Asymptotics for Homogeneity Tests Based on a Multivariate Random Effects Proportional Hazards Model

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1 Joural of Multivariate Aalysis 78, 8310 (001) doi jmva , available olie at httpwww.idealibrary.com o Asymptotics for Homogeeity Tests Based o a Multivariate Radom Effects Proportioal Hazards Model Lauret Bordes Uiversite de Techologie de Compie ge, Compie ge Cedex, Frace Lauret.Bordesutc.fr ad Daiel Commeges INSERM 330, Uiversite Victor-Segale Bordeaux, Bordeaux Cedex, Frace Daiel.Commegesu-bordeaux.fr Received Jue 3, 1999; published olie May 17, 001 D. Commeges ad H. Jacqmi-Gadda (1997, J. Roy. Statist. Soc. B 59, ) cosidered geeralized score tests of homogeeity that accommodate parametric ad semiparametric regressio models arisig from multivariate radom effects with commo variace ad kow correlatios. I this paper we give some sufficiet hypotheses uder which the asymptotic behavior of the test statistic is stadard ormal for the proportioal hazards model with right cesored data. Several examples of applicability are studied. 001 Academic Press AMS subject classificatios 6N; 6G0; 6H. Key words ad phrases coutig processes; frailty models; homogeeity test; multivariate radom effects; semiparametric proportioal hazards model. 1. INTRODUCTION A commo complicated feature of, e.g., reliability or survival aalysis may be the presece of uobserved heterogeeity betwee subjects (e.g. biological variatio ot explaied by available covariates i regressio models). Hece it is importat to both propose models that take ito accout such heterogeeity (like frailty models) ad test that such heterogeeity is ot preset for specific data (for a wide itroductio to these models ad tests see Sectio 1.1 i Ng ad Cook, 1999). Commeges ad JacqmiGadda (1997) proposed geeralized score test of homogeeity. The aim of these tests is to use frailties to itroduce specific deviatios from a uderlyig model, like the proportioal hazards model (PHM) ad the to test the absece of frailty. The structure of correlatio betwee frailties is kow ad allows to specify multivariate X Copyright 001 by Academic Press All rights of reproductio i ay form reserved.

2 84 BORDES AND COMMENGES effects like, for example, groups effects (see Commeges ad Aderse, 1995). The geeralizatio relatively to previous homogeeity tests is due to cosiderig a correlated (rather tha shared) radom effects model o oe had, ad cosiderig a geeral likelihood, which ca be a partial likelihood, o the other had thus homogeeity testig i a semiparametric cotext ca be doe. Li (1997) cosidered a more geeral parametric case i which the covariace of the radom effects depeds o a vector of parameters, while keepig the ull hypothesis to be a idepedece hypothesis. We will ot treat such a extesio here because the asymptotic is more complex tha i the semi-parametric case that we cosider. I Commeges ad JacqmiGadda (1997) the statistics were claimed to be asymptotically stadard ormal but coditios for asymptotic ormality were expected to be o trivial i the case of the PHM. Sice these homogeeity tests are based o the asymptotic behavior of the score statistic, it is clearly ecessary to give coditios for which asymptotic ormality of the score statistic is achieved; this is the aim of this paper. Suppose we observe (X i, $ i, z i )(1i) where X i =T i 7 C i is the miimum of the failure time T i ad the cesorig time C i, $ i =1(T i C i ) is the cesorig idicator (1( } ) is the idicator fuctio) ad z i # R p is a vector of covariates. Suppose that the coditioal hazard fuctio i of T i give z i is give by i (s)= 0 (s) exp(;$z i += i ), s0, (1) where 0 is the ukow baselie hazard fuctio of the PHM, ; # R p is the ukow regressio parameter ad exp(= i ) is the o observable frailty for the ith subject. Cosider ow the coutig processes N i (t)=1(x i t; $ i =1). It is well kow that for multiplicative itesity models with idepedet cesorig assumptio, processes M i (t)=n i (t)& t Y i (s) 0 (s) exp(;$z i += i ) ds, 0 t0, where Y i (t)=1(x i t), are martigales with respect to the right-cotiuous filtratio (F t ) t0 F t =_[N i (s), Y i (s), = i, z i ;1i; 0st], t0, with F t /F where (0, F, P) is the uderlyig complete probability space. Usig the above model Commeges ad JacqmiGadda (1997) proposed a score statistic to test the ull hypothesis that all the = i are ull. Notig = i =% 1 v i it is posed that E(v i )=0, E(v i )=1 ad E(v iv j )=w ij (kow)

3 ASYMPTOTICS FOR HOMOGENEITY TESTS 85 wheever i{ j, the the ull hypothesis reduces to %=0. They obtaied the followig stochastic itegral represetatio for the score at time t, where H i (s)= T () (;, t)= t H i (s) dm i (s), 0 (M j(s&)&p j (s))(w ij & k=1 w jk p k (s)), with p i (s)=exp(;$z i ) Y i (s)s (0) (s), S (0) (s)= exp(;$z j) Y j (s) ad the martigale residuals M i are defied by M i(t)=n i (t)& t 0 p i (s) dn j (s)=m i (t)& t p i (s) dm j (s). 0 The predictible variatio process (T () (;, }))(t) oft () (;, t) is defied by (T () (;, }))(t)= t H (s) Y i i(s) exp(;$z j ) 0 (s) ds. 0 I the followig sectio are give the mai results about the asymptotic behavior of suitable stadardizatio of T () whe both the regressio parameter ; is kow ad estimated. Sectio 3 deals with several examples were assumptios of Sectio are met. Sectio 4 is devoted to proofs while relegatig some techical developmets i Sectio 5.. MAIN RESULTS All covergece results are give with respect to tedig to ifiity. Deote by a & = w ij a ormalizig sequece ad $ ij =1(i= j)..1. The Regressio Parameter ; Is Kow I this subsectio we assume that parameter ; is kow; cosequetly i the sequel we will delete parameter ; from our otatios ad we will deote exp(;$z i )byr i for simplicity. A1. a. Let { #(0,+) be a real such that 0 is bouded o [0, {]. b. There exists a fiite costat R # R + such that 1 r R. A. lim 1 a 3 max 1i w ij =0.

4 86 BORDES AND COMMENGES A3. There exists a determiistic fuctio # defied o [0, {] such that } a w ij Y i(s) r i 4 1+$ ij j (s)&#(s) } wp 0, where the processes 4 i (})= } 0 r i Y i (s) 0 (s) ds are compesators of the coutig processes N i ( } ) uder the ull hypothesis. A4. For k=0, 1, there exist determiistic fuctios s (k) defied o [0, {] such that t #[0,{]} Y j (t) r k+1 j &s (k) (t) } wp 0, moreover, s (0) is bouded away from 0 o [0, {]. Theorem.1. Let us assume that assumptios A1A4 are satisfied ad '=lim + a T exists. The the statistic () ({) is asymptotically ((T () (})) ({)) stadard ormal. Moreover we have 1 (a T () (})) ({) w P! ({)=4 { 0 \ #(t)&' (s(1) (t)) s (0) (t) A 0 (t) + 0(t) dt. Remark.1. that Assumptio A1-a could be relaxed by choosig { such 0<{< {t0; A 0(t)= t 0 (s) ds<+ =... The Regressio Parameter ; Is Ukow We cosider the followig assumptios B1. Let us defie ad S (k) (;, t)= 1 S (3+k) (;, t)= 1 0 exp((k+1) ;$z j ) Y j (t), for k=0, 1,, z j exp((k+1) ;$z j ) Y j (t), for k=0, 1, S (5) (;, t)= 1 z j exp(;$z j ) Y j (t), where z =zz$ isap_p real matrix for z a p_1 real vector.

5 ASYMPTOTICS FOR HOMOGENEITY TESTS 87 There exists a eighborhood B of ; 0 ad real, vector ad matrix fuctios s (k) ad s (3+k) (k=0, 1, ) defied o B_[0, {] such that ad S (k) (;, t)&s (k) (;, t) w P 0, (;, t)#b_[0, {] &S (k) (;, t)&s (k) (;, t)& w P 0, (;, t)#b_[0, {] where &x&=max 1ip, 1jq x ij if x deotes a p_q real matrix with etries x ij. Moreover, s (0) is bouded away from 0 o B_[0, {] ad we have for all (;, t)#b_[0, {] ; s(0) (;, t)=s (3) (;, t) ad ;;$ s(0) (;, t)=s (5) (;, t). B. We have 1 &z &C<+. B3. The family of fuctios s (0) (},t), s (3) (},t) ad s (5) (},t), 0t{ is equicotiuous at ; 0. B4. The matrix 7 0 (; 0, {)= { 0 \ s(5) (; 0, t) s (0) (; 0, t) & \ s(3) (; 0, s (0) (; 0, t)+ + s(0) (; 0, t) 0 (t) dt, is positive defiite. From ow o, we replace ; by a estimator ; i T () (;, t). Theorem.. Uder A1-a, A ad B1B4, if '=lim + a exists ad ; is the Cox partial likelihood estimator of ; 0, a T () (;, {) is asymptotically ormal with zero mea ad variace _ ({) give by _ ({)=! ({)&\$({) 7 &1 0 (; 0, {) \$({), where the vector fuctio \ is defied by \({)=' 1 { 0 \ s(3) (t) s(1) (t) s (0) (t) &s(4) (t) + da 0(t). Remark.. Note that if lim + a =0, we get from Theorem. that statistics a T () (;, t) ad a T () (; 0, t) are asymptotically equivalet. For computatioal aspects see Commeges ad JacqmiGadda (1997).

6 88 BORDES AND COMMENGES 3. SPECIAL CASES I this sectio we give sufficiet coditios for which assumptios of Sectio may be satisfied. The, several examples are reviewed. Theorem 3.1. Suppose i the PHM that (T i, C i, z i ) (,..., ) are idepedet, idetically distributed replicates of (T, C, z), where the failure ad cesorig times variables T ad C are absolutely cotiuous positive radom variables coditioally idepedet give z. Let X i =T i 7 C i, $ i =1(T i C i ), N i (t)=1(t i t; $ i =1) ad Y i (t)=1(x i t). Assume the covariate vector z is costat (i time) ad bouded. Assumptios A1 ad B1B4 will be satisfied if P(Y i ({)>0)>0 ad 7 0 (; 0, {) is positive defiite. Moreover, if lim + a =0, assumptio A3 is satisfied too. Example 3.1 (Ucorrelated radom effects). This is the situatio where w ij =$ ij for 1i, j. It is easy to see that i such a case, assumptio A3 follows immediately from B1. The, uder assumptios of Theorem 3.1 without lim + a =0 but with lim + a =1 we get A1A3 ad B1B4. Theorem. may be applied. This test may be viewed as a goodessof-fit test for the PHM. Example 3. (Weak correlatio of radom effects). This is the situatio where w ij =$ ij +$ ij for 1i, j ($ i&$ ij ). It is easy to show that A is satisfied for >34. The we have lim + a =1 ad if B1 is satisfied it is easy to see that #=s () A 0 i A3. The, uder assumptios of Theorem 3.1 (without lim + a =0), Theorem. may be applied with '=1. Example 3.3 (Groups of same size). This is the situatio where there are k groups of size p. We have =kp ad w ij =1 if idividuals i ad j belog to the same group ad w ij =0 otherwise; this test shows up group effects. It is easy to see that lim 1 a 3 max 1i w ij =0. If p + as + we have lim + a =0 ad the, if assumptios of Theorem 3.1 are true, Theorem. may be applied. If p is fixed the lim + a =1p. Suppose that idividuals are umbered such that idividuals 1,..., p belog to the first group,

7 ASYMPTOTICS FOR HOMOGENEITY TESTS 89 idividuals p+1,..., p belog to the secod group ad so o. We ca write a w Y ij i(t) r i 4 1+$ ij j (t)=a 0 (t) t (s, t) 0 (s) ds, 0 where (s, t)= 1 k k Z i(s, t) with idepedet ad idetically distributed (i.i.d.) processes Z i 's defied by Z i (s, t)= 1 ip p j=(i&1) p+1 ip l=(i&1) p+1 r 1+$ jl j r l Y j (t) Y l (s). Sice the processes Z i 's are uiformly bouded o [0, {] by the strog law of large umbers for i.i.d. processes (see e.g. HoffmaJorgese, 1994) we get the strog (almost-surely) uiform covergece of to its mea o [0, {] ; assumptio A3 follows. Agai, assumptios of Theorem 3.1 without lim + a =0 allows the use of Theorem.. For groups of various sizes, a sufficiet coditio to get A is k &1 lim \ i+ where i is the size of the ith group. max i =0, 1ik 4. PROOFS I the sequel we delete ; from our otatios wheever it is equal to ; 0, we use the otatio $ i&$ ij ad we recall that covergece results are give with respect to tedig to ifiity. Proof (of Theorem.1). We have a T () (t)=a t 0 M j (s&) \w ij& k=1 w jk p k (s) + dm i(s) (a) +a t 0 &a t 0 (M j(s&)&m j (s&)) \w ij& p j (s) \w ij& k=1 k=1 w jk p k (s) + dm i(s). w jk p k (s) + dm i(s) (b) (c)

8 90 BORDES AND COMMENGES Let us ote respectively a T () b ad a T () c processes are martigales ad (a T () b a { 0 \ 4RA 0 ({) a\ max (M j(s&)&m j (s&)) \w ij& 1i } the terms i (b) ad (c). These (})) ({) is equal to k=1 w jk p k (s) ++ d4 i (s) w ij (M j(s)&m j (s)) }+ =o P (1), from Lemma 5.1(iv); it follows by the Leglart iequality that Likewise we have (a T () c (})) ({)=a a T ()(t) b wp 0. t #[0,{] { 0 \ 4RA 0 ({) a\ max 1i p j (s) \w ij& k=1 w jk p k (s) ++ d4 i (s) w ij p j (s) + =o P (1), from Lemma 5.1(iii); it follows by the Leglart iequality that a T () (t) c wp 0. t #[0,{] We have the proved that a T () (t) is asymptotically equivalet to a T () (t)=a t H 0 i(s) dm i (s) (defied by the right had side of (a)) uiformly o [0, {], so we shall apply the Rebolledo theorem (see e.g. Aderse et al., p. 8384) to a T (). First ote that from Lemma 5.1(ii), (a T () = (})) (t)=4a t 0 H i (s) 1(a H i(s) >=) d4 i (s) w P 0, ad the, the first assumptio of the Rebolledo theorem is satisfied. To fiish the proof we have to show that there exists a determiistic fuctio v such that (a T () (})) (t) v(t) for t #[0,{]. For this we ca see that (a T () (})) (t)=4v () () 1 (t)&4v (t),

9 ASYMPTOTICS FOR HOMOGENEITY TESTS 91 where, usig the fact that Y i (s) M i (s&)=(&1) 4 i (s) for =1,, we have V () 1 (t)=a =a t Y i (s) r i M j (s&) M k (s&) w ij w ik da 0 (s) k=1 0 t Y i (s) r i M j (s&) M k (s&) w ij w ik $ jkda 0 (s) (3a) k=1 0 +a t Y i (s) r i w (M (s&)&4 ij j j(s)) $ ijda 0 (s) 0 (3b) +a t Y i (s) r i w ij 41+$ ij j (s) da 0 (s) (3c) 0 ad V () (t)=a t 0 \ p i (u) w ij M j (u&) + S (0) (u) da 0 (u) = a t 0 \ + a t 0 \ Y i (u) r i w ij M j (u&) + Y i (u) r i w ij $ i jm j (u&) + 0 (u) s (0) (u) du \ S (0) (u) & 1 s (u)+ da 0(u) (0) (4a) (4b) & a t 0 +a t 0 \ k=1 Y i (u) r i 4 i (u) Y j (u) r j w jk $ jkm k (u&) 0(u) du (4c) s (0) (u) 1 Y i (u) r i 4 i (u) + 0 (u) s (0) (u) du (4d) Sice '=lim + a exists, Y i(s) 4 i (s)=y i (s) r i A 0 (s), from A3 ad A4 we get immediately that 4_(3c)&4_(4d) coverges to! (t)=4 t 0 \ #(u)&' (s(1) (u) A 0 (u)) s (0) (u) + 0(u) du. We shall achieve the proof by showig that terms (3a), (3b), (4a), (4b) ad (4c) ted to 0 i probability.

10 9 BORDES AND COMMENGES From A1A we get M i (s) 1+RA 0 ({) ad we kow that w ij 1. It follows that } a k=1 a 3 R(1+RA 0 ({)) Y i (s) r i M j (s&) M k (s&) w ij w ik $ jk} ad from calculatios of the appedix, both coditios (i) ad (ii) of Propositio 5.1 are satisfied. Hece term (3a) teds to 0 i probability. Let us remark that M j=m j &4 j are idepedet martigales uiformly bouded by (1+RA 0 ({)) +RA 0 ({)(+RA 0 ({)), thus } a Y i (s) r i w ij M j(s&) $ ij} a (+RA 0({)). The above result with calculatios of the appedix allow, oce agai, to use Propositio 5.1 ad so we get that term (3b) teds to 0 i probability. Term (4a) is bouded by u #[0,{] } S (0) (u) & 1 s (u)} a (0) t 0 \ Y i (s) r i w ij M j (s&) + da 0 (s). The remum teds to 0 i probability from A4 ad the we show for the remaider term (i a similar fashio as for (3a) ad (3b)) that coditios (i) ad (iii) of Propositio 5.1 are satisfied. Thus (4a) teds to 0 i probability. Sice 0 s (0) is bouded o [0, {] we apply Propositio 5.1 to (4b) ad (4c) as above to show that these terms ted to 0 i probability. Theorem.1 is proved. Now to prove Theorem. we eed the followig lemma. Lemma 4.1. Uder assumptios A1-a, A ad B1B, for all t #[0,{] we have a T () (;, t)=a T () (; 0, t)+a (; &; 0 )$ ; T () (; 0, t)+o P (1), if ; satisfies the asymptotic coditio 1 (; &; 0 )=O P (1).

11 ASYMPTOTICS FOR HOMOGENEITY TESTS 93 Proof (of Lemma 4.1). By a secod order Taylor expasio of T () (;, t) aroud ; 0 we get for t #[0,{] a T () (;, t)=a T () (; 0, t)+a (; &; 0 ) ; T () (; 0, t) (; &; 0 )$ a ;;$ T () (;*, t) 1 (; &; 0 ), (5) where ;* belogs to the lie segmet with extremities ; 0 ad ;. Let B(; 0, =) be the ball subset [; # R p ; &;&; 0 &=] of R p. Sice ; is cosistet, ;* is cosistet too, therefore we have just to show that ;*#B(; 0, =) " a ;;$ T () (;*, t) "=o P(1). We have the followig decompositio a ;;$ T () (;, t) =& a & a t 0 t z i 0 H i (;, s)z i 4 i (;, ds)& a ;$ H i(;, s) 4 i (;, ds)+ a t 0 t 0 ; H i(;, s) z$ i 4 i (;, ds) ;;$ H i(;, s) M i (;, ds). From A1, B ad for ; # B(; 0, =), there exists a fiite costat K such that " a t H i (;, s) z i 4 i (;, ds) " 0 K max 1i the, from Lemma 5. (iv) we get a H i (;, s), (s, ;)#[0,{]_B(; 0, =) (6) ; # B(; 0, =) " a t 0 H i (;, s) z i 4 i (;, ds) "=o P((a ) &1 )=o P (1). Similarly, usig Lemma 5. we show that the three other terms i the right had side of (6) coverge i probability to 0, which achieves the proof of Lemma 4.1. K

12 94 BORDES AND COMMENGES Proof (of Theorem.1). Uder regularity coditios A1-a ad B1B4 the Cox estimator ; of ; 0 satisfies 1 (; &; 0 )=O P (1) (see Aderse ad Gill, 198), therefore Lemma 4.1 may be applied. Let U () (; 0, } ) be the martigale process defied by U () (; 0, t)= t 0 \ z i& S (3) S (s)+ dm i(s). (0) Usig our Theorem.1, Theorem 8..1 i Flemig ad Harrigto (1991) ad the Rebolledo theorem we get that the process V () (})= \ a T () (; 0,}) &1 U () (; 0,})+ is asymptotically gaussia if there exists a determiistic vector fuctio defied o [0, {] such that for all t #[0,{] (a T () (; 0,}), &1 U () (; 0,})) (t) coverges poitwize i probability to that fuctio. By the stadard techique we used i the proof of Theorems.1 we get where (a T () (; 0,}), &1 U () (; 0,})) (t) = &1 a t 0 \ z i& S(3) S (s)+ H i(s) d4 (0) i (s) w P \(t)=&' 1 t 0 \ &\(t), s(4) (u)& s(1) (u) s (0) (u) s(3) (u)) da 0 (u). (7) Now we ca use the followig represetatio (see e.g. Flemig ad Harrigto, 1991, p. 99) for a T () (;, {), a T () (;, {)= \1, &1 a ; T () (; 0, {) I () (;*, {) + V() ({)+o P (1), where ;* is o a lie segmet betwee ; ad ; 0 ad I () (;*, {) w P 7 &1 0 (; 0, {). The, by the Slutsky theorem ad the asymptotic ormality of V () ({) we get that a T () (;, {) is asymptotically ormal with variace _ ({)=(1, \$({) 7 &1 0 (; 0, {)) \! ({) &\({) =! ({)&\$({) 7 &1 0 (; 0, {) \({), &\({) 7 0 (; 0, {)+\ 1 (; 0 0, {)\({)+ 7 &1

13 ASYMPTOTICS FOR HOMOGENEITY TESTS 95 where! is defied i Theorem.1. This achieves the proof of Theorem (.). K Proof (of Theorem 3.1). For A1 ad B1B4 it is eough to follow the lies of the proof of Theorem i Flemig ad Harrigto (1991). For A3 we ca write where Let us defie where a & =a & (s, t)=a (s, t)=a 1(t)= t (s, t) 0 (s) ds, 0 w ij r ir j Y i (t) Y j (s) 4 $ ij i (t). w ij $P ijr i r j Y i (t) Y j (s), &. Usig B ad lim + a =0, it is easy to see that (s, t)& (s, t) w P 0. (s, t)#[0,{] Moreover we have E( (s, t))=k(t) K(s), where K(t)=E(exp(;$ 0 z)1(t 7 Ct))=G(t) E(exp(;$ 0 z&a 0 (t) exp(;$ 0 z)), where G is the survival fuctio of C. By the Lebesgue Theorem the fuctio K is cotiuous ad o-icreasig. Now, with lim + a =0 straightforward calculatios lead to lim 0 E(( (s, t)&k(s) K(t)) )=0, the, for all (s, t)#[0,{] we have (s, t) w P K(s) K(t). By Propositio (5.) we get the expected result. K 5. TECHNICAL RESULTS Lemma 5.1. Uder assumptios A1, A ad A4 we have (i) (ii) max 1i a 1 w ijm j (s) =o P (1). max 1i a 1 H i(s) =o P (1).

14 96 BORDES AND COMMENGES (iii) max 1i (a ) 3 w ij p j (s)=o P (1). (iv) max 1i a 3 w ij (M j(s)&m j (s)) =o P (1). Proof. (i) First ote that a 1 w ijm j are martigales o [0, {] with a1 w ij M j (}) (s)ra 0({) a max 1i w ij =o(1), from A. The result follows immediately by applyig the Leglart iequality (see e.g. Aderse et al., 1993, p. 86). (ii) We have just to ote that for H i(s) max 1i } w ij M j (s) }, ad the apply (i). (iii) The result follows immediately form A1, A ad A4. (iv) Processes a 3 w ij (M j(t)&m j (t))=&a 3 are martigales ad we have t k=1 0 \ w ij p j (s) + dm k(s) a3 w ij (M j(})&m j (})) (t) RA 0 ({)(a ) 3 from (iii), therefore \ max 1i w ij p j (s) + =o P (1), max 1i w ij (M j(})&m j (})) ({) wp 0. The result follows by applyig the Leglart iequality. K Lemma 5.. Uder assumptios A1A ad B1B we have for B(; 0, =)/B (=0)

15 ASYMPTOTICS FOR HOMOGENEITY TESTS 97 (i) there exists a positive fiite real costat K such that for (s, ;)# [0, {]_B(; 0, =) max \" ; p i(;, s) ", " ; p i(;, s) "+Kp i(;, s). (ii) We have max 1i (s, ;)#[0,{]_B(; 0, =) " w ij p j (;, s) "=o P((a ) &3 ), where is the operator idetity, or ; (iii) max 1i (s, ;)#[0,{]_B(;0, =) w ijm j(;, s) =o P ( &1 a &3 ). (iv) max 1i (s, ;)#[0,{]_B(;0, =) H i (;, s) =o P ( &1 a &3 ). Proof. (i) Sice we have ;;$. ; p i(;, s)=p i (;, s) \z i& z j p j (;, s) +, j the result follows from B. The same method ca be applied to ;;$ p i (;, s). (ii) This is a straightforward cosequece of (i) ad A-b. (iii) We have w } ij M j (;, s) } = } w ij M j (; 0, s)+ s w ij (exp(;$ 0 z j )&exp(;$z j )) Y j (u) 0 (u) du }. 0 From A1, B ad the fact that ; # B(; 0, =), there exists a fiite positive costat K such that the last expressio is bouded by max 1i } From Lemma 5.1(i) ad A-b we get w ij M j (; 0, s) max }+K 1i w ij. max 1i (s, ;)#[0,{]_B(; 0, =) } w ij M j (;, s) }=o P( &1 a &3 ). (8)

16 98 BORDES AND COMMENGES We have also 1 a 3 & 1 a 3 & 1 a 3 w ij M j(;, s)= 1 a 3 s k=1 0 \ s k=1 0 \ w ij M j (;, s) w ij p j (;, u) + M k(; 0, du) w ij p j (;, u) + (e;$ (9a) (9b) 0 z k &e ;$z k ) Y k (u) 0 (u) du. (9c) From (8) the right had side i (9a) is a o P (1). Moreover (9b) is a martigale with covariatio process bouded by K a\ 3 max 1i (s, ;)#[0,{]_B(; 0, =) w ij p j (;, s) +, where K is a positive fiite costat which arises from A1, B ad the fact that ; belogs to B(; 0, =); hece, from (ii) the last expressio has a stochastic order o P ( a 3 (a ) &3 )=o P (1), the, by the Leglart iequality we get that 1 a 3 max 1i (s, ;)#[0,{]_B(; 0, =) } s k=1 0 \ w ij p j (;, u) + M k(; 0, du) } teds to 0 i probability. Now, for ; # B(; 0, =), (9c) is bouded by K$(a ) 3 max 1i (s, ;)#[0,{]_B(; 0, =) } w ij p j (;, s) }, where K$ is a fiite costat. Therefore, by (ii) the last expressio has stochastic order o P (1) ad the result (iii) follows. (iv) Straightforward i view of (ii), (iii) ad the fact that p j (;, s)=1. K The followig propositio is a corollary of a result by Hellad (1983) (see e.g. Aderse et al., 1993, p. 86). Propositio 5.1. Let [X (s), 0s{<+] 1 be a sequece of processes defied o the same probability space. We cosider the followig coditios (i) for all, X (s) is almost surely bouded. (ii) E[X (s)] 0. (iii) E[ X (s) ]0.

17 ASYMPTOTICS FOR HOMOGENEITY TESTS 99 The, if (i) is satisfied with (ii) or (iii) we have t #[0,{] } t X (s) ds } wp 0. 0 Proof. We give the proof whe coditios (i) ad (ii) are satisfied sice (ii) implies (iii). Let (k ) 1 be a sequece of fiite real umbers such that X (s) k almost surely. We may pose (k ) 1 o decreasig ad k 1 for all. Let=>0 be a fixed real; we put C 0 =k 0 where 0 satisfies E [X (s)]<=c 0, 0, which is possible from (ii). Let, from (i), for < 0 I [ X (s) >C]=0 for all C>C 0 ad the we have max 1< 0 E[ X (s) I [ X (s) >C]]=0. Moreover, for all C>C 0 E[ X (s) I [ X (s) >C]] E [X (s)] C =. Thus we have proved that for all s0 lim C + E[ X (s) I [ X (s) >C]]=0. 1 The sequece (X ) 1 is equi-itegrable. Now, from (ii) it follows that for ay, X (s) P 0. From the CauchySchwarz iequality we have E[ X (s) ](E[X (s)])1 ( E[X (s)])1 1 for all greater tha a give $ 0 (from (ii)), thus, from (i) we have for ay 1 ad E[ X (s) ]max(1, max k j ). 1 j$ 0 Now we ca apply a result of Hellad (1983) which says E _ t #[0,{] } t X (s) ds 0. 0 }&

18 100 BORDES AND COMMENGES By the Bieayme Tchebycheff iequality we achieve the proof of the propositio. Remark 5.1. Nothig is chaged i the proof of Propositio (5.1) if X is multiplied by a bouded positive fuctio defied o [0, {]. The followig propositio is stochastic versio of the well kow Dii theorem. Propositio 5.. Let (F ) 1 be a sequece of processes defied o A=> p [a j, b j ]/R p (with &<a j <b j <+ for,..., p) ad F a cotiuous determiistic fuctio defied o A. Suppose that F ad F (1) are (almost-surely for the F ) o-icreasig with respect to each of their compoets o A. If F coverges poitwise i probability to F o A, therefore F (x)&f(x) w P 0. x # A Proof. We give the proof for p=1; calculatios for p1 are straightforward. Let =>0 ad '>0 be two reals. Sice F is cotiuous o a compact subset [a, b] ofr it is uiformly cotiuous o [a, b]. The there exists $=$(=)>0 such that F(s)&F(t) <=. [(s, t)#[a, b] ; s&t <$] Let a=t 0 <t 1 <}}}<t k =b be a partitio of [a, b] such that t i+1 &t i < $ for i=0, 1,..., k&1 (k=k(=)). We have It follows that F (t)&f(t) =+ max F (t i )&F(t i ). t #[a, b] 0ik P( F (t)&f(t) >=)P( max F (t i )&F(t i ) >=4). (10) t #[a, b] 0ik Now, for 0ik there exists i =(i, =, ') such that for all i P( F (t i )&F(t i ) >=4)'k. The, for $=max 0ik i we have for all $ P( max F (t i )&F(t i ) >=4)'. 0ik

19 ASYMPTOTICS FOR HOMOGENEITY TESTS 101 By (10) we get for all $ P( F (t)&f(t) >=)'. t #[a, b] The proof is complete. K Term (3a) APPENDIX Straightforward but paiful calculatios lead to v () (s)=e 1 _\ a R (1+RA 0 ({))) 4 a{ k=1 R (1+RA 0 ({))) 4 + k=1 R (1+RA 0 ({))) 4 +a\ 4 max 1i It follows from A that Y i (s) r i w ij w ik $ jkm j (s&) M k (s&) + & k=1 w ij w ik + w ij +1 k=1 l=1 { 4a +16a4 \ max 1i w ij w ik l=1 w lj = { 4a \ +16a4 max 1i w ij + 3 =. k=1 w ij w ik w lj w lk = w ij + w ij + w ij w ik w jk v (s)=o(a )+o \ a4 a 3 + +o \ a4 3 a + =o(1). 9 Hece we proved that coditio (ii) of Propositio 5.1 is satisfied for term (3a).

20 10 BORDES AND COMMENGES Term (3b) First, it is easy to show that v () (s)=e _\a R (+RA 0 ({)) 4 a 4 \ Y i (s) r i w ij$ ijm j(s&) + Therefore, as for the above lies we get w ij + & k=1 v () (s)=o(a)+o \ a4 a+ =o(1). 3 Thus coditio (ii) of Propositio 5.1 is satisfied. w ij w ik +. ACKNOWLEDGMENTS The authors gratefully ackowledge the Editor, associate editor, ad referees for their valuable commets that substatially improved the presetatio of the paper. REFERENCES 1. P. K. Aderse, O. Borga, R. D. Gill, ad N. Keidig, ``Statistical Models Based o Coutig Processes,'' Spriger-Verlag, New York, P. K. Aderse ad R. D. Gill, Cox's regressio model for coutig processes A large sample study, A. Statist. 10 (198), D. Commeges ad H. Jacqmi-Gadda, Geeralized score test of homogeeity based o correlated radom effect models, J. R. Statist. Soc. B 59 (1997), D. Commeges ad P. K. Aderse, Score test of homogeeity for survival data, Lifetime Data Aal. 1 (1995), T. R. Flemig ad D. P. Harrigto, ``Coutig Processes ad Survival Aalysis,'' Wiley, New York, I. S. Hellad, Applicatios of cetral limit theorems for martigales with cotiuous time, Bull. It. Statist. Ist. 50 (1983), J. Hoffma-Jorgese, ``Probability with a View Toward Statistics,'' Probability Series, Chapma 6 Hall, Lodo, X. Li, Variace compoet testig i geeralised liear models with radom effects, Biometrika. 84 (1997), E. T. M. Ng ad R. J. Cook, Adjusted score tests of homogeeity for Poisso processes, J. Amer. Statist. Assoc. 94 (1999),