Estimating and Testing the Variation of a Binary Treatment Effect Over Clusters in a Proportional Hazard Frailty Model

Abstract

This article proposes a parametric (e.g., piecewise exponential) relative risk frailty model to test and estimate treatment-by-center effects for clustered time-to-event data, where the unobserved center and treatment-by-center effects are assumed to follow a bivariate normal distribution. We use an adaptive Gaussian quadrature numerical integration method to compute the full marginal likelihood. Under regularity conditions, the full likelihood estimators are consistent and the estimating procedure yields a valid likelihood ratio test statistic that asymptotically follows a mixture χ² distribution for treatment-by-center effects. Simulation results suggest the distribution of the likelihood ratio statistic for testing treatment-by-center effects generally approximates the mixture χ² distribution well when the study contains 100 or more centers. We use the proposed test and estimating procedure to analyze a dataset from the national kidney transplant registry.

Key Words:

• Adaptive Gaussian quadrature
• Kidney transplant
• Laplace approximation
• Survival analysis

Acknowledgments

The authors thank the Associate Editor and two referees for their constructive comments and suggestions that have improved the presentation and the quality of the article.