5 Case of a bounded number of transitions

We now assume that the number of transitions is bounded by a finite number K. For each subject i = 1,■ ■ ■ ,n, the observation ends at time ti = ^2Xk(i) if K(i) = K or if JK(i) is an absorbing state, and at time tj where there is a right censoring in the K (i)th visited state, K (i) < K.

Using notations in (14), the likelihood term relative to the initial state j may be written l0n(j)= N°(j, n) \og(pj), the terms relative to the fully observed sojourn times in state j is n(j) = £ { N j,jn) \og(pjj,) j'=i n K

+ ££Nik(j,j')[log(X\jj'(Xk(i))) - A\jj'(Xk(i))] }, i=1 k=1

Inference for a general semi-Markov model and a sub-model 239 and the term relative to the censored sojourn times in state j is lCn (j ) = £ NjogiE Pj (i))}].

The score equations for pjj' and Ajj' do not lead to explicit solutions because they involve the survival function Fj and the transition function F j/1 j. We define estimators pn,jj' and An, | jj/ by plugging in the score equations the Kaplan-Meier estimator of F j and the estimator of Fj/1 j given by Gill [GILL80],

We obtain the estimators pn,j

1 = N(j,j',n) + Nc(j,j', n) P1n'jj' Nnc(j, n)+ Nc(j, n) , dN (y,j,j',n)

with

? c(y,j,j' ,n) = £ YC(y,j) Fj(X *(i)), i=1 Fnj (X*(i))

The variable (n

1/2 (Pnj - Pj))j' and the process (n 1/2 (An | jr -

are asymptotically Gaussian, on every interval [0, t] such that fT (F°f I j G )-1 dA° l j is finite [PONS04].

6 A Test of the Hypothesis of Independent Competing Risks.

In the ICR case, the initial probabilities jointly with the survival functions Fijj' of the sojourn times conditional on states on both ends, are sufficient to determine completely the law of the process. In the general case, however, the two sets of parameters pjji and F| jji are independent and may be modeled separately. Our aim is to derive a test of the hypothesis of Independent Competing Risks (ICR):

H0 : The process is ICR H1 : The process is not ICR

The Kaplan-Meier estimator Fnj of Fj, given in (16), and the estimator Fn,ji lj of Fjij, given in (17), are consistent and asymptotically Gaussian both under Ho and under Hi. It is also true for the straightforward estimator Fn,j of the initial probabilities. From those estimators, one may derive general estimators of the transition probability pjji and of the survival function F j of the time elapsed between two successive jumps in states j and j'. For these estimators, we shall use the same notations as the estimators of pjji and F j defined in section 5, though they are now given by pn,jji = max Fnjij (t) (19)

pn,jjl

In the independent competing risk model, the transition probability Fjij satisfies (10) and thus may be estimated as where

Hj" pn,jj" Jo j,,=ji j (t)= f 1{Y (s,j,n) > 0} dN^J\J'f (22)

is the estimator of the cumulative hazard function Anjilj in the general model. A competitor to pn,jji is deduced as pRCji = max Fj (t). (23)

Let njO be the mean number of sojourn times in state j for subject i.

Proposition 2. If pj > 0 and f0°°{G°(s)F°(s)}-1 dA°(s) < <x>, then \/n(pn,jji — p0ji ) is asymptotically distributed as a normal random vector with mean 0, variances and covariances t

ajf = ~o =0—=0— [(Fj' Ij(s) —Pf Ij) "=0-

2 1 H 1 \(~ûP , ^ o snfi , ^ o ,dF°(s) an'f = Z0 s [(Fj' Ij(s) - Pf IJ )(Fj" I J(s) - Pf I J

+(F% j (s) - p 0 ' I j} dFf" jj (s) + (Fj" j (s) - pf" j )} j (s)].

Moreover, \/n(p>RR'Cj, — pf j,) is asymptotically distributed as a centered Gaussian variable.

Estimators of the asymptotic variance and covariances of (pn,jj')j'eJ(j) may be obtained by replacing the functions Ff, Fjf,| j and Af,| j by their estimators in the general model, (16), (17) and (22). Due to their intricate formulas, it seems difficult to use an empirical estimator of the asymptotic variance of PR(jj' and a bootstrap estimator should be preferred. Asymptotic confidence intervals for pf j, at the level a are deduced from the (1 — a/2)-quantile ca of their boostrap distributions, In,jj' (a) in the general case and IR(jj' (a) under the null hypothesis of Independent Competing Risks.

A test of the Independent Competing Risks hypothesis may be defined by rejecting H0 if Injj' (a) and iR'j' (a) are not overlapping for some j'. As the

Lamctci^ 0

with critical region estimators of the parameters p0j' are not independent, the level a* of this test

Rnj(a) = rÇ=1Rnjr (a), where Rnjf (a) = {InJj, (a) n If (a) = 0} satisfies a* > 1 - (1 - a)m.

7 Proofs

Let Tnj = arg maxt Fn,j (t). The asymptotic behavior of pn,jj' is derived fiom theorem 3 in Gill [GILL80] which states the weak convergence of the process

(Vn(Fu,j' I j (t A Tnj ) - Ff I j (t A Tnj ))j' ej (j), Vn(F n,j (t A Tnj ) - FJ (t A Tnj ))t> 0

to a Gaussian process defined, for continuous transition functions Ff I j, as jft Ff I j (s) dVjj' (s) Fo f dVj (s) + f F01J (s) dVj (s) \Jo EYi(s,j) fIj()Jo EYi(s,j)+Jo EYi(s,j) ,

where Vjj' ,j,j' G {1, 2, ■ ■ ■ , m} is a multivariate Gaussian process with independent increments, having mean 0 and covariances varj (t))

cov(Vjj, (t), Vjj" (t)) =0 if j' = j" and covj (t), Vjj (h)) =0 if j = j or ti = t, and Vj = £j, Vjj,.

As EYi(s,j) = n0G0(s)F0(s), it follows that y/n(j>n,jj> — p0j,) is asymptotically distributed as

oo and ajj, is the variance of A — B + C. The covariance ajj,, is obtained by similar calculations, but the covariance between the corresponding terms A(jj') and A(jj") is zero.

From (21), the asymptotic Gaussian distribution of \/n(pRCjj' — pj0j') is a consequence of the asymptotic behavior of the estimators Fnj and Fn,j'\j and

Inference for a general semi-Markov model and a sub-model 243 of the estimator An,j'| j given by (22), using again theorem 3 in Gill [GILL80]. Limiting covariances.

The limiting covariance of \/n(p>RR(Cj' — pj,') may be calculated using the following expressions

, 1 r 1 ( —j „ dFj(s j' (t) = -j —j —j / n {(Fj' | j (s) — Fj' | j (t))2 j 3 nj Jj GAs)FAs)V JU JU Fj(s)

+{Fj(s) + 2(Fj' j (s) — Fj' j (t))} dfr j (s)}, o 1 t'1 1 ( — j —0—0 —o dFjj(s)

3) = ^ ^ {(F°' | 3 (s) — Fj' 13 WXj J (s) — 3 3 (t)) j nj Jj G°(s)F°(sy F°(s)

(F°°'13 (s) — Fj'| J (t)) Fj"J (s) + (F°°J (s) — F0°^J (t)) dFJ'|

cj (t) = limn Cov{Vn(Fn,j (t) — Fj (t)), ^(Fnjf (t) — fJ, (t))}

J1,(t) = limVar^{Fri,j(t-^ Fn^j(t) — Fj(t-) fl Fjj(t)}

= 1im { n Pn^j (t)}2{ E Var ^n^j (t) — jj (t)}{ F3)- }2 jl =j' j2=j' F j21 j(t)

xCov{Vn(f n ,j2|j (t) — Fj2|j (t)), Vn{Fn ,j3\j (t) — Ffj (t))}

+ H Cov{Mf nj (t-) — F0 (t)), jn(fnMj (t) — jj (t))}

j2=j' j'3=j' ,32 F ,2 \j F j3 \j (t) j2=j' Fj2j (t)

and, for any sequence An, converging to A,, lim VarJn((JJ Anj - JJ A() = E JJ A2' lim nVar(Anj - A() j j j j'=j

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