ESPERANZA PEDAGÓGICA

The terminal event in an illness-death process is often subject only to administrative censoring. Progression may be right-censored at an earlier time (Example1.28), yield- ing what we call “doubly right-censored” data. Bebchuk and Betensky (2001,2002) combine local likelihood and multiple imputation to estimate the marginal distribu- tion of the event times. They further constructk-sample tests on the basis of a Markov Cox-type model with constant baseline transition intensities and time-dependent covariates (Bebchuk and Betensky2005). Yuan et al. (2012) apply Bayesian methods to estimate covariate effect on the marginal distribution of event times within the family of parametric accelerated failure time models (1.19). Ke et al. (2011) devise a generalized Kaplan-Meier estimator for the subdistribution function of the exit time from the initial state.

This chapter considers the estimation of parameters specifying a Markov model with Cox-type transition intensities (Figure3.1). Using the Fisher scoring algorithm

0 Entry 1 Progression 2 Death α01(t∣z) =λ01(t)exp(zT01θ) α02(t∣z) =λ02(t)exp(zT02θ) α12(t∣z) =λ12(t)exp(z12Tθ) figure 3.1 A Markov progressive illness-death model.

proposed by Kalbfleisch and Lawless (1985), Jackson’s (2011)msmpackage for r readily

handles the time-homogeneous case,λhj(t) =λ_hj, and can extend to piecewise expo-

nential intensities by way of time-dependent covariates. A relatively flexible piecewise exponential estimator is developed via sieve maximum likelihood.

3.1 model and observation scheme

Consider a trivariate processN= (N₀₁,N₀₂,N₁₂)counting the 0→1, 0→2 and 1→2

3.1 model and observation scheme

inf{t∶N_hj(t) =1}denote theh→ jtransition time andY_h(t) =1− ∑_j≠hN_hj(t−)be

theh→ jat-risk process. Assume that eachN_hjhas intensity processY_hα_hjwith

αhj(t∣Z) =λ_hj(t)exp(Z_hj⊺θ), (3.1)

whereZhjis a transition-type–specificdz-vector based on the fixed covariateZ,θis a

dz-variate regression parameter andΛhj=

∫

λ_hjis a nondecreasing baseline intensity

function. The parameterθis common to each of the transition intensities, butZhj

can be suitably constructed fromZto give type-specific covariate effects (Andersen et al.1993, pp. 478–80).

Over the finite interval[0,τ], suppose that observation of N is subject to the

right-censoring times 0<C≤D≤τ;N₀₁is known on[0,C]andN₀₂+N₁₂on[0,D].

LetS=T₀₁∧T₀₂denote the exit time from the initial state andT=T₀₂∧T₁₂the time

of death. PutU =S∧C,V =T ∧D,∆₀ =1(S <C)and∆₂= 1(T <D). Then the

transition times(T₀₁,T₀₂,T₁₂)for an observationX= (U,V,∆₀,∆₂,Z)are available

when ∆0 = ∆₂ = 1. In general the progression status 1(S < T)is known only if S

occurs beforeC; that is,∆0=1. Otherwise(U,V)is a “potential” censoring interval

forT01. Figure3.2illustrates this notation for two individuals,iand j. Exact data are

available for subject j. The observation foriis doubly right-censored.

0 Ci Ui=Ci,∆i0=0 Ti 01 T₀₂j Uj=Vj=T₀₂j ,∆₀j =∆₂j =1 Ti 12 Vi =T12i ,∆2i =1 Di Cj=Dj τ figure 3.2 Observation of an illness-death process under double right censoring.

LetAhj

∫

αhjforh≠ jandA_hh= − ∑_j≠hAhj. Then from Theorem1.4the transi-

tion probabilities are

Phh(s,t∣z) =exp{

∫

t s Ahh(dy∣z)}, (3.2) where P01(s,t∣z) =

∫

t s P00(s,y∣z)A01(dy∣z)P11(y,t∣z). (3.3)

Assume the following basic condition.

Then the realizationX=x = (u,υ,δ₀,δ₂,z)has density

pθ,Λ(x) =P₀₀(0,u∣z)[α₀₁(u∣z)P₁₁(u,υ∣z)α₁₂(υ∣z)δ2]δ01(u<υ) × [P₀₁(u,υ∣z)α₁₂(υ∣z)δ2 +P₀₀(u,υ∣z)α₀₂(υ∣z)δ2]1−δ0

×α₀₂(υ∣z)δ0(1−1(u<υ))δ2 (3.4)

with respect to a dominating measureνdetermined by the distribution of(C,D,Z).

3.1 remark. The expression in (3.4) is the same likelihood function obtained under (conditionally) independent censoring byCandD(Example1.1). Such a mechanism permits dependence on the observed history. The stronger requirement inb1simpli- fies the derivation of asymptotic properties. It is plausible under an intent-to-treat analysis in whichC represents loss to follow-up forS,Zadequately explains varia- tion inC, andDis an administrative censoring time forT. This precludes censoring individuals at the time of a change in treatment due to toxicity or need for addi- tional therapies—a scenario that likely induces dependent censoring (Fleming et al. 2009). ◽

3.2 sieve maximum likelihood estimation

LetXi = (U_i,V_i,∆i₀,∆i₂,Z_i),i =1, . . . ,n, beniid observations of Xfrom(θ₀,Λ₀),

Λ0 = (Λ0_hj) for h ≠ j. Note that _Pnlogp_θ,Λ a priori maximizes to infinity; with,

say,Λ02 continuously differentiableα02(V_i ∣ Z_i)can be made arbitrarily large and

A00(V_i ∣Z_i)close to zero at anyV_i =T₀₂i <C_i. The usual way out is to replaceα_hjby

the jump discontinuities ∆Ahj. However consider an individualihaving unknown

progression status(∆₀i =1)but known survival time(∆i₂=1):

Ui =C_i <V_i =S_i <D_i.

Suppose(L,V_i]is a subinterval of(U_i,V_i]containing no other observation times

from the sample. Surely we needΛ02(V_i) +Λ₁₂(V_i) −Λ₀₂(L) −Λ₁₂(L) >0, but the

available data are insufficient to jointlyestimate Λ02(V_i) − Λ₀₂(L) and Λ₁₂(V_i) −

Λ12(L). Thus no unbiased semiparametric estimator of(θ,Λ)exists. This problem is

evaded by employing the method of sieves (Section1.2.2) on the basis of the following assumptions.

b2 There exist 0 <σ <τand 0<M < ∞such that 1/M <Λ0_hj(σ−) <Λ0_hj(τ) < M,

3.2 sieve maximum likelihood estimation

b3 Letnhjdenote the number of individuals in the sample with{T_hj< ∞}observed

exactly,h≠ j. Then there existq_hj>0 such thatn_hj/n→q_hjasn→ ∞.

LetH = (H_hj)denote the set ofΛ = (Λ_hj)with eachΛ_hj ∶ [0,τ] → [0,M]cadlag

and nondecreasing. Any finite-dimensional approximation toHwhose size increases withnis asieve. Throughout consider the piecewise exponential sieve given by the set of piecewise linear interpolants ofΛ∈H.

3.2 definition. For each h → j,h ≠ j, let T_hj,n be a set containing the K_hj,n =

O(nκ), 0<κ<1, points in(0,τ)from the partition

0=t₀<t₁< ⋯ <t_{Kh j,n} <t_{Kh j,n+1}=τ

constructed so that every subinterval [t_k−1,t_k) contains at least one exact h → j

transition time observed in X1, . . . ,Xn and maxk(t_k −t_k−1) = O(n−κ). For every

Λhj∈H_hjletΛ_hj,n denote the piecewise linear interpolant

Λhj,n(t) = ∑ tk∈Th j,n Ik(t) {[1− Lk (0,t) Lk(0,τ) ]Λ_hj(t_k−1) + [Lk (0,t) Lk(0,τ) ]Λ_hj(t_k)}, (3.5)

whereIk(t) =1_[tk−1,tk)(t)andL_k(s,t)is the length of[tk−₁,t_k) ∩ [s,t). ◽

Let Ahj(s,t ∣ z) = A_hj(t ∣ z) −A_hj(s ∣ z). Then from b3, Λ₀₂ and Λ₁₂ are jointly

estimable by maximizing log likn(θ,Λ) = n ∑ i=1logpθ,Λ (X_i) = −A₀₁(U_i ∣Z_i) −A₀₂(U_i ∣Z_i) +∆₀i1(U_i <V_i)[logα₀₁(U_i ∣Z_i) −A₁₂(U_i,V_i ∣Z_i) +∆i₂logα₁₂(V_i ∣Z_i)] + (1−∆₀i)log[P₀₁(U_i,V_i ∣Z_i)α₁₂(V_i ∣Z_i)∆i2+P₀₀(U_i,V_i ∣Z_i)α₀₂(V_i ∣Z_i)∆2i] −∆₀i(1−1(U_i <V_i))∆i₂logα₀₂(V_i ∣Z_i). (3.6)

over the sieve Hn = (H_hj,n),H_hj,n = {Λ_hj,n ∶ Λ_hj ∈ H_hj}. LetΘ denote the set of

all possibleθ. Then the piecewise exponentialsieve maximum likelihood estimator (smle) satisfies

log likn(θˆ_n, ˆΛ_n) = max

θ∈Θ,Λ∈Hnlog likn

(θ,Λ). (3.7)

This optimization problem is well-defined and has finite dimension. Its solution is characterized by the score equations

∂

∂

∂λhj(t_k)log likn

(θˆ_n, ˆΛ_n) =0, t_k∈T_hj,n,

which can be solved using a self-consistency algorithm. For fixed Λ it is straight- forward to show that the log likn(θ,Λ)is strictly concave in θ, unlessZ_i = 0, for

everyi=1, . . . ,n. Uniqueness of the smle forΛis relatively difficult to establish. To

safeguard against the potential for non-convexity or multiple stationary points in the objective function, standard methods such as the examination of different starting values and profile plots of the log-likelihood (e.g. Lawless2003, p. 556) can be applied here. Further details on computation are deferred to Section3.4.

3.3 asymptotic properties

Under some regularity conditions the sieve maximum likelihood estimator(θˆ_n, ˆΛ_n)

globally converges to the truth(θ₀,Λ₀)slower than the parametric rate √

n, but ˆθnis

asymptotically efficient at(θ₀,Λ₀). Proofs are constructed by adapting results from

Section2.3. 3.3.1 Consistency

The smle is asymptotically unbiased by application of Theorem1.16. The conditions needed for this result can be verified along the same lines as Section2.3.1, though some adjustments are needed to accommodate the sieve estimator. These easily follow by adaption of Y. Zhang et al.’s (2010) proof of consistency.

b4 θ0lies in the interior ofΘandΘis a compact subset ofRdz.

b5 The distributions forCandDhave support contained in[σ,τ]such that P(C=

D=τ∣Z) >0, almost surely.

b6 The distribution ofZhas supportZ=supp(F_Z)on a bounded subset of_Rdz.

b7 For eachh≠ j, P(Z⊺_hja≠c) >0 for everya∈_Rdz andc∈R.

3.3 theorem.Under the above conditions∥θˆ_n−θ₀∥ + ∥Λˆ_n−Λ₀∥₂→as 0, where ∥Λˆ_n−Λ₀∥₂= ∑ h≠j [

∫

τ σ ∣Λˆhj,n−Λ 0 hj∣2(u)du] 1/2

3.3 asymptotic properties

Proof. We verify the conditions of Theorem1.16with criterion function mθ,Λ=log pθ,Λ

+p₀

2 .

Fromb2,His assumed bounded on[0,τ]. Withb4,Θ×H_n is a compact parametric

class for eachnwith bracketing number

N[ ](ε,Θ×H_n,L_r(_Pn)) ≲ (diamΘ/ε)d(M/ε)Kn, (3.8)

where Kn = ∑_h≠jK_hj,n. The corresponding bracketing integral converges and, by

Theorem1.12,Θ×H_nisP-Donsker. Suppose that(θ_L,θ_R)and(Λ_L,Λ_R)is a bracket

for(θ,Λ). By conditionsb2andb4and the mean value theorem

∫

τ

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