Implementation in R

The proposed model can easily be implemented in R using the gls function from the nlme package.

FIGURE 17.18

SAS covariance matrix estimates for individual-level surrogacy.

Common Parameter for Histology in the Wildtype Group

We adopt a dummy coding for the variables of interest as shown in the partial print of the MriHistData.dummy data object (Figure 17.20). The model can be fitted using the following R code:

library(nlme)

fit <- gls(response~-1+endpoint+mu_SURRO_wt4+mu_SURRO_wt6+ mu_SURRO_wt8+mu_SURRO_wt10 +beta_TRANS_TRUE_2 +alpha_TRANS_TRUE_4+alpha_TRANS_TRUE_6+alpha_TRANS_TRUE_8 +alpha_TRANS_TRUE_10+beta_TRANS_SURRO_2

FIGURE 17.19

Disease-level surrogacy: GLM regression model output.

+beta_TRANS_SURRO_4 +beta_TRANS_SURRO_6 +beta_TRANS_SURRO_8 +beta_TRANS_ SURRO _10, data=MriHistData.dummy,

correlation=corSymm(form = ~ 1| animalid ), weight=varIdent(form=~1|endpoint*genotype))

By specifying endpoint in the right-hand side of the formula, we allow for a common parameter estimate for histology (true endpoint) in wildtype as well as a parameter for MRI (surrogate endpoint) at 2 months for wildtype mice. The variables alpha_TRANS_TRUE_2- alpha_TRANS_TRUE_10 correspond to втвб while beta_TRANS_SURRO_2- beta_TRANS_SURRO_10 correspond to 01-05 in (17.1).

The argument correlations . . allows for the specification of correlated outcomes within a subject. Further, we specify an unstructured correlation using the corSym construct. In order to define heterogeneous variances, that is, endpoint and genotype-specific variance covariance matrices as defined in (17.3), the argument weight=varIdent(...) is used. The output for the disease progression parameters is shown in Figure 17.21.

The estimated correlation estimates for disease-level surrogacy (17.4) are shown in the panel below.

Transgenic.covmat

Marginal variance covariance matrix [,1] [,2]

[1,] 1.00000 0.13295 [2,] 0.13295 1.00000

Wildtype.covmat <- getVarCov(fit, individual=4) cov2cor(Wildtype.covmat)

FIGURE 17.20

Partial print of the MriHistData. dummy data object.

FIGURE 17.21

R gls output for the surrogacy model.

Marginal variance covariance matrix [,1] [,2]

[1,] 1.00000 0.13355 [2,] 0.13355 1.00000

To obtain the disease-level surrogacy, a linear regression is fitted to the disease progression effects and the model R² obtained.

summary(lm(alpha~beta, data=fixedEffectsProcessed))$r.squared [1] 0.9101327