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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.

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Il Do HA, Jianxin PAN, Seungyoung OH, and Youngjo LEE Variable selection methods using a penalized likelihood have been widely studied in various statistical models. However, in semiparametric frailty models, these methods have been relatively less studied because the marginal likelihood function involves analytically intractable integrals, particularly when modeling multicomponent or correlated frailties. In this article, we propose a simple but unified procedure via a penalized h-likelihood (HL) for variable selection of fixed effects in a general class of semiparametric frailty models, in which random effects may be shared, nested, or correlated. We consider three penalty functions (least absolute shrinkage and selection operator [LASSO], smoothly clipped absolute deviation [SCAD], and HL) in our variable selection procedure. We show that the proposed method can be easily implemented via a slight modification to existing HL estimation approaches. Simulation studies also show that the procedure using the SCAD or HL penalty performs well. The usefulness of the new method is illustrated using three practical datasets too. Supplementary materials for the article are available online.

This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore PrefaceSurvival or time-to-event data arise in various research areas such as medicine, epidemiology, genetics, engineering, econometrics, and sociology. Survival data have unique features including incomplete observation such as censoring and/or truncation. Use of semi-parametric models and potential correlation among time-to-events from the same cluster can make the statistical inference further complicated.Broad classes of multivariate models using random effects have been developed. For inferences about unobserved random variables, the hierarchical (or h-)likelihood has been proposed by Lee and Nelder (1996). This book presents recent works on h-likelihood for the analysis of survival data. The h-likelihood method has been used to make inferences on the random effects models, especially for the frailty model for time-to-event data, where the frailties are treated as unobserved yet realized in the data. The h-likelihood allows an extension to the frailty models under competing risks as well as to the models for joint outcomes, e.g., longitudinal and event time outcomes. The h-likelihood method estimates the population parameters and the random effects simultaneously, with the random effects being updated from the observed data. This book covers the state-of-the-art h-likelihood methods, which include interval estimation of the individual frailty and variable selection of the covariates in the general class for the frailty models with or without competing risks. A beauty of the h-likelihood is that once the statistical model is specified parametrically or nonparametric...