# Validation Using a Joint Model for Continuous and Binary Endpoints

Similar to the previous section, we assume an underlying latent normally distributed surrogate endpoint, Sj, and an observed surrogate given by (Van Sanden et al., 2012):

A joint model is assumed for the latent surrogate variable *S _{i}j* and the true endpoint

*Tij*,

FIGURE 12.37

*Descriptive plots for the prostate dataset. Panel a: Patients distribution by treatment arms across trials. Panel b: Histogram for the continuous surrogate endpoint. Panel c: Scatter plot between the survival time (ignoring censoring) true endpoint and the continuous surrogate endpoint.*

FIGURE 12.38

*Prostate Data. Surrogacy measures.*

where

FIGURE 12.39

*Evaluation of trial-level surrogacy for the prostate dataset. Treatment effects upon the true endpoints (log-hazard ratio) versus treatment effects upon the continuous surrogate. The size of the circle per trial is proportional to the sample size of the trial.*

Model formulation for the observed binary outcome Sj and the Tj are given

by

The correlation between the measurement of the response variables can be modeled directly using the covariance matrix of the residuals, specified in (12.40), and a measure for individual-level surrogacy is given by

)

Note that for this surrogacy setting the measure for individual-level surrogacy is the adjusted association between the true endpoint and the latent surrogate endpoint.

Trial-level surrogacy is estimated using a second-stage model for the parameter estimates of the treatment effects for the surrogate and true endpoints (Section 12.3.2).

FIGURE 12.40

Data *structure for some selected patients.*