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).

Theoretical Background

Assume now that Sj and Tj 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 (Sj, Tj) is expressed as

where (F_{Si}j, F_{T} j) denote marginal survivor functions and C_{g} is a copula, i.e., a distribution function on [0, 1]^{2} with в G R^{1}. The marginal survivor functions can be specified by using, e.g., the proportional hazard model:

where A_{Si}(t) and A_{T}j(t) are trial-specific marginal baseline hazard functions and Oj and в 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).

When the hazard functions are specified, estimates of the parameters of the model defined by (5.1)-(5.3) can be obtained by using maximum likelihood. Alternately, as suggested by Shih and Louis (1995), the marginal models (5.2)- (5.3) can be fitted first, and then the copula parameter в can be estimated from a profile-likelihood.

Note that different copulas may be used in (5.1), depending on assumptions made about the nature of the association between the surrogate and the true endpoint; such assumptions are generally unavailable, in which case the best fitting copula may be chosen. We will consider three particular copulas:

The Clayton copula The copula function (Clayton, 1978) has the following form:

It implies a positive, “late” dependence, i.e., mainly for large failure-times. The strength of the association decreases with decreasing в and reaches independence when в ^ 1.

The Hougaard copula The copula function (Hougaard, 1986) is given by

It induces a positive, “early” dependence, i.e., mainly for early failure-times. The strength of the association decreases with increasing в and reaches independence when в ^ 1.

The Plackett copula The copula function (Dale, 1986) is defined as follows:

where

and в G [0, +то]. Parameter в has an interesting interpretation as the constant global cross-ratio (Dale, 1986). The value of в = 1 corresponds to independence.