A Joint Model for Survival Endpoints

As explained in Chapter 5, a joint survival function of (Sij, Tj) can be written

as

where FSij and FTij denote marginal survival functions for both endpoints (overall survival and progression free survival) and Cg is a copula, i.e., a

FIGURE 12.27

Individual-level surrogacy measure with 95%C.Itwo-stage approach.

FIGURE 12.28

Kaplan-Meier estimates at 2 years on the true endpoint versus at 1 year on the surrogate endpoint.

bivariate distribution function on [0,1]2 which allows correlated probabilities to be modeled. In Chapter 5, we use marginal survival functions given by

where Aand ATi are trial-specific marginal baseline hazard functions and a and в are trial-specific treatment effects. At the second stage, a joint model is formulated to the treatment effects

FIGURE 12.29

Kaplan-Meier estimates for the treated group for a selected trial.

where the second term on the right-hand side of (12.24) is assumed to follow a zero mean normal distribution with a covariance matrix given by

The quality of the surrogate S at the trial level is assessed based on the coefficient of determination given by:

To assess the quality of the surrogate at the individual level, a measure of association between Sj and Tj, calculated while adjusting the marginal distributions of the two endpoints for both the trial and treatment effects, is needed. Burzykowski et al. (2001) proposed to used Kendall’s т as it depends only on the copula function Cg and is independent of the marginal distribution of Sj and Tj:

It describes the strength of the association between the two endpoints remaining after adjustment, through the marginal models (12.23), for the trial and the treatment effects.

FIGURE 12.30

Surrogacy measures at trial and individual levels with the 95%C.I.

 
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