So the method based on the analysis of multiple correlation allows to reveal the most significant differential diagnostic signs and to obtain the decision rule of differential diagnosis. The discriminant analysis is successfully used in prediction of insult outcome when different methods of its treatment were used. It can also predict the patient survivability when operated for renal cancer and to determine the survival time for patients with renal cancer having metastases in different organs [For88]. The accuracy is within the range from 68-73% (with survivability prognosis) to 90% (with operation outcome prognosis).

There are cases in diagnosis when relying upon the number of indirect signs it is necessary to evaluate the most important sign to detect which is very difficult. It can be done with the use of regression analysis. Such approach helps to determine the degree of anatomical lesion based on the indirect diagnostic signs, to evaluate the complication probability, survival time, biological age.

The discriminant and regression analyses are based on the assumption that the statistical data correspond to the normal distribution law. Meanwhile there is a great number of data that either cannot be subjected to the analysis with the help of normal distribution curve or do not satisfy the main prerequisites necessary for its use. To analyze such data the multi-modal distribution laws [CZ85] and mathematical apparatus of catastrophe theory can be used allowing to reveal the most significant factors of multivariate population of statistical data and to detect geometrically the critical region where the qualitative changes in the investigated objects occur.

The process of penetration of mathematical methods into theory and practice of medicine is natural. The analysis of literature published during the last decades shows that the number of works devoted to this problem is still increasing. The wide and methodologically substantiated application of mathematical methods in different fields of health service makes it possible to put the medical information processing on principally new basis.

The most significant are information systems based on the principle of gathering the multiple case records into the large database. The database means the system of information storage, processing and analysis consisted of sign population among certain patients. For example, in CHD patients, the first stage for creating such a base provides the signs collection in the patient according to the formalized case record and input of this information into computer. The second stage is the information analysis with the help of mathematical techniques and sampling of decision rule of differential diagnosis, disease prognosis and patient treatment. The third stage is the decision making based on the created decision rules in CHD patient and diagnosis making with recommendations concerning the methods of adequate therapy. The fourth stage is the storage and updating of database in computer.

The result of database formation is the construction of mathematical methods capable to reflect the patient specific state. It is usually directed towards development of individual therapy and creation of algorithm of patient treatment methods and rehabilitation.

[CZ85] Cobb, L., Zacs, S. : Applications of catastrophe theory for statistical modeling in biosciences. IASA., 80, N 392, 793-802, (1985) [For88] Forges, F. et al.: Prognostic factors of metastatic renal carcinoma: a multivariate analysis . Seminars Oncol., 4, N 3, 149-154 (1988)

[OW61] Overall, J.E., Williams, C.M.: Models for medical diagnosis . Behav.

Science, 6, N 2, 134-146 (1961) [VMO93] Vander Poel, H.C., Mulders, P.F., Oosterhof, C.O. et al. : Prognostic value of karyometric and clinical characteristics in renal carcinoma. Cancer, 72. N 9, 2667-2674 (1993)

[BR04] Balakrishnan, N., Rao, C.R.: Handbook in Statistics: Advances in Survival Analysis, 23, Elsevier, New York (2004).

[VN93] Voinov, V.G., Nikulin, M. Unbiased Estimators and Their Applications, 1, Univariate case. Kluwer Academic Publishres: Dordrecht, (1993).

[VN96] Voinov, V.G., Nikulin, M.: Unbiased Estimators and Their Applications, 2, Multivariate case. Kluwer Academic Publishres: Dordrecht, (1996).

[GN96] Greenwood, P.E., Nikulin, M.: A Guide to Chi-Squared Testing. John Wiley and Sons, New York, (1996).

[BS83] Bolshev, L.N., Smirnov, N.V. Tables of Mathematical Statistics, Nauka, Moscow, (1983).

[Zac71] Zacks, Sh.: The Theory of Statistical Inference, Nauka, Moscow, (1971).

[KK02] Kleinbaum, D.G., Klein, M.: Lodistic Regression, Springer, New York, (2002).

Degradation-Threshold-Shock Models

Axel Lehmann

Otto-von-Guericke-University Magdeburg Institute of Mathematical Stochastics PF 4120, D-39016 Magdeburg, Germany [email protected]

Summary. This paper deals with the joint modeling and simultaneous analysis of failure time data and degradation and covariate data. Many failure mechanisms can be traced to an underlying degradation process and stochastically changing covariates. We consider a general class of reliability models in which failure is due to the competing causes of degradation and trauma and express the failure time in terms of degradation and covariates. We compute the survival function of the resulting failure time and derive the likelihood function for the joint observation of failure time data and degradation data at discrete times.

Key words: Degradation process; degradation-threshold-shock model; dts-model; traumatic event; threshold; first passage time

This paper deals with the joint modeling and simultaneous analysis of failure time data and covariate data like internal degradation and external environmental processes. Many failure mechanisms in engineering, medical, social, and economic settings can be traced to an underlying degradation process and stochastically changing covariates that may influence degradation and failure.

Most items under study degrade physically over time and a measurable physical deterioration almost always precedes failure. The level of deterioration of an item is represented by a degradation process. In engineering applications, degradation may involve chemical changes brought about by corrosion and electro-migration or physical changes due to wearing out and fracturing, whereas degradation may be characterized by markers of health status and quality of life data in medical settings. Frequently, an item is regarded as failed and switched off when degradation first reaches a critical threshold level.

Moreover, in most practical applications items or systems operate in heterogeneous environments and loads, environmental stresses, and other dynamically changing environmental factors may influence their failure rate.

When it is possible to measure degradation as well as covariates, an alternative approach to reliability and survival analysis is the modeling of degradation and environmental factors by stochastic processes and their failure-generating mechanisms. This stochastic-process-based approach shows great flexibility and can give rise to new or alternative time-to-failure distributions defined by the degradation model. It provides additional information to failure time observations and is particularly useful when the application of traditional reliability models based only on failure and survival data is limited due to rare failures of highly reliable items or due to items operating in dynamic environments.

Two relevant stochastic models relating failure to degradation and other covariates have evolved in the theoretical and applied literature, threshold-models and shock-models. A threshold-model supposes that the item or system fails whenever its degradation level reaches a certain critical deterministic or random threshold. In a shock-model the item or system is subjected to external shocks which may be survived or which lead to failure. The shocks usually occur according to a Poisson process whose intensity depends on degradation and environmental factors. It appears that Lemoine and Wenocur [LW85] may have been the first to combine both approaches by considering two competing causes of failure: degradation reaching a threshold, and occurrence of a traumatic event like a shock of large magnitude severe enough to destroy the item. So, the failure time of an item is the minimum of the moment when degradation first reaches a critical threshold and the moment when a censoring traumatic event occurs.

We call this class of reliability models which consider failure due to the competing causes of degradation and trauma degradation-threshold-shock-models (DTS-models). Singpurwalla [Sin95] and Cox [Cox99] give detailed reviews on stochastic-process-based reliability models including DTS-models.

In this paper, we derive an expression for the survival function of the failure time in a general DTS-model and consider certain classes of submodels. For the joint observation of failure time data and degradation data at discrete times the likelihood function is given.

The intension of this paper is to give a general framework for dealing with DTS-models. To apply the DTS-model to real data situations it is of course neccessary to specify the degradation and covariate processes. In applied literature degradation processes are frequently described by a general path model, i.e., by a stochastic process that depends only on a finite dimensional random variable (see [ME98]), or by a univariate process with stationary independent increments. In the context of DTS-models, Bagdonavicius, Haghighi and Nikulin [BHN05] consider a general path model with time dependent covari-ates and multiple traumatic event modes. Several processes with stationary independent increments have been used in degradation models. Frequently degradation is related to external covariates through a random time scale describing slowing or accelerating degradation in real time. The time scale and the intensity of traumatic events may depend on possibly time-varying covariates, for instance on different stress levels, to model the influence on failure of a dynamic operating environment of the item and to cover non-linear degradation behavior.

Wiener diffusion processes have found application in Doksum and Hoyland [DH92], Doksum and Normand [DN95], Lawless, Lu and Cao [LHC95], Whit-more [Whi95], Whitmore and Schenkelberg [WS97] and Whitmore, Crowder and Lawless [WCL98]. Degradation models based on the gamma process were considered by Wenocur [Wen89] and, in the context of a DTS-model, by Bag-donavicius and Nikulin [BN01] together with a random time scale depending on covariates. Lehmann [Leh04] considers a DTS-model with a Levy degradation process and a random time scale and extends this DTS-model to the case of repairable items by a marked point process approach.

Suppose that the degradation level of some item is described by a stochastic process X = {X(t) : t G R+}, defined on a fixed probability space (f, F, P). Let Xt = {X(s) :0 < s < t} denote the path of X on [0,t] and FX = a(Xt) the history of all paths of X up to time t. For simplicity, we do not consider an external covariate process for the present.

An item is regarded as failed when the degradation process reaches a critical threshold level X* which is possibly random but independent of X. Additionally to failures which are immediately related to degradation, an item can also fail when a traumatic event like a shock of large magnitude occurs although the degradation process has not yet reached the threshold. We model such a censoring traumatic event as the first point of a doubly stochastic Poisson process & = {&(t) : t G R+} with a stochastic intensity n(t,X(t)) that may depend on time t and on the degradation level X(t). That means, given a known path x(^) of X, & is an nonhomogeneous Poisson process with intensity n(t,x(t)). Hence, the failure time of an item is defined as the minimum of the nontraumatic failure time D = inf{t > 0 : X(t) > X*} caused by degradation and the traumatic failure time C = inf{t > 0 : &(t) = 1} caused by a traumatic event. Given the degradation path Xt up to time t, the conditional survival function of C is

Thus, the survival funtion of T is

where 1(D > t) denotes the indicator function of the event {D > t}.

We call this model degradation-threshold-shock-model (DTS-model). Supposed that D has a failure rate A(t) and that C has a deterministic intensity «(t), (3) simplifies to

P(T > t) = exp {— jt k(s) ds^ P(D > t) = exp {— J^ (K(s) + A(s)) d^ .

To find an expression of the survival function and the failure rate of T in the general case we use a theorem given by Yashin and Manton [YM97]:

Theorem 1 (Yashin, Manton). Let Z and Z be stochastic processes influencing a failure rate a(t, Z, Z) and satisfying measurability conditions such that, for t > 0

and let T be related to Zt and by

P(T > 11 Zt, Zt) = exp (-J^t a(u, Z, £) du) . (4)

If the trajectories of Z are observed up to t, then

The random failure rate a(t, Z, Z) may depend on either the current values Z(t) and Z(t) or on the trajectories Zt and Zt up to t. If the covariate process Z does not appear in (4), i.e. a = a(t,Z), the statement of Theorem 1 obviously reads

P(T > t) = exp ^- J^ E[a(u, Z) | T > u] du) (5)

(see [Yas85]). Although we observe that

P(T > 11 Xt) = E[1(D > t)1(C > t) | Xt] = 1(D > t)P(C >t I Xt) is not of the form (4), Theorem 1 can be used to compute P(T > t).

Theorem 2. Let the traumatic failure time C has the stochastic failure rate n(t,X(t)) with E /0 k(s,X(s))ds < to for all t > 0 and assume that, given X* = x*, the nontraumatic failure time D have the conditional failure rate \(t,x*) with E ft \(s,X*) ds < to for all t > 0. Then,

P(D >t)= exp X(s) ds^j , where the failure rates k and X are given by K(t) = E[n(t,X(t)) | C > t] and X(t) = E[X(t, X*) I D > t]. The survival function of T can be expressed as

P(T >t) = exp J^ (k(s) + X(s)) ds where n(t) = E[n(t,X(t)) | T > t] is the failure rate of a traumatic event if a nontraumatic event has not occurred.

f. Let H(t) = 1(D < t) and Ht = {H(s) : 0 < s < t}. Since D is a F^-stopping time we have a(Ht) C F^ for all t > 0 and, therefore,

P(C>t I Ht, Xt) = P(C>t I Xt) = exp J k(s, X(s)) ds^ . (6)

First, apply (5) with £t = Xt to the second equation of (6) to get the survival function P(C > t) = exp fO K(s)dsj and then, with £t = X*, to P(D >

11 X*)=exp (- ft X(s, X*)ds^ to show P(D > t) = exp (- ftQ X(s)ds^). Moreover, applying Theorem 1 with Zt = Ht and £t = Xt to (6) we obtain

P(C>t I Ht) = exp (-J^ K*(s,Hs)ds^j where k* (t,Ht)=E[n(t,X(t)) I Ht, C > t]. Obviously, on {D > t}, we have K*(t, Ht) = K(t) and

Thus, we conclude

P(T >t) = P(C >t I D > t) P(D >t)= exp (k(s) + X(s)) ds^j .

In the following theorem an expression is derived for the density of the degradation process X(t) conditioned on the event that no failure has occurred up to the moment t. We assume that X(t) possesses a density fX(t) with respect to some dominating measure v, usually the Lebesgue measure or the counting measure. However, in the following we will write dx instead of v(dx) regardless of the nature of v. Further, we assume that for all t > 0 and tk = (to,...,tk) £ Rk+1 with 0 < t0 < ... < tk < t and xfc = (xo,... ,xk) £ Rk+1 the conditional joint density g(t, tk, Xk; x*) with

P(D >t, Xk £ dxk I X* = x*)= g(t, tk, Xk; x*)dxk (7)

Of course, g must satisfy g(t, tk, xk; x*) = 0 if min(xo,...,xk) > x*. If the paths of X are increasing then, obviously, g(t, tk, xk; x*) = fxk (xk) P(X(t) < x* I Xk = xk).

for xo < ... < xk < x* and g(t, tk, xk; x*) =0 otherwise.

If X is a Wiener process with drift with drift coefficient fj,, variance coefficient a and initial value X(0) = 0, then k g(t, tk, xk; x*) = JJ go (tj - tj-i,xj - xj-i; x*) Fo(t - tk ,x* - xk)

j=i for min(xi,..., xk) < x* where to = xo = 0 and

1 fx - ft go(t,x; x*) = l(x* > x)—^ip yjl V aVt

at for all t > 0 (see [KL98]). The function Fo is the survival function of an Inverse Gaussian distribution, i.e.,

-x - ft aVt where p and N denote the pdf and the cdf of a standard normal random variable.

Theorem 3. Let the traumatic failure time C has the stochastic failure rate n(t,X(t)) with E fo k(s,x(s))ds < to for all t > 0 and assume (7) for all t > 0, tk = (to,...,tk) £ Rk+1 with 0 < to < . .. < tk < t and xk = (xo,. .., xk) £ Rk+1. Then,

P(T >t, Xk £ dxk) = exp (- J^ K(s, xk(s))ds^ g(t, tk, xk)dxk, where k(s, xk(s)) = E[k(s, X(s)) | T > s, Xk(s) = xk(s)] with k(s) = max{j > 0 : tj < s} for 0 < s < t and g(t, tk, xk) =E[g(t, tk, xk; X*)].

Proof. For all u > 0, let H(u) = 1(D < u) and Hu = {H(s) : 0 < s < u}. Set tk+1 = to and let XA (t) = £k=0 1(tj < t < tj+i)X(tj) denote the process of discrete observations of X Defining Zu = (Hu,XA) = {(H(s),XA(s)) : 0 < s < u} we have rt

P(C>t | Zt, Xt) = P(C>t | Xt) = exp J K(s, X(s)) ds^) since a((t) C . Applying Theorem 1 with £t = Xt to (8) we obtain P(C>t I Zt) =P(C>t I Ht, XtA) = exp (—J K*(s, Hs,Xf) ds^) where k* (s,Hs,XA) = E[K(s,X(s)) I Hs,XA,C>s]. for all 0 < s < t. Since K*(s,Hs,XA) = K(s, Xk(s)) on {D > t}, we finally conclude

P(T >t, Xk e dxk) = P(C>t I D > t, Xk = xk) P(D >t, Xk e dxk)

The class of DTS-models contains two important subclasses, degradation-threshold-models (DT-models) and degradation-shock-models (DS-models).

In a degradation-threshold-model only nontraumatic failures can occur, i.e., the traumatic event intensity k is equal to zero and the failure time T = D = inf{t > 0 : X(t) > x*} is the first passage time of X to the threshold x*, which is assumed to be nonrandom in this subsection.

If degradation is modeled by a one dimensional Wiener process with drift X(t) = x + fit + aW(t) where W denotes a standard Brownian motion, then it is well known, that T ~ IG (z -x, (x a.2x)^J is Inverse Gaussian distributed if x < x*. In general, T has an upside bathtub failure rate, but it has an essentially increasing failure rate (IFR) if (x* — x)/a ^ 1 and a decreasing failure rate (DRF) if (x* — x)/a < 1 (see Fig. 1).

For an increasing degradation process X the survival function of T is given by P(T > t) = P(X(t) < x*). If, for instance, X is a homogeneous Poisson process with rate A > 0 then T ~ Ga([x*],A) follows a gamma distribution, which has an increasing failure rate. If X is an increasing jump process, then T has always increasing failure rate average (see [SS88]): Theorem 4 (Shaked, Shantikumar). If X is an increasing jump process, then T has increasing failure rate average (IFRA). If X is an increasing Levy process with a Levy measure v which has a decreasing density, then T has increasing failure rate (IFR).

2.2 Degradation-Shock-Models

The class of degradation-shock-models is characterized by the absence of a critical threshold X* which can be described formally by X* = to. Here the failure time is given by T = C = inf{t > 0 : &(t) = 1}, i.e., only traumatic failures can occur. Hence, by Theorem 2 we have

with K(t) = «(t) = E [k(s, X(s)) | C >t]. For a positive increasing Levy degradation process X with Levy measure v and drift rate ¡i and an intensity K(t,X(t)) = jX(t) that depends proportionally on degradation, Kebir [Keb91] proved exp ( —Y J X(s) ds

= exp ¡Y2 — Y Jo Jo [1 — e~s1 v(dx)d^j i.e., T has the increasing failure rate K(t) = Y(lt + /(^[l — e-tx] v(dx)).

Applying Kebirs formula to a homogeneous Poisson process with rate A > 0 we see that T follows a Makeham distribution with the survival function

For a DS-model with degradation modeled by a Wiener process with drift X(t) = x + ft + aW(t) and a quadratic intensity n(t,X(t)) = (X(t))2,

Wenocur [Wen86] computed the survival function of T as

= exp! — ^t + (— — ^ tanh(at) + 2Xf (sech(at) — 1)1 v/sech(at). [ a2 \a3 a J a2 J

Some failure rates of T are shown in Fig. 2. If a tends to zero the distribution

of T converges to a "generalized" Weibull distribution with form parameter three:

3 Maximum Likelihood Estimation

Suppose that n independent items are observed in [0,t*] with identically distributed degradation processes Xj, traumatic event processes failure times Ti and thresholds X*. We assume that X* and (Xiare independent for all i = l,...,n. The ith item is observed at planned inspection times

0 < ti o < tn < ■ ■ ■ until t*. Let xij = Xi(tij) denote the observed degradation levels. If a failure occurs in [0,t*], the observation of Xi will be stopped after this event. That means, in each interval (tij-i,tij] we observe either a failure at ti G (tij-i,tij] and the degradation level Xi(ti) or we observe the degradation level Xj = Xi(tij) at tj (and xi = Xi,(t*) at t*) under the condition that degradation has not yet exceeded the threshold. For the ith item let li = li(t*) = max{j > 0 : tj < min(Ti,t*)}, i.e., li + 1 is the number of observed degradation levels without failure in [0,t*). Further, let Ti = min(Ti,t*) be the observable censored failure time and

Vi = Vi(t*) = < 1, if Di < Ci, Di < t* (nontraumatic failure in [0,t*]), [ -1, if Ci < min(Di,t*) (traumatic failure in [0,t*])

an observable failure mode indicator.

with Xu = (Xi(tio), ■ ■ ■,Xi(tUi)). For ki = li + 1 set tik. = %, Xik. = Xi and Xik = (Xii,Xi(ti)). By fD(t | ti, x*) we denote the conditional density of the nontraumatic failure time given {X* = x*} and given l + 1 observations of the degradation process without reaching the threshold up to ti < t:

fD(t I ti, xi; x*)dt = P(D G dt | D> ti, Xti = xi; X* = x*)

and by fx* the density of the random threshold X*. Dropping the subscript

1 the likelihood of the data is according to Theorem 3

P(T G dt,v = 0, Xk G dxk) = l(t = t* )P(T >t*, Xk G dxk)

= 1(t = t*) exp {—j k(s, Xk(s))ds^ g(t*, tk, Xk) dxk, if no failure has occured in [0,t*],

P(T G dt,v = 1, Xk G dxk) = P(C > t,D G dt, Xk G dxk)

with

P(C>t I D = t, Xk = xk) = P(C>t I D = t, Xk = xk, X* = xk)

P(D e dt, Xk e dxfc) = P(D e dt | D> ti, Xi = xi, X* = xk)

P(D > ti, Xi e dxi IX* = Xk) P(X* e dxk) = Id (t I ti, xi; Xk )dt g (ti, ti, xi; Xk )dxi fx * (xk)dxk, if a nontraumatic failure has occured first in [0, t*], and, finally,

if a traumatic failure has occured first in [0,t*].

Thus, the complete likelihood function for the observation of n independent items in [0,t*] is given by

Based on this complex likelihood structure the Maximum Likelihood estimators of model parameters have to be found numerically in general. Explicit estimators of the degradation parameters in a special DT-model based on the Wiener process were given in Lehmann [Leh01].

The DTS-model can be easily extended to the case that m different modes of traumatic events are considered such that traumatic failures of mode i occur due to a point process If all these point processes are doubly stochastic Poisson processes conditionally independent given the degradation path X(■) and adapted to appropriate filtrations with intensities Ki(t, X(t)), then the Theorems 2 and 3 remain valid if we replace k(■) by m=i Ki(■)

Additionally to the degradation process, which is an internal covariate, one can consider an external covariate process Z = {Z(t) : t e R+} which describes the dynamic environment and may influence degradation and the intensity of traumatic events. Since such covariate processes like loads, stresses

or usage measures can often be completely observed, one is interested in the conditional distribution of degradation and failure time given the covariate history Zt = {Z(s) : 0 < s < t} up to some time t. If the failure rate X(t, Zt, X*) of D depends on the covariate Z and the threshold X* and if the intensity of traumatic events n(t, Z(t), X(t)) depends on the environment and on the degradation level, all concerned theorems and formulas remain valid if all probabilities and expectations are additionally conditioned on Zt. For instance, the survival function of T given in Theorem 2, but conditioned on Zt now, is with conditional failure rates K(t,Zt) = E\k(î,Z(t),X(t)) | Zt,T > t] and

References

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[BHN05] Bagdonavicius, V., Haghighi, F, Nikulin, M.S.: Statistical analysis of general degradation path model and failure time data with multiple failure modes. (accepted for publication in Communications in Statistics in 2005) [BN01] Bagdonavicius, V., Nikulin, M.S.: Estimation in degradation mod els with explanatory variables. Lifetime Data Analysis, 7, 85-103

[Cox99] Cox, D.R.: Some remarks on failure-times, surrogate markers, degradation, wear and the quality of life. Lifetime Data Analysis, 5, 307-314 (1999)

[DH92] Doksum, K.A., Hoyland, A.: Models for variable-stress accelerated life testing experiment based on a Wiener process and the inverse Gaussian distribution. Technometrics, 34, 74-82 (1992) [DN95] Doksum, K.A., Normand, S.L.T.: Gaussian models for degradation processes - part I: Methods for the analysis of biomarker data. Lifetime Data Analysis, 1, 131-144 (1995) [KL98] Kahle, W., Lehmann, A.: Parameter estimation in damage processes: Dependent observations of damage increments and first passage time. In: Kahle, W., von Collani, E., Franz, J., Jensen, U. (eds) Advances in Stochastic Models for Reliability, Quality and Safety, 139-152. Birkhauser, Boston (1998)

[Keb91] Kebir, Y.: On hazard rate processes. Naval Res. Logist. Quart., 38, 865-876 (1991)

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