Accelerated Failure Time Models: A Review

Transcription

1 International Journal of Performability Engineering, Vol. 10, No. 01, 2014, pp RAMS Consultants Printed in India Accelerated Failure Time Models: A Review JEAN-FRANÇOIS DUPUY * IRMAR/INSA of Rennes, F Rennes, FRANCE (Received on August 19, 2013, revised on August 26, 2013) Abstract: In classical life data analysis, one typically collects failure-time data by operating a set of units under usual (or design) stress conditions. But in reliability engineering, due to a variety of reasons such as cost and time constraints, one often wishes to collect the data more quickly than is allowed under the normal operating conditions. This can be achieved by applying higher-than-usual levels of stresses to the units, resulting in accelerated life testing data. In this paper, we provide a short review of the methods and models used to analyze such data. We concentrate on accelerated failure time models and on the related statistical inference. We describe some open questions and future research directions. Keywords: Accelerated failure time models, accelerated life testing, parametric and semiparametric models, statistical inference 1. Introduction Failure time regression data result from observing the failure times of units functioning under various values of explanatory variables (also called stresses, covariates or regressors), such as voltage, load, temperature, pressure Based on these data, one main objective of failure time data analysis is to estimate the reliability characteristics of the units (such as the reliability function, mean time to failure, failure time quantiles ) under some covariate values lying within the range of the operating conditions. In case of highly-reliable units however, the failures are rare. Indeed, highlyreliable units (such as those encountered in the nuclear, aeronautic or electronic fields for example) are designed to operate without failing during years or decades, while their design and manufacturing may allow a few months only to conduct the reliability testing and validation. One way to obtain the desired reliability information is thus to test the units at higher-than-usual levels of stresses and to infer, or extrapolate, the reliability characteristics of the units at use conditions. Obviously, this extrapolation is only possible if appropriate models relating the failure times of the units and the accelerating factors (stresses) are available. A huge amount of literature has been devoted so far to the design, modelling, and analysis of accelerated life testing experiments, and numerous contributions have been made to the development of statistical models that relate the accelerating variables to the reliability characteristics of the units under study (such models are called accelerated life models), statistical methods for planning accelerated life testing experiments, statistical methods for estimating reliability characteristics from a sample of accelerated life testing data. In particular, numerous plans of experiments for accelerated life testing have been proposed, most of them being designed for one-dimensional and two-dimensional stresses. Let x 0 < x 1 < < x k be a set of accelerated stresses (with x 0 being the usual stress). The constant-stress plan of experiments consists in testing k groups of units at distinct stress levels (that is, the n i units of the i-th group are tested under the stress x i ). In a step-stress plan, n units are placed on test at an initial low stress level and if a unit does *Corresponding author s Jean- François.Dupuy@insa-rennes.fr 23

2 24 Jean- François Dupuy not fail before some pre-specified time point t 1, the stress is increased and so on. All units are thus tested under the step-stress x 1, 0 t < t 1 x x(t) = 2, t 1 t < t 2 x k, t k 1 t < τ where τ denotes the end of the test. Progressive-stress plans of experiments have also been proposed, where the stress applied to the units increases continuously in time. Various modifications of these plans have been suggested. For example, assume that the failure times under the usual stress x 0 take large values, so that most of the failures occur after the endpoint τ of the experiment (as is the case with highly-reliable items). Then two groups of units may be tested: the first group of n 1 units is tested under a constant accelerated stress x 1 and the second group of n 2 units is tested under a step-stress scheme, including the usual stress x 0 : x(t) = x 1, 0 t < t 1 x 0, t 1 t < τ Under the step-stress, the units use much of their resource until t 1 under the accelerated stress x 1, which implies that failures will occur in the time interval [t 1, τ) under the usual stress x 0. This latter plan may further be modified by allowing the change-stress time t 1 to be random (one usual choice is to take t 1 as the moment when failures start to occur). We refer the reader to [1] for a detailed exposition of plans of experiments in accelerated life testing. A non-exhaustive bibliography includes [2, 3, 4, 5, 6, 7], see also the references therein. In the paper [8] (published in this special issue), the author considers the issue of accelerated life testing with competing risks, while in [9] (also published in this special issue), the authors investigate optimum 3-steps stress-test plans. Based on these plans, one can estimate the reliability characteristics of the tested units, provided that an appropriate model (called an accelerated failure time model) is available to relate the failure times of the units and the accelerating factors or stresses. The purpose of this paper is to give an overview of some useful accelerated failure time models and of the related inference. In Section 2, we describe the most common parametric and semiparametric accelerated failure time models. In Section 3, we give a brief overview of the statistical inference in these models, with emphasis on the estimation issue. In Section 4, we provide a review of some recent advances in the modelling and analysis of accelerated life data. Some open questions and promising future research directions are also mentioned. 2. Accelerated failure time models Let X denote the time to failure of some unit and Z = Z 1,, Z p be a p- dimensional vector of time-independent stresses applied to the unit. The accelerated failure time model relates the unit lifetime distribution to the explanatory variables or stresses by simply making a linear regression for the log-transformed event time log(x) given Z (see [10] for example): log(x) = Z T β + ε where β = ( β 1,, β p ) T is a p-dimensional vector of unknown regression parameters, T is the transpose sign and ε is an error term with unspecified distribution.

3 Accelerated Failure-Time Models: A Review 25 While this expression takes the familiar form of a general linear regression model, the sense of accelerated failure time models is best seen, however, when they are expressed in terms of the hazard (or rate) function of X, which is defined as follows. Let F Z (t) = P(X t Z), S Z (t) = 1 F Z (t) = P(X > t Z), f Z (t) = F Z (t) be the conditional (given the stress Z) cumulative distribution, survival and density functions of X respectively. The conditional hazard function of X is given by: λ Z (t) = f Z(t) S Z (t) = f Z(t) 1 F Z (t) = lim 1 P(X [t, t + h) T t, Z). h 0 h Noting that: F Z (t) = P(X t Z) = P( Z T β + ε log(t) Z) = P(e ε t exp(z T β) Z) = F e ε(t exp(z T β)) where F e ε 1 S e ε denotes the cumulative distribution function of e ε, we obtain: λ Z (t) = F e ε t exp Z T β = 1 F e ε t exp Z T β f e ε t exp ZT β 1 F e ε t exp Z T β exp(zt β). Therefore, letting λ e ε be the hazard function of e ε, the conditional hazard function of X given Z is: λ Z (t) = λ e ε(t exp(z T β)) exp(z T β). Equivalently, using a simple change of variable, the conditional survival function for X given Z is given by: t 0 S Z (t) = exp λ Z (u)du = S e ε(t exp(z T β)) (1) From λ Z (t) and S Z (t), one clearly sees that the stress Z acts multiplicatively on the time t, so that the effect of Z is to decelerate or accelerate the time to failure of the unit. Time-dependent stresses Z( ) = Z 1 ( ),, Z p ( ) can be introduced in accelerated failure models (e.g., [11, 12]) by letting the survival function of X be defined as: t S Z (t) = S e ε exp(z T (u)β) du (2) 0 Obviously, this expression reduces to (1) when the stresses are fixed over time. Time-dependent regression coefficients can also be introduced to accommodate a differential effect of the stress across time. Such a model is defined by: t 0 S Z (t) = S e ε exp Z T (u)β(u) du (3) with β( ) = β 1 ( ),, β p ( ) T a vector of p unknown regression functions. The one-dimensional functions β i ( ) (i = 1,, p) are usually taken of the form β i ( ) = β i + γ i g i ( ) where the g i ( ) are specified deterministic functions (or realizations of predictable random processes), see [1]. By specifying the distribution of ε (up to a finite number of unknown scalar parameters) in the models (1)-(3), one can derive various useful parametric accelerated failure time models. One first common class of models is obtained by letting S e ε belong to a given scale-shape class of survival functions: S e ε(t) = G t η ν (η, ν > 0). Classical examples include G(t) = e t, G(t) = (1 + t) 1, G(t) = 1 Φ(log t)

4 26 Jean- François Dupuy (where Φ denotes the distribution function of the standard Gaussian distribution), which yields the Weibull, loglogistic and lognormal distributions respectively. These families of distributions however do not give -shaped hazard functions. The generalized Weibull distribution, which allows various forms for the hazard rate (constant, decreasing, increasing, -shaped, -shaped) has thus been proposed. Its survival function is given by: S e ε(t) = exp{1 (1 + (t/φ) ν ) γ } (η, ν, φ > 0). If the baseline hazard function S e ε is left unspecified, one obtains a semiparametric accelerated failure time model. A synthesis of all these models can be found in [1, 13, 14, 15, 16], see also the references therein. Note that an appealing variant of the model (1) was proposed in [17]: λ Z (t) = λ e ε(t exp(z T β 1 )) exp(z T β 2 ). This model contains both the accelerated failure time model (when β 1 = β 2 ) and the celebrated proportional hazards model (when β 1 = 0) (see [18, 19]) as special cases. In particular, it allows to choose between the proportional hazards model and the accelerated failure time model which one is the more appropriate for a given data set. 3. Statistical Inference in Accelerated Failure Time Models 3.1 Parametric Accelerated Failure Time Models The statistical inference in parametric accelerated failure time models of the form λ Z (t) = λ e ε(t exp(z T β)) exp(z T β) usually relies on the maximum likelihood method (see [1] for a detailed account), as we describe now. In what follows, we assume that λ e ε is parameterized by a finite-dimensional parameter μ, we denote by θ = (β T, μ T ) T the full vector of unknown parameters, and we note λ Z (t, θ) = λ Z (t). In accelerated failure time experiments, the data often arise in the form of rightcensored observations that is, one eventually only observes a lower bound of the failure time of interest. As mentioned above, X denotes the time to failure of a unit and Z( ) = Z 1 ( ),, Z p ( ) denotes a p-vector of possibly time-varying explanatory variables (stresses) applying to this unit. Let C be a positive random variable (the censoring time) and assume that X and Care independent given Z( ). Suppose that the data consist of n independent replicates T i, Δ i, Z i ( ), i = 1,, n of T, Δ, Z( ) wheret = min(x, C), = 1(T C) and 1( ) is the indicator function. Then the likelihood score function for θ is: n log λ Zi (T i, θ) T i U(θ) = Δ i λ Z i (u, θ) du. i=1 θ 0 θ The maximum likelihood estimator θ = β T, μ T T is the solution of the estimating equation U(θ) = 0. Then n 1/2 θ θ is asymptotically distributed as a Gaussian law with mean zero and a covariance matrix that can be consistently estimated. From this, one can easily derive confidence intervals and tests of hypothesis for the components of θ (and in particular, for the regression parameter β which is usually the parameter of interest of the model). One can also deduce estimates of the survival function, quantiles, and mean time to failure of X given Z (see [1]). Residual diagnostic plots and goodness-of-fit criteria are particularly useful for evaluating the adequacy of a fitted model. We refer the interested reader to [1, 20, 21] for a detailed discussion of this topic.

5 Accelerated Failure-Time Models: A Review Semiparametric Accelerated Failure Time Models When the error distribution is not parameterized, one obtains a semiparametric accelerated failure time model. Various methods have been proposed to estimate the parameter of interest of such models, namely the regression parameter β (we consider here the case where the explanatory variables are constant in time and refer the interested reader to [1] for a detailed treatment of a more general setting). One proposed estimation method directly builds on the linear regression formulation of the accelerated failure time model (see [22, 23]). Letting e i (β) = log T i + Z T i β (for i = 1,, n) one classical way to estimate β is to form the logrank statistic based on (e i (β), Δ i, Z i ), i = 1,, n, namely: n W(β) = Δ i Z i Z β, e i (β) i=1 where Z (β, u) = n i=1 1(e i(β) u)z i n. i=1 1(e i (β) u) The estimator β is chosen as the value which minimizes W(β) and it follows from the fact that the random vector n 1 W(β) is asymptotically zero-mean Gaussian that n 1/2 β β is asymptotically Gaussian with a covariance matrix that can be consistently estimated (see [24, 25, 26]). Similar to the parametric case, one can then construct confidence intervals and tests of hypothesis for β and deduce estimates of the survival function, quantiles, and mean time to failure of X given Z (see [1]). Note that Bayesian methods have also been developed in the accelerated failure time models (see [27, 28, 29] and the references therein) but they will not be described here. 4. Discussion: Some Recent Advances and Open Questions in Accelerated Failure Time Modeling The previous two sections have briefly described the general formulations of accelerated failure time models and provided a description of the most used estimation procedures in both parametric and semiparametric models. These procedures are theoretically wellestablished and are implemented in dedicated softwares. Accelerated failure time models are thus now more and more used in practice (and in particular, as an alternative to the well-known proportional hazards model). They are present in both reliability engineering and the medical field (see for example [30, 31]). Some remarkable recent advances have extended the range of application of these models. For example, several authors have considered the fitting of accelerated failure time models with degraded data or model information: the problem of missing censoring indicators was addressed in [32] and the issue of misspecification of the fitted model was considered in [33]. The problem of estimating an accelerated failure time model when some covariates are subject to measurement error was studied in [34]. In [35], the authors investigated the issue of variable selection in a high-dimensional setting. Numerous open questions and problems however still remain unsolved. We briefly mention a few promising research directions. First, there is a strong need for developing adapted tools for clustered observations (clustered data arise when the failure times of the units are not independent, which is likely to be the case when the units belong to a same system). Developing appropriate tools for fitting accelerated failure time models under competing risks is also an open question which has attracted little attention until now. Finally, despite some recent contributions, the issue of model selection in a high-

6 28 Jean- François Dupuy dimensional setting still requires a careful attention, motivated by the availability of more and more covariate information in reliability experiments. All these topics are open to methodological developments and fruitful applications. References [1] Bagdonavicius, V., and M.S. Nikulin. Accelerated Life Models: Modeling and Statistical Analysis. Chapman & Hall, London, [2] Meeker, W. Q. A Comparison of Accelerated Life Test Plans for Weibull and lognormal Distributions and Type-I Censoring. Technometrics, 1984; 26(2): [3] Nelson, W.B. Accelerated Testing, Statistical Models, Test Plans and Data Analysis. John Wiley & Sons, New York, [4] Miller, R. and W.B. Nelson. Optimum Simple Step-Stress Plans for Accelerated Life Testing. IEEE Transactions on Reliability, 1983; R-32(1): [5] Bai, D.S., M.S. Kim, and S.H. Lee. Optimum Simple Step-Stress Accelerated Life Tests with Censoring. IEEE Transactions on Reliability, 1989; 38(5): [6] Fard, N. and C. Li. Optimal Simple Step Stress Accelerated Life Test Design for Reliability Prediction, Journal of Statistical Planning and Inference, 2009; 139(5): [7] Hunt, S. and X. Xu. Optimum Design for Accelerated Life Testing with Simple Step Stress Plans. International Journal of Performability Engineering, 2012; 8(5): [8] Haghighi, F. Accelerated Test Planning with Independent Competing Risks and Concave Degradation Path. International Journal of Performability Engineering, 2014;10(1): [9] Chandra, N., M. A. Khan, and M. Pandey. Optimum Test Plan for 3-Step, Step-Stress Accelerated Life Tests. International Journal of Performability Engineering,2014;10(1): [10] Martinussen, T., and T. Scheike. Dynamic Regression Models for Survival Data. Springer, New York, [11] Lin, D.Y., and Z. Ying. Semiparametric Inference for the Accelerated Life Model with Time- Dependent Covariates. Journal of Statistical Planning and Inference, 1995; 44(1): [12] Tseng, Y.-K., F. Hsieh, and J.-L. Wang. Joint Modelling of Accelerated Failure Time and Longitudinal Data. Biometrika, 2005; 92(3): [13] Bagdonavicius, V., and M. S. Nikulin. Semiparametric Models in Accelerated Life Testing. Queen s Papers in Pure and Applied Mathematics, Kingston, [14] Meeker, W.Q., and L.A. Escobar. Statistical Methods for Reliability Data. Wiley, New York, [15] Meeker, W. Q., and L. A. Escobar. A Review of Accelerated Test Models. Statistical Science, 2006; 21(4): [16] Viertl, R. Statistical Methods in Accelerated Life Testing. Vandenhoeck and Ruprecht, Göttingen, [17] Chen, Y., and N. Jewell. On a General Class of Semiparametric Hazards Regression Models. Biometrika, 2001; 88: [18] Cox, D.R. Regression Models and Life Tables (with discussion). Journal of the Royal Statistical Society Series B, 1972; 34: [19] Andersen, P.K., O. Borgan., R.D. Gill., and N. Keiding. Statistical Models Based on Counting Processes. Springer-Verlag, New York, [20] Lin, D.Y., and C.F. Spiekerman. Model Checking Techniques for Parametric regression with Censored Data. Scandinavian Journal of Statistics, 1996; 23: [21] Nikulin, M.S., and Q.X. Tran. On Chi-Squared Testing In Accelerated Trials. International Journal of Performability Engineering, 2014;10(1): [22] Buckley, J., and I. R. James. Linear Regression with Censored Data. Biometrika, 1979; 66: [23] Ritov, Y. Estimation in a Linear Regression Model with Censored Data. The Annals of Statistics, 1990; 18: [24] Tsiatis, A.A. Estimating Regression Parameters Using Linear Rank Tests for Censored Data. The Annals of Statistics, 1990; 18:

7 Accelerated Failure-Time Models: A Review 29 [25] Wei, L.J., Z. Ying, and D.Y. Lin. Linear Regression Analysis of Censored Survival Data Based on Rank Tests. Biometrika, 1990; 77: [26] Ying, Z. A Large Sample Study of Rank Estimation for Censored regression Data. The Annals of Statistics, 1993; 21: [27] Kuo, L., and B. Mallick. Bayesian Semiparametric Inference for the Accelerated Failure- Time Model. The Canadian Journal of Statistics, 1997; 25: [28] Walker, S., and B.K. Mallick. A Bayesian Semiparametric Accelerated Failure Time Model. Biometrics, 1999; 55(2): [29] Zhang, J., and H. Zhang. A Semiparametric Bayesian Estimation Method of the Accelerated Failure Time Model. Advances and Applications in Statistical Sciences, 2010; 1(2): [30] Lambert, P., D. Collett, A. Kimber, and R. Johnson. Parametric Accelerated Failure Time Models with Random Effects and an Application to Kidney Transplant Survival. Statistics in Medicine, 2004; 23(20): [31] Zou, Y., Zhang, J., and Qin, G. A SemiparametricAccelerated Failure Time Partial Linear Model and its Application to Breast Cancer. Computational Statistics & Data Analysis, 2011; 55(3): [32] Li, X., and Q. Wang. The Weighted Least Square Based Estimators with Censoring Indicators Missing at Random. Journal of Statistical Planning and Inference, 2012; 142(11): [33] Hattori, S. Testing the No-Treatment Effect Based on a Possibly Misspecified Accelerated Failure Time Model. Statistics and Probability Letters, 2012; 82(2): [34] He, W., G. Yi, and J. Xuong. Accelerated Failure Time Models with Covariates Subject to Measurement Error. Statistics in Medicine, 2007; 26(26): [35] Wang, X., and L. Song. Adaptive Lasso Variable Selection for the Accelerated Failure Models. Communications in Statistics, Theory and Methods, 2011; 40(24): Jean-François Dupuy is Professor at the Institut National des Sciences Appliquées de Rennes (France), specialized in statistics and stochastic modelling. He is a member of the Statistics team at the Mathematics Research Institute of Rennes (IRMAR, UMR CNRS 6625). He obtained his Ph.D. at the university Paris Descartes in 2002 and his Accreditation to Supervise Research at the university Toulouse III Paul Sabatier (France) in He has been an Associate Professor at the university Toulouse III Paul Sabatier ( ) and a Professor at La Rochelle university, France ( ). His research focuses on statistical methods and models for reliability and survival analysis (with a particular interest in missing data problems, non-parametric testing, cure rate models and competing risks), extreme value theory, regression modelling.