Frailty modelling for survival data from multi‐centre clinical trials

Abstract

Summary Despite the use of standardized protocols in, multicentre, randomised clinical trials (RCTs), outcome may vary between centres. Such heterogeneity may alter the interpretation and reporting of the treatment effect. Below, we propose a general frailty modelling approach for investigating, inter alia, putative treatment-by-centre interactions in time-to-event data in multi-centre clinical trials. A correlated random effects model is used to model the baseline risk and the treatment effect across centres. It may be based on shared, individual or correlated random-effects. For inference we develop the hierarchical-likelihood (or h-likelihood) approach which facilitates computation of prediction intervals for the random effects with proper precision. We illustrate our methods using disease-free time-to-event data on bladder cancer patients participating in an European Organization for Research and Treatment of Cancer (EORTC) trial, and a simulation study. We also demonstrate model selection using h-likelihood criteria.

A visualizing method for investigating individual frailties using frailtyHL R-package

“…Although the results of this paper can be extended to non-normal frailties (e.g. gamma frailty), for simplicity, we assume a normal distribution v i ∼ N (0, θ) with θ = σ 2 , which is useful for modelling multi-component frailties (Ha et al, 2007) including multilevel (nested) structures and/or correlated frailties including negative correlation (Rondeau et al, 2008;Ha et al, 2011).…”

“…Thus Nelder's (1996, 2001) h-likelihood procedure for hierarchical generalized linear models (HGLMs) can be extended to the frailty models based on h * . Accordingly, given frailty parameter θ, the maximum h-likelihood estimators of τ = (β T , v T ) T are obtained by solving the joint estimating equations, ∂h * /∂τ = ∂h/∂τ | λ0=λ0 = 0; in particular,v(θ) is the solution to ∂h * /∂v = 0 for a given θ andv(θ) ≈ E θ (v|y, δ) as n * = min 1≤i≤q n i → ∞ (Lee and Nelder, 2009;Ha et al, 2011). Furthermore, for the estimation of θ we use an adjusted profile h-likelihood, p β,v (h * ), defined by…”

“…In frailty models (2.1), as in HGLMs (Lee and Nelder, 2009;Kim et al, 2011), location parameters (β, λ 0 , v) and frailty parameters θ are asymptotically orthogonal (Ha and Lee, 2003;Ha et al, 2011). For a moment, assume that θ is known.…”

“…centers, patients) in order to understand and interpret the variability in the data (Vaida and Xu, 2000). For example, despite the use of standardized protocols in multicenter randomized clinical trials, outcome may vary between centers (Rondeau et al, 2008;Ha et al, 2011). Such heterogeneity may alter the interpretation and reporting of the treatment.…”

“…Such heterogeneity may alter the interpretation and reporting of the treatment. For purpose of this analysis, the interval estimation of frailty is more useful than the inference of variance components of frailty (Vaida and Xu, 2000;Lee and Nelder, 2009;Ha et al, 2011). Plots based on these confidence intervals are also useful for investigating such heterogeneity.…”