Analysis for a Surrogacy Setting with Two Survival Endpoints

Analysis strategies for a surrogacy setting in which the two endpoints are time-to-event variables were discussed in Chapter 5. In this section, we focus on three applications. In the first two, the analysis is based on a two-stage approach (see Section 5.3) while the third application is based on the joint modeling of two time-to-event endpoints (see Section 5.2).

For illustration, we used the ovarian study with overall survival and progression-free survival as the true and surrogate endpoints, respectively. Overall, 1153 patients from 39 trials are included in the analysis (569 in the treatment arm and 584 in the control arm). Throughout this section, we briefly discuss the technical details and we refer to Chapter 5 for an elaborate discussion about the modeling approaches and surrogacy measures. A partial print of the data is given in Figure 12.19. The data for each patient appear as a single line.

A Two-Stage Approach (I)

As pointed out in Chapter 5, the first-stage model consists of trial-specific Cox proportional hazard models for the two endpoints given by

where Si0 (t) and Tj0(t) are the trial-specific baseline hazard functions, and Zj is a treatment indicator for the jth individual in the ith trial. The parameters в and Oj are the trial-specific treatment effects.

One way to account for variation in trial size is to use the number of patients in each trial in a weighted linear regression of the form

A second approach to account for the variability between trials is to use a robust sandwich estimator of Lin and Wei (1989) for the covariance matrix of the parameter estimates for treatment effects in (12.18) and follow the approach proposed by van Houwelingen et al. (2002). The two approaches are implemented in the macro %TWOSTAGECOX discussed below. As before, the coefficient of determination from (12.19) is used as a trial-level surrogacy measure. A leave-one-out analysis can be performed in order to assess model accuracy.

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