except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this pubUcation, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

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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.

The validation of surrogate endpoints has been studied by Prentice (1989, Statistics in Medicine 8, 431-440) and Freedman, Graubard, and Schatzkin (1992, Statistics in Medicine 11, 167-178). We extended their proposals in the cases where the surrogate and the final endpoints are both binary or normally distributed. Letting T and S be random variables that denote the true and surrogate endpoint, respectively, and Z be an indicator variable for treatment, Prentice's criteria are fulfilled if Z has a significant effect on T and on S, if S has a significant effect on T, and if Z has no effect on T given S. Freedman relaxed the latter criterion by estimating PE, the proportion of the effect of Z on T that is explained by S, and by requiring that the lower confidence limit of PE be larger than some proportion, say 0.5 or 0.75. This condition can only be verified if the treatment has a massively significant effect on the true endpoint, a rare situation. We argue that two other quantities must be considered in the validation of a surrogate endpoint: RE, the effect of Z on T relative to that of Z on S, and gamma Z, the association between S and T after adjustment for Z. A surrogate is said to be perfect at the individual level when there is a perfect association between the surrogate and the final endpoint after adjustment for treatment. A surrogate is said to be perfect at the population level if RE is 1. A perfect surrogate fulfills both conditions, in which case S and T are identical up to a deterministic transformation. Fieller's theorem is used for the estimation of PE, RE, and their respective confidence intervals. Logistic regression models and the global odds ratio model studied by Dale (1986, Biometrics, 42, 909-917) are used for binary endpoints. Linear models are employed for continuous endpoints. In order to be of practical value, the validation of surrogate endpoints is shown to require large numbers of observations.

10013), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

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Non-Gaussian outcomes are often modeled using members of the so-called exponential family. Notorious members are the Bernoulli model for binary data, leading to logistic regression, and the Poisson model for count data, leading to Poisson regression. Two of the main reasons for extending this family are (1) the occurrence of overdispersion, meaning that the variability in the data is not adequately described by the models, which often exhibit a prescribed mean-variance link, and (2) the accommodation of hierarchical structure in the data, stemming from clustering in the data which, in turn, may result from repeatedly measuring the outcome, for various members of the same family, etc. The first issue is dealt with through a variety of overdispersion models, such as, for example, the beta-binomial model for grouped binary data and the negative-binomial model for counts. Clustering is often accommodated through the inclusion of random subject-specific effects. Though not always, one conventionally assumes such random effects to be normally distributed. While both of these phenomena may occur simultaneously, models combining them are uncommon. This paper proposes a broad class of generalized linear models accommodating overdispersion and clustering through two separate sets of random effects. We place particular emphasis on so-called conjugate random effects at the level of the mean for the first aspect and normal random effects embedded within the linear predictor for the second aspect, even though our family is more general. The binary, count and time-to-event cases are given particular emphasis. Apart from model formulation, we present an overview of estimation methods, and then settle for maximum likelihood estimation with analytic-numerical integration. Implications for the derivation of marginal correlations functions are discussed. The methodology is applied to data from a study in epileptic seizures, a clinical trial in toenail infection named onychomycosis and survival data in children with asthma. 1 2 MOLENBERGHS, VERBEKE, DEMÉTRIO AND VIEIRA