# The Reduced Fixed-Effects Model

## Model Formulation

The reduced fixed-effects model approach assumes common intercepts for S and T in (12.1). Hence, trial-specific and are replaced by and , respectively. The full fixed-effects model in (12.1) can be rewritten as

The term is dropped from the second-stage model, which implies that trial- level surrogacy is assessed using the coefficient of determination obtained for the model

Individual-level surrogacy can be assessed using the adjusted association in (12.3).

## The SAS Macro %CONTCONTRED

The SAS macro %CONTCONTRED can be used to fit the reduced joint model specified in (12.6). For the ARMD data we use

FIGURE 12.9

*Surrogacy measures with 95% C.I; reduced fixed effects model.*

%CONTCONTRED(data=armd,true=diff52,surrog=diff24,trt=treat, trial=center,patid=patientId,weighted=1, looa=1)

The specification of the macro’s arguments is the same as the specification presented in Section 12.2.

## Data Analysis and Output

Surrogacy measures obtained from the reduced fixed-effect model are shown below. Similar to the results presented in Section 12.3.1, the surrogacy measures *R2 _{ndiv}* = 0.5318 (0.4315,0.6321) and R

_{t}

^{2}rial(r) = 0.6585 (0.4695,0.8476) indicated that visual acuity after 24 weeks after starting the interferon-a treatment is a surrogate of moderate value for the visual acuity at 52 weeks after starting the interferon-a treatment. Trial-specific parameter estimates for treatment effects are shown in Figure 12.10. The regression line fitted at the second stage is added. The circle sizes in the plot are proportional to the number of patients from a given trial. Similar to the analysis presented in the previous section, if the argument looa=1 is used, a “leave-one-out” analysis is performed.