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Evaluation of Surrogate Endpoints from an InformationTheoretic PerspectiveAriel Alonso Abad KU Leuven, Belgium IntroductionBuyse et al. (2000) developed an elegant formalism to assess the validity of surrogate endpoints in a metaanalytic context. However, these authors only considered the simplest setting where both endpoints are Gaussian random variables measured crosssectionally. Over the years, work was done to extend the metaanalytic methodology to other outcome types and in the preceding chapters, some of these methods were described in detail. Assessing surrogacy in more complex scenarios raises a number of difficult challenges. First, one needs to deal with highly complicated hierarchical models that frequently suffer from severe numerical issues. Second, based on the outputs of these models, one needs to define meaningful metrics to quantify surrogacy at both trial and individual levels. If one is willing to consider only a linear relationship between the expected causal treatment effect on the surrogate and the true endpoint, then the methodology described in Chapter 4 can be applied in a straightforward fashion to quantify triallevel surrogacy. At the individual level, however, abandoning the realm of normality has much deeper implications. Indeed, within the metaanalytic paradigm, several individuallevel metrics of surrogacy have been proposed. For instance, in the binarybinary setting, Renard et al. (2002) assumed that the observed dichotomous outcomes emerge from two latent and normally distributed variables (S,T). Essentially, it is assumed that the surrogate (true endpoint) takes value 1 when the corresponding latent variable is positive and 0 otherwise. In this framework, using a bivariate probit model, these authors defined individual level surrogacy as R2_{ndiv} = corr(S, T)^{2}, i.e., the squared correlation at the latent level. Alternatively, they also defined R2_{ndiv} = ^, the global odds ratio between both binary endpoints estimated from a bivariate PlackettDale model. When the true endpoint is a survival time and the surrogate is a longitudinal sequence, Renard et al. (2003), using Henderson’s model, proposed to study the individuallevel surrogacy based on a time function defined as R2_{n}di_{v}(t) = corr[Wi(t), W_{2}(t)]^{2}, where (Wi(t),W_{2}(t)) is a latent bivariate Gaussian process. Burzykowski et al. (2001) approached the case of two failuretime endpoints based on copula models and quantified the individual level surrogacy using Kendall’s т. Spearman’s rho has also been used in this scenario. In addition, using multivariate ideas, the socalled R was proposed to evaluate surrogacy when both responses are measured longitudinally (Alonso et al., 2006). The R^ coefficient quantifies the association between both longitudinal sequences, after adjustment by treatment and trial, and is defined using the covariance matrices emanating from a longitudinal model that characterizes the joint distribution of both endpoints. Furthermore, R can be incorporated into a more general framework allowing for interpretation in terms of canonical correlations of the error vectors, based on which, a family of individuallevel parameters can be defined (Alonso et al., 2006). These examples underscore a limitation of the metaanalytic methodology so far: different settings require different definitions. Even though it seems logical that different settings require different approaches, some of these metrics are defined at a latent level, which hampers their clinical interpretation. Actually, as previously illustrated, the metaanalytic methodology may lead to different metrics even within one single setting and these metrics do not necessarily have the same interpretation or lead to the same conclusion. Clearly, this could bring problems when different surrogates need to be compared. In addition, the need for complex hierarchical models hinders the use of the methodology and causes important numerical issues. In the next section, a unified approach to the validation of surrogate endpoints will be introduced based on information theory. Furthermore, it will be argued that this approach may help to overcome some of the aforementioned problems. 
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