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IV Additional Considerations and Further TopicsSurrogate Endpoints in Rare DiseasesWim Van der Elst Hasselt University and Janssen Pharmaceutica, Belgium Geert Molenberghs Hasselt University nad KU Leuven, Belgium CONTENTS
IntroductionEvaluating a surrogate endpoint typically requires a large amount of data, and this is particularly so when the contemporary multipletrial surrogate endpoint evaluation methods are used. The need for a large sample size is an issue in all clinical trials (e.g., an increased study duration and cost, a higher probability of missing values due to study dropout, and so on), and it is particularly problematic in clinical trials in rare diseases (or in clinical trials in small patient subgroups that are identified within the context of a common disease). Indeed, in rare diseases the number of patients that are available for study participation is typically substantially smaller compared to what is the case in nonrare disease clinical trials. Yet, surrogate endpoints may be particularly useful in rare disease clinical trials (Korn et al., 2013). Indeed, the use of a surrogate endpoint may result in a smaller sample size that is needed to show the effectiveness of a new treatment, because (i) the surrogate endpoint may be an event that occurs more frequently than the true endpoint, and/or because (ii) the treatment effect on the surrogate endpoint may be larger compared to the treatment effect on the true endpoint (in particular when the surrogate endpoint is closer to the treatment in terms of time and the biological mechanism that is being targeted (Korn et al., 2013). As detailed in Part II, the contemporary multipletrial surrogate evaluation methods usually involve the fitting of complex hierarchical models. For example, when both S and T are normally distributed endpoints, a linear mixed effects model (see Eq. (4.1)) is fitted to estimate the trial and individuallevel surrogacy coefficients. When the number of available patients is small, fitting such linear mixedeffects models is often problematic. Indeed, when the data are sparse, the NewtonRaphson (or quasiNewton)based iterative process that is typically used to fit the model (i) may not converge at all, or (ii) may converge to values that are close to or outside the boundary of the parameter space. In the latter case, the estimated variance components are close to zero or negative, which may lead to a nonpositivedefinite variancecovariance matrix of the random effects D (see (4.2)). A nonpositivedefinite D matrix is not an issue when one is merely interested in the fixedeffect components, because the marginal model can still be used to make valid inferences regarding the fixedeffect parameters as long as the overall V matrix is positivedefinite (with V = ZiDZi + Ej, where Z_{i} are matrices of known covariates associated with the random effects (Verbeke and Molenberghs, 2000; West et al., 2007)). However, in a surrogate evaluation context, interest is in the random effects rather than in the fixed effects. In the latter setting, it is paramount that D is positivedefinite, as this condition will guarantee that R_{rial} is within the unit interval. The remainder of this chapter is organized in the following way. In Section 15.2, a simulation study is conducted to examine which factors affect convergence of linear mixedeffects models. Based on the results of these simulations, a multiple imputation (MI)based strategy that can be used to overcome model convergence problems is detailed in Section 15.3. As an alternative to overcome model convergence issues, a twostage modelfitting approach can be used (for details, see Section 4.3). Section 15.4 provides a formal basis for the latter approach. Notice that the current chapter focuses on the setting where both S and T are normally distributed endpoints, but similar strategies are conceivable in other settings as well. 
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