Implementation in SAS
In this section, we discuss the implementation of the joint model discussed in Section 17.4 in SAS using data from the motor cortex with a pair of MRI-AK
Individual-level surrogacy in 23 regions of the brain (pT and pw, respectively, for each MRI parameter and histology percentage of area stained). White fill: surrogacy measure not computed.
(biomarker) and GFAP percentage of area stained (true endpoint).
The joint model, discussed in Section 17.4, was fitted using procedure MIXED in SAS 9.4. For each subject, measurements for both MRI and histology were available. Hence, data for MRI and histology parameters for a single subject appeared in subsequent rows. A partial print of the data is given below:
proc print data=MriHistData; run animalid age genotype response endpoint 1 2 TRANSGENIC 0.001128658 MRI 1 2 TRANSGENIC 0.126652588 Histology 2 2 WILDTYPE 0.189814745 Histology 2 2 WILDTYPE 0.001124777 MRI 17.7.2 Common Parameter for Histology in the Wildtype Group
As mentioned in Section 17.4, from a biological point of view, it is assumed that histology values of wildtype mice should remain constant between the age of 2 and 10 months, since there is no significant disease pathology (due to aging) progression. Hence, in the model for histology, wildtype mice have a single parameter which does not change with age. In SAS, this can be achieved by defining a common parameter CommonInt for histology in wildtype using the following code:
data MriHistData; set MriHistData; commonInt=age;
if treatment=’WILDTYPE’ and endpoint=’Histology’ then commonInt=0; run;
The association between MRI and histology is modeled using the REPEATED statement. The option GROUP=genotype allows for genotype-specific covariance matrices (17.3). The estimated disease effects on both MRI and histology are output by passing the solution option and stored in a dataset named
fixedeffects using the SOLUTIONF option in the ODS OUTPUT command. The complete SAS code used to fit the joint model is:
proc mixed data=surrogate;
class genotype animalid endpoint commonint(ref=’0’); model response=commonint*endpoint commonint*endpoint*genotype / solution noint;
repeated endpoint / sujbect=animalid*commonint
type=un group=genotype R=1,4 rcorr=1,4; ods output solutionf=fixedeffects; run;
Parameter estimates for the disease progression effects (in both endpoints) are shown in Figure 17.17. The individual-level surrogacy for wildtype and transgenic mice is computed using the estimated covariance matrices (Figure 17.18). To compute disease-level surrogacy, a linear regression model is fitted using PROC GLM in SAS.
proc glm data=fixedeffectsProcessed; model effectstrue=effectssurrogate; run;
In the above code, effectstrue corresponds to the estimated disease effects on the true endpoint /3, while effectssurrogate corresponds to the estimated disease effects on the surrogate endpoint a. The reported measure for disease- level surrogacy corresponds to R2 obtained from the linear regression model (Figure 17.19).
Age-Specific Parameters for Histology in the Wildtype Model
Rather than assume a common histology parameter in wildtype mice, an age-specific parameter can be specified by substituting commonint with age. In order to obtain age and genotype-specific estimates for the disease effect on both MRI and histology, we include in the model an interaction term age*endpoint*genotype.
proc mixed data=MriHistData; class genotype animalid endpoint age; model response=age*endpoint age*endpoint*genotype / solution noint;
repeated endpoint / subject=animalid*age type=un
group=genotype R=1,4 rcorr=1,4; ods output solutionf=fixedeffects;
SAS fixed-effects parameter estimates.