the ratio of the hazards converges to unity as time increases. This is in contrast to the proportional hazards model which constrains the ratio of the

Transcription

1 MLE IN THE PROPORTIONAL ODDS MODEL BY S.A. MURPHY, A.J. ROSSINI AND A.W. VAN DER VAART Pennsylvania State University, University of South Carolina and Free University Amsterdam We consider maximum likelihood estimation of the parameters in the proportional odds model with right censored data. The estimator of the regression coeæcient is shown to be asymptotically normal with eæcient variance. The maximum likelihood estimator of the unknown monotonic transformation of the survival time converges uniformly at a parametric rate to the true transformation. Estimates for the standard errors of the estimated regression coeæcients are obtained by diæerentiation of the proæle likelihood and are shown to be consistent. A likelihood ratio test for the regression coeæcient is also considered.. Introduction Since Pettitt è982è and Bennett è983abè generalized the proportional odds model to the survival analysis context, much eæort and research has gone into proposing reasonable estimators for the regression coeæcient in this model. For example, inference for this model has been considered by Pettitt è984è, Bickel è986è, Dabrowska & Doksum è988è, Cuzick è988è,wu è995è, Cheng et al. è995è, Shen è995è and Huang and Rossini è995è, among others. However, to date, no estimation method has been shown to produce eæcient estimators for the proportional odds model with right censored data and more than one covariate or continuous covariates. We demonstrate that Bennett's è983aè semiparametric maximum likelihood estimation method produces an eæcient estimator for the regression coeæcient. We also provide methods for conducting inference about the regression coeæcient. The survival function, S Z, given the vector of covariates, Z, is parameterized by,logitès Z ètèè = Gètè+Z T æ; where logitèxè = logèx=è, xèè. The unknown parameters are G, a baseline log odds of failure at time t, and æ, a p-vector of regression coeæcients. In the two sample problem, this model constrains the ratio of odds of survival to be constant with time. Consequently, Research partially supported by NSF grant DMS and NIDA grant A P5 DA 75-. MSC 99 subject classiæcations: 62G5, 62G2, 62F25. Keywords and phrases: Survival analysis, proæle likelihood, semiparametric model.

2 the ratio of the hazards converges to unity as time increases. This is in contrast to the proportional hazards model which constrains the ratio of the hazards to be constant with time while the odds ratio tends to zero or inænity. Bennett è983aè suggests the use of this model to demonstrate an eæective cure. In this case the mortality rate of a disease group would approach the mortality ofacontrol group as time progresses. In a similar fashion, the morbidity rate for a group halting a deleterious habit, such as smoking, might be compared with the morbidity rate of similar never-smokers; this model can be used to demonstrate that the morbidity rates converge with time. When G is strictly increasing, the proportional odds model is a linear transformation model; that is, for T the survival time, GèT è=,z T æ+ log æ; where log æ is distributed according to a standard logistic distribution. If instead, æ is distributed as a standard exponential then the proportional hazards model results. To allow for unobservable heterogeneity in a proportional hazards model, one may put log æ = log æ + log æ 2 where æ ;æ 2 are independent, with æ distributed according to a Gamma distribution with mean one and variance,, and æ 2 following a standard exponential distribution. The term log æ represents unobserved heterogeneity. This is the gamma frailty model of Clayton and Cuzick è985è. If the variance,, is set to one then log æ + log æ 2 has a standard logistic distribution, thus the proportional odds model results. So the proportional odds model may be considered an alternative to the proportional hazards model within the class of transformation models, and it may also be considered as a proportional hazards model with unobservable heterogeneity. More results on transformation models can be found in Bickel, Klassen, Ritov and Wellner è993è. Consideration of the linear transformation model lead Cheng et al. è995è to estimate æ by an estimating equation. Cheng et al. show that the resulting ^æ is consistent and asymptotically normal. Both Dabrowska & Doksum è988è and Wu è995è provide estimators in the two sample problem. Cuzick è988è, in considering estimation in the linear transformation model, proposes a method based on ranks, and Pettitt è984è, minimizes a marginal likelihood of the ranks in order to estimate the regression coeæcient. Shen è995è, Huang and Rossini è995è and Rossini and Tsiatis è996è use sieve maximum likelihood, for right censored failure times, interval censored failure times and current status data respectively. Bennett's estimator of æ is the maximum proæle likelihood estimator of æ èg is proæled outè. We demonstrate that the proæle likelihood for æ can be treated much as a parametric likelihood for æ. In particular, the maximum proæle likelihood estimator of æ is consistent, asymptotically normal and eæcient. Diæerentiation of the proæle likelihood yields consistent estimators of the eæcient information matrix. Additionally, a proæle likelihood ratio statistic can be compared to percentiles of the chi-squared distribution to produce asymptotic hypothesis tests of the appropriate size. Section 2 contains a description of the asymptotic 2

3 results and Section 3 gives a numerical comparison of the maximum proæle likelihood estimator with the estimator of Cheng et al. è995è. For the convenience of the reader all of the proofs are in Section Results As in Bennett è983aè, it is convenient to use a slight reparametrization, Hètè = e Gètè, so that H is the baseline odds of failure. The parameter, H, is a nondecreasing, rightcontinuous function with left-hand limits, from the positive real line to the positive real line, and Hèè =. In terms of the reparametrization, S Z ètè = e,zt æ Hètè+e,ZT æ : Wewant to allow S Z to be either a continuous or discrete or possibly a mixed distribution. Specify the conditional probability density with respect to the sum of the Lebesgue and the counting measure, by e,zt æ èhètè+e,ztæ èèhèt,è+e,ztæ è hètè; where Hèt,è is the left-hand limit of H at t. If H is absolutely continuous, then h is the derivative of H; if H is discrete, then hètè = æhètè = Hètè, Hèt,è. Bennett è983aè assumes that S Z is continuous, so that the denominator reduces to the square of Hètè + e,ztæ. We consider the use of this model for right censored data. That is, the observations are n i.i.d. copies of X =èy =T^C; æ;zè, where T denotes the failure time, and C is a censoring time. Conditional on Z, T is independent of C. The censoring indicator, æ, is one if T C and zero otherwise. Then the contribution of X is è! æ è!,æ e,ztæ è, F C èy,jzèè p H;æ èxè = èhèyè+e,ztæ èèhèy,è+e,ztæ è hèyè e,zt æ Hèyè+e,zT æ f Cèyjzè f Z èzè; where F Z is the marginal distribution of Z, F C is the conditional distribution of C given Z, and lowercase letters denote the respective densities. If H is known to be absolutely continuous, then, as in the case of density estimation, there is no maximizer of the likelihood. However, if H is known to be discrete then a maximizer exists èas is proved belowè. To entertain a maximum likelihood estimator, we take the contribution to the likelihood for one observation to be, likèx; H; æè = è! æ è e,zt æ èhèyè+e,ztæ èèhèy,è+e,ztæ è æhèyè 3 e,zt æ HèYè+e,ZT æ!,æ ;

4 where we have replaced h by æh. An alternative approach is to use a sieve; see Shen è995è and Huang and Rossini è995è for examples using the proportional odds model. The maximum likelihood estimator of H will be a nondecreasing step function with steps at the observed failure times. The proæle log likelihood for æ is given by Prlik n èæè = P n i= log likèx i; ^Hæ ;æè, where ^H æ maximizes the log likelihood for a æxed æ. The maximum proæle likelihood estimator, ^æ, maximizes Prlik n èæè. This estimator is a function of the survival times only through their ranks. To see this, order the observed survival times from smallest to largest so that t èè ét è2è é::: ét èkè, where k is the number of unique observed survival times. Assign the ranks through k to these times. Censored times are assigned the rank corresponding to the largest t èiè less than or equal to the censored time. The proæle likelihood is the same, whether we use the Y i 's, or replace them by their ranks. For a æxed æ, the log likelihood evaluated at ^Hæ is maximal. The log likelihood with Y i 's replaced by their ranks can attain the same maximum by setting æ ^Hæ èi j è equal to æ ^Hæ èy j è, where i j is the rank of Y j. The same argument can be made in the opposite direction; hence the proæle likelihood is invariant under any rank preserving transformation, and subsequently ^æ is a function of the Y i 's only through their ranks. In the theorems below we make the following assumptions. For a ænite time point, assume P ëc ë =PëC=ëé. That is, the study ends at time, and any remaining live individuals are considered censored at time. Also assume that, on the average, some individuals are at risk at time, that is P ët éëé. Finally, for any possible covariate pattern, the chance of observing a survival time should be positive, i.e. P ët CjZë é almost surely under F Z. In order for æ to be identiæable, we need to assume that the covariance matrix of Z is positive deænite. Two technical conditions èthat should not be necessaryè are that æ is known to lie in a compact set, say B, and that the support of Z is bounded. Since the maximum proæle likelihood estimator of æ is also the æ-coordinate of the maximum likelihood estimator è ^H; ^æè, the existence and consistency results can be expressed in terms of the joint maximum likelihood estimators. THEOREM 2.. èexistenceè. There exists a pair èh; æè that maximizes Q n i= likèx i; H; æè over the set of nondecreasing functions H with Hèè = by B. If we constrain Hèè to be bounded by a constant, then the continuity of the likelihood in H and æ can easily be seen to imply the existence of the maximum likelihood estimators. As a result, the proof of è2.è needs to show only that the likelihood evaluated at any sequence H m with H m èè diverging to inænity as m increases will not approach the maximum value of the likelihood. This proof is omitted, as it is similar to the proof that the estimator of Hèè does not diverge as n increases; this is done in Theorem 2.2. Denote the supremum norm on the interval ë;ëbykæk and the Euclidean norm by kæk 2. 4

5 THEOREM 2.2. èconsistency and Asymptotic Normalityè. The maximum likelihood estimator is consistent; k ^H,H k and k ^æ,æ k 2 converge almost surely to zero as n 7!. Additionally if æ is in the interior of B, then p nk ^H, H k is bounded in probability and p nè ^æ, æ è converges in distribution to a p-variate normal distribution with mean zero and eæcient variance, æ,. In the estimation, the parameter space for H is not assumed to be bounded; that is, the parameter space for H is noncompact. Additionally the extension of the likelihood used here will not exist for a true continuous H. This combination makes the consistency proof for maximum likelihood estimators appear problematic. However one may use ideas developed in Murphy è994è and extended by Parner è996aè to show that the form of the likelihood forces the estimator, ^H to be bounded and that in the limit, the extension of the likelihood does perform as a likelihood. To derive the asymptotic distribution of the estimators we must verify that an analog to the information matrix ènow an operatorè is continuously invertiable and that the score equations are asymptotically normal. In the latter veriæcation we use results from empirical process theory. In the proof of the preceding theorem, we give an asymptotic normality result for p nè ^H, H è considered as a functional on the space of uniformly bounded functions of uniformly bounded variation. This implies, for example, that the asymptotic distribution of p nè ^Hètè, H ètèè is normal. It turns out that, in general, the eæcientvariance, æ, èsee è4.6è, è4.7è and è4.9èè, is not expressible in an explicit form. However, this will not be a problem as we will still be able to consistently estimate æ. Just as one may use the second derivative of the log likelihood to estimate the information matrix in parametric model, one may expect that the second derivative of the proæle log likelihood may be used to estimate the information matrix for æ. However, since ^Hæ is not an explicit function of æ, we are unable to diæerentiate the proæle log likelihood explicitly in æ to form an estimator of æ. Instead, we numerically diæerentiate this function, using forward ænite diæerences for the oæ-diagonal elements, and a combination of a forward and backward ænite diæerences for the diagonal elements. Let h ;:::;h p be random variables, all converging in probability to zero as n increases. Denote the p-vector with a one in the ith location and zeros elsewhere by e i. THEOREM 2.3. èestimation of the Standard Errorsè. For i 6= j let,^æ ij = Prlik n è nh i h ^æ + h i e i + h j e j è, Prlik n è ^æ + h i e i è, Prlik n è ^æ + h j e j è + Prlik n è ^æè j and put,^æ ii = nh 2 Prlik n è ^æ + h i e i è, 2Prlik n è ^æè + Prlik n è ^æ, h i e i è : i If, for each i; j, h i P! and both h i =h j and è p nh i è, are bounded in probabilityasn!, then ^æ ij converges in probability to the èi; jèth component ofæ. 5

6 In the simulations, we use h i = maxèj ^æ i j; è æ signè ^æ i è= p n or h i = signè ^æ i è= p n. The proof of this theorem is given in Murphy and van der Vaart è996cè. Other, more complex, numerical methods could be considered. One possibility is to æt a higher order spline to the proæle log likelihood, and to diæerentiate the spline. However the simulations below demonstrate that this simple approach works well. In the following theorem we verify the use of the proæle likelihood ratio statistic for hypothesis testing and for obtaining, via inversion, conædence intervals. For example, consider a test of H : æ j = æ j versus H : æ j 6= æ j. Denote the maximum likelihood estimator of èh; æè under the null hypothesis by è^h ;^æ è, ie., ^H ^æ is ^H. THEOREM 2.4. èlikelihood Ratio Inferenceè. Under H : æ j = æ j the proæle likelihood ratio statistic, lrt n èæ j è=2 Prlik n è ^æè, Prlik n è ^æ è has an asymptotic chi-squared distribution with one degree of freedom. The proof of the above theorem can be found in Murphy and van der Vaart è996bè and a general discussion of results of this type is contained in Murphy and van der Vaart è996aè. 3. Computation and Simulation Results The likelihood can be written as è ny i=!è e,zt i æ HèY i è+e,zt i æ! æi æhèy i è HèY i,è+e,zt i æ : The presence of the terms æhèy i è forces the estimator of H to have positive jumps at the observed failure times. Further considerations èsee e.g. the representation for ^H belowè show that ^H does not have jumps at any other points. Thus, the number of parameters is p plus the number of observed failure times. If the largest Y,sayY ènè, is an observed failure time and there are no censorings at this time, then æhèy ènè è appears in the likelihood only through the term æhèy ènè è=, HèY ènè è+e,zt ènè ææ. This term is increasing in æhèy ènè è èwhen keeping HèY ènè,è æxedè, and maximally one for æ ^HèYènè è=. By our stipulation that P èt é è é, the largest Y will be censored at with probability tending to one. Hence in our proofs we need not worry about this case. In practice the largest Y may be an observed failure time, and then the likelihood is given by ny i= Y i éy ènè è!è! e,zt i æ æi æhèy i è HèY i è+e,zt i æ HèY i,è+e,zt i æ 6 ny i= Y i =Y ènè e,zt i æ C HèY ènè,è+e æ æ i A:,ZT i

7 We see that all observed failure times equal to the largest Y enter the likelihood as if they were censored failure times. So when the largest Y is an observed failure time, we set æ ^HèYènè è equal to positive inænity, and for estimation of the remaining jumps of H we act in the computations as if all observed failure times equal to the largest Y are censored. In the computations the number of parameters is p + k, where k is the number of observed failure times not equal to the largest Y. To maximize the log likelihood, we use the IMSL routine, UMIAH, which employs a modiæcation of the Newton-Raphson algorithm. This routine requires both the speciæcation of the gradient and the Hessian. The gradient is a p + k dimensional vector found by diæerentiating the log likelihood with respect to the p-dimensional vector æ and with respect to the k jumps, æh. Similarly, the Hessian is a p + k æ p + k matrix. In a problem with two covariates, estimation of the parameters requires one maximization, estimation of the standard errors requires æve maximizations, and a likelihood ratio test requires an additional maximization. However, the routine works quite fast as can be seen from the CPU times in Table. These are the average times to simulate a sample, calculate the estimators and their standard errors, and to perform a likelihood ratio test for the ærst regression coeæcient. In the simulations we compared the proæle likelihood approach with the estimating equation approach of Cheng et al. They allow for a variety of weight functions. It was our experience that the method using the suggested optimal weight function occasionally lead to computational diæculties. When there were no diæculties, the results were virtually identical to using a weight function of one. As a result we used Cheng et al.'s estimator based onaweight function of in the following comparisions. We considered three sample sizes, 5, and 2, along with two levels of censoring, è and 2è censoring. There are two covariates: the ærst covariate is a Bernoulliè.5è variable and the second covariate is either a uniform, standard exponential or gamma variable. The gamma variable is standardized so as to have mean and variance equal to one but skewness equal to four. This is twice the skewness of a standard exponential which has mean and variance equal to one but skewness equal to two. When the censoring is independent, the failure times are censored at a æxed time point. Furthermore, we considered two types of dependent censoring. In the ærst type of dependent censoring, the censoring varied by the covariate pattern, but was at a æxed time point corresponding to è or 2è censoring, conditionally on the covariate pattern. In the second type of dependent censoring, the failures with the ærst covariate equal to zero were not censored, and the failures with the ærst covariate equal to one were censored at a æxed time which corresponded to 2è censoring. Two possibilities were considered for the function H: Hètè =tand Hètè =t 2. The two regression coeæcients were set to either or. To compare the two methods, we calculated the mean square errors of the estimators ècheng et al.'s estimator and the proæle estimatorè, and constructed Wald 95è conædence 7

8 intervals for the regression coeæcients. In general, the mean square errors of the estimators of æ are comparable. It may be that the Cheng et al. estimator is eæcient for this combination of simulated response distribution and covariate distribution. For the estimators of æ 2, the results depend on the distribution of the covariate. For a uniformly distributed covariate the estimators were comparable, whereas for an exponential distribution the proæle likelihood estimators appears to be between è and 3è more eæcient. Both methods produce conædence intervals with slightly lower conædence than 95è. However, the conædence level of the Wald interval based on the proæle estimator is rarely signiæcantly diæerent from 95è, whereas the conædence of the Wald interval based on Cheng et al.'s estimator is more often signiæcantly diæerent from 95è. Tables and 2 give the results of six and nine representative simulations, respectively. Each simulation is composed of samples, ^æ =è;è and Hètè =t. Our example is from the Veterans Administration lung cancer trial èprentice, 973è. This data set has been analyzed by many authors; in particular, Bennett è983abè, Pettitt è984è and Cheng et al. è995è æt a proportional odds model. All of these authors use the subgroup of 97 patients with no prior therapy. The response is patient survival time and the four covariates are performance status èpsè and a factor with four levels, èlarge, adeno, small and squamous tumor typesè. Cheng et al. present a table comparing their estimates and estimated standard errors with the other two methods. Estimates, standard errors and likelihood ratio test p-values for the proæle method are given in Table 3. The likelihood ratio tests are for the null hypothesis that the parameter is zero. Figure gives a contour plot of 2 times the proæle log likelihood ratio for the coeæcient of small èvs. large tumor typeè and PS. Each contour corresponds to a constant value of 2 times the log likelihood, maximized over the entire parameter space minus 2 times the log likelihood, maximized over H and the two remaining regression coeæcients. The dark contour is at 6, the 95th percentile of a chi-squared on two degrees of freedom. As a result, the collection of covariate values corresponding to the interior of the dark contour forms a 95è conædence region for small tumor type and PS. Figure 2 gives a similar plot for adeno èvs. large tumor typeè and PS. Both plots illustrate the degree to which the data indicates that the parameters are nonzero. Additionally both plots lend support to the asymptotic normality results given in the last section. 8

9 4. Proofs We denote expectation with respect to the empirical distribution of the data by Pn and denote expectation with respect to the true underlying distribution of the data by P H ;æ or, more brieæy, byp. In general, for a function g of the data, X, and estimators, è ^H; ^æè, PnëgèX; H; æèë evaluated at èh; æè = è ^H; ^æè is written as P nëgèx; ^H; ^æèë and likewise for expectation with respect to P. The supremum norm on the interval ë;ë is denoted by jj æ jj. è4:è We ærst show that the maximum likelihood estimator, ^H, satisæes the equation, ^Hètè = Z t W n èu; ^H; ^æè dg n èuè; where è=w n èèuè and G n èuè are nondecreasing functions in u deæned by W n èu; H; æè =Pn IfY ug HèYè+e,ZT æ + æify éug HèY,è+e,ZT æ ; G n èuè=pnæify ug: The equations è4.è are a reexpression of the likelihood equations for H. To derive these equations, deæne a path through ^H, indexed by æ, as dhæ ètè = è + æh ètèèd ^Hètè, where h is an arbitrary nonnegative bounded function. Since the log likelihood is maximized at è ^H; ^æè over the whole model, it is maximized at æ = when evaluated on the submodel given by èh æ ; ^æè. The derivative ofthe log likelihood with respect to æ evaluated at æ = yields a score function for H in the ëdirection" h, given by R Y `Hæ èxèëh ë=æh èy è, h dh ær Y, HèY è+e,zt æ, h dh HèY,è+e,ZT æ : The preceding argument shows that Pn` ^H;^æèXèëh ë = for all h. Putting h èuè =Ifu tgand changing the order of integration results in è4.è. In a similar fashion, deæne a path through ^æ, indexed by æ, asæ æ =^æ+æh 2, where h 2 is a æxed vector in R p. Diæerentiation of the log likelihood with respect to æ and evaluation at æ = yields the score function for æ in the direction h 2, h T 2 `2HæèXè =,h T 2 è, Z e,zt æ HèYè+e,ZT æ, æe,z HèY,è+e,ZT æ If ^æ belongs to the interior of B, then Pn`2 ^H;^æèXè =. We shall repeatedly use the following lemma, taken from Chapter 2. of van der Vaart and Wellner è996è. The general deænition of a Donsker class can be found in this reference as well. However, we shall need only the result that the class BV M of all functions f:ë;ë! R that are uniformly bounded by a constant M and are of variation bounded by M is Donsker èfor each æxed M é è. T æ We note that every Donsker class F with integrable envelope function x 7! sup f jfèxèj, in particular a uniformly bounded class, is Glivenko-Cantelli. This means that sup f2f jpnf, Pfj!, almost surely. 9! :

10 LEMMA 4.. For a Lipschitz function : R k 7! R and classes F i of functions f i : X 7! R let èf ;:::;F k è, denote the set of all functions x 7!, f èxè;:::;f k èxè æ as f i ranges over F i, for each i. If each class F i is Donsker with integrable envelope function, then the class èf ;:::;F k è is Donsker, provided that it consists of square-integrable functions. One use of Lemma 4. is in the analysis of random step functions of the form Z æ where H n and æ n could be random. BV M W n èu; H n ;æ n è dg nèuè; The functions u 7! W n èu; H; æè are contained in for some M and uniformly bounded away from zero as H ranges over the set H of nonnegative, nondecreasing elements of BV M and æ n ranges over B. Suppose that H n is contained in H and converges uniformly to a function H æ almost surely and æ n converges to an element æ æ of B almost surely. Then two applications of Lemma 4. yield and æ Z æ W n èu; H n ;æ n è dèg nèuè,gèuèè sup u2ë;ë;h2h;æ2b æ æ as! æ æ Wn èu; H; æè, Wèu; H; æè as! ; where W èu; H; æè and G are the expectations of W n èu; H; æè and G n, respectively, evaluated at H and æ. The above displays, combined with the dominated convergence theorem, yield è4:2è æ Z æ W n èu; H n ;æ n è dg nèuè, Note that H ètè = R t Wèu;H ;æ è, dgèuè. LEMMA 4.2. Z æ W èu; H æ ;æ æ è dgèuè æ ææ as! : Both H and æ are identiæable. If H is absolutely continuous with respect to H, Hèè = and p H;æ èxè =p H ;æ èxè almost everywhere under P H ;æ, then H = H on ë;ë and æ = æ. Proof. Considering the densities p H;æ = p H ;æ on æ =,we see that è4:3è dh èyè =e zt èæ,æ è dh, æ, æ Hèyè+e,z T æ Hèy,è+e,zT æ, H èyè+e,zt æ æ, æ; H èy,è+e,zt æ for almost every èy; zè such that PèC yj Z = zè é. If H has a jump at its left endpoint t æ = infft: H ètè é g, then we can insert y = t æ and obtain that for F Z -almost every z. T æ æhèt æ è æh èt æ è = æhètæ è+e,z æh èt æ ; è+e,zt æ This implies that e zt èæ,æè = æhèt æ è=æh èt æ è and hence that the variable z T èæ, æè is degenerate. Thus, æ = æ under our assumption that cov Z is nondegenerate.

11 If H does not jump at t æ, then there exists a sequence, fy m g m converging to t æ for which è4.3è holds for almost every z in the set A m = æ z:pèc y m jz = zè é æ. Now, A m " A = æ z:pèc é t æ jz = zè é æ, which has probability under F Z by assumption. The limit as m! of the right hand side of è4.3è when evaluated at y m is e zt èæ,æè. Therefore the left-hand side must converge as well and the limit is independent of z. Again we obtain that æ = æ. Now onæ= and the set of z for which P èc jzè é, èhètè+e,zt æ èèhèt,è+e,z T æ è èdh=dh èètè = èh ètè+e,zt æ èèh èt,è+e,zt æ è ; for almost every t 2 ë;ë under H. By assumption, the set of z for which this holds has positive measure and hence is nonempty. Integrating both sides from to u, yields Hèuè+e,zT æ = for all u. Therefore Hèuè =H èuè for all u. Proof of Consistency in Theorem 2.2. H èuè+e,zt æ First we show that the sequence ^Hèè is bounded, almost surely. Then, using è4.è we show that the sequence è ^H; ^æè is relatively compact. Finally, using the above lemma on identiæability, we prove that any convergent subsequence of è ^H; ^æèmust converge to èh ;æ è. Deæne a random step function ~Hètè = Z t W n èu;h ;æ è dg nèuè: Note that ~ H relates to Gn in a similar manner as ^H, but with Wn evaluated at èh ;æ è, rather than at è ^H; ^æè. Since H is bounded on ë;ë and æ is restricted to a compact set, the functions u 7! W n èu; H ;æ è are uniformly bounded away from zero and inænity and of uniformly bounded variation. By the argument in the beginning of the appendix, we see that k ~ H, H k converges almost surely to zero. Since è ^H; ^æè maximizes the log likelihood, h i è4:4è log likèx; ^H; ^æè,log likèx; H;æ ~ è : Pn The idea is to show that if ^Hèè diverges to inænity on a set of positive measure then the above diverges to minus inænity resulting in a contradiction. This step is omitted; see Murphy, Rossini and van der Vaart è996bè for the details. Thus the sequence ^Hèè is almost surely bounded as n!. For æ ranging over B and for H and H 2 ranging over the set of bounded èby a constant Mè monotone step functions on ë;ë all of which satisfy y 7! æh =æh 2 èyè è with variation bounded by M, 2è bounded above by M, and 3è bounded below by a constant mé, the

12 set of functions x 7! log likèx; H ;æè,log likèx; H 2 ;æ è is Glivenko-Cantelli. Since ^H and ~H satisfy all these constraints as n!, almost surely, è4.4è implies that è4:5è P hlog likèx; ^H; ^æè, log likèx; ~ H;æ è i,oèè; Now æx an! in the underlying probability space such that ^Hèè is bounded by a constant M for every n, such that G n f, Gf! uniformly in the functions of variation a:s:: bounded by M and such that the preceding display is valid for this!. u 7! W n èu; ^H; ^æè are decreasing and bounded away from zero and inænity. The functions By Helly's lemma and the compactness of B, every subsequence of n has a further subsequence along which ^æ! æ æ for some æ æ and W n èu; ^H; ^æè! W æ èuè for every u and some monotone function W æ. Then, uniformly in t, ^Hètè = Z t W n èu; ^H; ^æè dg n èuè = by the dominated convergence theorem.! Z t Z t dgèuè+oèè W n èu; ^H; ^æè W æ èuè dgèuè =:Hæ ètè; Hence ^H! H æ uniformly on ë;ë and H æ is absolutely continuous with respect to G and thus also H. Consequently, log likèx; ^H; ^æè, log likèx; ~ H;æ è! log likèx; H æ ;æ æ è,log likèx; H ;æ è; every x: By the dominated convergence theorem and è4.5è, we conclude that the last function has a nonnegative mean under P, which is the Kullback-Leibler divergence of the measures P H æ ;æ æ and P. Hence H æ = H and æ æ = æ by Lemma 4.2. Since every subsequence of n contains a further subsequence for whichè^h; ^æè converges uniformly to èh ;æ è, we have convergence for the entire sequence. Uniform convergence of ^H follows from a lemma stated in Chung è974, pg. 33è, since, for any u a jump point of H,wehaveæ^Hèuè converges to æh èuè ènote that æ ^Hèuè =Wn èu;^h; ^æè, æg n èuèè. Let h be a bounded function of bounded variation and h 2 a p-dimensional vector. For h = èh ;h 2 è, denote the information operator by èhè = è èhè; 2 èhèè where èhè is a bounded function of bounded variation and 2 èhè is a p dimensional vector. The name information operator comes from the fact that P ë`h ;æ èxèëh ë+h T 2`2H ;æ èxèë 2 = R èhèèuèh èuèdh + h T 2 2èhè. The forms are: èhèèuè =h èuèwèu;h ;æ è,p " IfY ug R Y h dh è4:6è h T 2 P IfY ug èh èy è+e,zt æ è 2 + éugr Y, èh èy è+e,zt æ è 2 +æify h dh èh èy,è+e,zt æ è 2 æify éug e èh èy,è+e,zt æ è,zt æ Z 2 2 è +

13 and è4:7è R Y, èh èy è+e,zt æ è 2 + æ h dh èh èy,è+e,zt æ è 2 2 èhè =P "è R Y h dh P! e,zt æ Z H èy è èh èy è+e,zt æ è 2 + æh èy,è e èh èy,è+e,zt æ è,zt æ ZZ T h 2 2 : è + Deæne the space BV to be the set of uniformly bounded functions of bounded variation, equipped with the norm kæk BV, which is deæned as the maximum of the supremum norm and the total variation norm both on the interval ë;ë. We equipped the space BV æ R p with the norm kækequal to the maximum of kæk BV and the Euclidean norm. LEMMA 4.3. The linear operator, :BV æ R p 7! BV æ R p is onto and continuously invertible. Proof. The operator can be written as the sum of A + K of two operators, where Ah = èh W èæ; H ;æ è, ; h 2 è. The operator Aèhè is continuously invertible èwith inverse A, èhè =èh Wèæ;H ;æ è, ; h 2 èè, since W is bounded away from zero. Since we can write in the form AèI +A, Kè, it suæces by Theorem 4.25 in Rudin è973è to show that A, K is compact and that A + K is one-to-one. The ærst is true if K is compact. Consider K. Since a bounded linear operator with ænite-dimensional range is compact, we need only show that the operator K :BV 7! BV, given by K èh èèuè =P " IfY ug R Y h dh éugr Y, èh èy è+e,zt æ è 2 +æify h dh èh èy,è+e,zt æ è 2 is compact. Thus, given a sequence of functions h n with kh n k BV, we must show that there exists a subsequence and an element g 2 BV such that kk h n, gk BV!. Now K is a linear operator with kk h k BV C R jh j dh for every h, and a æxed constant C. Hence, it suæces to show that there exists a subsequence of h n that converges in L èh è. This is an easy consequence of Helly's lemma. We can split h n in its positive and negative variation, and select a subsequence along which both parts converges pointwise. Then h n converges to the diæerence of the limits in L èh èby the dominated convergence theorem. To prove that is one-to-one, we prove that if kèhèk = for an h with a bounded norm then khk =. Suppose that kèhèk =, then R h èhèdh + h T 2 2èhè =, and both èhè and 2 èhè are identically zero. As in the proof of Lemma 4.2, consider two cases, corresponding to æh èt æ è equal to zero or positive, in order to prove that h 2 is zero èin Lemma 4.2 the goal was to prove that æ = æ è. We omit the details, as they are very similar to the method given in Lemma 4.2. Since = èh ; è = P ë`h æ èxèëh ë, we have that when P èc = jzè é, R h dh H èè+e ZT æ =: 3 è

14 Since P èc = jzè é on a set of positive probability, we may use the above along with a proof that h is either nonnegative or nonpositive, to infer that h is almost everywhere èa:e:è equal to zero. This plus the form of equation è4.6è and h 2 = implies that h èuèw èu; H ;æ è is identically equal to zero and thus h is identically equal to zero. To prove that h is either nonnegative or nonpositive, we use èh ; è =. restricting ourselves to the portion of the sample space for which æ = and Z is in a set A of positive probability for which P ëc = jzë é, we have that R T R h èt è= h T, dh èh èt è+e,zt æ è 2 + h dh èh èt,è+e,zt æ è 2 for almost every T 2 ë;ë and Z 2 A. Rearranging terms results in, R T, è! h èt è= h dh + H ètè+e,zt æ : H èt,è+e,zt æ H èt,è+e,zt æ Replace h by a function g which has both right and left hand limits èg = h a:e:h è, given by, gètè = R t, è! h dh H èt,è+e,zt æ + H ètè+e,zt æ H èt,è+e,zt æ for an arbitrary z in A. The function g can be expressed as a sum of integrals with respect to H, gètè = Z t R u, Z gdh t, èh èu,è+e,zt æ è 2dH èuè+ R u, gdh èh èu,è+e,zt æ è 2dH èuè: A proof by contradiction suæces to show that g must be either nonnegative or nonpositive. Since g is a:e: equal to h,wehave that h is a:e: èh è nonnegative or nonpositive. Write the inverse of as ~. So ~èhè =è~ èhè;~ 2 èhèè with ærst component a function in BV and second component a p dimensional vector. Denote the p dimensional vector with a one in the i location and zeros elsewhere by e i. Deæne the p æ p matrix, è4:8è æ, =è~ 2 è;e è;:::;~ 2 è;e p èè : The matrix is symmetric and Lemma 4.3 implies that h T 2 æ, h 2 = h T 2 ~ 2 è;h 2 è é for h 2 nonzero; hence æ, is invertible. By LEMMA 4.4. The eæcient score function for the estimation of æ is the vector ~`èxè =`2H æ èxè,`h æ èxèëg æ ë where `H æ is applied component wise to the p dimensional vector of functions g æ and, g æ =,æ ~ è;e è. ~ è;e p è C A: 4

15 Proof. We must show that ~` is orthogonal to the score for H, given by `H æ èxèëg ë for any bounded function g. Consider e T i æ, P ~``H æ èxèëg ë which is equal to, P æ e T i æ,`2h æ èxè, `H æ èxèëe T i æ, g æ ë æ `H æ èxèëg ë Now e T i æ, =~ 2 è;e i è T so the above is equal to P æ ~2 è;e i è T`2H æ èxè+`h æ èxèë~ è;e i èë æ `H æ èxèëg ë: This is equal to R g è~è;e i èèdh =. Proof of Asymptotic Normality in Theorem 2.2. In the following, ëuniform boundedness" refers to the norm kæk on BV æ R p equal to the maximum of kæk BV and the Euclidean norm. Let G n = p nèpn, P è be the empirical process of the observations. è4:9è We prove that p n Z h dè ^H, H è+h T 2è^æ,æ è è~ 2 èhèè T `2H æ èxè+`h æ èxèë~ èhèë =Gn + o P èè; where the remainder term converges to zero in probability uniformly over h 2 BV M æ R p. Since Lemma 4.3 implies that ~èhè is uniformly bounded for all h 2 BV æ R p which are uniformly bounded, setting h = results in p nh T 2 è ^æ, æ è=gn ~ 2 è;h 2 è T`2H æ èxè+`h æ èxèë~ è;h 2 èë + o P èè; which is asymptotically normal with mean zero and variance h T 2 ~ 2 è;h 2 è = h T 2æ, h 2 for each h 2 in R p. Indeed, we may set h 2 equal to each of the unit vectors, e i to get that p nè ^æ, æ è = G næ,=2 ~`èxè +op èè, where ~` is the eæcient score function as given in Lemma 4.4. Alternatively, wemay put h 2 = in è4.9è to get p nk ^H, H k p n = sup sup kh k BV æ æg n kh k BV Z æ h dè ^H, H è æ ~ 2 èh ; è T `2H æ èxè+`h æ èxèë~ èh ; èëæ ææ + op èè; which is bounded in probability, since the integrand with h varying over BV belongs to a Donsker class. Setting h æ 2 =~ 2 èhè and h æ =~ èhè, we have that =Pn èh æ 2è T`2^H^æèXè+`^H^æèXèëh æ ë =P èh æ 2è T`2H æ èxè+`h æ èxèëh æ ë 5 :

16 Furthermore, simple algebra using the consistency of è ^H; ^æè, the assumption that Z has bounded support and the fact that h = èh æ è shows that è4:è P èh æ 2è T `2 ^H ^æèxè+`^h^æèxèëh æ ë =,, P Combination of the three equations yields that p n Z Z èh æ 2è T `2H æ èxè+`h æ èxèëh æ ë h dè ^H, H è, h T 2 è ^æ, æ è+o P, k^h,h k 2 +k^æ,æ k 2 2æ : h dè ^H, H è+h T 2è^æ,æ è =Gn èh æ 2è T`2^H^æèXè+`^H^æèXèëh æ ë + p no P, k^h,h k 2 +k^æ,æ k 2 2æ : The set F of functions of the type x 7! h T 2 `2Hæèxè+`Hæ èxèëh ë, with h 2 and æ varying in a compact set in R p, h varying over BV M, and H varying in the set of nonnegative nondecreasing functions with Hèè 2H èè, is Donsker and uniformly bounded. function x 7! èh æ 2è T `2 ^H ^æèxè+`^h^æèxèëh æ ë belongs to F with probability converging to one. Furthermore, è ^H; ^æ;h;h ;æ ;hè=p h T 2`2^H^æèXè+`^H^æèXèëh ë, h T 2 `2H æ èxè, `H æ èxèëh ë converges to zero almost surely, uniformly over bounded h, by the consistency of è ^H; ^æè and the dominated convergence theorem. Hence, by the asymptotic uniform equicontinuity of the empirical process, p Z n h dè ^H, H è+h T 2è^æ,æ è =Gn èh æ 2è T`2H æ èxè+`h æ èxèëh æ ë + o P èè + p no P, k^h,h k +k^æ,æ k 2 æ : Take the norm left and right, and conclude ærst that p nk ^H, H k and p nk ^æ, æ k 2 are bounded in probability, and next that the remainder term in the display iso P èè. The 2 6

17 Table : Average CPU Time for the Proæle Estimator and the Ratio of Mean Square Errors Design CPU Time èproæleè a MSE èproæleèèmse èchengè Uniform Z 2, n = and, æ 33. è independent censoring æ 2. Uniform Z 2, n = and, æ è independent censoring æ 2.4 Uniform Z 2, n = and, æ 34. è dependent censoringèè æ 2. Uniform Z 2, n = and, æ 38.2 è dependent censoringè2è æ 2.3 Exponential Z 2, n = and, æ è dependent censoringèè æ 2.87 Exponential Z 2, n = and, æ è dependent censoringè2è æ 2.9 a Average CPU Time in Seconds on a Sun Sparc Station, 32mb RAM, 264mb swap space 7

18 Table 2: Error rates of Likelihood Ratio Test a and Wald Conædence Intervals èproæle, Cheng et al. è995èè Design LRT èproæleè 95è CI èproæleè 95è CI èchengè Uniform Z 2, n = and, æ è independent censoring æ æ Uniform Z 2, n = and, æ.67 æ.64 æ.68 æ 2è independent censoring æ 2.66 æ.74 æ Uniform Z 2, n = and, æ è dependent censoringèè æ Uniform Z 2, n = and, æ è dependent censoringè2è æ Exponential Z 2, n = and, æ è dependent censoringèè æ æ Exponential Z 2, n = and, æ è dependent censoringè2è æ 2.65 æ.76 æ Uniform Z 2, n = 2 and, æ è dependent censoringè2è æ Exponential Z 2, n = 2 and, æ æ è dependent censoringè2è æ æ Gamma Z 2, n = 2 and, æ è dependent censoringè2è æ æ a Type I error is.5 æ Error rates signiæcantly diæerent from.5 8

19 Table 3: Analysis of the Veteran Administration Lung Cancer Data Covariate Estimator Estimated Standard Error LRT p-value PS æ,8 adeno vs. large small vs. large squamous vs. large

20 Figure. 2*Profile Log Likelihood Ratio for PS and Small Tumor Type 2 PS Small vs Large Tumor Type

21 Figure 2: 2*Profile Log Likelihood Ratio for PS and Adeno Tumor Type 2 PS Adeno vs Large Tumor Type

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