Acknowledgements

Research of Thomas Mazzuchi and Raymond Spring was supported by Contract DAAK60 -8'7-K-OOOlID of U. S. Army Raearch Office. Refik Soyer's �rch was supprted by Contract DAALOI-8'7-K-0056 of U. S. Army Research Office end Contract NOO!�-S--K-2020 or Office of Naval Research.

Thomas A. Mazzuchi KSLA MSE Department Badhuisweg 0131 CM Amsterdam, Netherlands

Bioaraphies

Tom Mazzuchi received his B.A. in mathematics from Gettysburg College, Gettysburg, Pennsylvania, in 1978 and his M.s. and D.Sc. in operations research from George Washington University, Washington, D. C. in 1979 and 1982 respectively. Currently he is a �rch scientist at the Shell Research Labs in Aaurterdam. Dr. Mazzuchi's research interests are in soft were reliability, accelerated life testing, failure rate estimation, reliability demonstration, and reliability estimation with covariate information.

Refik Soyer

Department of Manaa_ent Science School of Government and BuBiness Administration George Washington UniverBity Washincton, D. C. 20052

Rafik Soyer received his B.A. in economics from Bopzici Univ..,!'SitYi I!!t!!!!bu� Turkey" !n 1978, hie MSe. in operations research from Sussex UniverBity, Briahton, England, in 1979, and D.sc. in operations r .... ch from George Washington University, Washington, D. C. in 1985 respectively. Currently he is an UBistant professor in the Department of Manaaement Science at George Washington UniverBity and a member of the Institute tor Reliability and Risk Analyais at Gaorge Washington UniverBity. Dr. Soyer'. research interests are in applications of the Kal_n filter, BOftware reliability, accelerated lifa testina. and Bayesian statistics.

Raymond V. Spring

U. S. Army Natick Research, Development, and Engineering Center Natick, Massachusetts 01160

Raymond Spring received his B.A. in education and his M.S. in applied statistics from the University of Maryland, College Park, Maryland, in 1968 and 1970 respectively. He has worked as a research statistician for the Bureau of Census and currently works as research statistician for the U. S. Army Natick Research, Development, and Engineering Center. Mr. Sprina's research interests are in experimental design, nonorthoaonal ANDV A, and reliability.

256 1989 PROCEEDINGS Annual RELIABILITY AND MAINTAINABILITY Symposium