Validation Using Joint Modeling of a Time-to- Event and a Binary Endpoint
The setting we consider in this section consists of a binary surrogate endpoint and a time-to-event true endpoint. A bivariate copula model, proposed by Burzykowski, Molenberghs, and Buyse (2004), is formulated for the true endpoint Tj and a latent normally distributed variable Sj. The binary surrogate is defined by
For the surrogate endpoint, Sj, a logistic regression model is assumed The marginal cumulative distribution function of Sj, given Zj = z, is denoted
To model the effect of treatment on the marginal distribution of Tj, Burzykowski, Molenberghs, and Buyse (2004) proposed to use a proportional hazard model of the form
where Д are trial-specific treatment effects and A* (t) is a trial-specific baseline hazard function. The marginal cumulative distribution function of Tj, with Zj = z, is denoted by FTij (t; z). As pointed out in Chapter 5, the joint cumulative distribution of Tj and Sj, given Zj = z, is generated by one parameter copula function Cg:
Here, Cg is a distribution function on [0,1]2 with в G R1.
The two-stage approach proposed by Burzykowski, Molenberghs, and Buyse (2004) consists of maximum likelihood estimation for в and the trial- specific treatment effects a and в at the first stage, while at the second stage, it is assumed that
The second term of the right-hand side of (12.32) is assumed to follow a bivariate normal distribution with mean zero and covariance matrix given by
Hence, similar to Section 4.3.2, trial-level surrogacy is estimated by:
To assess the quality of the surrogate endpoint at the individual level, a measure of association between Sj and Tj is needed. Burzykowski, Molenberghs, and Buyse (2004) proposed to use the bivariate Plackett copula. This particular choice was motivated by the fact that, for the Plackett copula, the association parameter в takes the form of a (constant) global odds ratio.
For a binary surrogate, it is just the odds ratio for responders versus nonresponders (assuming k = 2 indicates the response).