# Validation Using a Joint Model for Two Binary Endpoints

To extend the methodology used for continuous endpoints to the case of binary endpoints, Renard et al. (2002) adopted a latent variable approach, assuming that the observed binary variables (Sj ,Tj) are obtained from dichotomizing unobserved continuous variables (Sj*,Tj*). The realized value of *Sj* (Tj) equals 1 if Sj > 0 (Tj > 0), and 0 otherwise. It is assumed that the latent variables, representing the continuous underlying values of the surrogate and the true endpoints for the jth subject in the ith trial, follow a random-effects model at latent scale given by:

Here, p_{s} and are fixed intercepts, *a* and *в* are fixed treatment effects, m_{Si} and m_{Tj} are random (i.e., trial-specific) intercepts, a* and b* are random treatment effects, and ?«.. and eV. are error terms. The random effects are zero-mean normally distributed with covariance matrix D. The error terms are assumed to follow a bivariate normal distribution with zero-mean and covariance matrix given by

The model formulated in (12.44) leads to a joint probit model:

where Ф denotes the standard normal cumulative distribution function. Similar to the normal-normal setting, a reduced fixed-effects model in which the random intercepts and slopes are excluded and assuming common intercepts can be formulated as

Individual-level surrogacy can be estimated using the adjusted association based on the covariance matrix (12.45). This implies that for the binarybinary setting, this level of surrogacy should be interpreted at the scale of the linear predictors. Trial-specific treatment effects upon the true and the surrogate endpoints can be used in the second stage to fit a linear regression model of the form

As in previous sections, the trial sizes are used as weights in order to account for the variability due to difference in sample sizes. The trial-level surrogacy measure is equal to the coefficient of determination for model (12.48).