The two-stage approach for two normally distributed endpoints was discussed in Chapter 4. Briefly, the first stage consists of a joint model for the surrogate and true endpoints given by

where p_{Si} and p_{Ti} are trial-specific intercepts for S and T, a_{i} and Д are trial- specific treatment effects on the surrogate and the true endpoint, respectively. The error terms, ?_{Si}j and ?_{Ti}j, are bivariate normally distributed with zero mean and covariance matrix given by

As pointed out in Chapter 4, individual-level surrogacy is assessed by the squared correlation between S and T after adjusting for trial-specific treat-

FIGURE 12.2

A leave-one-out evaluation procedure.

ment effects, that is,

For the full fixed-effects model, the trial-level surrogacy is estimated using the coefficient of determination obtained by fitting the following model:

where /?*, Ms*, and Sj are the parameter estimates obtained from the joint model specified in (12.1). A weighted analysis, which takes into account the sample size in each trial (see also Section 4.3.4), can be used as well. The standard error for R?_{ndiv} and R^_{rial} can be calculated using the delta method.