Hjm1leE0 amtjmwmi zi

where Em—i is the conditional expectation of Z0 with respect to lm—i(dzo\wm—i, zi). Under the assumption of the proportional hazard model, we have

Pr(Tm >t\Zi ,Wm—i)= Fm(t,Jm—i \Wm—i,Zi) =E

m exp — ep Zh / ah(u)du h:h=(Jm-i,l)eEo J(Tm-i,t]

In addition

Pr[Jm = j\Tm = t, Wm—i, Zi] = qm (j\Tm = t, Wm—i, Zi)

aJm-ij (t)Em—i[epT Zjm-l'j \Tm > t,Wm—i,Zi ]

El ajm_ui(t)Em—i[epTZjm-i'l\Tm > t, Wm—i,Zi]

and am(t,jm\Wm-i,Zi) =\im1Pr(Tm e [t,t + s],Jm = j\Tm = t,Wm-!,Z1]

= l(Tm > t > Tm-1 )ajm_ltj(t)Em-i[e/T\Tm > t,Wm-i,Zi]

The cumulative intensity of the process [Nh(t) : h e E0] with respect to the marginal filtration o~(Zi) ® Ft is given by

Thus marginal process has a compensator of a different form than the original process.

Next we consider the MAR condition 2.2. It is equivalent to the following two conditions.

(i) For m > 0, the conditional distribution of the missing data indicator satisfies

= vm(r\wm,z(r)) if jm eA,tm < t, m > 1 = vm(r,T\wm,z(r)) if jm eT,tm < t < Tm+i,m > 0

for some functions (vm,vm) depending only on the sequence wm and the observed covariate z(r) = (zo(r),zi), but not the missing covariates. In addition,

r r where the sums extend over possible values of the missing data indicators. (ii) The parameters of the conditional distribution of the missing data indicators, (vm,vm > 0) are non-informative on the parameters of the underlying model of interest.

The joint density of the vector (V, Z0(R), R), V = [N. (t), (Je, T^)^\ zi] is given by vm(r\wm,z(r))pm(wm\z(r))~p0(dz0 (r)\zi,jo)

if jm e A,tm < T, N.(t)= m > 1 Vm(r,T\wm,z(r))pm(T, wm\z(r))~p0(dzo(r)\zi,jo)

Here ~^o(dzo(r)\zi, jo) is the marginal conditional distributions of the covariate Zo(r) given (Zi,Jo). In addition, z(r) = (zo(r),zi) and for any sequence 0 = to <ti <t2 ... < tm, we have m

Pm(wm\z(r)) = JJ fl (ti,ji\wi-i,z(r)) if jm eA,tm < T,m > 1

Pm(T,Wm\z(r)) = Fm+i(r, jm\wm, z(r)) JJ fi(ti ,ji\w—1 ,z(r))

if jm eT,tm < T <tm+1,m > 1 = fi(t, jo\wo, z(r)) if T<ti,m = 0 .

The function pm(wm\z(r)) is the joint conditional subdensity density of a sequence Wm = ((Ti, Ji) : t = 0,..., m) terminating in an absorbing state, and evaluated conditionally on the initial state and the covariate z(r) in the marginal model obtained by integrating out the covariate z(r). Similarly, the function pm(T,wm\z(r)) is the joint marginal conditional subdensity of survival in a transient state.

If parameters of the functions vm,Fm do not depend on the Euclidean parameter of the Markov chain model, then in Section 2.3, the complete data likelihood is of the form n

Lik(v,V) = n v(Vk ,Zo,k (Rk ))p(Rk ,Vk ,Zo,k (Rk); 4) . (10)

For each subject, Z(R) = (Z0(R),Z1), v(V,Zo(R)) = (vn{t )(R\Wn {t ),Z(R)))1(jn-(t )€A) x (Fn.(t)(R,t\Wn,t),Z(R)))1jn■(t)eT)

x yFN. (t ) + 1(t,Jn. (t ); 4\Wn, (t ),Z(R))j x Fo(Zo(R)\Z1, Jo) , where Fo(:\Z1, Jo) is the marginal conditional density of Zo(R) given (Z1, Jo).

In the case of randomly censored data, the likelihood factorization (10) can be derived in an analogous fashion. We omit the details.

Acknowledgement. Research supported by the National Cancer Institute grant 1-R01-96-CA65595-02.


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