A Continuous (Normally-Distributed) and a Failure-Time Endpoint

Tomasz Burzykowski

Hasselt University, Belgium, and IDDI, Belgium

Introduction

In this chapter, we focus on the case when the surrogate is a continuous, normally distributed endpoint and the true endpoint is a failure-time endpoint. Note that the described approach is applicable also in the reverse case, i.e., with a failure-time surrogate and a continuous true endpoint.

Theoretical Background

Assume that the true endpoint T is a failure-time random variable and the surrogate S is a normally distributed, continuous variable. For each of j = 1,... ,щ patients from trial i (i = 1,... ,N) we thus have quadruplets

(Xij, Aij ,Sij ,Zj), where X_{i}j is a possibly censored version of survival time Tij and Aij is the censoring indicator assuming a value of 1 for observed failures and 0 otherwise.

As in Chapter 4, we consider the two-stage approach with model (4.10) replaced by a copula-based model (see, e.g., Section 5.2) for the true endpoint T_{i}j and the continuous variable S_{i}j. Toward this aim, various copula functions can be used (see Section 5.2).

The marginal model for S_{i}j is the classical linear regression model:

where e_{i}j is normally distributed with mean zero and variance a^{2}.

For Tij, the proportional hazard model is used:

where Д are trial-specific effects of treatment Z and A_{i}(t) is a trial-specific baseline hazard function.

If a parametric (e.g., Weibull-distribution-based) baseline hazard is used in (7.2), the joint distribution function defined by the copula and the marginal models (7.1) and (7.2) allows constructing the likelihood function for the observed data (Xj = xj, A_{i}j = S_{i}j, S_{i}j = s_{i}j, Z_{i}j = z_{i}j) and obtaining estimates of the treatment effects a_{i} and p_{i}.

The quality of the surrogate at the individual level can be evaluated by using Kendall’s т or Spearman’s p (see Section 5.2). The quality at the trial level can be evaluated by considering the correlation coefficient between the estimated treatment effects a_{i} and p_{i}. Note that, in this step, the adjustment for the estimation error, present in a_{i} and /3_{i}, should be made. Toward this aim, model (5.13)-(5.14) (see Section 5.2) can be used.

If the individual-level association is not of immediate interest, one may base analysis on the marginal models (7.1) and (7.2), without specifying the baseline hazards in the latter. When fitting the models, it is worth estimating the variance-covariance matrix of the estimated treatment effects a_{i} and /3_{i }while taking into account the association between S and T. Toward this aim, an estimator similar to the one proposed in Appendix 6.5 can be used (see also Appendix 7.5).