A Continuous (Normally Distributed) and a Survival Endpoint

Model Formulation

The two-stage approach for a continuous (normally distributed) and a survival endpoints was discussed in Chapter 7. It is assumed that the true endpoint, T, is a failure-time random variable and the surrogate, S, is a normally distributed continuous variable.

Briefly, the first stage consists of the classical linear regression model for

FIGURE 12.36

Data structure.

the surrogate given by

where e®j is normally distributed with mean zero and variance of. The proportional hazard model for the true endpoints given by

where в* are trial-specific effects of treatment Z and A® (t), is a trial-specific baseline hazard function.

If a parametric (e.g., Weibull-distribution-based) baseline hazard is used in (12.37), the joint distribution function defined by the copula and the marginal models (12.36) and (12.37) allows us to construct the likelihood function for the observed data and obtain estimates of the treatment effects and в*.

Individual-level surrogacy can be evaluated by using Kendall’s т or Spearman’s p (see Section 5.2). Trial-level surrogacy is assessed using the correlation coefficient between the estimated treatment effects and в®.

 
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