A joint frailty‐copula model for meta‐analytic validation of failure time surrogate endpoints in clinical trials

Abstract

In a meta‐analysis framework, the classical approach for the validation of time‐to‐event surrogate endpoint is based on a two‐step analysis. This approach often raises estimation issues. Recently, we proposed a one‐step validation approach based on a joint frailty model. This approach was quite time consuming, despite parallel computing, due to individual‐level frailties used to take into account heterogeneity in the data at the individual level. We now propose an alternative one‐step approach for evaluating surrogacy, using a joint frailty‐copula model. The model includes two correlated random effects treatment‐by‐trial interaction and a shared random effect associated with the baseline risks. At the individual level, the joint survivor functions of time‐to‐event endpoints are linked using copula functions. We used splines for the baseline hazard functions. We estimated parameters and hazard function using a semiparametric penalized marginal likelihood method, considering various numerical integration methods. Both individual‐level and trial‐level surrogacy were evaluated using Kendall's tau and coefficient of determination. The performance of the estimators was evaluated using simulation studies. The model was applied to individual patient data meta‐analyses in advanced ovarian cancer to assess progression‐free survival as a surrogate for overall survival, as part of the evaluation of new therapy. The model showed good performance and was quite robust regarding the integration methods and data variation, regardless of the surrogacy evaluation criteria. Kendall's Tau was better estimated using the Clayton copula model compared to the joint frailty model. The proposed model reduces the convergence and model estimation issues encountered in the two‐step approach.

Conditional copula models for correlated survival endpoints: Individual patient data meta-analysis of randomized controlled trials

“…This assumption is expected to hold in randomized controlled trials and thus imposed in the literature of surrogate endpoint validation. [13][14][15][16][17][18][19][20]…”

“…That is, the treatment may change the correlation structure for endpoints. We shall show that this simplifying assumption was imposed for the classical copula model, 14 the joint frailty-copula models, 20,31 and the joint frailty models. 17,19 In the copula literature, the simplifying assumption refers to a case, where a conditional copula for two random variables do not depend on the variables that are conditioned on.…”

“…Meta-analytical datasets (individual-patient data from multiple clinical trials) have been utilized to estimate Kendall's tau in order to measure the quality of surrogate endpoints for OS at individual level. 13,14,[17][18][19][20][21] Any valid surrogate is required to have Kendall's tau close to 1 (desirably around 0.9 as quoted in Burzykowski et al 14 ).…”

“…In the classical copula model of Burzykowski et al 14 Kendall's tau is defined through a copula function that describes residual association between PFS and OS given a treatment indicator and a trialspecific baseline hazard function. In mixed-effect models, [17][18][19][20][21] Kendall's tau is defined given a treatment indicator and trial-specific random effects.…”

“…Therefore, it is natural to introduce the between-trial heterogeneity through a vector of trial-level random effects U . [17][18][19][20][21][30][31][32][33] Frailty is an example of random effect. [34][35][36][37] In meta-analysis, the variables S, T, and Z are sampled within a trial (conditional on a trial's random-effects U = u).…”