Bayesian Inference of Interval-Censored Survival Data - Interval-Censored-Time-to-Event Data: Methods and Applications - page 170

7.2.2 Cox Model with Time-Varying Regression Coecients

A Cox model with time-varying coecients has the form

h(tjx i ) = h 0 (t) expfx 0 i (t)g: (7.1)

When (t) = , the model in Equation (7.1) reduces to the standard Cox

model. For subject i, the likelihood of observing the interval data [Li, i ;R i ) is
given by

Pr(T i 2 [L i ;R i )jx i ;) = S(L i jx i ;) S(R i jx i ;); (7.2)

where S(tjx i ;h 0 ;) = expfH(tjx i ;)g is the survival function and

H(tjx i ;) = R t

0 h 0 (u) expfx 0 i (u)gdu is the cumulative hazard function. Note
that when r i = K + 1, S(R i jx i ;) = 0. Let = (t);(t) denote the set
of all model parameters. Using Equation (7.2), the likelihood function for the

observed data D obs is given by

h
S(L i jx i ;) S(R i jx i ;)

i

Y

L(jD obs ) =

i=1

h i Y

r i =K+1

= Y
r i K

S(a ` i jx i ;) S(a r i jx i ;) S(a ` i jx i ;): (7.3)

Following Sinha et al. (1999) and Wang et al. (2011a), we assume that on grid

G,

(t) = (a k ); a k1 t < a k ; (7.4)

for k = 1;:::;K. From Equation (7.3), it is clear that we do not need to specify

(t) and h 0 (t) for t a K as there is no data over the time interval [a K ;a K+1 ).
In the subsequent sub-sections, various Bayesian models are considered for

modeling (t) and h 0 (t).

7.2.3 Piecewise Exponential Model for h 0

For interval-censored data, Sinha et al. (1999) assumed a piecewise exponential

model for h 0 (t). Specifically, we have

h 0 (t) = k ; a k1 t < a k ; (7.5)