The validation of surrogate endpoints has been studied by Prentice (1989). He presented a definition as well as a set of criteria, which are equivalent only if the surrogate and true endpoints are binary. Freedman et al. (1992) supplemented these criteria with the so-called 'proportion explained'. Buyse and Molenberghs (1998) proposed replacing the proportion explained by two quantities: (1) the relative effect linking the effect of treatment on both endpoints and (2) an individual-level measure of agreement between both endpoints. The latter quantity carries over when data are available on several randomized trials, while the former can be extended to be a trial-level measure of agreement between the effects of treatment of both endpoints. This approach suggests a new method for the validation of surrogate endpoints, and naturally leads to the prediction of the effect of treatment upon the true endpoint, given its observed effect upon the surrogate endpoint. These ideas are illustrated using data from two sets of multicenter trials: one comparing chemotherapy regimens for patients with advanced ovarian cancer, the other comparing interferon-alpha with placebo for patients with age-related macular degeneration.

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.

The validation of surrogate endpoints has been studied by Prentice, who presented a definition as well as a set of criteria that are equivalent if the surrogate and true endpoints are binary. Freedman et al. supplemented these criteria with the so-called proportion explained. Buyse and Molenberghs proposed to replace the proportion explained by two quantities: (1) the relative effect, linking the effect of treatment on both endpoints, and (2) the adjusted association, an individual-level measure of agreement between both endpoints. In a multiunit setting, these quantities can be generalized to a trial-level measure of surrogacy and an individual-level measure of surrogacy. In this paper, we argue that such a multiunit approach should be adopted because it overcomes difficulties that necessarily surround validation efforts based on a single trial. These difficulties are highlighted.

Quantitative risk assessment involves the determination of a safe level of exposure. Recent techniques use the estimated dose-response curve to estimate such a safe dose level. Although such methods have attractive features, a low-dose extrapolation is highly dependent on the model choice. Fractional polynomials, basically being a set of (generalized) linear models, are a nice extension of classical polynomials, providing the necessary flexibility to estimate the dose-response curve. Typically, one selects the best-fitting model in this set of polynomials and proceeds as if no model selection were carried out. We show that model averaging using a set of fractional polynomials reduces bias and has better precision in estimating a safe level of exposure (say, the benchmark dose), as compared to an estimator from the selected best model. To estimate a lower limit of this benchmark dose, an approximation of the variance of the model-averaged estimator, as proposed by Burnham and Anderson, can be used. However, this is a conservative method, often resulting in unrealistically low safe doses. Therefore, a bootstrap-based method to more accurately estimate the variance of the model averaged parameter is proposed.

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