M k x M kAxM kB x

and the asymptotic distribution of the vector (LR\, LR2)' will be derived from he asymptotic propertie n-1/2M 2A,nB1/2M 2b )'.

the asymptotic properties of the martingales vector (n—1/2M1A,nB1/2Mib ,

2.1 Preliminary results for the martingales under H0

Rebolledo's theorem ensures the convergence of the vector (n-1/2M 1a, n-1/2Mib,n—1/2M2A,n-1/2M2b)' to a Gaussian process (miA,miB,m2A, m2B)', with null expectation and with variances vj(x) where vkj (x) = Jim^ E^n- 1M kj (x)|

The covariance between m^A and m^B is null because subjects in group A are independent of those in group B. But we have to express the covariance between m1j and m2j for j = A, B. Let v12j (x1,x2) = E {m1j (x1)m,2j — )} .

By definition, v12j(x1 ,x2)= lim E{n-1Mj(x{)M2j — )}

A development of dM1ji(u)dM2ji(v) shows that E {dM1ji(u)dM2ji(v)} is equal to

E {dN1ji(u)dN2ji(v)} - dA1(u)E {Yji(u)dNji(v)} -d,A2(v)E {Y2ji(v)dN1ji(u)} + dA1(u)dA2(v)E {Yji(u)Y2ji(v)} .

The first term is

Pr {Tj e [u,u + du\,T2ji e [v,v + dv],Cj > Tji, C2ji > T2ji} , that is equal to f (u,v)G(u,v)dudv.

The second term is

-dA1(u)Pr {Tji > u,Cj > u,T2ji e [v,v + dv],C2ji > T2ji}

that equals G(u,v)F(u,dv)dA1(u), and symmetrically the third term equals G(u, v)F(du, v)dA2(v).

The last term is simply G(u,v)dA1(u)dA2(v)F(u,v). Finally,

Let n(u,v) = F(u,v)G(u,v) = E {Yji(u)Y2ji(v)}. Then (3) can be written j as u,v)

that allows to establish a consistent estimator of each fraction in the above expression. The first numerator can be consistently estimated by

an estimator of the second one is

and symmetrically F(du,v)G(u,v) has for consistent estimator

Weighted Logrank Tests With Multiple Events 383 last, the denominator can be estimated by n ^2Yiji(u)Y2ji(v) = n ir(u,v).

Finally, noticing that dMiji(u)dM2ji(v) is null if subject i is not at risk in u for event 1 and in v for event 2, a consistent estimator of (3) is

Yiji(u)Y2ji(v) j j f d,Niji(u)d,N2ji(v) - dh(u)Yijl(u)dN2ji(v)

r(u,v) -dA2(v)Y2ji(v)dNiji(u) + dAi(u)dA2(v)r(u,v) j '

with Ak the Nelson-Aalen estimate ([NEL69], [AAL78]) of the common cumulative hazard function for event k, k = 1, 2:

Introducing the martingale residuals

o the above estimator is simply

E {dMiji(u)dM2ji(v)} = Yiji(u)Y2ji(v) jj jr dMiji(u)dM2ji(v).

Consequently, an appealing estimator of vi2j (du,dv) is nj vi2j(du, dv) = n-iJ2 E {dMji(u)dM2ji(v)} (5)

where n- Y,Mi(u)dM2ji(v), j=A i=i rj (u,v) = E Yiji(u)Y2ji(v).

2.2 Asymptotic distribution of (LR-^^LR^)' under H0

The above results will help us to establish the asymptotic distribution of the vector (LRi, LR2)' under H0, and more particularly to express the covariance between its two components. But first consider

J0 I yk yk under H0, LRJ is an asymptotic equivalent to (2) because of the convergence in distribution of the martingales to Gaussian processes, and because of uni-

form convergence of WkYk Ykj — wkpjyk ykj to 0 for k =1, 2 and j = A, B. Then the asymptotic distribution of (LRi,LR2)' will be the asymptotic distribution of (LR*,LR2)', which is easier to derive because the only random terms in LR* and LR* are the martingales.

LR*k can be written fT / (nA \1/2 ykB -1/2^ (nB \i/2 ykA -1/2 jtt wk< Pb ( — nA dM kA — PA[--nB dM kB

Since (n-1/2MiA,n-1/2Mi— ,n—1/2M2A,n-1/2M2—)' converges in distribution to the Gaussian process (m1A,m1B,m2A,m2B)' previously defined, and since nA/n and n— /n have limits pa and p—, we can conclude that (LR*, LR*)' converges in distribution to the Gaussian vector f0T wi |p— P-2 dmiA — PAP—2 ^^dmi—

This vector has null expectation because (mkA, mk—)' is centered for k =1, 2.

Let S2 = (a*k'), k = 1, 2, k' = 1, 2, be the variance covariance matrix of (LR2, LR*)'. The variance of LR2k is:

J0 yk f 2 ykAykB ia

Weighted Logrank Tests With Multiple Events 385 The covariance between LRJ and LRJ is:

Jo Jo Vi V2

So the asymptotic distribution of the vector (LR\,LR^)' is a Gaussian law, with null expectation and with variance covariance matrix S*. A consistent estimator of the variance is simply

Jo Y k which is the classical marginal variance estimator of the logrank statistic. And inserting (5) into (6), we obtain a consistent estimator of a*2 :

Ô-12 = —[ ( Wl(u)W2(v)(u)(v)v12A(du, dv) n Jo Jo Y i Y2

Jo Jo I [Y1B (u)Y2B (v)rA(u,v) + Y a(u)Y2a(v)tb (u,v)\

Notice that the above formula allows to find the variance estimator akk. Actually, write an as j- T Ç T ( Wk (u)Wk' (v)T, B=AT,ZI dMkjl(U)dMk,jl(V) Ï

and let k = k'; in this case, YlB=A 1 dMkjl(u)dMkjl(v) is null unless u = v, and then rj(u,u) = Ykj(u) and r(u,u) = Yk(u). So

Since

/T w2 (_2 _ _2 _ ^ ^ r ^ i wr {ykBYkA + YkAVkBj d,Ak{ 1 - dAkj (8)

J0 Y k as we supposed Ak continuous on [0,rj.

Therefore, the asymptotic distribution of the vector (LR1, LR2)' is a Gaussian law, with null expectation and with variance covariance matrix consistently estimated by U = (akk>), k = 1, 2, k' = 1,2, where akk> is defined by (7). A test statistic for H0 : {AA = AB = A1, AA = AB = A2 on [0, r^ is then

which asymptotically follows under Ho a chi-square distribution with two degrees of freedom.

2.3 What if the joint censoring distributions or the joint survival functions differ in groups A and B under H0 ?

If one suspects that the censoring distribution in group A differs from the one in group B, or that a copula model for the joint survival function is not appropriate, the above results have to be modified. For j = A, B, let Fj and Gj be the joint survival functions of {(Tiji,T2ji),i = 1,...,nj} and

{(Ciji,C2ji),i = 1, ...,nj} respectively, and let Fjk and Gjk be the respective marginal survival function of {Tkji,i = 1, ...,nj} and {Ckji,i = 1, ...,nj} for event k. Let also fj denote the density function of {(Tiji,T2ji),i = 1, ...,nj}. Then (n—/2M 1A,nB1/2M1B,nA1/2M2A,nB1/2M2B)' converges in distribution to a centered Gaussian process with variances vkj (x) = Vkj (u)dA k (u), o and with covariance between n- 1/2 M kj and n- 1/2 M fj for k = 1, 2 and j = A, B now equal to

vi2j(xi,x2)= / E {dMlji(u)dM2ji(v)} Jo Jo fXl ¡'X2 G ( )< fj (s,u)dsdu + dA i(s)Fj (s,du)

Weighted Logrank Tests With Multiple Events 387 Estimating consistently E {dMiji(u)dM2ji(v)} by

E {dM1ji(u)dM2ji(v)} = ^fUYj^ j dMiji(u)dM2ji(v), a consistent estimator of vyij (du, dv) is now vi2j(du, dv) = n-1 jt £ dMijt(u)dM2jt(v)

= n-1J2 dMiji(u)dM2ji(v), i=i with the martingale residuals Mkjl defined by (4).

Then the vector (LRi, LR2)' converges in distribution to a Gaussian law, with null expectation and with variance covariance matrix S = (ffkk'), k = 1, 2, k' = 1, 2, being consistently estimated by S = (akk'), k = 1, 2, k' = 1, 2, with &kk' now expressed as n-i r r Wk(u)Wk' (v) f YkB (u)Yk'B (v)J2nAi dMkAi(u)dMk'Ai(v) \ n Jo Jo Yk(u)Yk'(v) +YkA(u)Yk'A(v) Y^Bi dMkBi(u)dMk'Bi(v) \ '

Notice that for k = k', £n= 1 dMkji(u)dMk/ji(v) is no more equal to zero if u = v; more precisely, for u < v, it equals nj

J2dMkji(u)dMk'ji(v) = dAk(u)Ykj(v) jdAk(v) — dAj(v)|

where

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