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Proportional hazards modelling in reliability assessment
Reliability Engineering 11 (1985) 175-183
Proportional Hazards Modelling in Reliability Assessment A. Bendell Statistics and Operational Research, Dundee College of Technology, Bell Street, Dundee DD1 I H G and Services Ltd, Rowanbank, 7 Riverside Road, Wormit, Fife DD6 8LR, Great Britain (Received: 23 July, 1984)
1 INTRODUCTION There has been much recent interest in the reliability field in the application of proportional hazards modelling. A number of practitioners and academics have recently been advocating its use, and reviewing the basic theory. The technique is, however, in the early stages of application in reliability studies, and no really satisfactory examples of its application to reliability data are to be found in the open literature. An implication is that the enormous potential of this approach for reliability analysis, as well as the particular shortcomings and ditiiculties of applying it in this field, are not well-understood in the reliability community as a whole. It is the purpose of this paper to start to get the reliability community to exploit the potential of this technique fully. Proportional hazards modelling identifies the effects of various explanatory variables or factors which may be associated with variations in the length of life of equipment. Factors such as temperature, pressure, material, use, etc., may be included. Repairable as well as nonrepairable systems may be studied. Data may be censored or uncensored. The This paper is based upon a lecture read to the Reliability Engineering 11 course, 21-25 May, 1984, at the Lucas Institute for Engineering Production, University of Birmingham, UK. The illustrative data sets employed in that lecture are omitted due to their sensitixTe nature. A case study illustrating the special features of proportional hazards modelling in reliability will be published in due course. 175
Reliability Engineering 0143-8174/85/$03.30 ~• Elsevier Applied Science Publishers Ltd, England, 1985. Printed in Great Britain
176
A. Bendell
technique is a powerful method for decomposing the variation in life lengths into orthogonal factors, identifying the significant ones and reconstituting the model for prediction purposes. The technique largely owes its origins to the seminal paper which Professor D. R. Cox read to the Royal Statistical Society in March, 1972. At that time, he described the probable applications of proportional hazards modelling as being qn industrial reliability studies and in medical statistics'. From its introduction the technique has generated a great amount of interest, although until very recently mainly in medical fields. A vast array of research papers have dealt with theoretical considerations and developments of the basic model. Why is it, however, that recorded applications of the model to date have until recently been almost entirely associated with medical data ? The answer must be very largely because of the familiar lack of qualified statisticians working in practical reliability. The recent attempts at proportional hazards modelling in the reliability literature very much reinforce this view. The papers have appeared to date almost entirely in Conference Proceedings or internal reports, with the implication that they were not initially refereed; e.g. refs 1-7. The data contained in these papers is diverse, including motorettes, ° marine gas turbines and ships' sonar, 4 valves in Light Water Reactor nuclear generating plants 1 and aircraft engines. 7 However, the proportional hazards models employed are, in general, basic and are not developed to take specific account of the complexities of structure arising in reliability data. Further, the data sets chosen to illustrate the methodology in these papers have often been inappropriate for the analysis carried out. Indeed, sometimes the data have so little structure or the models are so inappropriate that proportional hazards modelling yields almost no useful information: see, for example, ref. 4. It is also essential that assessments of the appropriateness and the fit of the proportional hazards model and the explanatory variables are made, but these have not always been reported in the recent reliability literature, possibly giving a misleading impression; see ref. 7. Apart from these Conference papers and internal reports, there is very little in the current open literature dealing specifically with proportional hazards modelling for reliability data. The major books such as those by Kalbfleisch and Prentice 8 and Lawless 9 are more general and deal largely in terms of medical data. However, an important discussion of some aspects of proportional hazards modelling in reliability is contained in a recent review paper 1° and in the contributions by its various discussants.
Proportional hazards modelling
2
177
T H E BASIC M O D E L
The fundamental equation on which the proportional hazards model is based is an assumed decomposition of the hazard function for an item of equipment into the product of a base-line hazard function and an exponential term incorporating the effects of a number of explanatory variables or covariates, z l , . . . , z k. That is
h(t; zl, z2,.. .,Zk) =ho(t ) exp(fllZl + . . .+ flkZk)
(1)
The fli are unknown parameters of the model, defining the effects of each of the explanatory variables. These need to be estimated from the data and tested to see whether each explanatory variable really has an effect in explaining the variation in observed failure times. These effects are assumed to act multiplicatively on the hazard rate, so that for different values of the explanatory variables, that is say different temperature and pressure conditions, etc., the hazard functions are proportional to each other over all time t. It is this property that gives the technique its name. The baseline hazard function ho(t ) represents the hazard function that the equipment would experience if the covariates all take the baseline value zero, which may correspond, depending on what the covariate is, either to a natural zero such as a zero temperature, or an arbitrary zero corresponding, say, to an arbitrary time point from which to measure installation date. There are essentially two ways of modelling this baseline hazard. Either a parametric regression model may be assumed so that, say, a Weibull hazard function may be assumed for ho(t), or a distribution-free approach may be taken under which no particular form is assumed for ho(t ), but its form is estimated from the data. The latter approach was first suggested by Cox, 1~ is more common, and is sometimes taken as synonymous with the term proportional hazards modelling. Indeed, it is a major advantage of proportional hazards modeling that one does not need to assume a specified form for the baseline hazard when this may be difficult due to the confusing effects of the covariates. Instead, the model provides a distribution-free estimate of the baseline hazard function which can be used to check the appropriateness of standard distributional forms. The fact that in the author's experience standard forms such as the Weibull or constant hazard rate of the exponential distributions often do then turn out to be appropriate is highly reassuring. However, it is not necessary to assume a
178
A. Bendell
apriori.
given form In a number of the recent reliability papers this has been done, although sometimes with a little justification. 2'7 The remainder of this paper will largely be concerned with the distribution-free approach. The detail of the methodology is not developed here, but the interested reader is referred to ref. 8. The usual method first iteratively estimates the effects of the covariates/~1,/32,. •., flk using the so-called method of partial likelihood. Essentially, this technique employs the usual method of scoring based upon a Taylor expansion for each step in the iteration, starting with initial values of zero. Once the estimates converge, which by experience is usually within four or five iterations, tests of whether each explanatory variable has any significant effect are based upon the asymptotic Normality of the estimates. Thus, either Normal tests or the related chi-squared tests may be used. Chi-squared tests can also be conducted for the explanatory ability of the whole set of covariates included, based upon the likelihood ratio statistic, which compares the likelihood under the fitted parameter values and under the assumption that they are all zero, or upon the sums of squares of the standardized estimates. Other methods of testing fit are also available, e.g. ref. 12. A stepwise procedure may be incorporated into the proportional hazards approach, whereby in its backwards version non-significant explanatory factors are excluded one at a time and the model re-run until all factors are significant. Alternatively, a forward approach may be taken, whereby new variables are sequentially forced into the model. Having estimated ill, f12,. •.,/3~ the method obtains a distribution-free estimate of the baseline hazard function based upon discrete hazard contributions at each of the times at which failures were actually observed to occur. This can then be compared with standard distributional forms such as the Weibull. The effect upon the survivor function of different values for the explanatory variables is a power one:
R(I; were
Z l , Z 2 , . . . , Z k ) ~-- [ R o ( t ) ]
exp~zllJ~ + "
+ zk/~k)
Ro(t) is the baseline survivor function: Ro(t)=exp[- il ho(x)dx1
It should be noted that since the above approach assumes a continuous model, ties in the data create an initial problem, and various approximations can be employed to eliminate this. It is also noteworthy
Proportional hazards modelling
179
that the approach is essentially a regression one, so that the usual difficulties of regression analysis apply. Most noteworthy in the current context is probably the problem of multi-collinearity. So far the treatment has only been discussed of times to failure of nonrepairable items, or times to failure of repairable ones in which subsequent failure times are identically distributed or perhaps where only the times to first failure are studied, Ascher, 4 Dale 6 and Ascher in the discussion of Lawless 1° propose the method introduced in ref. 13 in connection with medical analysis for the analysis of repairable systems. The problem is that this approach is not the uniformally appropriate approach to the analysis of repairable systems. The variations of reliability data that can arise are highly diffuse, and in the author's experience the assumptions implicit in the models of Prentice et aL 13 and the purposes for which they are formulated, are not reflected in the usual forms of reliability data. Alternative more-appropriate models can be developed for such given data sets with very little modification, and experience of this approach to date has been most rewarding. Some indications of the adaptation to the special features of reliability data are given in the next section. The paper by Prentice at aL 13 essentially introduces four different extensions of the basic proportional hazards model [eqn (1)], based upon stratifying the data into strata corresponding to the various inter-failure times in the life cycle for each component. This effectively removes the proportionality assumption between the subsequent inter-repair periods for a given item of equipment so that the baseline hazard function is different within each strata. The four models correspond to (i) either basing the baseline hazard function in eqn (1) on the time since the component was new or upon the time since it was last repaired and (ii) either assuming that the effects of the covariates are the same within every strata, or allowing their effects to vary between the strata representing the number of previous repairs the item of equipment has experienced. It is interesting that in analysing his example data set Ascher 4 in fact only considers the one model in which the baseline hazard depends upon the time since the last repair, and in which the effects of the covariates are assumed to be identical within every stratum. His results are uninspiring and not a good advertisement for the methodology. One must wonder about the appropriateness of the particular model he has chosen. In contrast, Dale 6 appears to obtain much more satisfactory results using a parametric model in a complex example involving environmental testing.
180
,4. Bendell
3
M O D E L L I N G RELIABILITY DATA
It is, of course, impossible here to enumerate all the complexities of structure arising in reliability data and the adaptations to the basic model necessary to appropriately deal with them. However, instead it is attempted to convey some flavour of c o m m o n data structures and the associated analysis. At the same time, it is also attempted to warn the unwary of the complexity that may be present in their data, and of the dangers of automatic non-seeing utilization of the proportional hazard method or software. The emphasis, as in so much of modern statistical reliability assessment, is on careful exploratory analysis. The debate about interactions in proportional hazards modelling is started, and attention is drawn to some of the problems in reliability data currently being tackled. Let us begin with the enthusiasm in the recent reliability literature about the paper by Prentice et al. 13 It is an important paper, but its importance must be kept in proportion. The structure of the model considered there is essentially one that allows for reliability growth or decay in subsequent inter-repair periods to the same item of equipment. It is the belief of the present author that it is almost always worth looking for such structure, along with many other aspects, in preliminary exploratory analysis of reliability data. However, whether it should be included in a proportional hazards model must depend upon what the major aspects of the data one wishes to model are, whether there is clear evidence that this particular feature is or may be important, the extent of the data and the purpose of the analysis. For example, often in reliability data subsequent repairs may not be causally related, nor to the same subassembly of the item of equipment, so that subsequent inter-repair periods should not experience such reliability decay or growth. Indeed, sometimes one has no choice but to work with times between failures within a group of different items of equipment. It is important to identijy the appropriate structure o f the data as the basis f o r your model. In varied applications of proportional hazards modelling and other techniques in reliability studies, the author has come across a large proportion of data sets in which there are no such trend effects, or insufficient data to investigate them. Concentration upon this assumed trend effect can divert attention from, and even remove the possibility of testing for, more-appropriate structure in the data. This question of appropriate structure really centres about what the appropriate underlying point processes for failures of
Proportional hazards modelling
181
repairable systems are. For example, the appropriate point process model to build might be a series of subsequent failure times for each item of equipment, or for each cause for each item, or for each cause and environment for each item. If one has no choice but to work with a group of different items of equipment within which failures to individual items are not labelled, then the appropriate point process model might be the series of subsequent failure times for the group, for each cause over the gfoup, for each cause and supplier combination within the group, etc. The possibilities are limitless and heavily dependent upon the context in which the data arise. The appropriate time metric for the baseline hazard is, similarly, data-dependent. However, having identified the appropriate structure in the data, the adaptation of the basic proportional hazard model to incorporate it is usually straightforward, although possibly complex to implement on a computer package. Usually, a separate strata (i.e. a separate baseline hazard function) should correspond to each separate point process, except for the classification by item. Thus, if the appropriate structure of the data is a separate stream for each cause for each item of equipment, then there should be a separate stratum for each cause. In data sets suggesting a large number of such natural strata, the stratification in terms of reliability growth/decay might not be worthwhile. Another aspect of the modelling of repairable systems data concerns the treatment of competing causes of failure. If repair restores a unit to 'as good as new', the failure time on one cause represents a censoring time on other causes. Again, to reiterate the point, the implementation and adaptation oJproportional hazards models must depend on the structure of the data. Interactions in proportional hazards modelling are an interesting area where developments are currently going on at Dundee. Hitherto, they appear to have been neglected in proportional hazards modelling, the covariates being treated as acting orthogonally. Interactions may be defined simply as products of other covariates, and initial application of this approach has yielded promising results and invaluable information about the physical nature of the failure processes in the data. One does, however, have to be wary of introducing multi-collinearities, particularly with dummy (i.e. 0-1) variables which are essential if one has ordinal or nominal data. This, at present, causes some procedural problems in what interactions to include. One more complex problem for the future is the 'nested nature' of much
182
A. Bendell
reliability data: items may fail, assemblies within items, sub-assemblies, all the way down to component level. Such data arise in practice, but whilst a failure may represent a censoring time to all lower levels, the form of analysis for such data is not clear.
4
P R O G R A M S A N D PACKAGES
A number of programs and packages are now available to implement the proportional hazards model, but their application to reliability data is not straightforward, particularly for repairable systems. Perhaps the most widely accessible routines although not necessarily easiest to implement nor complete, are within the B M D P and SAS packages. Alternatively, a paper by Whitehead 14 explains how to implement proportional hazards modelling using GLIM, but this is probably not to be recommended unless you are already familiar with GLIM. Preston and Clarkson 15 also describe a program S U R V R E G for proportional hazards modelling. In the major book by Kalbfleisch and Prentice 8 Fortran programs for the proportional hazards model are provided, although these contain some errors. Currently, at Dundee, modified and extended versions of these programs are being worked with and developed further. These have now been implemented in a number of organisations.
5
CONCLUSIONS
Proportional hazards modelling is very new in reliability. It has an enormous potential and can give invaluable information, but it must be used carefully.
REFERENCES 1. Davis, H. T., Campbell, K. and Schrader, R. M. Improving the analysis of LWR component failure data, Los Alamos Scientific Laboratory Report, LA- UR 80-92, 1980. 2. Nagel, P. M. and Skrivan, J. A. Software reliability: repetitive run experimentation and modelling, Boeing Computer Sere'ices Co. Report, BCS-40366, NASA Report No. CR-165836, 1981.
Proportional hazards modelling
183
3. Booker, J., Campbell, K., Goldman, A. G., Johnson, M. E. and Bryson, M.C. Application of Cox's proportional hazards model to light water reactor, Los Alamos Scientific Laboratory Report LA-8834-SR, 1981. 4. Ascher, H. Regression analysis of repairable systems reliability. In Electronic Systems Effectiveness and Life Cycle Costing (Skwirzynski, J. K., Ed.), pp. 118-33, Springer-Verlag, 1983. 5. Jardine, A. K. S. Component and system replacement decisions. In Electronic Systems Eff~,ctiveness and Life Cycle Costing (Skwirzynski, J. K., Ed.), pp. 647-54, Springer-Verlag, 1983. 6. Dale, C. Application of the proportional hazardsmodel inthe reliability field, Proc. Fourth Nat. Reliab. Conf., pp. 5B/1/1-9, 1983. 7. Jardine, A. K. S. and Anderson, M. Use of concomitant variables for reliability estimations and setting component replacement policies, Proc. 8th Adt'ances Reliab. Techn. Syrup., pp. B3/2/1-6, 1984. 8. Kalbfleisch, J. D. and Prentice, R. L. The Statistical Analysis of Failure Time Data, Wiley, New York, 1980. 9. Lawless, J. F. Statistical Models and Methods Jot Lij~,time Data, Wiley, New York, 1982. 10. Lawless, J. F. Statistical methods in reliability (with discussion), Technometrics, 25 (1983), pp. 305 35. 11. Cox, D. R. Regression models and life tables (with discussion). J. R. Stat. Soc., Ser. B, 34 (1972), pp. 187-202. 12. Andersen, P. K. Testing goodness of fit of Cox's regression life model, Biometrics, 38 (1982), pp. 67-77. 13. Prentice, R. L., Williams, B. J. and Peterson, A. V. On the regression analysis of multivariate failure time data, Biometrika, 68 (1981), pp. 373-9. 14. Whitehead, J. Fitting Cox's regression model to survival data using GLIM, Appl. Statist., 29 (1980), pp. 268-75. 15. Preston, D. L. and Clarkson, D. B. SURVREG: an interactive program for regression analysis with censored survival data, Conf. Proc. Statist. ~Ass., Section on Statistical Computing, pp. 195-7, 1980.
Proportional Hazards Modelling in Reliability Assessment A. Bendell Statistics and Operational Research, Dundee College of Technology, Bell Street, Dundee DD1 I H G and Services Ltd, Rowanbank, 7 Riverside Road, Wormit, Fife DD6 8LR, Great Britain (Received: 23 July, 1984)
1 INTRODUCTION There has been much recent interest in the reliability field in the application of proportional hazards modelling. A number of practitioners and academics have recently been advocating its use, and reviewing the basic theory. The technique is, however, in the early stages of application in reliability studies, and no really satisfactory examples of its application to reliability data are to be found in the open literature. An implication is that the enormous potential of this approach for reliability analysis, as well as the particular shortcomings and ditiiculties of applying it in this field, are not well-understood in the reliability community as a whole. It is the purpose of this paper to start to get the reliability community to exploit the potential of this technique fully. Proportional hazards modelling identifies the effects of various explanatory variables or factors which may be associated with variations in the length of life of equipment. Factors such as temperature, pressure, material, use, etc., may be included. Repairable as well as nonrepairable systems may be studied. Data may be censored or uncensored. The This paper is based upon a lecture read to the Reliability Engineering 11 course, 21-25 May, 1984, at the Lucas Institute for Engineering Production, University of Birmingham, UK. The illustrative data sets employed in that lecture are omitted due to their sensitixTe nature. A case study illustrating the special features of proportional hazards modelling in reliability will be published in due course. 175
Reliability Engineering 0143-8174/85/$03.30 ~• Elsevier Applied Science Publishers Ltd, England, 1985. Printed in Great Britain
176
A. Bendell
technique is a powerful method for decomposing the variation in life lengths into orthogonal factors, identifying the significant ones and reconstituting the model for prediction purposes. The technique largely owes its origins to the seminal paper which Professor D. R. Cox read to the Royal Statistical Society in March, 1972. At that time, he described the probable applications of proportional hazards modelling as being qn industrial reliability studies and in medical statistics'. From its introduction the technique has generated a great amount of interest, although until very recently mainly in medical fields. A vast array of research papers have dealt with theoretical considerations and developments of the basic model. Why is it, however, that recorded applications of the model to date have until recently been almost entirely associated with medical data ? The answer must be very largely because of the familiar lack of qualified statisticians working in practical reliability. The recent attempts at proportional hazards modelling in the reliability literature very much reinforce this view. The papers have appeared to date almost entirely in Conference Proceedings or internal reports, with the implication that they were not initially refereed; e.g. refs 1-7. The data contained in these papers is diverse, including motorettes, ° marine gas turbines and ships' sonar, 4 valves in Light Water Reactor nuclear generating plants 1 and aircraft engines. 7 However, the proportional hazards models employed are, in general, basic and are not developed to take specific account of the complexities of structure arising in reliability data. Further, the data sets chosen to illustrate the methodology in these papers have often been inappropriate for the analysis carried out. Indeed, sometimes the data have so little structure or the models are so inappropriate that proportional hazards modelling yields almost no useful information: see, for example, ref. 4. It is also essential that assessments of the appropriateness and the fit of the proportional hazards model and the explanatory variables are made, but these have not always been reported in the recent reliability literature, possibly giving a misleading impression; see ref. 7. Apart from these Conference papers and internal reports, there is very little in the current open literature dealing specifically with proportional hazards modelling for reliability data. The major books such as those by Kalbfleisch and Prentice 8 and Lawless 9 are more general and deal largely in terms of medical data. However, an important discussion of some aspects of proportional hazards modelling in reliability is contained in a recent review paper 1° and in the contributions by its various discussants.
Proportional hazards modelling
2
177
T H E BASIC M O D E L
The fundamental equation on which the proportional hazards model is based is an assumed decomposition of the hazard function for an item of equipment into the product of a base-line hazard function and an exponential term incorporating the effects of a number of explanatory variables or covariates, z l , . . . , z k. That is
h(t; zl, z2,.. .,Zk) =ho(t ) exp(fllZl + . . .+ flkZk)
(1)
The fli are unknown parameters of the model, defining the effects of each of the explanatory variables. These need to be estimated from the data and tested to see whether each explanatory variable really has an effect in explaining the variation in observed failure times. These effects are assumed to act multiplicatively on the hazard rate, so that for different values of the explanatory variables, that is say different temperature and pressure conditions, etc., the hazard functions are proportional to each other over all time t. It is this property that gives the technique its name. The baseline hazard function ho(t ) represents the hazard function that the equipment would experience if the covariates all take the baseline value zero, which may correspond, depending on what the covariate is, either to a natural zero such as a zero temperature, or an arbitrary zero corresponding, say, to an arbitrary time point from which to measure installation date. There are essentially two ways of modelling this baseline hazard. Either a parametric regression model may be assumed so that, say, a Weibull hazard function may be assumed for ho(t), or a distribution-free approach may be taken under which no particular form is assumed for ho(t ), but its form is estimated from the data. The latter approach was first suggested by Cox, 1~ is more common, and is sometimes taken as synonymous with the term proportional hazards modelling. Indeed, it is a major advantage of proportional hazards modeling that one does not need to assume a specified form for the baseline hazard when this may be difficult due to the confusing effects of the covariates. Instead, the model provides a distribution-free estimate of the baseline hazard function which can be used to check the appropriateness of standard distributional forms. The fact that in the author's experience standard forms such as the Weibull or constant hazard rate of the exponential distributions often do then turn out to be appropriate is highly reassuring. However, it is not necessary to assume a
178
A. Bendell
apriori.
given form In a number of the recent reliability papers this has been done, although sometimes with a little justification. 2'7 The remainder of this paper will largely be concerned with the distribution-free approach. The detail of the methodology is not developed here, but the interested reader is referred to ref. 8. The usual method first iteratively estimates the effects of the covariates/~1,/32,. •., flk using the so-called method of partial likelihood. Essentially, this technique employs the usual method of scoring based upon a Taylor expansion for each step in the iteration, starting with initial values of zero. Once the estimates converge, which by experience is usually within four or five iterations, tests of whether each explanatory variable has any significant effect are based upon the asymptotic Normality of the estimates. Thus, either Normal tests or the related chi-squared tests may be used. Chi-squared tests can also be conducted for the explanatory ability of the whole set of covariates included, based upon the likelihood ratio statistic, which compares the likelihood under the fitted parameter values and under the assumption that they are all zero, or upon the sums of squares of the standardized estimates. Other methods of testing fit are also available, e.g. ref. 12. A stepwise procedure may be incorporated into the proportional hazards approach, whereby in its backwards version non-significant explanatory factors are excluded one at a time and the model re-run until all factors are significant. Alternatively, a forward approach may be taken, whereby new variables are sequentially forced into the model. Having estimated ill, f12,. •.,/3~ the method obtains a distribution-free estimate of the baseline hazard function based upon discrete hazard contributions at each of the times at which failures were actually observed to occur. This can then be compared with standard distributional forms such as the Weibull. The effect upon the survivor function of different values for the explanatory variables is a power one:
R(I; were
Z l , Z 2 , . . . , Z k ) ~-- [ R o ( t ) ]
exp~zllJ~ + "
+ zk/~k)
Ro(t) is the baseline survivor function: Ro(t)=exp[- il ho(x)dx1
It should be noted that since the above approach assumes a continuous model, ties in the data create an initial problem, and various approximations can be employed to eliminate this. It is also noteworthy
Proportional hazards modelling
179
that the approach is essentially a regression one, so that the usual difficulties of regression analysis apply. Most noteworthy in the current context is probably the problem of multi-collinearity. So far the treatment has only been discussed of times to failure of nonrepairable items, or times to failure of repairable ones in which subsequent failure times are identically distributed or perhaps where only the times to first failure are studied, Ascher, 4 Dale 6 and Ascher in the discussion of Lawless 1° propose the method introduced in ref. 13 in connection with medical analysis for the analysis of repairable systems. The problem is that this approach is not the uniformally appropriate approach to the analysis of repairable systems. The variations of reliability data that can arise are highly diffuse, and in the author's experience the assumptions implicit in the models of Prentice et aL 13 and the purposes for which they are formulated, are not reflected in the usual forms of reliability data. Alternative more-appropriate models can be developed for such given data sets with very little modification, and experience of this approach to date has been most rewarding. Some indications of the adaptation to the special features of reliability data are given in the next section. The paper by Prentice at aL 13 essentially introduces four different extensions of the basic proportional hazards model [eqn (1)], based upon stratifying the data into strata corresponding to the various inter-failure times in the life cycle for each component. This effectively removes the proportionality assumption between the subsequent inter-repair periods for a given item of equipment so that the baseline hazard function is different within each strata. The four models correspond to (i) either basing the baseline hazard function in eqn (1) on the time since the component was new or upon the time since it was last repaired and (ii) either assuming that the effects of the covariates are the same within every strata, or allowing their effects to vary between the strata representing the number of previous repairs the item of equipment has experienced. It is interesting that in analysing his example data set Ascher 4 in fact only considers the one model in which the baseline hazard depends upon the time since the last repair, and in which the effects of the covariates are assumed to be identical within every stratum. His results are uninspiring and not a good advertisement for the methodology. One must wonder about the appropriateness of the particular model he has chosen. In contrast, Dale 6 appears to obtain much more satisfactory results using a parametric model in a complex example involving environmental testing.
180
,4. Bendell
3
M O D E L L I N G RELIABILITY DATA
It is, of course, impossible here to enumerate all the complexities of structure arising in reliability data and the adaptations to the basic model necessary to appropriately deal with them. However, instead it is attempted to convey some flavour of c o m m o n data structures and the associated analysis. At the same time, it is also attempted to warn the unwary of the complexity that may be present in their data, and of the dangers of automatic non-seeing utilization of the proportional hazard method or software. The emphasis, as in so much of modern statistical reliability assessment, is on careful exploratory analysis. The debate about interactions in proportional hazards modelling is started, and attention is drawn to some of the problems in reliability data currently being tackled. Let us begin with the enthusiasm in the recent reliability literature about the paper by Prentice et al. 13 It is an important paper, but its importance must be kept in proportion. The structure of the model considered there is essentially one that allows for reliability growth or decay in subsequent inter-repair periods to the same item of equipment. It is the belief of the present author that it is almost always worth looking for such structure, along with many other aspects, in preliminary exploratory analysis of reliability data. However, whether it should be included in a proportional hazards model must depend upon what the major aspects of the data one wishes to model are, whether there is clear evidence that this particular feature is or may be important, the extent of the data and the purpose of the analysis. For example, often in reliability data subsequent repairs may not be causally related, nor to the same subassembly of the item of equipment, so that subsequent inter-repair periods should not experience such reliability decay or growth. Indeed, sometimes one has no choice but to work with times between failures within a group of different items of equipment. It is important to identijy the appropriate structure o f the data as the basis f o r your model. In varied applications of proportional hazards modelling and other techniques in reliability studies, the author has come across a large proportion of data sets in which there are no such trend effects, or insufficient data to investigate them. Concentration upon this assumed trend effect can divert attention from, and even remove the possibility of testing for, more-appropriate structure in the data. This question of appropriate structure really centres about what the appropriate underlying point processes for failures of
Proportional hazards modelling
181
repairable systems are. For example, the appropriate point process model to build might be a series of subsequent failure times for each item of equipment, or for each cause for each item, or for each cause and environment for each item. If one has no choice but to work with a group of different items of equipment within which failures to individual items are not labelled, then the appropriate point process model might be the series of subsequent failure times for the group, for each cause over the gfoup, for each cause and supplier combination within the group, etc. The possibilities are limitless and heavily dependent upon the context in which the data arise. The appropriate time metric for the baseline hazard is, similarly, data-dependent. However, having identified the appropriate structure in the data, the adaptation of the basic proportional hazard model to incorporate it is usually straightforward, although possibly complex to implement on a computer package. Usually, a separate strata (i.e. a separate baseline hazard function) should correspond to each separate point process, except for the classification by item. Thus, if the appropriate structure of the data is a separate stream for each cause for each item of equipment, then there should be a separate stratum for each cause. In data sets suggesting a large number of such natural strata, the stratification in terms of reliability growth/decay might not be worthwhile. Another aspect of the modelling of repairable systems data concerns the treatment of competing causes of failure. If repair restores a unit to 'as good as new', the failure time on one cause represents a censoring time on other causes. Again, to reiterate the point, the implementation and adaptation oJproportional hazards models must depend on the structure of the data. Interactions in proportional hazards modelling are an interesting area where developments are currently going on at Dundee. Hitherto, they appear to have been neglected in proportional hazards modelling, the covariates being treated as acting orthogonally. Interactions may be defined simply as products of other covariates, and initial application of this approach has yielded promising results and invaluable information about the physical nature of the failure processes in the data. One does, however, have to be wary of introducing multi-collinearities, particularly with dummy (i.e. 0-1) variables which are essential if one has ordinal or nominal data. This, at present, causes some procedural problems in what interactions to include. One more complex problem for the future is the 'nested nature' of much
182
A. Bendell
reliability data: items may fail, assemblies within items, sub-assemblies, all the way down to component level. Such data arise in practice, but whilst a failure may represent a censoring time to all lower levels, the form of analysis for such data is not clear.
4
P R O G R A M S A N D PACKAGES
A number of programs and packages are now available to implement the proportional hazards model, but their application to reliability data is not straightforward, particularly for repairable systems. Perhaps the most widely accessible routines although not necessarily easiest to implement nor complete, are within the B M D P and SAS packages. Alternatively, a paper by Whitehead 14 explains how to implement proportional hazards modelling using GLIM, but this is probably not to be recommended unless you are already familiar with GLIM. Preston and Clarkson 15 also describe a program S U R V R E G for proportional hazards modelling. In the major book by Kalbfleisch and Prentice 8 Fortran programs for the proportional hazards model are provided, although these contain some errors. Currently, at Dundee, modified and extended versions of these programs are being worked with and developed further. These have now been implemented in a number of organisations.
5
CONCLUSIONS
Proportional hazards modelling is very new in reliability. It has an enormous potential and can give invaluable information, but it must be used carefully.
REFERENCES 1. Davis, H. T., Campbell, K. and Schrader, R. M. Improving the analysis of LWR component failure data, Los Alamos Scientific Laboratory Report, LA- UR 80-92, 1980. 2. Nagel, P. M. and Skrivan, J. A. Software reliability: repetitive run experimentation and modelling, Boeing Computer Sere'ices Co. Report, BCS-40366, NASA Report No. CR-165836, 1981.
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3. Booker, J., Campbell, K., Goldman, A. G., Johnson, M. E. and Bryson, M.C. Application of Cox's proportional hazards model to light water reactor, Los Alamos Scientific Laboratory Report LA-8834-SR, 1981. 4. Ascher, H. Regression analysis of repairable systems reliability. In Electronic Systems Effectiveness and Life Cycle Costing (Skwirzynski, J. K., Ed.), pp. 118-33, Springer-Verlag, 1983. 5. Jardine, A. K. S. Component and system replacement decisions. In Electronic Systems Eff~,ctiveness and Life Cycle Costing (Skwirzynski, J. K., Ed.), pp. 647-54, Springer-Verlag, 1983. 6. Dale, C. Application of the proportional hazardsmodel inthe reliability field, Proc. Fourth Nat. Reliab. Conf., pp. 5B/1/1-9, 1983. 7. Jardine, A. K. S. and Anderson, M. Use of concomitant variables for reliability estimations and setting component replacement policies, Proc. 8th Adt'ances Reliab. Techn. Syrup., pp. B3/2/1-6, 1984. 8. Kalbfleisch, J. D. and Prentice, R. L. The Statistical Analysis of Failure Time Data, Wiley, New York, 1980. 9. Lawless, J. F. Statistical Models and Methods Jot Lij~,time Data, Wiley, New York, 1982. 10. Lawless, J. F. Statistical methods in reliability (with discussion), Technometrics, 25 (1983), pp. 305 35. 11. Cox, D. R. Regression models and life tables (with discussion). J. R. Stat. Soc., Ser. B, 34 (1972), pp. 187-202. 12. Andersen, P. K. Testing goodness of fit of Cox's regression life model, Biometrics, 38 (1982), pp. 67-77. 13. Prentice, R. L., Williams, B. J. and Peterson, A. V. On the regression analysis of multivariate failure time data, Biometrika, 68 (1981), pp. 373-9. 14. Whitehead, J. Fitting Cox's regression model to survival data using GLIM, Appl. Statist., 29 (1980), pp. 268-75. 15. Preston, D. L. and Clarkson, D. B. SURVREG: an interactive program for regression analysis with censored survival data, Conf. Proc. Statist. ~Ass., Section on Statistical Computing, pp. 195-7, 1980.