Semiparametric Methods of Time Scale Selection

Transcription

1 1 Semparametrc Methods of Tme Scale Selecton Therry Duchesne Unversty of Toronto, Toronto, Canada Abstract: In several relablty applcatons, there may not be a unque plausble scale n whch to analyze falure In ths paper, I consder semparametrc methods of tme scale selecton I propose a rank-based estmator of the tme scale parameters that can readly handle censored observatons I llustrate how to assess the form of the tme scale through generalzed resduals I also gve deas for nonparametrc estmaton of the tme scale Keywords and phrases: collapsble model, generalzed resduals, deal tme scale, survval analyss, quantle regresson, rank regresson, relablty, usage path 11 Introducton In many relablty applcatons, there may not be a unque plausble tme scale n whch to assess performance or model falure For example, should we measure the age of an automoble n calendar tme snce purchase or n cumulatve mleage, or perhaps even n a functon of these two measures [Lawless et al (1995)]? Should we measure the age of an arcraft n calendar tme, cumulatve flght tme, or number of landngs [Kordonsky and Gertsbakh (1993, 1995ab, 1997)]? Should we measure the lfetme of mners exposed to asbestos dust n bologcal age or n cumulatve exposure [Oakes (1995)]? Even though t s possble to model most processes wth respect to chronologcal tme, there are some advantages n adoptng an alternatve scale t n certan contexts [Farewell and Cox (1979), Duchesne and Lawless (2000)] To name a couple: 1 the falure mechansm tself may depend prmarly on t, so that t s n some way scentfcally suffcent For example, the probablty that an 1

2 2 Therry Duchesne tem fals after some chronologcal tme x may depend only on ts age n the alternatve scale t at tme x; 2 the effect of treatments or covarates may be most smply expressed on some scale other than real tme In ths paper, I present some semparametrc methods of nference for a specal class of models emphaszng smple tme scales ntroduced by Oakes (1995) In Secton 2, the concept of deal tme scale s presented and a class of smple models s consdered I derve semparametrc estmators of the tme scale parameters for these models n Secton 3 Semparametrc model assessment from generalzed resduals s llustrated through an example n Secton 4 I provde deas for completely nonparametrc estmaton of the deal tme scale n Secton 5 Some concludng remarks are gven n Secton 6 12 Modelng falure Let x denote chronologcal tme from an orgn that usually corresponds to the brth or ntroducton of a unt nto servce Let X represent the random varable of chronologcal tme or age at falure for a unt Defnton 121 A usage measure s a tme-varyng covarate, say z(x), such that z(x) s left-contnuous; z(x) s non-decreasng n x; z(0) = 0; z( ) s external, e ts hstory s determned ndependently of the falure tme X Examples of usage measures nclude the cumulatve mleage on an automoble, the cumulatve number of landngs of an arcraft and the cumulatve amount of exposure of a rat to a potental carcnogen Now suppose that for each unt under study, the usage path (or hstory) s avalable, e we are gven Z = {z 1 (x),, z p (x); x 0} for every unt, where z 1 (x),, z p (x) are p usage measures Let Z represent the set of all possble usage paths We conceve of a tme scale (TS) as a functon of chronologcal tme and usage measures Formally, we consder a TS to be a non-negatve-valued functonal Φ : IR + Z IR + that maps (x, Z) to Φ[x, Z] and such that Φ[x, Z] s non-decreasng n x for all Z Z We use the notaton t Z (x) = Φ[x, Z] to

3 Semparametrc Methods of Tme Scale Selecton 3 emphasze that Φ[x, Z] s the tme (age) n the Φ-scale of a unt on path Z at chronologcal tme x Usage paths are assumed to vary from unt to unt n some populaton The fact that they are external allows us to nvestgate the dstrbuton of tme to falure condtonal on the realzed path Z Some advantages of such a condtonal approach s that t s vald for experments where the usage hstores are fxed by study desgn, and a sngle model may be used for populatons wth dssmlar usage accumulaton tendences The followng defnton of an deal tme scale has been consdered by several authors 1 [Cnlar and Ozekc (1987), Kordonsky and Gertsbakh (1993, 1995ab, 1997), Bagdonavčus and Nkuln (1997a) and Duchesne and Lawless (2000)] and t relates an deal tme scale to the condtonal probablty of survval gven the usage hstory: Defnton 122 Let T = Φ[X, Z] represent the random varable of tme to falure measured n scale Φ Then Φ defnes an deal tme scale (ITS) f for all Z Z and all t 0 such that t = Φ[x, Z] for some x 0, Pr [X > x Z] = Pr [T > t] (11) = G(t), where G( ) s a strctly decreasng survvor functon Ths defnes an ITS as any tme scale that s a one-to-one functon of the condtonal survvor functon Pr [X > x Z] A consequence of Defnton 122 that wll be useful for nference later on s that f Φ s an deal tme scale, then Pr [T > t Z] = Pr [T > t], whch means that the falure tmes n the ITS, though functonally dependent of Z, are statstcally ndependent of the usage paths Let us now consder the cases where Pr [X > x Z] s contnuous and dfferentable n x for all x 0 and Z Z Then one approach to modelng s to ncorporate the effect of Z on X through the condtonal hazard functon: { x } Pr [X > x Z] = exp h(u Z(u)) du, (12) where Z(x) = {z 1 (u),, z p (u); 0 u < x} We can rewrte (12) so as to emphasze an ITS: Pr [X > x Z] = G(Φ[x, Z]), (13) where G s a strctly decreasng survvor functon and Φ[x, Z] s an ITS for Pr [X > x Z] The two models descrbed n (12) and (13) are equvalent and the relatonshp between the ITS n (13) and the condtonal hazard functon n (12) s gven by 0 h(x Z(x)) = h G (Φ[x, Z]) d Φ[x, Z], (14) dx 1 These authors do not necessarly use the term deal tme scale, however

4 4 Therry Duchesne where h G (u) = d ln G(u)/du s the hazard functon correspondng to survvor functon G Generally, models wth a smple form for h(x Z(x)) wll have a complex form for Φ[x, Z] and vce-versa In ths paper, we consder a class of models for whch Φ[x, Z] s of a smple form Ths class of model was ntroduced by Oakes (1995) and s defned as follows: Defnton 123 The model Pr [X > x Z] s collapsble n x, z 1 (x),, z p (x) f Pr [X > x Z] = f(x, z 1 (x),, z p (x)), x 0, Z Z, where f : IR (p+1)+ IR + s non-decreasng n x, z 1 (x),, z p (x) Ths defnton means that for a collapsble model, the ITS s wll be of the form Φ[x, z 1 (x),, z p (x)], e the probablty of survvng past a certan tme x depends on the value of the usage measures at x only, not on ther past hstory Ths s a rather strong assumpton and we wll see how t can be verfed n Secton 4 13 Semparametrc nference Usually, we specfy an ITS up to a fnte vector of unknown parameters that are to be estmated from observed data Of specal nterest s the case where we observe (x 1, Z 1, δ 1 ),, (x n, Z n, δ n ), where δ = 1 f x s an observed falure tme and δ = 0 f x s a rght-censorng tme, = 1, 2,, n, and where we postulate the model Pr [X > x Z] = G(Φ[x, z 1 (x),, z p (x); η]), (15) where η s a fnte vector of unknown parameters and G s a strctly decreasng survvor functon left unspecfed Oakes (1995) consdered nference n the case where G was specfed up to a fnte vector of parameters Kordonsky and Gertsbakh (1993, 1995ab, 1997) propose a method of estmaton for η when G s left unspecfed n (15) Let t Z (x ; η) = Φ[x, z 1,,, z n, ; η], = 1,, n Then Kordonsky and Gertsbakh propose to use the value of η that mnmzes the squared sample coeffcent of varaton of the t Z (x ; η) s as an estmate of η Ths method s numercally smple, yelds consstent estmators n several frameworks, and a tme scale wth small coeffcent of varaton may be desrable n some engneerng applcatons However, Kordonsky and Gertsbakh (1995b, 1997) have to assume a parametrc form for G when some observatons n the sample are censored Defnng a censored nonparametrc verson of the coeffcent of varaton s also a problem, as extrapolaton of the

5 Semparametrc Methods of Tme Scale Selecton 5 tal of the dstrbuton s needed when the largest tme n the sample s a censorng tme Moreover, t s not clear that ths method s effcent, as nvestgated by Duchesne and Lawless (2000) As an alternatve, we can adapt a rank-based method derved for the accelerated falure tme model proposed by Robns and Tsats (1992) and Ln and Yng (1995) Ths method shows good effcency propertes and can readly handle censorng Furthermore, the estmatng functons obtaned are derved drectly from the condtonal probablty model gven n (15) and n cases where they have a smple form, they have a nce nterpretaton that relates to the ndependence of the falure tmes n the deal tme scale and the usage paths It s convenent to ntroduce some further notaton Let X, = 1,, n denote a sequence of n ndependent falure tmes Under rght-censorshp, we observe X = mn(x, C ) and δ = I[X C ], = 1,, n, where C s the censorng tme for the th tem and I[ ] s the ndcator functon Fnally, we assume that gven Z, X and C are ndependent, = 1,, n Under model (15), the log-lkelhood for η s gven by l(η) = 0 {( ) ln λ[t Z ] + ln t Z dn (x) (16) } Y (x)λ[t Z ]t Z dx, where λ[u] s the hazard functon correspondng to G(u), N (u) s a countng process takng value 0 f u < X and 1 f u X, Y (u) s 1 f ndvdual s at rsk of falng at real tme u, 0 otherwse, and t Z = dt Z /dx Assumng that the usual regularty condtons hold, we can take the dervatve wth respect to η under the ntegral sgn and obtan the followng score functon: U(η) = λ [t Z ] t Z + t Z / η 0 λ[t Z ] η t dn (x) Z ( Y (x) λ (t Z ] t Z t η Z (17) ) } +λ[t Z ] t Z dx η Followng Ln and Yng (1995), let us now replace the unknown cumulatve hazard functon by ts Nelson-Aalen estmator (n scale t Z ) and the dervatve of the unknown hazard functon by zero After rearrangng the terms, we get Ũ(η) = ln t Z η 0 x=x 1 [t,η,z ] ( ) dñ(t; η) Ỹ(t; η) d Λ[t; η], (18)

6 6 Therry Duchesne where x 1 [t, η, Z ] s the value of x such that t Z = t, and where Ñ (t; η) = Ỹ (t; η) = { 0, f t < tz (X ; η) 1, f t t Z (X ; η) { 1, f ndvdual s at rsk of falng at tme t n scale tz 0, otherwse Λ[t; η] = value of Nelson-Aalen estmator n scale t Z at tme t Notce that the score (18) s of the form δ ( Q Q ), where Q = ln t Z / η and Q s the average of the Q s stll at rsk when tem fals (n scale t Z ) Ths means that score (18) s an unbased estmatng functon However ths score s not contnuous, so we defne the estmator of η to be arg mnη Ũ(η)tŨ(η), the mnmum beng taken over all η n some compact regon [Ln and Yng (1995)] Asymptotc propertes of estmators of ths type are dscussed n Robns and Tsats (1992), Yng (1993), Ln and Yng (1995), Bagdonavčus and Nkuln (1997b) and Bordes (1999) In patcular, we have that a relatvely accurate approxmate (1 α) confdence regon for η s gven by {η : Ũ(η)t V 1 (η)ũ(η) χ2 k;1 α}, (19) where k = dm(η), χ 2 k;1 α k degrees of freedom and s the 1 α quantle of a ch-square dstrbuton on V (η) = 0 [ nj=1 Ỹj(t; η)q 2 j (t; η) nj=1 Ỹj(t; η) (110) ( nj=1 Ỹj(t; ) 2 η)q j (t; η) nj=1 Ỹj(t; dñ(t; η), (111) η) where Q j (t; η) = ln t Z j η x=x 1 [t,η,z j ] and, for some vector u, u 2 denotes the outer product uu t Duchesne (1999) argued that two condtons were necessary for precse estmaton of η wth ths and any nference method: large varablty n the observed usage paths Z, and a dstrbuton Pr [X > x Z] wth small varaton Through smulatons, he observed that when these condtons were not met, all

7 Semparametrc Methods of Tme Scale Selecton 7 the estmators consdered were based and naccurate For most of the models he consdered, the rank-based estmator seemed to be more effcent than the mnmum coeffcent of varaton estmator Now, as an llustraton, let z(x) = θx and put t Z = (1 η)x+ηθx, e usage s acccumulated at a constant rate, θ, and the ITS s a lnear combnaton of real tme and usage Assumng no censorng and substtutng those values nto (18), we obtan Ũ(η) = θ () 1 1 η + ηθ () ( 1 1 n 1 ) n 1 1, (112) n + 1 where () s the label of the th tem to fal n scale (1 η)x + ηθx Notce that ths last score s the one we would get f we were to test no assocaton between the falure tmes n scale (1 η)x+ηθx and the covarates (θ 1)/(1 η +ηθ) usng rank regresson wth exponental scores As θ completely defnes the usage path n ths case, ths s consstent wth the defnton of an ITS whch mples that the falure tmes n the ITS are ndependent of the usage hstores 14 Semparametrc model assessment Supposng that a collapsble model s approprate s a strong assumpton In ths secton, we propose a smple graphcal method based on generalzed resduals to assess the adequacy of a specfed tme scale We use data provded by Kordonsky and Gertsbakh (1995a) to llustrate the method As was mentoned earler, the falure tmes n the ITS are ndependent of the usage hstores Ths means that f we could plot these tmes aganst some features of the usage paths, we should observe only scatter, wthout any type of trend Unfortunately, unless the value of η s known, we do not observe the falure tmes n the ITS Nevertheless, f we have a good estmate of η, say ˆη, the generalzed resduals t Z (x ; ˆη), = 1, 2,, n wll be close to the falure tmes n the ITS We can thus plot these generalzed resduals aganst some features of the usage hstores; f we observe any type of trend n any of the plots, ths s evdence that t Z s not an ITS To llustrate ths method, let us ft two dfferent tme scales to the data provded by Kordonsky and Gertsbakh (1995a) Frst, let us ft a lnear scale of the form t Z = (1 η)x + ηz(x), where x s the number of low stress cycles and z(x) = θx s the number of hgh stress cycles Usng the rank-based estmator of Secton 13, we obtan ˆη = 0868, whch s very close to 0871, the value obtaned wth the mnmum coeffcent of varaton method The 95% confdence nterval derved from equaton (19) s (0844, 0910) Now, let us ft a multplcatve scale of the form t Z = x η z(x) 1 η The estmates of η obtaned wth the rank-based and mnmum coeffcent of varaton methods

8 8 Therry Duchesne are respectvely 0800 and 0804 The 95% confdence nterval for η from (19) s (0662, 0930) Fgure 1 shows a plot of the generalzed resduals aganst the path parameter θ for both scales There does not seem to be any apparent trend n the plot for the lnear scale, but there s a defnte quadratc trend n the plot for the multplcatve scale The lnear scale seems to be more approprate n ths case t(eta) Lnear scale path t(eta) Multplcatve scale path Fgure 1 Generalzed resdual plots: falure tmes n scales 0132x z(x) (left) and x 0200 z(x) 0800 (rght) versus the proporton of hgh stress cycles 15 Nonparametrc age curves The nce nterpretaton of a model collapsble n x, z 1 (x),, z p (x), s that the tems travel on a path that crosses age curves For example f a model s collapsble n x and z(x), and f a lnear scale of the form 02x + 08z(x) s deal, then all the ponts n the postve quadrant on a lne parallel to the lne 02x + 08z = c have the same age n the ITS, e Pr [X > x Z] s constant over these lnes An nterestng consequence of ths s that f we could draw the level curves of Pr [X > x Z] nonparametrcally, we would have a pcture of the form of the ITS functon Φ[x, z 1 (x),, z p (x)] Lttle work has been done n ths area In the case where the usage paths can be completely descrbed by a fnte vector of parameters, say θ, we can try to adapt some of the nonparametrc quantle regresson algorthms descrbed n the lterature [Lejeune and Sarda (1988), Cole and Green (1992)] However for the smple lnear and multplcatve scales descrbed above, these nonparametrc

9 Semparametrc Methods of Tme Scale Selecton 9 quantle regresson methods exhbt some serous edge effects that dstort the pcture of the deal tme scale functon We can adopt another approach to nonparametrc estmaton of the deal tme scale when the usage paths are such that we have several observatons dstrbuted among a few groups of smlar paths Then we can group those smlar paths together, estmate a gven quantle of X n each group of paths, then draw segments between the estmated quantles These segments should form a curve that wll be representatve of the form of the deal tme scale To llustrate, consder the followng two models: Pr [X > x Z ] = G(x + z (x)) and Pr [X > x Z ] = G(x 2 + z (x) 2 ), and assume that z (x) = θ x, where the θ s vary n [0, ) wthn the populaton of tems We generated a sample of sze 400 from both models wth G beng the Webull(shape=5, scale=9) and atan(θ) Unform(0,π/2) Then for both samples, we splt the postve quadrant n 4 slces, estmated the quntles of X n each slce, then drew lnes that lnked correspondng quntles together The results are shown n Fgure 2 x y(x) T~Webull(5,9), T=X+(Theta)X atan(theta)~u(0,p/2), sample sze=400 Estmated quntles True quntles Regon delmters Mean paths x y(x) ~Webull(5,9), T=X^2+[(Theta)X atan(theta)~u(0,p/2), sample sze=400 Estmated quntles True quntles Regon delmters Mean paths Fgure 2 Age curves drawn nonparametrcally The two plots n Fgure 2 clearly dentfy the lnear and quadratc forms of the deal tme scales However, these two samples were ncer than the samples we are lkely to encounter n practce; we had a large number of observatons and the usage paths were ncely spread over the whole postve quadrant Fne tunng ths age curve drawng method s requred Perhaps ntroducng smoothng or combnng ths method wth a quantle regresson algorthm could yeld good nonparametrc estmates of the ITS

10 10 Therry Duchesne 16 Concluson The selecton of an deal collapsble tme scale s not a trval exercse When a parametrc form for the tme scale has been selected, semparametrc methods based on ranks allow for precse parameter estmaton, even under rghtcensorng, provded that enough varablty n the covarate hstores s observed and that these covarates are good predctors of falure Supposng that a smple tme scale s deal s a strong assumpton that should be verfed A smple graphcal method based on semparametrc generalzed resduals has been proposed More formal semparametrc methods would be desrable Nonparametrc estmaton of the deal tme scale has not yet been nvestgated thoroughly, but t seems to be possble n the case of collapsble models Ackowledgements I would lke to thank the Natural Scences and Engneerng Research Councl of Canada for ther fnancal support and professor Jerry Lawless for hs helpful advse and comments References 1 Bagdonavčus, VB and Nkuln, MS (1997a) Transfer functonals and semparametrc regresson models, Bometrka, 84, Bagdonavčus, VB and Nkuln, MS (1997b) Asymptotcal analyss of semparametrc models n survval analyss and accelerated lfe testng, Statstcs, 29, Bordes, L (1999) Semparametrc addtve accelerated lfe models, Scandnavan Journal of Statstcs, 26, pp Cole, TJ and Green, PJ (1992) Smoothng reference centle curves: the lms method and penalzed lkelhood Statstcs n Medcne, 11, Cnlar, E and Ozekc, S (1987) Relablty of complex devces n random envronments, Probablty n the Engneerng and Informatonal Scences, 1, Duchesne, T (1999) Multple Tme Scales n Survval Analyss, Doctoral dssertaton, Unversty of Waterloo

11 Semparametrc Methods of Tme Scale Selecton 11 7 Duchesne, T and Lawless, J (2000) Alternatve tme scales and falure tme models, Lfetme Data Analyss, to appear 8 Farewell, VT and Cox, DR (1979) A note on multple tme scales n lfe testng, Appled Statstcs, 28, Kordonsky, KB and Gertsbakh, I (1993) Choce of the best tme scale for system relablty analyss, European Journal of Operatonal Research, 65, Kordonsky, KB and Gertsbakh, I (1995a) System state montorng and lfetme scales I, Relablty Engneerng and System Safety, 47, Kordonsky, KB and Gertsbakh, I (1995b) System state montorng and lfetme scales II, Relablty Engneerng and System Safety, 49, Kordonsky, KB and Gertsbakh, I (1997) Multple tme scales and the lfetme coeffcent of varaton: Engneerng applcatons, Lfetme Data Analyss, 2, Lawless, JF, Hu, J and Cao, J (1995) Methods for the estmaton of falure dstrbutons and rates from automoble warranty data, Lfetme Data Analyss, 1, Lejeune, MG and Sarda, P (1988) Quantle regresson: a nonparametrc approach, Computatonal Statstcs and Data Analyss, 6, Ln, DY and Yng, Z (1995) Semparametrc nference for the accelerated lfe model wth tme-dependent covarates, Statstcal Plannng and Inference, 44, Oakes, D (1995) Multple tme scales n survval analyss, Lfetme Data Analyss, 1, Robns, J and Tsats, AA (1992) Semparametrc estmaton of an accelerated falure tme model wth tme-dependent covarate, Bometrka, 79, Yng, Z (1993) A large sample study of rank estmaton for censored regresson data, Annals of Statstcs, 21, pp 76 99