ABSTRACT
In this chapter, we focus on the case when both the surrogate and the true endpoint are failure-time endpoints. We mainly follow the two-stage approach proposed by Burzykowski et al. (2001).
Assume now that Sij and Tij are failure-time endpoints. Similar to the case with two normally distributed endpoints, the two-stage approach can be applied. In particular, model (4.10) is replaced by a model for two correlated failure-time random variables. Burzykowski et al. (2001) used single-parameter copulas toward this end (Clayton, 1978; Dale, 1986; Hougaard, 1986). In particular, the joint survivor function of (Sij , Tij) is expressed as
F (s, t) = P (Sij ≥ s, Tij ≥ t) = Cθ{FSij(s), FT ij(t)}, s, t ≥ 0, (5.1)
with
where (FSij , FT ij) denote marginal survivor functions and Cθ is a copula, i.e., a distribution function on [0, 1]2 with θ ∈ R1. The marginal survivor functions can be specified by using, e.g., the proportional hazard model:
FSij(s) = exp
{ − ∫ s 0
λSi(x) exp(αiZij)dx
} , (5.2)
FT ij(t) = exp
{ − ∫ t 0
λT i(x) exp(βiZij)dx
} , (5.3)
where λSi(t) and λT i(t) are trial-specific marginal baseline hazard functions and αi and βi are trial-specific effects of treatment Z on S and T , respectively, in trial i. The hazard functions can be specified parametrically or can be left unspecified as in the classical formulation of the model proposed by Cox (1972).
