An elaborate discussion about the reduced mixed-effects model is given in Chapter 4. Briefly, a joint model if formulated for the true and the surrogate endpoints in which trial-specific treatment effects are assumed to be random:

is used to estimate the surrogacy measures. Trial- and individual-level surrogacy measures are given, respectively, by (see 4.14 and 4.9 for more details):

The SAS Macro %CONTRANRED

The macro %CONTRANRED is used to conduct the analysis.

The macro’s arguments are presented in Section 12.2.

FIGURE 12.16

Surrogacy measures with their 95% C.I, reduced mixed-effects model.

Data Analysis and Output

Similar to full random effects models, convergence problems arise. Simulated data were used to generate numerical and graphical output. The following parameters were used to simulate the data: 1000 observations from 50 trials were generated from a multivariate normal distribution with the mean vector (p_{S}, а, в) = (5, 3, 5,4), and covariance matrices given by

Parameter estimates for trial- and individual-level surrogacy measures obtained for the reduced mixed-effects model are equal to R?2riai(r) = 0.8144

(0.7186, 0.9102) and R?_{1}^{2}ndiv = 0.6241 (0.5872, 0.6609), respectively (Figure 12.16). Figure 12.17 shows the empirical Bayes estimates for the random effects.