Validation of Surrogate end Points in Multiple Randomized Clinical Trials with Failure Time end Points

Abstract

Summary. Before a surrogate end point can replace a ®nal (true) end point in the evaluation of an experimental treatment, it must be formally`validated'. The validation will typically require large numbers of observations. It is therefore useful to consider situations in which data are available from several randomized experiments. For two normally distributed end points Buyse and co-workers suggested a new de®nition of validity in terms of the quality of both trial level and individual level associations between the surrogate and true end points. This paper extends this approach to the important case of two failure time end points, using bivariate survival modelling. The method is illustrated by using two actual sets of data from cancer clinical trials.

Pseudo‐likelihood estimation for a marginal multivariate survival model

“…Kendall's thus measures the association between both time points after adjustment for the covariates used in the model. Such a relationship is very simple for the Clayton and Hougaard copulas [11]. Precisely, one obtains = (Â − 1)=(Â + 1) for Clayton and = 1 − Â for Hougaard.…”

“…These data were studied by Nielsen et al [9], who proposed a shared gamma frailty model and by Parner [10], who proposed a composite likelihood method for the estimation of the frailty parameters and the standard deviations. We propose to use a Plackett-Dale model [11] for correlated survival time data with Weibull margins, as described in the next section.…”

“…Some possible options are the Clayton, Hougaard and Plackett copulas. Burzykowski et al [11] studied them in detail within the framework of surrogate endpoints. For the Clayton and Hougaard copulas, model (1) reduces to a proportional frailty model [12] with frailties generated, respectively, by the gamma and the positive stable distributions.…”

“…The marginal distributions of T 1 and T 2 do not a ect (11), and hence it follows that only depends on the copula function C T1T2 [27]. Kendall's thus measures the association between both time points after adjustment for the covariates used in the model.…”

“…Estimates and conÿdence intervals (using the delta method) are accordingly easily obtained. There is no closed form for Kendall's in the Plackett-Dale case and an estimate has to be obtained directly from (11). We have developed a SAS IML 8.02 macro to this e ect.…”