generalized martingale-residual processes for goodness-of-fit inference in cox's type regression...

7/31/2019 Generalized martingale-residual processes for goodness-of-fit inference in Cox's type regression models

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The Annals of Statistics1997, Vol. 25, No. 2, 683714

GENERALIZED MARTINGALE-RESIDUAL PROCESSES

FOR GOODNESS-OF-FIT INFERENCE IN COXS

TYPE REGRESSION MODELS

BY LESZEK MARZEC AND PAWE MARZEC

University of Wrocaw

In the paper a general class of stochastic processes based on the sums

of weighted martingale-transform residuals for goodness-of-fit inference ingeneral Coxs type regression models is studied. Their form makes the

inference robust to covariate outliers. A weak convergence result for such

processes is obtained giving the possibility of establishing the randomness

of their graphs together with the construction of the formal 2-type

goodness-of-fit tests. By using the Khmaladze innovation approach, a

modified version of the initial class of processes is also defined. Weak

convergence results for the processes are derived. This leads to the main

a pp li ca ti on w h ic h c on ce rn s t he f or m al c on st ru ct io n o f t he

KolmogorovSmirnov and Cramervon Mises-type goodness-of-fit tests.This is done within the general situation considered.

1. Introduction. We consider a general class of stochastic processes for

goodness-of-fit examination with the general Cox regression model of .Andersen and Gill 1982 and other semiparametric regression models. In the

.inference we use the robust version of Coxs 1975 estimatorthe maximum . .weighted partial likelihood estimator MWPLE of Bednarski 1993 . It should

. .be mentioned that Bednarskis 1993 idea of modifying the Cox 1975 partial . .likelihood function differs from that of Lin 1991 and Sasieni 1993a, b . In

this paper, we proceed with the idea of defining the weighted martingale-

transform residuals which can be treated as the robust version of Barlow and . w .Prentice 1988 residuals see also Fleming and Harrington 1991 ,

. .Henderson and Milner 1991 , Lin, Wei and Ying 1993 , Segal, James, .xFrench and Mallai 1995 .

Our general class of processes for checking the adequacy of the model is

based on the sums of weighted martingale-transform residuals. The classw . .contains, as special cases, the scores Schoenfeld 1980 , Arjas 1988 and Lin,

.xWei and Ying 1993 for one-parameter fixed-type processes. These previ-

ously considered processes have been mainly used to propose diagnosticw . .x 2graphical methods Arjas 1988 , Lin, Wei and Ying 1993 , the -type test

w .x .Schoenfeld 1980 or the KolmogorovSmirnov type test for the Cox 1972w . .model with a single time-independent covariate Wei 1984 , Haara 1987 ,

.xThernau, Grambsch and Fleming 1990 .

Received December 1995; revised June 1996.

AMS 1991 subject classifications. Primary 62F12; secondary 62G10, 62G20.

Key words and phrases. Censoring, counting process, Cox-regression, goodness-of-fit, martin-gale, residual, innovation process, weak convergence, robust inference, survival analysis.

683


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L. MARZEC AND P. MARZEC684

In this paper our main purpose is to provide the formal constructions of .goodness-of-fit tests of the KolmogorovSmirnov KS and Cramervon Mises

.CM type under a relatively weak set of restrictions concerning the generalsemiparametric models considered. In particular, repeated occurrences of the

.events of interest e.g., failures are possible. The asymptotic behavior ofthe processes based on sums of weighted martingale-transform residuals is

established in Theorem 3.1 of Section 3.1. The general structure of the

asymptotic limit corresponds to a zero-mean continuous Gaussian process

which only in special cases appears to be a transformed Brownian motion or

Brownian bridge, allowing then for the direct constructions of KS and CM-type

tests. In the general situation, however, the Gaussian limit process appears

to be neither a martingale nor a process with well-known properties. It .should be noted that for situations of such type, Lin, Wei and Ying 1993

proposed some numerical methods based on the given data and simulated

normal samples to derive p-values of tests.

In the present paper we propose an alternative approach leading to the

formal constructions of the critical regions of KS and CM-type goodness-of-fit

tests. To achieve this, we introduce in the second part of Section 3.1 the

appropriately modified version of the initial processes based on weighted

martingale residuals and establish the weak convergence of the latter pro- .cesses to Gaussian martingales Theorem 3.3 . As a result of this, we obtain

the convergence in distribution of the KS and CM-type test statistics to the

well-known functionals of the standard Brownian motion. The behavior of the

considered processes outside initial model specification together with the

power function performance of the tests are investigated in Section 4 . Some

optimality results are also reached. Comparison of some KS and CM-type

tests with respect to existing tests proposed for the Cox proportional hazards

model is provided in Section 5. This is done by using Monte Carlo simula-

tions. Section 6 is devoted to the generalizations of the asymptotic results of

Section 3 with respect to the semiparametric model with general relative risk

form and the multistates semiparametric regression model. All proofs are

provided in Section 7.

2. Notation and preliminaries.

Notation. We use the following notation, where xgR, gR p:n1

mk m1 TQ , x s Y x w Z x , x x Z x exp Z x , . . . . . . .k , l i i i i in

is1

k , ls 0 , 1 , 2 ; kq lF 2,

where m0s 1, m1s , m2s T, m1Zm1sZT;n

x

N x s w Z s , s dN s ; . . . . Hw i i0is1

x

M x sN x y n Q , s s ds; . . . .Hw w 0 , 0 0 00


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COXS TYPE REGRESSION MODELS 685

m2q , x q , x . .2 , 0 1, 0T

a , x s l , x y l , x ; . . .2 5q , x . q , x .0, 0 0, 0m2

Q , x Q , x . .2 , 0 1 , 0Ta , x sL , x y L , x ; . . . 2 5Q , x . Q , x .0, 0 0, 0

x

A , x s a , s q , s s ds; . . . .H 0 , 0 00

x

A , x s a , s dN s rn; . . .H w0

Tq , x q , x q , x . . .1 , 1 1 , 0 0 , 1

, x s y ; . 2q , x . q , x .0, 0 0, 0T

Q , x Q , x Q , x . . .1 , 1 1 , 0 0, 1 , x s y . 2Q , x . Q , x .0, 0 0, 0

T Tb , x s , x l , x ; b , x s , x L , x ; . . . . . .T

b , x s , x L , x ; . . .1 0x

B , x s b , s q , s s ds; . . . .H 0 , 0 00

x

B , x s b , s dN s rn; . . .H w0

m2q , x q , x . .0 , 2 0 , 1

, x s y ; . 2q , x . q , x .0, 0 0, 0m2

Q , x Q , x . .0, 2 0 , 1 , x s y ; . 2Q , x . Q , x .0, 0 0, 0

x

, x s , s q , s s ds; . . . .H 0 , 0 00

x

, x s , s dN s rn; . . .H w0

x s , s q , s s ds; . . . .Hk l k l 0 0, 0 0 0x

x s , s dN s rn; . . .Hk l k l w ww xx ,

, x s , y , x ; , x s , y , x y . . . . . . .

y

. 5 5C stands for the generalized inverse of the matrix C s c ; C si j<


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. w x4Let , FF, PP be a complete probability space and let FF: t g 0, be antincreasing right-continuous family of sub -algebras of FF. Counting pro-

cesses, local martingales, predictable processes, and so on are defined with .reference to these sub -algebras. Following Andersen and Gill 1982 , let

.Ns N , . . . , N , n G 1, be the multivariate counting process defined so that1 nw xN counts failures on the ith subject at times t g 0, . Thus N has compo-i

nents N which are right-continuous step functions, zero at time zero, withijumps of size q1 only, such that no two components jump simultaneously.

.Assume each N to be almost surely finite.i .The general version of Coxs regression model of Andersen and Gill 1982

.postulates that N i s 1, . . . , n has intensity process , that is,i i

t

w x1 M t sN t y s ds, t g 0, . . . .Hi i i0

is a local square integrable martingale, of the form

T w x2 t s Y t exp Z t t , t g 0, . . . . . .i i 0 i 0

Here is a column vector of p unknown regression coefficients, is an0 0 4arbitrary and unspecified baseline hazard function, Y is a predictable 0, 1 -i

valued process indicating that the ith individual is at risk when Y s 1 andi

Z is a p-variate column vector of processes which are assumed to beipredictable and locally bounded.

The MWPLE of is defined as a solution ofw 0

3 U , s 0, . .

where

n1 Q , s .t 0, 1U , t s w Z s , s Z s y dN s , . . . . . H i i i 5' Q , s .4 n . 0 0, 0is1

w xt g 0, .

p w xHere a weight function w: R = 0, R is assumed to be bounded andq

Borel-measurable. We use the estimator in further goodness-of-fit infer-w .ence for Coxs regression model 2 . Our bases are the weighted martingale

residuals of the form

t tw T M t s w Z s , s dN s y w Z s , s Y s exp Z s d s , . . . . . . . . .H Hi i i i i w i w

0 0

w xt g 0, , i s 1, . . . , n,

where

s y1 s s nQ , x dN x . . .Hw 0, 0 w w

0

is a weighted version of the standard estimator for the cumulative baseline . s . w xhazard function s s H x dx, s g 0, .0 0 0


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Now the generalization based on the approach of Barlow and Prentice .1988 leads to the weighted martingale-transform residuals

tw w w x5 t s s dM s , t g 0, , i s 1, . . . , n, . . . .Hi i i

0

where are predictable and locally bounded q-variate processes. Thei w . .weighted sum of , i s 1, . . . , n, of 5 is our main interest. It has thei

form

n1 T . w , s L , s d s . . . Hw w i'n 0 is1n 1 Q , sT . . 1, 0 w

s L , s w Z s , s s y dN s . . . . . . H w i i i' n Q , s0 .is1 0, 0 w

6 .

.The process L , is assumed to be a linear combination, with coefficients0which are random variables, of predictable and locally bounded q-variate

. .processes. If model 2 holds, , will fluctuate randomly around zero;w .we may replace dN by dM , where M is given by 1 , in the definition ofi i i

. . . , . Since under the correct model of 2 , , is a linear combina-0 0 .tion of local martingales, , expresses a balance between the suitable

wrestricted actual count of failures and the corresponding estimated collective

cumulative hazard.

A variety of examples of the processes and L appear in the definition ofi . . , of 6 which are of special interest. Most of them can be adoptedw

. . .from Crowley and Jones 1989, 1990 and Jones 1991 who considered 0, t . .with w' 1 p s 1 for the problem of testing s 0 p s 1 in Coxs regres-

. . < .sion model. Their general proposition for s of the form g Z s FF ,i s i sywhere g is an FF -measurable function, contains as special exampless sy

. . . pq 1 q s sg Z s , s with a Borel-measurable function g: R R ori i . . .. s s Z s yZ s with Z s Y Z rY and some downweighting func-i i i i i

p q .tion : R R . Note that the examples of g of the form g z, s sz, . . 4 p w x . . 4 pg z, s s I z, s g A , A ; R = 0, , g z, s s f z I z F x , x g R , . . .g z, s szf s lead, according to 6 with w' 1, L' 1, to the score process

w . . .xWei 1984 , Haara 1987 , Thernau, Grambsch and Fleming 1990 , Schoen- . . .felds 1980 process, Lin, Wei and Yings 1993 class and Sasienis 1993b

example, respectively. In the situation when has the status of the addi-i . . .tional model covariate, , of 6 with w' 1, L' 1 is the standardw

.score process based on the Cox-type alternative to 2 of the form si T T . w .Y exp Z q , / 0 see Tsatis, Rosner and Tritchler 1985 , Sludi i i 0

.x 4 4 .1991 . Note also that for sI i gJ , J; 1, . . . , n , the process of 6 is ai . wweighted version of the stratified process of Arjas 1988 see also Marzec and

. .xMarzec 1993 , Marzec 1993 . .By considering the weighting process L in the definition of 6 , we provide

on the one hand a possibility for the practitioner to exercise his choice . w .xconcerning the way in which model 2 is to be fitted cf. Sasieni 1993a . On


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the other hand, L can be used as the parameter in optimization problems . .see Section 4 . Following the Crowley and Jones 1989, 1990 examples of L

.that include for p s 1, p s q L' 1 and the Gehan-type weight L s Yrn,i . . .one may consider more general special forms L , s s l s or L , s s

. .l s K , s , G 1, where l is a deterministic function and for a measurable . . ..function W, K , s s Y s W , Z s rn is the predictable and locallyi i

bounded process. Note that to reflect a decreasing willingness to rely on the . . w .model 2 assumption as time progresses, one can put L , s s K , y

.x K , s , gN. .Finally, by considering the weight function w in 6 , we provide a possibil-

wity of making the inference robust to the violations of measured covariates cf. .xBednarski 1993 . In this paper we assume, for simplicity, that the weight . 4function w z, s takes values from 0, 1 . In particular, it corresponds to the

.situation when w z, s censors large values of covariates. Then the resultingestimators and processes should become generally robust to covariate out-

.liers. Obviously, the choice w z, s ' 1 leads to considerations based on theclassical MPLE.

3. Asymptotic properties of residual process. In this section we .investigate the asymptotic performance of the process of 6 together with its

modification useful for constructing the formal KS and CM-type goodness-of-fittests. The process is given in Section 3.1, the modification in Section 3.2.

3.1. Weak convergence results. The following assumptions are used in . .order to establish the weak convergence of the process , of 6 .w

.CONDITION A. H s ds-.0 0

CONDITION B. There exists a neighborhood B of and the functions q ,0 k, lk, l s 0,1,2, k q l F 2 such that

sup Q , s y q , s s o 1 , . . .k , l k , l pgBw xsg 0,

. w x .where q , s are continuous in gB, uniformly in s g 0, , q , arek, l k, lw x . . bounded on B= 0, , q , is bounded away from zero, r q ,0, 0 0, 0

. . . . . w xs s q , s , r q , s s q , s , s g 0, .0, 1 0, 1 0, 2

CONDITION C. There exists ) 0 such that

1Tsup w Z s , s Y s Z s q s I Z s )y Z s 4 . . . . . . 4 .k k k k 0 k k'n k , s

s o 1 . .p


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.Note that in view of 7 the covariance function of the limit process ofTheorem 3.1 has the form

T y1Cov s , t s A , min s, t yB , s , B , t , . . . . . . .0 0 0 0

w xs, t g 0, .

Obviously, Theorem 3.1 establishes the asymptotic randomness in the graph .based on the trajectory of , which can be informally used for modelw

w .xchecking cf. Arjas 1988 . On the other hand, it gives a possibility ofconstructing some formal goodness-of-fit tests. Indeed, by using Theorem 3.1

one can obtain that the statistic

2 , t .w28 K t s , t g 0, . . yT

A , yB , t , B , t . . . .w w w w

has a limiting 2 distribution with one degree of freedom. Note that follow- . .ing the considerations of Schoenfeld 1980 , Hjort 1990 , McKeague and

. . 2Utikal 1991 , Marzec 1993 and using the above theorem, other -typew .xtests may also be constructed see also Li and Doss 1993 . Theorem 3.1

provides also a tool for the formal constructions of KS and CM-type goodness-of-fit tests under some model restrictions.

.First, observe that if B , ' 0 then the limit process of the theorem0corresponds to a time transformed standard Brownian motion W. Then it can

be easily shown that

y1 r2 w xS sA , sup , t : t g 0, , . . 5w w

3r2 9 , t dA , t rA , , . . . .H w w w

0

2 2 , t dA , t rA , . . .H w w w

0

are asymptotically distributed as

1 1 2sup W s , W s ds, W s ds, . . .H H

0 0w xsg 0, 1

. .respectively. Note that B , ' 0 is fulfilled when, for example, Y , Z , 0 i i i . .are independent copies of Y, Z, , say, and and Y, Z are independent

processes. This is the common assumption in randomized clinical trials if whas a status of the added model covariate cf. Tsatis, Rosner and Tritchler

. .x .1985 , Slud 1991 . For other possibilities see Arjas 1988 and Marzec and .

Marzec 1993.

Moreover, both KS and CM-type test statistics can also be . .constructed in the one-parameter case. Given l , ' l , s Z , the0 0 i i


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limit process is a time-transformation of the Brownian bridge W0. Then

1r2 w xsup , t : t g 0, L , , , . . . 5w w w

3r2 , t d , t L , , , . . . .H w w w w

0

22 , t d , t L , , . . . .H w w w w

0

are asymptotically distributed as

1 1 20 0 0sup W s , W s ds, W s ds, . . .H H0 0w xsg 0, 1

respectively. The above construction of the supremum-type test based on the . .score process w' 1 corresponds to the well-known results of Wei 1984 ,

. .Haara 1987 and Thernau, Grambsch and Fleming 1990 obtained in thecase of time independent covariates. Obviously, large values of all mentioned

.test statistics are significant for the rejection of model 2 .

In a general situation, however, the limit process of Theorem 3.1 is

neither a Gaussian martingale nor a process with well-known properties,

which leads to some substantial problems in the direct construction of KS andCM-type tests. Hence in what follows, to omit these difficulties, we will use

.the innovation approach of Khmaladze 1981, 1988, 1993 and propose aformal method for the construction of tests of the above-mentioned type.

Now we make the following assumptions.

CONDITION A.

1lim s ds) 0 and lim y -. . .H 0 0

y 00

CONDITION B. The functions of Condition B satisfy

1sup Q , s y q , s s O , k , l s 0 , 1 , 2 , k q l F 2. . .k , l 0 k , l 0 p /'nw xsg 0, CONDITION C. There exists a constant C such that for each i,

w Z s , s s q Z s F Cw Z s , s . . . . . . .i i i i

. w . .CONDITION D. The matrices , s , s g 0, , , are nonsingular.0 0

CONDITION E. The functions of Condition E satisfy

1

sup L , s y l , s s O.

. .0 0 p /'nw xsg 0,


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5 .5 . . xCONDITION E. lim l , s r ys - for some g 0, 1 .0sy

. .REMARK. Note that if Y, Z , are i.i.d. replicates of Y, Z, , where Yi i icorresponds to the standard indicator risk process which is generated by a

random variable with continuous distribution function and possibly multi- 4plied by a 0, 1 random variable, Z and are left-continuous piecewise

constant processes according to a finite nonrandom division of the timew xinterval 0, , then Condition B is clearly satisfied. This can be established

.by applying the central limit theorem of Hahn 1978 . Conditions A and D .are technical. Note that in the case of i.i.d. observations N, Y, Z andi i i

. < . . . w .x4p s 1, Condition D is implied by Var Z Y w Z , exp Z ) 0.0Now we give some comments on Conditions E and E according to examples . . . . .a , b , c for the weighting process L , . If the function l of examples a

.and b satisfies E then obviously the same holds for the corresponding . . . .function l , s l k , where k , is the limit in probability of0 0 0

. .K , . Condition E, in view of the structure of K , , is analogical to0 0 . .Condition B. To discuss L , of the form of example c , we consider the0

. . w .case of i.i.d. observations N , Y , Z . Then l , s s E Y yi i i 0 .x .. .w .. ..xY s W , Z qEY s W , Z y W , Z s . If W is a constant0 0 0

< . .


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.The modified version of the process of 6 is defined as follows:

, t s , t . .w w

dN sy .Tt w y U , y U , s y , s b , s , . . . .H w w w w

n0

12 .

w xt g 0, ,

.where the process U is given by 4 .Then under Conditions A, A, B, B, C, D, E, E, E and F, we have the

following theorem.

. . w xTHEOREM 3.3. The process , of 12 converges weakly in D 0, asw . .n to a zero-mean continuous Gaussian martingale G of 10 .

3.2. KolmogorovSmirnov and Cramervon Mises type goodness-of-fit tests.k . . . p w xGiven m G 1, consider the weight functions w z, s , z, s gR = 0, ,

k . k . k . . .k s 1, . . . , m, with the corresponding quantities U , , , , ,k . k . k . k . k . . . . . .Q , , A , , a , , , N , and so on, denoted previously0, 0

. . . . . . .for m s 1 as U , , , , , Q , , A , , a , , , N andw 0, 0 w

so on, respectively, such that

k . l . 413 w w s 0 where k , l g 1, . . . , m , k/ l. .

k . . . 4 p w xNote that when w z, s sI z, s gA , A ;R = 0, , k s 1, . . . , m,k k 4are Borel-measurable subsets and A lA s for i/j, i, j g 1, . . . , mi j

.then 13 is satisfied. For some practical remarks concerning the partition of . .the covariate space, see Schoenfeld 1980 and Marzec 1993 .

Under the assumptions of Theorem 3.3, fulfilled according to each weightk . .w , k s 1, . . . , m, m G 1 and 11 we have the following theorem.

THEOREM 3.4.

k . k . w xsup , s : s g 0, . 5 m.~ a max Z , . D 11r2

k . k . 1FkFm A , .

k . k . k . k . , s dA , s . .H0 m.~ b max Z , . D 23r2

k . k .1FkFm A , . 2

k . k . k . k . , s dA , s . .H0 m.~ c max Z , . D 32

k . k .1FkFm

A , .


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wherem

m.P Z Fx s P sup W s Fx , . 41 5w xsg 0, 1

m1

m.P Z Fx s P W s ds Fx , . 4 H2 50

m1 2 m.P Z Fx s P W s ds Fx , x G 0 . 4 H3

50

and W is a standard Brownian motion.

Obviously, the above result is the basis for the formal construction of the

critical regions of KS and CM-type goodness-of-fit tests.

4. Power considerations. In this section, we briefly discuss results . . . .about the distributions of , and , of 6 and 12 outside modelw

conditions, which are relevant for power function considerations. At first we .derive the asymptotic distribution of , by considering a sequence ofw

.contiguous models alternatives indexed by n.Given n, we assume that under the alternative H n. the stochastic inten-A

n. n. n. n. n. n. . .sity process of N s N , . . . , N equals s , . . . , , where1 n 1 n n. T y1r214 t s Y t exp Z t q n g , t t , . . . . . .i i 0 i i 0 0

w x . .t g 0, , i s 1, . . . , n, / 0. Here g , i s 1, . . . , n are uniformlyi 0bounded predictable processes.

. .REMARK. Note that if g , i s 1, . . . , n are independent of , theni 0 0they may have the status of additional covariates not considered in the model . . ..2 . On the other hand, if g , sg , Z for a Borel measurablei 0 0 i

. .function then 14 describes the local misspecification of 2 based on the .nonlog linear hazards form. Indeed, if we consider a 2p q 1 -dimensional

T . .real function g , , z , where s corresponds to z, then 14 can0 0 0 . w be viewed in the Pitman-like framework as Y t exp g , qi 0 0

y1 r2 ..x . .n , Z t t . Here, by a Taylor expansion g , z , is approximatelyi 0 0 .the derivative of g , , z with respect to and evaluated at .0 0

Now assume that the asymptotic stability assumptions B, C, E, F are also . wvalid under the sequence of models given by 14 i.e., under the probability

.x .measures leading to the intensities of 14 and that under 14 ,

n1mk m1 TT , t s Y t w Z t , t t Z t exp Z t g , t . . . . . . . .k , l 0 i i i i 0 i i 0

nis1

.converges in probability to the bounded function t , t , uniformly ink, l 0w x . .t g 0, k q l F 1 or k s 0 and l s 2 . Moreover, let Z i s 1, . . . , n bei

uniformly bounded.

Then we have the following result.


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. .THEOREM 4.1. Under the sequence of local alternatives 14 , , ofw . w x6 converges weakly in D 0, as n to

. T s q l , s t , s y q , s . . . . .H 0 1 , 0 0 1, 0 0

0

=t , s rq , s s ds . . .0 , 0 0 0 , 0 0 0

T y1yB , , t , s y q , s t . . . .H0 0 0 , 1 0 0, 1 0 0 , 0

0

= , s rq , s s ds , . . .0 0, 0 0 0 5where is given in Theorem 3.1.

. . . .REMARK. If p s 1, l , s ' 1, g , s ' s then s0 i 0 i D .. .W A , q A , is the transformed Brownian motion with drift .0 0

This concerns the situation, for example, when is the indicator for thei .treatment group for the ith item and 14 describes the local alternative to

. wthe null hypothesis of 2 of the no treatment effect on survival cf. Lagakos . . .xand Schoenfeld 1984 , Tsatis, Rosner and Tritchler 1985 and Slud 1991 .

The result of Theorem 4.1 can be used to derive the optimal test within the2 2 . .class of -type texts indexed by L and based on the statistics K t of 8 .

Let q s 1. First observe that by using Theorem 4.1 it is easy to establish that . 2 . 2 . .under 14 , K t converges in distribution to U t , say, where U t y

1r2 . w .x4E t r Var t has the standard normal distribution. So the problem2 .of maximizing the local asymptotic power of the test based on K t with

2w .x w .xrespect to L is equivalent to that of maximizing E t rVar t .Note that by Theorem 4.1 we can write

t22E t s l s s s ds . . . .H 0 0 0

0

t Ty l s s s ds . . .H 0 0 0

0

2

y1= , s s ds , . . .H0 0 0 5

0

:Var t s l , l , . 0 0 :where , stands for a bona fide inner product of the form

t

:f, g s f s g s s s ds . . . .H 0 00

t Ty f s s s ds . . .H 0 0

0

ty1

= , g s s s ds . . . .H0 0 00


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and

l s s l , s , . .0 02

s s q , s y q , s rq , s , . . . .0 2 , 0 0 1 , 0 0 0 , 0 0

s s t , s y q , s t , s rq , s , . . . . .0 1 , 0 0 1, 0 0 0 , 0 0 0, 0 015 .

T T s s q , s y q , s q , s rq , s , . . . . .0 1 , 1 0 1 , 0 0 0 , 1 0 0 , 0 0

s s t , s y q , s t , s rq , s . . . . . .0 0 , 1 0 0 , 1 0 0 , 0 0 0 , 0 0

T . . :An inspection shows that E t s l , r y t r , where0 0 0 0 0 0

y1t m2

t s , y s s r s ds . . . . .H0 0 0 0 00

16 .

=

t s s ds y s s s r s ds . . . . . . .H H0 0 0 0 0 0

0 0

2w .x w .xThus by the CauchySchwarz inequality E t rVar t is maximal .Twhen l A r y t r . This leads to the optimal weight process0 0 0 0 0 0

T

L , s s s r s y t s r s , . . . . . .w

.where the forms of , , , correspond to that of , , , of 15 with 0 0 0 0 . . . .q , s , t , s replaced by Q , s , T , s . By replacing, ink , l 0 k , l 0 k , l w k , l w

. . .the form of 16 , the quantities , , , , , , s ds by0 0 0 0 0y1 . w .x . . , , , , , nQ , s dN s , respectively, t is defined. w 0, 0 w w

.EXAMPLES. a If sZ then s , s and the optimal l Ai i 0 0 0 0 0w . . xy1 . . r y H s s ds H s s ds.0 0 t 0 0 t 0 0

. 4 . .b When g 0, 1 , g , s as in the previous Remark, theni i 0 iw . s , s so th at the optim al l A 1 y , y0 0 0 0 0 0

t .m2 . . xw . . x .H s s r s ds H s s ds r . If, moreover, Y, Z , are0 0 0 0 t 0 0 0 0 i i i

. .independent replicates of Y, Z, and and Y, Z are independent, theni i i s 0 and consequently our l A 1.0 0

REMARK. One can use the methods of the proofs of Theorems 3.3 and 4.1 .to establish that under the sequence of local alternatives given by 14 , the

. . . .process , of 12 tends in distribution to G q , wherew

t T t s l ,s t ,s y q ,s t ,s rq ,s s ds . . . . . . .H 0 1, 0 0 1 , 0 0 0, 0 0 0 , 0 0 0

0

Tt

y t , u yq , u t , u rq , u u du . . . . .H H 0, 1 0 0 , 1 0 0 , 0 0 0, 0 0 0 50 s

y1 w x= , s b , s q , s s ds, t g 0, , . . . .0 0 0 , 0 0 0


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and G is the limiting process of Theorem 3.3. In view of the highly complex .form of , the optimization of the power functions of the KS and CM-type

tests of Theorem 3.4 becomes here an extremely difficult problem.

Now consider the case of a fixed alternative and assume that in the truew T .x . .state of affairs exp Z t t is replaced by a random function h t . This0 i 0 i

means that the true intensity process of N equals Y h , i s 1, . . . , n. Ouri i i .additional notation specifies Q t to be of the form corresponding tok , l

. w T .x . .Q , t but with exp Z t replaced by h t . Moreover, q t denotesk , l i i k , l . w .the limit in probability of Q t . It should be noted Hjort 1992 , Sasienik , l

.x .1993b that under a possibly misspecified Cox-type model of 2 , the MWPLE converges in probability to a p-vector of constants * depend-w

.ing on w , which is the unique solution to the system of p equations

17 q s y q *, s q s rq *, s ds s 0. . . . . .H 0 , 1 0 , 1 0 , 0 0 , 00

This can be proved by using the condition of the form corresponding to that of

Condition B but with replaced by *, where we also postulate the0 . . .assumptions for Q and q , quite analogous to these for Q , andk , l k , l k, l

.q , and the condition corresponding to D which concerns the positive-k , l

.definiteness of the matrix*

*

, .

Heret m2 2

* , t s q , s rq , s y q , s rq , s q s ds, . . . . . .H 0 , 2 0, 0 0 , 1 0 , 0 0 , 00

w xt g 0, .

.Under the assumptions defined analogously to A yE, E, F, but with s0 . w . . .and replaced by q s and * with * *, s s * *, y * *, s0 0, 0

xin the condition corresponding to D , under the requirement of the conver- . . .gence of Q to q of the form specified in B for Q , andk, l k , l k , l 0

. . . .q , , and in view of 2 , 12 and 17 , we have the following result.k , l 0

y1 r2 w x .THEOREM 4.2. Uniformly in t g 0, , n , t converges in proba-w .bility, as n to t , where

t T t s l *, s f *, s . . .H

0

T y1qu *, s * *, s b *, s q s ds, . . . .0, 0

18 .f *, s s q s rq s y q *, s rq *, s , . . . . .1 , 0 0 , 0 1 , 0 0 , 0

s

u *, s s q t y q *, t q t rq *, t dt . . . . . .H 0 , 1 0 , 1 0 , 0 0 , 00

y1 r2 . .REMARK. For n , t of 6 , the similar result holds if we only usewthe assumptions corresponding to B, E and F. Then the limit is also of the

. .form of 18 , where we put b *, s s 0.


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Now by using Theorem 4.2, we can state that the tests of KS and CM type .of Theorem 3.4 let m s 1 without loss of generality are consistent against

. .any model misspecification under which t of 18 is nonzero for some x . .t g 0, . By combining the two complex structures of 17 and 18 it seems

that the above condition is generally satisfied. We shall briefly examine this .for the special but most common in real applications situations of the

monotone departures from the proportional hazards assumption and for thew .misspecification based on the added covariate cf. Lin 1991 , Lagakos and

. . .xSchoenfeld 1984 , Struthers and Kalblfleisch 1986 , Slud 1991 .

. . .EXAMPLES. a Let Y, Z be independent replicates of Y, Z . Given thei i .null hypothesis of the form 2 with p s 1, let, under the alternative, the

. w . .x U .hazard rate structure be of the form h t s exp t Z t t , where isi i 0w .an unspecified strictly monotone function of t see Lin and Wei 1991 , Lin

. .x . .1991 , Lin, Wei and Ying 1993 . Since q , t rq , t increases in 0, 1 0, 0 . . . .we have that q t rq t s q *, t rq *, t at the point where0, 1 0, 0 0, 1 0, 0

. . . t s *. By monotonicity of and in view of 17 , this point is unique .and belongs to the interval 0, . Now observe that for the modified weighted

. . . score-type process , of 12 i.e., when sZ , we have that f *,w i i. . .0 / 0 here q rq s q rq and since u *, 0 s 0, the argument of0, 1 0, 0 1, 0 0, 0

.continuity automatically guarantees that t / 0 in a neighborhood of zero . . .assuming that l *, t q t / 0 for t near zero . For the general case of0, 0

. . . w x t sg Z t , t , where g : R= 0, R, consider the binary randomi i 4 4 . w .xvariable Z g 0, 1 such that P Z s 1 s , g 0, 1 Lin and Wei 1991 .

. .We have again that f *, 0 / 0 since otherwise 0 s *, which is impos-

sible. . . .b Let Y, Z , be independent replicates of Y, Z, . Suppose that thei i i

. w T . .x U .time hazard rate structure is of the form h t s exp Z t q t t ,i i i 0 . . ./ 0. Also assume that 0 and Z 0 are independent and 0 is the

. 4 . 4 . wbinary variable: P 0 s 1 s q s 1 yP 0 s 0 , q g 0,1 Lagakos and . . .xSchoenfeld 1984 , Tsatis, Rosner and Tritchler 1985 , Slud 1991 . Then

. w . x w . .xf *, 0 s exp y 1 r q q 1 y q exp . Obviously it is different from .zero and consequently we conclude that of 18 is nonzero in a neighbor-

hood of zero.

Now we shall discuss the problem of the optimality in the class of the . .KS-type tests m s 1, q s 1 of Theorem 3.4 a . Similarly to the previous

considerations, we take the weight process L as the parameter. Unfortu-

nately, the asymptotic distribution of the supremum-type test statistic is not

normal and consequently the asymptotic Pitman efficacy approach does not

work here. We therefore consider the approximate Bahadur efficacy approachw .x .see Bahadur 1960 . By Theorem 3.4 a , the KS type-test statistic has the

< . < w x4asymptotic distribution function G of sup W t : t g 0, 1 . Hence it satisfiesw .xthe required condition cf. Aki 1986 of the form

y1 2log 1 y G x s y2 x 1 q o 1 . .

as x. On the other hand, it can be easily obtained, by using Theorem 4 .2,


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that under the Cox model misspecification the supremum-type test statistic of . y1 r2Theorem 3.4 a when multiplied by n converges in probability as n

to b , wherel

t

w xsup l *, s f *, s q s ds : t g 0, . . .H 0, 0 50

19 b s . . l 1r2t 2l *, s *, s q s ds . . .H 0, 0

0

y1 2 . .Here fsfq u * , s q rq y q rq . Thus the approximate2, 0 0, 0 1, 0 0, 0 . 2Bahadur slope of the test statistic of Theorem 3.4 a is equal to b . Tol

maximize b2 with respect to the function l, note that by applying thel .CauchySchwarz inequality we have in view of 19 that

1r2t 2

b F f *, s r *, s q s ds . . . . 4Hl 0, 00

.This upper bound is attained for the weight function l* *, s , where l* A .fr. Obviously, the natural candidate for the estimator of l* *, s is

y1y1r2 L* , s s f , s q n U , s , s , s , s , . . . . . .w w w w w w

where

f , s s q s rq s y Q , s rQ , s , . . . . .w 1 , 0 0 , 0 1 , 0 w 0, 0 w2

, s s Q , s rQ , s y Q , s rQ , s . . . . . .w 2, 0 w 0, 0 w 1, 0 w 0, 0 w . . . .Here q s rq s denotes the estimator of q s rq s of the general 1, 0 0, 0 1, 0 0, 0

. . . . . . . . .form Y s w Z s , s s h s rY s w Z s , s h s , where h is definedi i i i i i i i .for the special misspecifications of the model 2 as follows:

EXAMPLES. Let U be an unspecified baseline hazard function.0

. . . . U .a h t s Z t , t t , where is a completely specified function.i i 0 . . .Then h t s Z t , t .

i i . . w .x U .b h t s exp h , Z t t , where h is a completely specified func-i i 0 . w .xtion. Then h t s exp h , Z t . Here denotes the MPLE of .i i

. . w T . T .x U .c h t s exp Z t q W t t , where W is the additional co-i i i 0 i T T . w . .x .variate. Then h t s exp Z t q W t , where , is the MPLE ofi w i w i w w

., based on the weighted partial likelihood function with covariates .Z , W .i i

The main drawback of the above optimality result is that it is not obvious .why l* *, s should generally satisfy a condition corresponding to E re-

quired for Theorem 4.2. Note, however, that when we confine ourselves in ourw xconsiderations to the time interval 0, y for some ) 0, that is, if we

. w x4consider the weight L

, s I sg

0,y

, the problem disappears. Thenw . w x4the optimal weight is of the form L* , s I s g 0, y .w


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. .REMARK. If under the null hypothesis of 2 , s 0 b s 0 , then one may .consider the supremum-type test based on S of 9 . Then in view of the

.Remark following Theorem 4.2, we have that the optimal weight L* , s sw . .f , s r , s .w w

5. Simulation and example. A Monte Carlo study was undertaken for

the special situation in which the counting processes jump at most once . To

reveal the general behavior of the KS and CM-type goodness-of-fit tests we

confined ourselves to the same hypothesis and alternatives as those consid- . .ered by Lin and Wei 1991 . Following Lin and Wei 1991 , the hazard

function under the null hypothesis H concerning the Cox proportional haz- . .ards model was of the form t, Z s exp 0.2Z . Z was taken to have the

standard normal law and the censorship was imposed by the generation ofw xindependent uniform random variables on the interval 0, 4 which was also

w xchosen as our 0, finite time interval. Similarly, the two alternatives A1 . .and A were specified to be of the form t, Z, Z* s exp 2Z q 0.5Z* and2

. t, Z s 1 q 0.5Z, respectively. In the case of A , Z* was generated indepen-1dently of Z and so that Z* took values y1 and q1 with probability 0.5. In

the case of A , Z was truncated at "1.98. Moreover, the censoring times2w x . w x .came from the uniform distributions on 0, 7 A and on 0, 5 A . For the1 2

robustness study, the covariates appearing in the definitions of the Cox-type 4estimator and tests were generated from Z q 5J with Jg 0, 1 being

4independent of Z and so that P Js 1 s 0.3. . .The KS and CM-type goodness-of-fit tests of Theorem 3 .4 a and b were

2 1 1 . w . .x chosen to have simple forms: sZ, w s 1, L s s Y y Y s KS , CM. 2 2 .tests and sZ*, w s 1, L s 1 KS , CM tests . In our study we also

considered the KS i and CM i tests being the counterparts of the KSi andw wi CM tests, respectively, defined with the use of nontrivial weight w sI Z F

4 .2.5 . For comparison, the rejection rates of the tests proposed by Cox 1972 , . . . Schoenfeld 1980 , Wei 1984 and Lin and Wei 1991 designated as Cox,

.Sch, Wei and LW in the tables were also presented. All of these previously

defined tests were proposed to assess the adequacy of the classical Coxs

model defined in survival analysis context. For Schoenfelds test, the time

axis and the covariate range were partitioned at the sample median of

survival times and at the mean of covariate, respectively. The levels of

significance of 0.01 and 0.05 were considered. The experiments were carried

out using sample sizes of n s 100 and 200. One thousand repetitions were

used in each instance. The results from these Monte Carlo studies are

summarized in Tables 1 and 2.

Table 1 shows that the KSi, KSi , CM i, CM i , i s 1, 2, tests maintain theirw wsizes near nominal levels, which reflects the appropriateness of approxima-

tions given in Theorem 3.4 for practical use. On the other hand, the omnibus

property of these tests is reflected by generally quite adequate powers for

detecting the two violations of Coxs model given by A and A . It should also1 2be noted that, as expected, the KS2, KS2 , CM 2 and CM 2 tests are highly

w wpowerful with respect to the alternative A . Moreover, the KS1 and CM11 w w


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TABLE 1

Empirical sizes and powers of goodness-of-fit tests

Test H A A1 2

s 0.01 s 0.05 s 0.01 s 0.05 s 0.01 s 0.05

n n n n n n

100 200 100 200 100 200 100 200 100 200 100 200

LW 0.012 0.009 0.058 0.044 0.095 0.099 0.221 0.254 0.149 0.258 0.290 0.447

Cox 0.015 0.014 0.061 0.059 0.018 0.020 0.074 0.093 0.020 0.022 0.085 0.125

Sch 0.

017 0.

009 0.

061 0.

045 0.

013 0.

024 0.

061 0.

089 0.

013 0.

013 0.

062 0.

056Wei 0.009 0.008 0.036 0.043 0.011 0.018 0.052 0.076 0.006 0.009 0.034 0.049

1KS 0.014 0.012 0.061 0.057 0.014 0.021 0.058 0.076 0.019 0.024 0.076 0.0861KS 0.012 0.011 0.051 0.056 0.023 0.031 0.076 0.095 0.227 0.353 0.412 0.577w2KS 0.006 0.009 0.041 0.046 0.890 0.998 0.971 0.999 0.010 0.012 0.051 0.0562KS 0.007 0.011 0.042 0.061 0.841 0.999 0.952 0.999 0.011 0.018 0.071 0.082w1CM 0.004 0.009 0.062 0.071 0.013 0.025 0.060 0.071 0.018 0.021 0.080 0.085

CM1 0.008 0.009 0.061 0.058 0.022 0.029 0.075 0.089 0.118 0.279 0.305 0.481w2CM 0.004 0.009 0.040 0.053 0.774 0.981 0.930 0.996 0.013 0.014 0.053 0.0522CM 0.008 0.011 0.055 0.063 0.841 0.984 0.902 0.999 0.012 0.025 0.049 0.079w

tests have significantly larger powers with respect to A then their counter-2parts KS1 and CM1. Table 1 shows that the good competitors, as compared to

the LW, Cox, Sch and Wei tests, are for A , the KS2, KS2 , CM 2, CM 2 tests1 w wand for A , the KS1 , CM1 tests. Table 2 shows that the weight w, which2 w wcensors large observations among the covariates, makes the tests robust to

covariate outliers. The empirical sizes are then more stable and closer to the

nominal significance levels. The same remark concerns the powers of the KS iwi .and CM tests is 1, 2 .w

TABLE 2

Empirical sizes and powers of goodness-of-fit tests with covariates subject to measurement error

Test H A A1 2

s 0.01 s 0.05 s 0.01 s 0.05 s 0.01 s 0.05

n n n n n n

100 200 100 200 100 200 100 200 100 200 100 200

1KS 0.037 0.031 0.098 0.089 0.129 0.165 0.207 0.311 0.009 0.012 0.052 0.0491KS 0.018 0.012 0.069 0.055 0.033 0.051 0.101 0.163 0.157 0.284 0.372 0.521w2KS 0.021 0.009 0.093 0.080 0.249 0.562 0.470 0.769 0.014 0.014 0.049 0.0632KS 0.013 0.008 0.054 0.046 0.701 0.912 0.795 0.939 0.012 0.016 0.068 0.089w1CM 0.029 0.021 0.099 0.079 0.024 0.070 0.092 0.208 0.012 0.012 0.045 0.0531CM 0.006 0.009 0.095 0.052 0.031 0.042 0.123 0.228 0.125 0.226 0.351 0.404w2CM 0.024 0.013 0.074 0.081 0.256 0.564 0.478 0.791 0.012 0.023 0.051 0.048

2CM 0.

012 0.

009 0.

054 0.

048 0.

635 0.

890 0.

710 0.

893 0.

015 0.

027 0.

057 0.

081w


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In the simulation study we considered the simplest examples of the

weights L discussed in Section 2. The resulting tests performed quite well in .general. Simulation indicated that the KS and CM-type tests with L s s

w . .xY y Y s , G 0, gN, or more generally with the weights discussedin Section 2, should contain representatives which are powerful for concrete

alternatives. Moreover, the simulation study confirmed that the true sizes of

the KS and CM-type tests for moderate sample sizes and under different

levels of censorship are indeed accurately approximated by the nominal

significance level based upon the asymptotic distribution results of Section 3.

We have also applied the KS1 and CM1 tests to the Stanford heart . .transplant data given in Miller and Halpern 1982 . Lin, Wei and Ying 1993

note that the fit of Coxs model to these data, using the single covariate

ageage at transplant, is not satisfactory. Our tests confirm this. We

obtained the p-values of the KS1 and CM1 tests of 0.025 and 0.004, respec- .tively. On the other hand, Lin, Wei and Ying 1993 note that considering the

.2two covariates age and age provides a satisfactory description of the data.Our tests also confirm this phenomenon. The p-values of the KS1 and CM1

tests are now equal to 0.412 and 0.526.

6. Generalizations. In this section we shall outline the extensions of .the results of Section 3 for the two generalizations of model 2 . Since the

concepts are similar to that of Sections 2 and 3, the proofs will be omitted and

only the main results will be presented. The first generalization discussed

concerns the semiparametric model which allows the stochastic intensity to

have the general form. The model postulates that the counting process Nihas the stochastic intensity

w x20 t s Y t exp h , Z t t , t g 0, , . . . . . .i i 0 i 0

where h is a completely specified Borel-measurable real-valued function, .. . ..h , Z and r h , Z are predictable and locally bounded pro-i i

.cesses. The above form corresponds to Coxs 1972 intuitive approach and for . T . .h , z s h z leads to the Prentice and Self 1983 model. The MWPLE

. is defined by the equality U , s 0, wherew, h h

n1 tU , t s w Z s , s . . .Hh i'n 0 is1

S , s .0, 1= h , Z s y dN s . . . .i i 5 S , s .0, 0

21 .

Here

mln1 mk

S , u s w Z u , u Y u u h , Z u . . . . . . .k , l i i i in

is1

=exp h , Z u , . .i


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4 .k, l g 0, 1, 2 , k q l F 2, and we also define s , u as the function beingk, l .the limit in probability of S , u , uniformly for in a neighborhood of k, l 0

w xand u g 0, . Then under some additional standard asymptotic stability and

regularity assumptions, including, for example, the asymptotic stability ofy1 r2 . .the observed information matrix n r U , for in a neighbor-h

w . .xhood of cf. Arjas and Haara 1988 , Marzec 1996 , we have the decompo-0sition

y1'22 n y s , U , q o 1 , . . . . .w , h 0 h 0 h 0 p

corresponding to the asymptotic normality of . In the above equality thew, h . .matrix , is of the form given by , but with q , q , qh 0 0 0, 0 0, 1 0, 2

.replaced by s , s , s , respectively. In the same manner, , ,0, 0 0, 1 0, 2 h . . . , s , A , s , B , s are defined as the counterparts of the previouslyh h h

. . . . .used quantities , , , s , A , s , B , s . Similarly, , ,h . . . . , s , b , s and so on are now defined analogously to , , , s ,h h .b , s and so on, but with Q replaced here by S . For the goodness-of-fitk, l k, l

. .inference with the model 20 we consider the process , and itsh w, h . . . . .modification , corresponding to , of 6 and , of 12 ,h w, h w w

respectively, which are now based on the newly defined quantities

S , S , U , , b , , put in the place of Q , Q , U, , b, , .0, 0 1, 0 h h h w, h h 0, 0 1, 0 w .Starting from 22 we present a list of assumptions that will guarantee

. .the analogous results previously obtained for , and , under thew wCox regression model. The counterpart of Theorem 3.1 states that under the

. .model 20 , the process , converges weakly to a zero-mean continu-w, hous Gaussian process, with covariance function of the form

T y1 w xA , min s, t yB , s , B , t , s, t g 0, . . . . . .h 0 h 0 h 0 h 0

The following conditions are sufficient for this result. Conditions A, E and F

remain unchanged. Conditions B and D are adapted to the present situation

as follows. The former concerns the quantities S and their correspondingk, l .limits s , while the latter relates to , .k, l h 0

Now Condition C should have the slightly stronger form

1 sup w Z s , s Y s h , Z s q s s o 1 . . . . . . . .i i k k p 5' n k , s

.The weak convergence of the process , to a zero-mean continuoush w, h . w xGaussian martingale with variance function A , t , t g 0, , that is, theh 0

counterpart of Theorem 3.3, can be established by using the following as-

sumptions: Conditions A, E, E, the counterparts of Conditions B and D, . . .where we only replace Q , q , , , by S , s , , ,k, l k, l 0 0 k, l k, l h 0

. , , respectively, the condition corresponding to Condition C of theh 0


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form: there exist a constant C and a neighborhood B of such that0

2 w Z s , s sup h , Z s q h , Z s . . . . . .i i 0 i2 gB

q h , Z s q Z s q s F Cw Z s , s . . . . . . .0 i i i i5The following assumption is also required: there exists a matrix-valued

w xfunction defined in a neighborhood of and on 0, which is the limit in0probability of the observed information matrix-valued function

y1 r2 . .n r U , x , uniformly in from a neighborhood of and x gh 0w x0, . Obviously, in the statement of the counterpart of Theorem 3.4, one

k . k . k . k .should replace A , by A , .h h .Another useful generalization of the model 2 is the following multistate

. wCoxs-type regression model of Andersen, Hansen and Keiding 1991 see . .xalso Andersen and Borgan 1985 , Marzec and Marzec 1996 . Given n

. w x 4processes X t , t g 0, , i s 1, . . . , n, with state space 1, . . . , M , MG 2,i .let N t be the observed number of direct transitions from h to j of Xh ji i

w x .during the time interval 0, and let Y t indicate if X was observed to beh i iin state h at time t y . The model of interest postulates that the multivariate

.counting process N : h, j s 1, . . . , M, h/j, i s 1, . . . , n has the stochastich ji .intensity process : h, j s 1, . . . , M, h/j, i s 1, . . . , n , whereh ji

T w x23 t s Y t exp Z t t , t g 0, . . . . . .h ji h i h j i h j0

.Here the parameters , correspond to , of model 2 and Z is theh j0 h j 0 0 icovariate process associated with the ith individual. In our present notation

Q h. and q h. are defined similarly to Q and q but with Y replaced herek , l k , l k, l k , l iby Y . Moreover, let U , , , , and so on be the counterparts of U, ,h i h j h j h j h j

h. h., and so on, obtained by substituting Q , , q , N in place of Q ,k , l h j0 k, l h ji k, l . , q , N respectively. Thus according to 4 the possibly weighted Cox type0 k, l i

.estimator of is defined by the equality U , s 0, h, j s 1, . . . , M,h j h j h j .h/j. The goodness-of-fit procedures for the model 23 are now based on the

. . .processes , : h, j s 1, . . . , M, h/j and , : h, j s 1, . . . , M,h j h j h j h j . . . . .h/j , where , and , have structures like , of 6h j h j h j h j w

. . .and , of 12 , respectively. They are constructed by using L , ,w h j h j h. h. .Q , Q , U , , b , N , instead of L , , Q , Q , U, , b, N,0, 0 1, 0 h j h j h j h ji h j w 0, 0 1, 0 i

. Then under the conditions analogous to that of Section 3 specified now .separately with respect to each pair of states h, j , h, j s 1, . . . , M, h/j,

one can obtain the multivariate generalizations of Theorems 3.1 and 3.3. In

their statements we have the asymptotic independence of the components of . . .the limiting Gaussian processes : h, j s 1, . . . , M, h/j and G :h j h j

.h,j s 1, . . . , M, h/j , where and G are the counterparts of and G ofh j h j . .Theorems 3.1 and 3.3, respectively, defined by using A , , B , h j h j h j h j

.and , . This leads to the obvious constructions of the KS andh j h j

CM-type goodness-of-fit tests. In the statement of Theorem 3.4, one should


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k . k . k . k . . . . .only replace , s , A , s by , s , A , s and considerh j h j h j h j .the maxima with respect to all pairs of states h, j , h, j s 1, . . . , M, h /j.

.Consequently we have M My 1 instead of m.

7. Proofs.

PROOF OF THEOREM 3.1. If we denote by

n Q , s .t 1, 0w xV , t s w Z s , s s y dN s , t g 0, , . . . . .H i i iQ , s .0 0, 0is1

then by Taylors expansion we have that

1 Tt , t s L , s V , ds . . .Hw w w'n 0

1 t T s , t y b , s dN s y . . . .H0 1 w w 0'n 0

1 T tq y L *, s V , ds . . . Hw 0 0' n 0

24 .

1 T t y y L *, s , s dN s y , . . . . .Hw 0 w w 0' n 0

5 5 5 54 5 5where max * y , y F y . By using Conditions A, B, E,0 0 w 0F, it can be easily shown that the third and fourth terms in the above

.expansion of , are asymptotically negligible. For the second term wewobtain, by using the decomposition

y1'n y s , U , q o 1 . . . .w 0 0 0 p

and Conditions A, B, D, E that

1 t T b , s dN s y . . .H 1 w w 0'n 01 t T y1

s b , s dN s , U , q o 1 . . . . .H 0 w 0 0 pn 0

25 .

T y1sB , t , U , q o 1 . . . . .0 0 0 p

.Here the last equality is a direct consequence of Lenglarts 1977 inequality . . . .since obviously U , s O 1 . Equations 24 and 25 , Condition E and0 p

again Lenglarts inequality lead to the final asymptotic representation of the

form

T y126 , t s , t yB , t , U , q o 1 , . . . . . . .w 1 0 0 0 0 p


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wheren1 t T

, t s l , s w Z s , s . . . .H1 0 0 i'n 0 is1Q , s .1 , 0 0

= s y dM s , . .i iQ , s .0, 0 0

27 .

w xt g 0,

. .and U , is given by 4 . By using the multivariate martingale central0w .xlimit theorem cf

.

Fleming and Harrington 1991 one can conclude that the . . ..p q 1 -variate local square integrable martingale , , U , con-1 0 0

w x. pq 1 . . ..verges weakly in D 0, to the process s , specified in1 2Theorem 3.1. This can be established in view of Conditions A, B, C, and E .

. ..Hence we have the weak convergence of , , U , . Now observe1 0 0 . . .. .that the process of 26 may be written as H , , U , q o 1 ,1 0 0 p

w x p w x .where H is a function from D 0, =R to D 0, of the form H x, u s .T .y1 w x px yB , , u. Note that H is continuous on C 0, =R . Hence0 0

w . xthe continuous mapping theorem cf. Billingsley 1968 , Theorem 5.1 com-pletes the proof. I

. .PROOF OF LEMMA 3.2. Obviously, by 7 and Conditions A, B, D, G of

.10 has mean zero and is a Gaussian process with continuous sample paths.

w . .x w xWe shall find the expression of Cov G t , G u , t, u g 0, . Let u F t and . . . . w xdenote by s s b , s q , s s , s g 0, . Then0 0, 0 0 0

u T y1EG t G u sE t u yE t y s , s s ds . . . . . . . . .H1 1 1 2 2 0

0

t T y1yE u y s , s s ds . . . . .H1 2 2 0

0

t T y1qE y s , s s ds . . . .H 2 2 0

0

28 .

u T y1= y y , y y dy . . . .H 2 2 0

0

sL yL yL qL say . .1 2 3 4 .By applying Fubinis theorem and 7 we obtain

L sA , u , .1 0u

T y1L sB , t , s s ds . . .H2 0 0

0

uT y1

y B , s , s s ds, . . .H 0 0029 .

uT y1

L sB , u , s s ds . . .H3 0 00

uT y1

y B , s , s s ds. . . .H 0 00


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To deal with L note that4

TE y s y y s , max s, y , . . . . . .2 2 2 2 0

u u uT y1 T y1

s , y y dy ds s B , y , y y dy. . . . . . .H H H0 0 00 s 0

Hence one can deduce that

uT y1

L s B , t qB , u , s s ds . . . .H4 0 0 00u

T y1y 2 B , s , s s ds. . . .H 0 0

0

. . . .The above equality together with 28 and 29 imply that EG t G u s .A , u , u F t and from the symmetry of considerations we finally have that0 .G is a zero-mean Gaussian process with independent increments and hence

a Gaussian martingale with respect to its natural filtration . This completes

the proof. I

PROOF OF THEOREM 3.3. Since by a Taylor series expansion,

T T U , y U , s y s U , y U , s y . . . .w w 0 0T

'y n y *, s , . .w 05 5 5 5 . .where * y F y , 12 and 26 lead to the decomposition0 w 0

t T , t s , t y U , y U , s y . . . . Hw 1 0 0 0

0

30 .dN sy .w

= , s b , s q o 1 , . . .w w pn

provided

T y1U , , B , t . . .0 0 0

dN sy .T t w 'y n y *, s , s b , s s o 1 . . . . . .Hw 0 w w p

n0

31 .

By using Conditions F, B, C and E, one can obtain that

1 32 sup b , s y b , s s O . . . .w 0 p /'nw xsg 0,

Moreover, by Lenglarts inequality and Conditions A and B we have

dN s .t w 33 sup b , s yB , t s o 1 , . . . .H w 0 pn0w xtg 0,


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which yields the equality

dN s .T t w 'n y b , s . .Hw 0 w

n0T

's n y B , t q o 1 . . .w 0 0 p34 .

T y1s U , , B , t q o 1 . . . . .0 0 0 p

. .Now observe that it is possible to replace the integrand *, s in 31 by . , s . This can be established by observing that under conditions B, B, C,w

'5 . .5 .sup *, s y , s s O 1r n and by considering the approxi-s gw 0, x w p . w xmation of the integral in 31 with respect to the intervals 0, y andn

x qy1r2 .y , separately, where s n with 0 - q - r 2 q 2 and n nw xgiven by Condition E. According to the interval 0, y we may confinen

y y1 . .ourselves to the set where , s s , s and use the fact thatw w y15 . 5 . . , s F O 1 1r . This follows by an application of the followingw n

facts that are technical but quite straightforward in establishing, in view of

Conditions B, A and D,

1 w xP det , s G det , s , s g 0, y 1, . . 5w 0 n2y1 y1c w x35 P n ys , s y , s F1, s g 0, y 1 . . . . 5w 0 n

as n ,

.where 0 - c - q. On the other hand, by using 32 , Condition E and the factthat under Conditions A and B,

qq 1r2 qq1r236 P C n FN yN y y F C n 1 as n . . . 41 w w n 2for some constants 0 -C -C , the desired approximation can be estab-1 2

x .lished on the interval y , . Thus 30 holds and the next step is to shownthat

dN sy .t wT U , y U , s y , s b , s . . . .H 0 0 w w n0

dN s .t wT y1s U , y U , s y , s b , s . . . .H 0 0 0 0

n0

37 .

q o 1 . .p

. . .By applying 32 we may replace b , s in the left-hand side of 37 byw .b , s provided we show that0

1 dN sy .w O U , y U , s y , s s o 1 . . . . .Hp 0 0 w p /' nn 0

This, however, can be established again by considering the above integralw x xseparately on the intervals 0, y and y , , respectively, and usingn n


27/32


. . 5 .5Condition C together with 35 and 36 . By Conditions A, E, l , s F0 . . xc ys , s F c for some , c , c ) 0 and s g y , . Hence in view1 0 2 1 2

. 5 .5 w x4 .of 35 , by the fact that sup U , s : s g 0, s O 1 and finally by0 papplying Lenglarts inequality, the quantity

y yn y1 U , y U , s y , s y , s . . . .H 0 0 w 00

=dN s .w

b , s .0n

is asymptotically negligible. On the other hand, by Conditions A, B, D,

dN s .wy1U , y U , s y , s b , s . . . .H 0 0 0 0

nyn

38 . l , s dN s . .0 w

F O 1 . .Hpys ny

n

. . .4Since for c s 1r n log n , P N sN y c 1 as n , and by then w w nMarkov inequality

yc l , s dN s . .n 0 wyr2 r2

P

G1

F O1 ,

.Hn n 5ys nyn

.the quantity in the left-hand side of 38 is asymptotically negligible. It is alsoy1 y . .asymptotically negligible if we replace , s by , s in this quan-0 w

tity. This follows by an application of the inequality

dN x .wU , y U , s y F 2C , . . H0 0 'w x ns,

which is valid under Condition C, and the facts thatp

1 dN x .w w xP det , s G det , s , s g y , 1, . . Hw 0 n 5 54 nw xs ,

y1dN x .y1 w

w xP , s F O 1 , s g y , 1 . . Hw p n 5 5nw xs , as n .

The above convergence can be established by using Condition B, D and in . w xview of Theorem F.2.a of Marshall and Olkin 1979 applied for s g y , n

w . . xto th e qu an tity de t H , x dN x rn which approximatesws, x 0 w w .xdet , s . Consequently,w

dN sy .wy1 U , y U , s y , s y , s b , s . . . . .H 0 0 w 0 0n0

s o 1 . .p


28/32


.Thus 37 follows and as a result of the above considerations we have that

t T , t s , t y U , y U , s y . . . . Hw 1 0 0 00

dN s .wy1= , s b , s q o 1 . . . .0 0 p

n

Now by using Lenglarts inequality and Conditions A, B, E, it can be easily

shown that

dM s .t wT y1sup U , y U , s y , s b , s s o 1 . . . . .H 0 0 0 0 p

n0w xtg 0,

and finally that

, t .wt T

s , t y U , y U , s . . .H1 0 0 00

39 .

y1= , s b , s q , s s ds q o 1 . . . . . .0 0 0 , 0 0 0 p

pq 1w x w xNow observe that the function h: D 0, D 0, defined by

. Th x , x sx y x yx s . . .

H1 2 1 2 2040 .y1

= , s b , s q , s s ds . . . .0 0 0 , 0 0 0pq 1w xis continuous on C 0, relative to the product Skorohod topology. This is

guaranteed by the fact that

y1

, s b , s q , s s ds-, . . . .H 0 0 0 , 0 0 00

. .in view of Conditions A, B, E. Obviously, by 39 and 40 we have that . . .. . , s h , , U , q o 1 . Since from the proof of Theorem 3.1w 1 0 0 p

. .. pq 1w xit follows that , , U , converges in D 0, to the continuous1 0 0 . .. .Gaussian process , specified by 7 , the continuous mapping theo-1 2

rem and Lemma 3.2 complete the proof. I

PROOF OF THEOREM 3.4. It could be easily obtained that under the asymp- k . .totic stability and regularity condition, A , converges in probability to

k . k . k . . .A , . Moreover, if k) 1, then by using, for each , , the0 . .asymptotic representation analogous to that of 39 and 13 , one can deduce

k . k . m . . w xthat , : k s 1, . . . , m converges weakly in D 0, to the process k . k . .. . k . .W A , : k s 1, . . . , m , where W , k s 1, . . . , m, are indepen-0dent standard Brownian motions. The direct application of the continuous

mapping theorem together with the scale-change property of Brownian mo-

tion completes the proof. I

PROOF OF

THEOREM

4.1. The proof is similar to that of Theorem 3.

1.

Firstobserve that in view of the asymptotic stability and regularity assumptions


29/32


. .the process U , given by 4 is asymptotically equivalent to the process0

q , s . . 0 , 1 0U , q t , s y t , s s ds, . . . .H0 0 0 , 1 0 0, 0 0 0

q , s .0 0 , 0 0

where

n Q , s . . 0 , 1 0y1r2 U , s n w Z s , s Z s y dM s . . . . .H

0 i i 1

Q , s .0 0 , 0 0is1

and

. n.M sN y s ds, i s 1, . . . , n, . . .Hi i i0

are mutually orthogonal local square integrable martingales. By repeating .the Andersen and Gill 1982 argumentation, one can easily deduce that .under 14 the MWPLE also converges in probability to . Thus by usingw 0

Taylors expansion we conclude that

y11r2 n y y

, . .w 0 0

=

q , s .0 , 1 0U , q t , s y t , s s ds . . . .H0 0 , 1 0 0, 0 0 0 5q , s .0 0 , 0 0

is asymptotically negligible. In view of the considerations similar to those ofy1 r2 . .Theorem 3.1, we obtain that under 14 the process n , is asymp-w

totically equivalent to the process

q , s . . 1 , 0 0T , q l , s t , s y t , s s ds . . . . .H0 0 1 , 0 0 0 , 0 0 0

q , s .0 0, 0 0

T y1

yB ,

, . .0 0 q , s .0 , 1 0

= U , q t , s y t , s s ds . . . . .H0 0 , 1 0 0, 0 0 0 5q , s .0 0 , 0 0 . . .Here the process , is defined in a similar manner as , of 270 i 0

.with M replaced by M . An application of Rebolledos 1980 theorem showsi i w . .x w . .xthat , , U , converges weakly to the process , given in0 0 1 2

Theorem 3.1. Thus the final result is a direct consequence of the application

of the continuous mapping theorem. I

PROOF OF THEOREM 4.2. The result of the theorem could be easily estab-

lished by mimicking the method of the proof of Theorem 3 .1. Therefore we

shall present only the main steps of the proof. Taylors series expansion of


30/32


. . .L , s and Q , s rQ , s at the point * together with thew 1, 0 w 0, 0 wasymptotic stability and regularity assumptions lead to the equality

1 Q *, s .t 1, 0T , t s L *, s Q s y Q s ds q o 1 . . . . . Hw 1 , 0 0 , 0 p' Q *, s .n 0 0, 0and finally to

1 q *, s .t 1, 0T , t s l *, s q s y q s ds q o 1 . . . . . . Hw 1 , 0 0, 0 p' q *, s .n 00, 0

y1 r2 t T . . . .Thus n , t converges in probability to H l *, s f *, s q s ds,w 0 0, 0w xuniformly in t g 0, . Similarly

1 q *, s .t 0, 1U , t s q s y q s ds q o 1 . . . . Hw 0, 1 0, 0 p' q *, s .n 0 0, 0

s u *, t q o 1 . . .p

By applying an assumption corresponding to B together with Lenglarts . .inequality, we conclude that U *, s O 1 and consequently thatp

1r2 . .n y * s O 1 . Then by using the counterparts of the steps of thew pproof of Theorem 3.3 conducted with the use of the assumptions preceding

Theorem 4.2, we obtain the following equalities:T

1 1 dN sy .t w U , y U , s y , s b , s . . . .H w w w w' ' nn n0

T1 1t

s U *, y U *, s y . .H ' 'n n0dN sy .w

= , s b , s q o 1 . . .w w pn

T1 1t

s U *, y U *, s y . .H ' 'n n0dN s .wy1

=* *, s b *, s q o 1 . . .pn

T1 1t

s U *, y U *, s y . .H ' 'n n0y1

=* *, s b *, s q s ds q o 1 . . . .0, 0 p

t Ts u *, y u *, s . .H

0

y1=* *, s b *, s q s ds q o 1 . . . . .0, 0 p

. . .By 17 , u *, s 0. Thus in view of 12 the proof is complete. I


31/32


Acknowledgments. The authors express their thanks to an Associate

Editor and the referees for their valuable comments and suggestions which

led to a substantial improvement of the original manuscript.

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