The key motivation to evaluate a surrogate endpoint is to be able to predict the treatment effect on the true endpoint \(T\) based on the treatment effect on \(S\) in a new trial \(i=0\).
When a so-called full (fixed or mixed) bi- or univariate model was fitted in the surrogate evaluation phase (for details, see BimixedContCont, BifixedContCont, UnimixedContCont and UnifixedContCont), this prediction is made as:
$$E(\beta + b_0 | m_{S0}, a_0) = \beta + \left(\begin{array}{c}
d_{Sb}\\
d_{ab}
\end{array}\right)^T \left(\begin{array}{cc}
d_{SS} & D_{Sa}\\
d_{Sa} & d_{aa}
\end{array}\right)^{-1} \left(\begin{array}{c}
\mu_{S0} - \mu_S\\
\alpha_0 - \alpha
\end{array}\right) $$
$$Var(\beta + b_0 | m_{S0}, a_0) = d_{bb} + \left(\begin{array}{c}
d_{Sb}\\
d_{ab}
\end{array}\right)^T \left(\begin{array}{cc}
d_{SS} & D_{Sa}\\
d_{Sa} & d_{aa}
\end{array}\right)^{-1} \left(\begin{array}{c}
d_{Sb}\\
d_{ab}
\end{array}\right),$$
where all components are defined as in BimixedContCont. When the univariate mixed-effects models are used or the (univariate or bivariate) fixed effects models, the fitted components contained in D.Equiv are used instead of those in D.
When a reduced-model approach was used in the surrogate evaluation phase, the prediction is made as:
$$E(\beta + b_0 | a_0) = \beta + \frac{d_{ab}}{d_{aa}} + (\alpha_0 - \alpha),$$
$$Var(\beta + b_0 | a_0) = d_{bb} - \frac{d_{ab}^2}{d_{aa}},$$
where all components are defined as in BimixedContCont. When the univariate mixed-effects models are used or the (univariate or bivariate) fixed effects models, the fitted components contained in D.Equiv are used instead of those in D.
A \((1-\gamma)100\%\) prediction interval for \(E(\beta + b_0 | m_{S0}, a_0)\) can be obtained as \(E(\beta + b_0 | m_{S0}, a_0) \pm z_{1-\gamma/2} \sqrt{Var(\beta + b_0 | m_{S0}, a_0)}\) (and similarly for \(E(\beta + b_0 | a_0)\)).