Strong consistency under proportional censorship when covariables are present

https://doi.org/10.1016/S0167-7152(98)00218-1Get rights and content

Abstract

Let (X,Y) be the variable of interest, where Y is the possibly right-censored lifetime and X is a p-dimensional covariate. We introduce a generalized ACL (Abdushukurov, Cheng, Lin) estimator FXY of FXY(x,y)=P(Xx,Yy) and we prove ∫ϕdFXY→∫ϕdFXY a.s. for any integrable function ϕ under an extended Koziol–Green model of proportional censorship.

Introduction

Censoring is a typical phenomenon in survival analysis. In a censored sample incomplete observations may occur. The general random censorship model (from the right) assumes that one observes the pair (Z,δ) where Z=min(Y,C) and δ=I(YC), Y being the lifetime with distribution function F and C being the censoring variable (independent of Y) with distribution function G. Inference under this model is carried out via Kaplan–Meier-based estimation.

The additional condition that led us to the Koziol–Green model (Koziol and Green, 1976) is1−G=(1−F)βforsomeβ>0.When F and G are continuous (as we will assume throughout the text), this proportional censorship model is characterized by the independence between Z and δ (Armitage, 1959; Sethuraman, 1965). Then letting H and θ be the distribution function of Z and the mean value of δ respectively, it is easily proved that1−F=(1−H)θ,where θ=1/(1+β). This relation motivates the ACL estimator Fn(y)=1−(1−Hn(y))θn (Abdushukurov, 1984; Cheng and Lin (1984), Cheng and Lin (1987)) based on the given sample (Z1,δ1),…,(Zn,δn), where Hn is the empirical distribution function of the Z's and θn is the sample mean of the δ's. The estimator Fn has been extensively analyzed in the literature.

Here we consider a situation in which the observed variable is the vector (X,Z,δ), where X is a covariate (in general p-dimensional) paired with the lifetime. We propose an extension of the Koziol–Green model based on the hypothesis(X,Z)isindependentofδ.Under (1.3) we have that Z is independent of δ and hence (1.1) and (1.2) hold. For identifiability reasons we assume the conditionP(Y⩽C|X,Y)=P(Y⩽C|Y).Condition (1.4) is discussed and used to get estimators under random censorship in Stute (1993). Under (1.4) we haveE(ϕ(X,Y))=P(δ=1|X,Y)ϕ(X,Y)1−G(Y)for a general FXY-integrable ϕ-function, FXY denoting the joint distribution function of (X,Y). ButP(δ=1|X,Y)ϕ(X,Y)1−G(Y)=EEϕ(X,Y)δ1−G(Y)X,Y=EEϕ(X,Z)δ1−G(Z)X,Y=Eϕ(X,Z)δ1−G(Z).Now we use the independence between (X,Z) and δ (and relations (1.1) and (1.2)) to concludeE(ϕ(X,Y))=E(θ(1−H(Z))θ−1ϕ(X,Z))under (1.3) and (1.4). Our estimator of E(ϕ(X,Y))≡S(ϕ)≡S is given byS̃n=∫∫ϕ(u,v)θn(1−Hn(v))θn−1HXZ(du,dv)where Hn=((n+1)/n)Hn is the empirical distribution function of the Z's and HXZ=((n+1)/n)HXZ is the empirical distribution function of the (X,Z)'s. By considering the indicator function ϕI((−∞,x]×(−∞,y]) we obtain for FXY the estimateFXY(x,y)=i=1n(n+1)−1θn(1−Hn(Zi))θn−1I(Xi⩽x,Zi⩽y).

In this paper we prove the strong consistency of (1.6) for each FXY-integrable function ϕ using martingale theory. As a consequence, we obtain strong consistency for the extended ACL-based integralS̃n=∫∫ϕ(u,v)FXY(du,dv),whereFXY(x,y)=i=1n1−Hn(Zi)+1nθn−(1−Hn(Zi))θnI(Xi⩽x,Zi⩽y)for the class of ϕ-functions that satisfy∫∫|ϕ(u,v)|(1−F(v))αFXY(du,dv)<∞forsomeα>0.The estimator FXY collapses to Fn when there is no X. Our results cover such a case. In this manner we obtain an alternative proof for a result on strong consistency of ∫ϕdFn given in Stute (1992).

When no covariables are present, condition (1.4) is superfluous, and (1.3) collapses to (1.1). Under our extended Koziol–Green model (1.3)–(1.4), ACL-based estimation is still consistent. Other authors have considered (conditional) extensions for the model, by assuming the independence between Z and δ conditionally on X. See Veraverbeke and Cadarso-Suárez (1997). Such ideas require smoothing methods for the estimation of the (conditional) lifetime distribution.

Section snippets

Main results

Now we give our results for both estimators S̃n and S̃n given in Section 1.

Theorem 2.1

For each FXY-integrable function ϕ we have S̃n→S a.s.

Theorem 2.2

For each function ϕ satisfying (1.8) we have S̃nS̃n→0 a.s.

Corollary 2.3

For each function ϕ satisfying (1.8) we have S̃n→S a.s.

An immediate consequence of our theorems is the strong consistency of FXY and FXY. As mentioned in Stute (1993), these results can be also used to get consistency of the estimated variance–covariance matrix of (X,Y). Consistent least-squares estimation in linear

Proofs

Firstly we show that, for proving Theorem 2.1, it suffices to establishS̄n=i=1n(n+1)−1θ1−in+1θ−1ϕ(X[i:n],Z(i:n))→Sa.s.,where Z(1:n)⩽⋯⩽Z(n:n) are the ordered Z-values and X[i:n] is the ith concomitant. Certainly, by the mean-value theoremθi=1n(n+1)−11−in+1θn−1ϕ(X[i:n],Z(i:n))−S̄n⩽|θn−θ|i=1n(n+1)−11−in+1θ̄i−1ln1−in+1ϕ(X[i:n],Z(i:n))with θ̄i between θn and θ, 1⩽in. Now by the SLLN given ϵ>0 we have |θnθ|<ϵ a.s. for big n. Hence1−in+1θ̄i−11−in+1θ−1−ϵ,1⩽i⩽na.s. for big n. Now write for 1⩽in

Concluding remarks

In this paper we introduce consistent estimators for quantities of interest in survival analysis when the lifetime is at risk of being censored from the right and an extended Koziol–Green model of proportional censorship holds. In order to get this, we analyze an approximated generalized ACL estimator FXY and we prove strong consistency for ∫ϕdFXY, ϕ denoting any FXY-integrable function. We use this result to get consistency when dealing with the actual ACL weights. We mention several

Acknowledgements

The authors thank to a Referee and an Associate Editor for their careful reading of the paper and their suggestions.

References (13)

There are more references available in the full text version of this article.

Cited by (0)

1

This work was partially supported by the project XUGA20701B96 of the Xunta de Galicia (Spain) and by the DGES grant PB95-0826.

View full text