Proportional hazards model with random effects

Abstract

We propose a general proportional hazards model with random effects for handling clustered survival data. This generalizes the usual frailty model by allowing a multivariate random effect with arbitrary design matrix in the log relative risk, in a way similar to the modelling of random effects in linear, generalized linear and non‐linear mixed models. The distribution of the random effects is generally assumed to be multivariate normal, but other (preferably symmetrical) distributions are also possible. Maximum likelihood estimates of the regression parameters, the variance components and the baseline hazard function are obtained via the EM algorithm. The E‐step of the algorithm involves computation of the conditional expectations of functions of the random effects, for which we use Markov chain Monte Carlo (MCMC) methods. Approximate variances of the estimates are computed by Louis' formula, and posterior expectations and variances of the individual random effects can be obtained as a by‐product of the estimation. The inference procedure is exemplified on two data sets. Copyright © 2000 John Wiley & Sons, Ltd.

Bayesian Accelerated Failure Time Model With Multivariate Doubly Interval-Censored Data and Flexible Distributional Assumptions

“…For example, Kooperberg and Clarkson (1997), Betensky et al (1999), and Goetghebeur and Ryan (2000) consider independent interval-censored data. Vaida and Xu (2000) offer an approach based on the proportional hazards linear mixed model with right-censored data.…”

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

“…Here, the variance estimate of frailty isσ 2 = 0.478 (with SE = 0.313). Note that although we report the SE of σ 2 , one should not use it for testing the absence of frailty σ 2 = 0 (Vaida and Xu, 2000). Such a null hypothesis is on the boundary of the parameter space, and hence, the critical value of an asymptotic (χ 2 0 + χ 2 1 )/2 distribution is χ 2 1,0.10 = 2.71 at the 5% level (Ha et al, , 2012b.…”

“…It is also important to investigate the potential heterogeneity in event times among clusters (e.g. 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. 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.…”

Conditional Transformation Models for Survivor Function Estimation

“…This trial has been analysed earlier using a Cox model [43][44][45][46]. Since the proportional hazards assumption is questionable for the different treatment groups, we re-analysed the data set using the CTM approach and allowed for non-proportional effects of the patient characteristics over time.…”