Product-limit estimators of the gap time distribution of a renewal process under different sampling patterns

Pro duct-limit estimators of the gap time distribution of a renew al pro cess under differen t sampling patterns Ric hard D. Gill Departmen t of Mathematics Univ ersity of Leiden Niels Keiding Departmen t of Biostatistics Univ ersity of Cop enhagen 1 F ebruary 2010 Abstract Nonparametric estimation of the gap time distribution in a simple re- new al pro cess may b e considered a problem in surviv al analysis under particular sampling frames corresp onding to ho w the renewal pro cess is observ ed. This note describ es sev eral suc h situations where simple pro duct limit estimators, though inefficien t, ma y still b e useful. 1 1 In tro duction This note is ab out t w o classical problems in nonparametric statistical analysis of recurrent ev en t data, both formalised within the framework of a simple, stationary renewal process. W e first consider observ ation around a fixed time p oin t, i.e., we observ e a backw ard recurrence time R and a forward recurrence time S . It is w ell kno wn that the nonparametric maxim um likelihoo d estimator of the gap- time distribution is the Cox-V ardi estimator (Co x 1969, V ardi 1985) deriv ed 1 Key w ords. Kaplan-Meier estimator, Cox-V ardi estimator, Laslett’s line segmen t prob- lem, nonparametric maxim um likelihoo d, Mark ov process 1 from the length-biased distribution of the gap time R + S . How ev er, Winter & F¨ oldes (1988) prop osed to use a pro duct-limit estimator based on S , with dela yed en try given b y R . Keiding & Gill (1988) clarified the relation of that estimator to the standard left truncation problem. Unfortunately this discussion w as omitted from the published v ersion (Keiding & Gill, 1990). Since these simple relationships do not seem to b e on record elsewhere, w e offer them here. The second observ ation scheme considers a stationary renew al pro cess ob- serv ed in a finite in terv al where the left endp oin t do es not necessarily corre- sp ond to an ev ent. The full lik eliho od function is complicated, and we briefly surv ey p ossibilities for restricting atten tion to v arious partial lik eliho o ds, in the nonparametric case again allowing the use of simple pro duct-limit esti- mators. 2 Observ ation of a stationary renew al pro cess around a fixed p oin t Win ter & F¨ oldes (1988) studied the following estimation problem. Consider n independent renewal pro cesses in equilibrium with underlying distribution function F , which w e shall assume absolutely contin uous with density f , minimal supp ort in terv al (0 , ∞ ), and hazard β ( t ) = f ( t ) / (1 − F ( t )), t > 0. The reason for our uncon ven tional choice β for the hazard rate b elonging to F will become apparen t later. Corresp onding to a fixed time, say 0, the bac kward and forw ard recurrence times R i and S i , i = 1 , ..., n , are observed; their sums Q i = R i + S i are length-biased observ ations from F , i.e., their den- sit y is prop ortional to tf ( t ). Let ( R, S, Q ) denote a generic triple ( R i , S i , Q i ). W e quote the following distribution results: let µ b e the expectation v alue corresp onding to the the distribution F , µ = Z ∞ 0 uf ( u )d u = Z ∞ 0 (1 − F ( u ))d u, then the join t distribution of R and S has density f ( r + s ) /µ , the marginal distributions of R and S are equal with density (1 − F ( r )) /µ , and the marginal distribution of Q = R + S has densit y q f ( q ) /µ , the length-biased density corresp onding to f . Win ter and F¨ oldes considered the pro duct-limit estimator 1 − e F ( t ) = Y i : Q i ≤ t  1 − 1 Y ( Q i )  2 where Y ( t ) = n X i =1 I { R i < t ≤ R i + S i } is the numb er at risk at time t . This estimator is the same as the Kaplan- Meier estimator for iid surviv al data Q 1 , . . . , Q n left-truncated at R 1 , . . . , R n (Kaplan & Meier 1958, Andersen et al. 1993). Winter & F¨ oldes sho wed that 1 − e F is strongly consisten t for the underlying surviv al function 1 − F . W e shall show how the deriv ation of this estimator follo ws from a simple Mark ov pro cess mo del similar to the one used by Keiding & Gill (1990) to study the random truncation mo del. First notice that the conditional distribution of Q = R + S given that R = r has densit y f ( q ) /µ (1 − F ( r )) /µ , r < q < ∞ that is, in tensity (hazard) f ( q ) / (1 − F ( q )), q > r , which is just the hazard β ( q ) corresponding to the underlying distribution F left-truncated at r . Now define corresponding to ( R, S, Q ) a sto c hastic pro cess U on [0 , ∞ ] with state space { 0 , 1 , 2 } by U ( t ) = ( 0 , t < R, 1 , R ≤ t < R + S, 2 , R + S ≤ t. Note that it tak es in successsion the v alues 0, 1 and 2. F or U ( t ) = 0, P  U ( t + h ) = 1   U ( u ) , 0 ≤ u ≤ t  = P  R ≤ t + h   R > t  = α ( h ) h + o ( h ) , where α is the hazard rate of the marginal distribution of R . F or U ( t ) = 1 (and hence R ≤ t ) P  U ( t + h ) = 2   U ( u ) , 0 ≤ u ≤ t  = P  R + S ≤ t + h   R = r ≤ t, R + S > t  = f ( t ) 1 − F ( t ) h + o ( h ) b y the abov e result on the conditional hazard of R + S giv en R . F or U ( t ) = 0, P  U ( t + h ) = 2   U ( u ) , 0 ≤ u ≤ t  = o ( h ) . 3 that is, in ten s ity (h aza rd)   1 hq H q  ª ¬ º ¼ , which is just the h azard   q E corresponding to the underlying distribution H . Now define for each i ( t h e i i s s u p p r e s s e d i n t h e notation) a stochastic process U o n > @ 0, f with state space ^ ` 0, 1 , 2 b y  0, 0 1, 2, . tR Ut R t R S RS t d  ° d  ® ° d ¯ We h a v e    ^ `  2, 0 PU t h U u u t o h  d d for , and for ( t h a t i s ,  0 Ut  1 Ut R tR S d  ) th is is ^`     , 1 ht PR S t h R R S t h o h Ht d   !   by the abov e result on the hazard of R SR  . That this depends on t b u t n o t o n R proves that U is a Markov process 0 1 2 DE  o  o with intensities     11 t tH t H r D f   dr ª ºª º ¬ ¼¬ ¼ ³ (the m argina l haza rd of R , equal to the residual m ean lifetime function of the underlying distribution H ) and 4 Figure 1: Inhomogenous 3-state Mark ov pro cess, 2 allow ed transitions Other transitions are impossible. That these conditional probabilities dep end on U ( t ) and t only , but not on U ( u ), u < t , pro ves that U is a Mark ov process with in tensities α ( t ) = 1 − F ( t ) R ∞ t (1 − F ( r ))d r (the marginal hazard of R , equal to the residual mean lifetime function of the underlying distribution F ) and β ( t ) = f ( t ) 1 − F ( t ) , see Figure 1. The Marko v pro cess framework of Keiding & Gill (1990) now indicates that (ignoring information ab out F in α , and just fo cussing on the transition with rate β ) the pro duct limit estimator 1 − e F is a natural estimator of the survivor function 1 − F of interest, and consistency and asymptotic normalit y ma y b e obtained as shown b y Keiding & Gill (1990, Sec. 5). Note that the bac kwards in tensity α ( t ) = α ( t ) P  U ( t ) = 0  P  U ( t ) = 1  = α ( t ) P  R > t  P  R ≤ t < R + S  = α ( t ) µ − 1 R ∞ t (1 − F ( r ))d r µ − 1 R t 0 R ∞ t − r f ( r + s )d s d r = 1 − F ( t ) R ∞ t (1 − F ( r ))d r R ∞ t (1 − F ( r ))d r R t 0 (1 − F ( t ))d r = 1 t , the b ackwar ds hazard-rate of a uniform distribution on a b ounded in terv al (0 , A ), A < ∞ . Since it has b een assumed that R has support in terv al (0 , ∞ ), this sho ws that the present mo del may not b e interpreted strictly as a left 4 truncation mo del, whic h would require that α ( t ) w as the bac kw ards hazard rate of some probability distribution on (0 , ∞ ). Ho wev er, this distinction is not imp ortant to our discussion. The fact that α ( t ) do es not dep end on F corresponds to Win ter and F¨ oldes’ statemen t that ( R , S ) con tains no more information than R + S about F . This already follows from sufficiency since the joint density of ( R, S ) is f ( r + s ) /µ . The lik eliho o d function based on observ ation of ( R 1 , S 1 ) , . . . ( R n , S n ) is µ − n n Y i =1 f ( r i + s i ) from which the NPMLE of F is readily derived as b F ( t ) = n X i =1 I  R i + S i ≤ t  R i + S i  n X i =1 1 R i + S i , that is the Cox-V ardi estimator in the terminology of Winter and F¨ oldes (Co x 1969, V ardi 1985). It follows that the estimator 1 − e F is not NPMLE. The imp ortan t dif- ference betw een the situation here and that of the random truncation model studied by Keiding & Gill (1990, Sec. 3) is that not only the in tensity β ( t ), but also α ( t ) dep ends only on the estimand F . As already men tioned, weak con v ergence of 1 − e F is immediate from Keiding & Gill (1990, Sec. 5). In particular, in order to ac hieve the extension to conv ergence on [0 , M ] it should b e required that Z ε 0 dΦ( s ) /ν 2 ( s ) < ∞ in the terminology of Keiding & Gill (1990, Sec. 5c), and using dΦ( t ) = β ( t )d t and ν 2 ( t ) = P  U ( t ) = 1  = Z t 0 1 − F ( s ) µ 1 − F ( t ) 1 − F ( s ) d s = t µ (1 − F ( t )) , the integrabilit y condition translates in to Z ε 0 β ( t ) P  U ( t ) = 1  d t < ∞ or finiteness of E (1 /X ) where X has the underlying (“length-unbiased”) in terarriv al time distribution F . It ma y easily b e seen from Gill et al. (1988) that the same condition is needed to ensure w eak con vergence of the Cox- V ardi estimator. 5 A v ariation of the observ ation sc heme of this section would b e to al- lo w also right censoring of the S i . This can b e immediately included in the Mark ov-process/counting pro cess approach leading to the inefficient pro duct- limit t yp e estimator 1 − e F ; the delay ed-en try observ ations S i are sim ulta- neously right-censored. See V ardi (1985, 1989) and Asgharian et al. (2002) for treatment of the full non-parametric maximum lik eliho o d estimator of F , extending the Cox-V ardi estimator to allo w righ t censoring. Other ad ho c estimators and the ric h relationships with a n umber of other imp ortan t non-parametric estimation problems are discussed b y Den by and V ardi (1985) and V ardi (1989). 3 Observ ation of a stationary renew al pro cess in a finite in terv al W e consider again a stationary renewal pro cess on the whole line and as- sume that w e observe it in some interv al [ t 1 , t 2 ] determined indep enden tly of the pro cess. Nonparametric estimation of the gap time distribution F w as definitively discussed b y V ardi (1982) in discrete time and b y So on & W o o dro ofe (1996) in con tinuous time. Co ok & La wless (2007, Chapter 4) surv eyed the general area of analysis of gap times emphasizing that the as- sumption of indep endent gap times is often unrealistic. W e shall here nevertheless work under the assumption of the simplest p ossible mo del as indicated ab o v e. Because the nonparametric maximum lik eliho o d estimator is computationally inv olv ed it may sometimes b e useful to calculate less efficient alternatives, and there are indeed such p ossibilities. Under the observ ation sc heme indicated ab o ve w e may ha ve the follo wing four types of elemen tary observ ations 1. Times x i from one renew al to the next, contributing the density f ( x i ) to the likelihoo d. 2. Times from one renewal T to t 2 , which are right-censored versions of 1., con tributing factors of the form (1 − F ( t 2 − T )) to the lik eliho o d. 3. Times from t 1 to the first renew al T (forw ard recurrence times), con tribut- ing factors of the form (1 − F ( T − t 1 )) /µ to the lik eliho o d. 4. Kno wledge that no renewal happ ened in [ t 1 , t 2 ] , actually a right-censored v ersion of 3., con tributing factors of the form R ∞ t 2 − t 1 (1 − F ( u ))d u/µ to the 6 lik eliho o d. McClean & Devine (1995) studied nonparametric maximum lik eliho o d estimation in the conditional distribution giv en that there is at least one renew al in the interv al, i.e., that there are no observ ations of t yp e 4. Our interest is in basing the estimation only on complete or righ t-censored gap times, i.e., observ ations of type 1 or 2. When this is p ossible, we ha v e simple pro duct-limit estimators in the one-sample situation, and we ma y use well-established regression mo dels (such as Cox regression) to account for cov ariates. P e ˜ na et al. (2001) assumed that observ ation started at a renew al (thereby defining aw a y observ ations of t yp e 3 and 4) and gav e a comprehensiv e discussion of exact and asymptotic prop erties of pro duct-limit estimators with comparisons to alternativ es, building in particular on results of Gill (1980, 1981) and Sellke (1988). The crucial point here is that calendar time and time since last renewal b oth need to b e tak en in to account, so the straigh tforward martingale approach displa y ed by Andersen et al. (1993) is not a v ailable. Pe˜ na et al. also studied robustness to deviations from the assumption of indep endent gap times. As noted by Aalen & Huseby e (1991) in their attractive non-technical discussion of observ ation patterns, observ ation do es how ev er often start b e- t ween renew als. (In the example of Keiding et al. (1998), auto insurance claims were considered in a fixed calendar p erio d). As long as observ ation starts at a stopping time, inference is still v alid, so by starting observ ation at the first renew al in the interv al we can essen tially refer bac k to P e ˜ na et al. (2001). A more formal argument could b e based on the concept of the A alen filter , see Andersen et al. (1993, p. 164). The resulting pro duct-limit estimators will not b e fully efficien t, since the information in the backw ard recurrence time (types 3 and 4) is ignored. It is imp ortant to realize that the v alidit y of this w ay of reducing the data dep ends critically on the indep en- dence assumptions of the mo del. Keiding et al. (1998), cf. Keiding (2002) for details, used this fact to base a go o dness-of-fit test on a comparison of the full nonparametric maxim um lik eliho o d estimator with the pro duct-limit estimator. Similar terms app ear in another mo del, called the L aslett line se gment pr oblem (Laslett, 1982). Supp ose one has a stationary P oisson pro cess, with in tensity µ , of p oin ts on the real line. W e think of the real line as a calendar time axis, and the p oin ts of the Poisson process will b e called pseudo r enewal times or birth times of some population of individuals. Supp ose the individ- uals ha ve indep enden t and identically distributed lifetimes, each one starting at the corresp onding birth time. The corresp onding calendar time of the end of each lifetime can of course b e called a de ath time . Now supp ose that al l 7 we c an observe are the intersections of individuals’ lifetimes (though t of as time segments on the time axis) with an observ ational window [ t 1 , t 2 ]. In particular, w e do not know the curren t age of an individual who is observed aliv e at time t 1 . Again we ha ve exactly the same four kinds of observ ations: 1. Complete pr op er lifetimes corresp onding to births within [ t 1 , t 2 ] for whic h death o ccurred b efore time t 2 . 2. Censored pr op er lifetimes corresp onding to births within [ t 1 , t 2 ] for whic h death o ccurred after time t 2 . 3. Complete r esidual lifetimes corresp onding to births whic h o ccurred at an unkno wn momen t b efore time t 1 , and for whic h death o ccurred after t 1 and b efore time t 2 . 4. Censored r esidual lifetimes corresp onding to births which o ccurred at an unkno wn momen t b efore time t 1 , for whic h death occurred after time t 2 , and whic h are therefore censored at time t 2 . The numb er of at least partially observ ed lifetimes (prop er or residual) is random, and P oisson distributed with mean equal to the in tensit y µ of the underlying Poisson process of birth times, times the factor t 2 − t 1 + Z ∞ 0 (1 − F ( y ))d y . This provides a fifth, “Poisson”, factor in the nonparametric lik eliho o d func- tion for parameters µ and F , based on all the av ailable data. Maximizing o v er µ and F , the mean of the P oisson distribution is estimated by the observ ed n umber of partially observed lifetimes. Thus w e find that the pr ofile likeli- ho o d for F , and the mar ginal likeliho o d for F based only on contributions 1.–4., are prop ortional to one another. Nonparametric maximum likelihoo d estimation of F w as studied by Wij- ers (1995) and v an der Laan (1996), cf. v an der Laan & Gill (1999). Some of their results, and the calculations leading to this likelihoo d, w ere survey ed b y Gill (1994, pp. 190 ff.). The nonparametric maxim um likelihoo d estima- tor is consisten t; whether or not it con verges in distribution as µ tends to infinit y is unkno wn, the mo del has a singularity coming from the v anishing probabilit y density of complete lifetimes just larger than the length of the observ ation windo w corresp onding to births just b efore the start of the ob- serv ation window and deaths just after its end. V an der Laan show ed that 8 a mild reduction of the data by grouping or binning leads to a m uc h b et- ter b eha ved nonparametric maximum lik eliho o d estimator. If the amoun t of binning decreases at an appropriate rate as µ increases, this leads to an asymptotically efficien t estimator of F . This pro cedure can b e though t of as r e gularization , a pro cedure often needed in nonparametric in verse statistical problems, where maximum lik eliho o d can b e to o greedy . Both “unregularized” and regularized estimators are easy to compute with the EM algorithm; and the sp eed of the algorithm is not so painfully slo w as in other inv erse problems, since this is still a problem where “ro ot n ” rate estimation is possible. The problem allo ws, just as w e ha ve seen in earlier sections, all the same inefficien t but rapidly computable pro duct-limit type estimators based on v arious marginal lik eliho o ds. Moreov er since the direction of time is basically irrelev an t to the mo del, one can also lo ok at the pro cess “bac kw ards”, leading to another plethora of inefficient but easy estimators. One can even com bine in a formal wa y the censored surviv al data from a forward and a bac kward time p oin t of view, which comes do wn to coun ting all uncensored observ ations t wice, all singly censored once, and discarding all doubly censored data. (This idea was essentially suggested muc h earlier b y R.C. P almer and D.R. Co x, cf. Palmer(1948)). The attractiv e feature of this estimator is again the ease of computation, the fact that it only discards the doubly censored data, and its symmetry under rev ersing time. The asymptotic distribution theory of this estimator is of course not standard, but using the nonparametric delta method one can fairly easily giv e form ulas for asymptotic v ariances and co v ariances. In practice one could easily and correctly use the nonparametric b o otstrap, resampling from the partially observ ed lifetimes, where again a resampled complete lifetime is en tered t wice in to the estimate. The Laslett line segment problem has rather imp ortan t extensions to ob- serv ation of line segments (e.g., crac ks in a rock surface) observ ed through an observ ational window in the plane. Under the assumption of a homogenous P oisson line segment process one can write down nonparametric lik eliho o ds, maximize them with the EM algorithm; it seems that regularization ma y w ell b e necessary to get optimal “ro ot n ” b ehaviour but in principle it is clear ho w this might b e done. Again, we hav e the same plethora of ineffi- cien t but easy pro duct-limit t yp e estimators. V an Zwet (2004) studied the b eha viour of suc h estimators when the line segmen t pro cess is not Poisson, but merely stationary . The idea is to use the Poisson pro cess likelihoo d as a quasi lik eliho o d, i.e., as a basis for generating estimating equations, whic h will b e unbiased but not efficient, just as in parametric quasi-likelihoo d. V an Zw et sho ws that this pro cedure works fine. Coming full circle, one can apply these ideas to the renewal pro cess we first describ ed in this section, and the 9 other mo dels describ ed in earlier sections. All of them generate stationary line segment processes observ ed through a finite time window on the line. Th us the nonparametric quasi-lik eliho o d approac h can b e used there to o. Since in the renewal pro cess case w e are ignoring the fact that the in tensity of the point pro cess of births equals the in verse mean life-time, we do not get full efficiency . So it is disputable whether it is worth using an inefficient ad-ho c estimator whic h is difficult to compute when we hav e the options of So on and W o o dro ofe’s fully efficient (but hard to compute) full nonparamet- ric maximum likelihoo d estimator, and the man y inefficient but easy and robust pro duct-limit type estimators of this pap er. 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