Data Analysis and Output
Due to convergence problems with the full random effects, a simulated set of data was used to generate numerical and graphical outputs. 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
Surrogacy measures with their 95% C.I., full mixed-effects model.
Trial-specific random-effects plot.
(ps, Мт, а, в) = (5, 5, 5, 5), and covariance matrices given by
As shown in Figure 12.13, the surrogacy measures are equal to R?ndiv = 0.6260 (0.5893, 06628) and Rtlia = 0.7655 (0.6483, 0.8828). Figure 12.14 shows the empirical Bayes estimates for the trial-specific random treatment effects.
Covariance matrices, full mixed-effects model.
SAS Code for the Full Mixed-Effects Model
The full mixed-effects model can be fitted using procedure MIXED. Possible code is given by
proc mixed data=dataset covtest; class endp patid trial;
model outcome = endp endp*treat / solution noint; random endp endp*treat / subject=trial type=un; repeated endp / type=un subject=patid(trial); ods output solutionF=fix CovParms=covar SolutionR=eb; run;
The data structure and variables are identical to those outlined in Section 12.3.1. The MODEL statement defines the 4 fixed effects in the mean structure, (pT, , а, в), while the RANDOM statement defines the structure of the
covariance matrix D for the random effects, and the REPEATED statement builds up the error covariance matrix E in (12.2). The estimated covariance matrices are shown in Figure 12.15. The lower panel presents the parameter estimates of the covariance matrix D (given in 4.2).
The trial-level surrogacy measure is estimated by (4.4):
The estimated covariance matrix for the residuals (defined in (12.2)) is given by
and individual-level surrogacy is derived according to (4.9):