# Two Continuous Endpoints: The Reduced Fixed- Effects Model

In Chapter 4, we discussed the reduced fixed-effects model for two continuous endpoints. The model can be formulated as:

For the reduced fixed-effect model, trial-level surrogacy is assessed using the coefficient of determination obtained by fitting a linear regression model of the form:

where *(3®* and a® are the trial-specific estimated treatment effects upon Tj and *Sij*, respectively. The error terms, e®, are normally distributed with mean zero and a constant variance. Individual-level surrogacy is assessed by the squared correlation between S and T after adjusting for trial-specific treatment effects and is given by:

Once the variables specification is complete (see the left panel in Figure 14.1), one can choose the model to be fitted using the command bar in the upper part in Figure 14.1. The Surrogate Shiny App produces a default output shown in Figure 14.2. For the ARMD study, R?_{ndiv} = 0.5318 (0.4315, 0.6321) and Retrial = 0.6585 (0.4695, 0.8476). If other statistics are of interest, one can use the R package Surrogate to produce them.

The reduced fixed-effects model specified in (14.1) and in the Surrogate Shiny App in Figure 14.1 is identical to the model fitted using the function BifixedContCont below:

Sur<-BifixedContCont(Dataset=ARMD, Surr=Diff24, True=Diff52,

Treat=Treat, Trial.ID=Center,

Pat.ID=Id,Model="Reduced", Weighted=TRUE)