Consistent Variance Estimation in Semiparametric Models with Application to Interval-Censored Data - Interval-Censored-Time-to-Event Data: Methods and Applications - page 245

restriction of s to I Kt is a polynomial of order m for m K; (ii) for m 2
and 0 m 0 m 2, s is m 0 times continuously dierentiable on [a;b]. This
definition is phrased after Stone (1985), which is a descriptive version of Schu-

maker (1981), page 108, Definition 4.1. According to Schumaker (1981), page

117, Corollary 4.10, there exists a local basis b n fb t ; 1 t q n g, so called
B-splines, for S n (D n ;K n ;m), where q n K n + m. These basis functions are
nonnegative and sum up to 1 at each point in [a;b], and each b t is 0 outside
the interval [d t ;d t+m ].

9.4.1 Cox Model for Interval-Censored Data

Interval-censored data occur very frequently in long-term follow-up studies for

an event time of interest. With such data, the exact event time T is not observ-

able; it is only known with certainty that T is bracketed between two adjacent

examination times, or occurs before the first or after the last follow-up exami-

nation. Let (L;R) be the pair of examination times bracketing the event time

T. That is, L is the last examination time before and R is the first examina-

tion time after the event. If 0 < L < R < 1, then T is interval-censored. If

the event occurs before the first examination, then T is left-censored. If the

event has not occurred after last examination, then T is right-censored. Such

data are called \Case 2" interval-censored data. Nonparametric estimation of

a distribution function and its smooth functionals with interval-censored data

has been studied by Groeneboom and Wellner (1992) and Geskus and Groene-

boom (1996). A systematic treatment of interval-censored data can be found
in Sun (2006).

In this example, we consider the B-splines sieve MLE of the Cox propor-

tional hazards model for interval-censored data. With the proportional hazards
model, the conditional hazard of T given a covariate vector Z 2 R d is propor-

tional to the baseline hazard. In terms of cumulative hazard functions, this

model is

(tjz) = 0 (t)e 0 0 z ;

(9.13)