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The Endpoint Dimension: Univariate versus Bivariate Models

The error terms for S and T may be assumed to be independent rather than dependent by fitting univariate instead of bivariate (mixed- or fixed-effects) models. This consideration seems odd at first sight, because it is natural to assume that S and T are correlated in a surrogate marker evaluation context. Nonetheless, making the simplifying assumption that the error terms are uncorrelated is not necessarily a problem. Indeed, the explicit consideration of the bivariate nature of the endpoints is mainly of importance to obtain the E matrix, which is mainly of interest to estimate R?ndiv, and often the focus of the analysis is on trial-level surrogacy rather than on individual-level surrogacy. It has been shown that the Rrial values that are obtained by using mixed- effects univariate and mixed-effects bivariate models are similar (Tibaldi et al., 2003), and the R(rial values that are obtained by using fixed-effects univariate and fixed-effects bivariate models are identical (Johnson and Wichern, 2007). Moreover, if interest is also in R?ndiv, this quantity can always be estimated by computing the squared correlation of the residuals of the fitted univariate models (Molenberghs et al., 2010), or by using the information-theoretic approach (see Chapter 9 and 10).

The Measurement Error Dimension: Weighted versus Unweighted Models

When the bivariate mixed-effects model is not used, one is confronted with measurement error because the treatment effects on S and T in the different trials are affected by sampling variability. The magnitude of this error likely depends on the trial size. Therefore, a straightforward approach to address this issue is to use weighted regression with trial sizes as the weights in the Stage 2 model (Burzykowski, Molenberghs, and Buyse, 2005; Tibaldi et al., 2003). Notice that the measurement error dimension is not relevant when a bivariate mixed-effects model is used, because it is automatically accounted for and therefore explicit corrections are not needed.

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