Frailty modelling for survival data from multi-centre clinical trials

Multilevel joint frailty model for hierarchically clustered binary and survival data

Hierarchical data arise when observations are clustered into groups. Multilevel models are practically useful in these settings, but these models are elusive in the context of hierarchical data with mixed multivariate outcomes. In this article, we consider binary and survival outcomes and assume the hierarchical structure is induced by clustering of both outcomes within patients and clustering of patients within hospitals which frequently occur in multicenter studies. We introduce a multilevel joint frailty model that analyzes the outcomes simultaneously to jointly estimate their regression parameters and explicitly model within-patient correlation between the outcomes and within-hospital correlation separately for each outcome. Estimation is facilitated by a computationally efficient residual maximum likelihood method that further predicts cluster-specific frailties for both outcomes and circumvents the formidable challenges induced by multidimensional integration that complicates the underlying likelihood. The performance of the model and estimation procedure is investigated via extensive simulation studies. The practical utility of the model is illustrated through simultaneous modeling of disease-free survival and binary endpoint of platelet recovery in a multicenter allogeneic bone marrow transplantation dataset that motivates this study.

Goodness-of-fit tests in proportional hazards models with random effects

This paper deals with testing the functional form of the covariate effects in a Cox proportional hazards model with random effects. We assume that the responses are clustered and incomplete due to right censoring. The estimation of the model under the null (parametric covariate effect) and the alternative (nonparametric effect) is performed using the full marginal likelihood. Under the alternative, the nonparametric covariate effects are estimated using orthogonal expansions. The test statistic is the likelihood ratio statistic, and its distribution is approximated using a bootstrap method. The performance of the proposed testing procedure is studied through simulations. The method is also applied on two real data sets one from biomedical research and one from veterinary medicine.

Sample size estimation for cancer randomized trials in the presence of heterogeneous populations

A key issue when designing clinical trials is the estimation of the number of subjects required. Assuming for multicenter trials or biomarker-stratified designs that the effect size between treatment arms is the same among the whole study population might be inappropriate. Limited work is available for properly determining the sample size for such trials. However, we need to account for both, the heterogeneity of the baseline hazards over clusters or strata but also the heterogeneity of the treatment effects, otherwise sample size estimates might be biased. Most existing methods account for either heterogeneous baseline hazards or treatment effects but they dot not allow to simultaneously account for both sources of variations. This article proposes an approach to calculate sample size formula for clustered or stratified survival data relying on frailty models. Both theoretical derivations and simulation results show the proposed approach can guarantee the desired power in worst case scenarios and is often much more efficient than existing approaches. Application to a real clinical trial designs is also illustrated.

A general class of small area estimation using calibrated hierarchical likelihood approach with applications to COVID-19 data

The direct estimation techniques in small area estimation (SAE) models require sufficiently large sample sizes to provide accurate estimates. Hence, indirect model-based methodologies are developed to incorporate auxiliary information. The most commonly used SAE models, including the Fay-Herriot (FH) model and its extended models, are estimated using marginal likelihood estimation and the Bayesian methods, which rely heavily on the computationally intensive integration of likelihood function. In this article, we propose a Calibrated Hierarchical (CH) likelihood approach to obtain SAE through hierarchical estimation of fixed effects and random effects with the regression calibration method for bias correction. The latent random variables at the domain level are treated as 'parameters' and estimated jointly with other parameters of interest. Then the dispersion parameters are estimated iteratively based on the Laplace approximation of the profile likelihood. The proposed method avoids the intractable integration to estimate the marginal distribution. Hence, it can be applied to a wide class of distributions, including generalized linear mixed models, survival analysis, and joint modeling with distinct distributions. We demonstrate our method using an area-level analysis of publicly available count data from the novel coronavirus (COVID-19) positive cases.

Preoperative serum CA19-9 should be routinely measured in the colorectal patients with preoperative normal serum CEA: a multicenter retrospective cohort study

Objective

Whether preoperative serum carbohydrate antigen 19–9 (CA19-9) is an independent prognostic factor and there are interactions of serum CA19-9 with carcinoembryonic antigen (CEA) on the risk of recurrence in colorectal cancer (CRC) patients are still not clarified.

Methods

Consecutive patients with CRC who underwent curative resection for stage II-III colorectal adenocarcinoma at five hospitals were collected. Based on Cox models, associations of preoperative CA19-9 with recurrence-free survival (RFS) and overall survival (OS) were evaluated in patients with or without elevated CEA, and interactions between CEA and CA19-9 were also calculated. Restricted cubic spline (RCS) curves were used to evaluate the associations between preoperative CA19-9 and CRC outcomes on a continuous scale.

Results

A total of 5048 patients (3029 [60.0%] men; median [interquartile range, IQR] age, 61.0 [51.0, 68.0] years; median [IQR] follow-up duration 46.8 [36.5–62.4] months) were included. The risk of recurrence increased with the elevated level of preoperative CA19-9, with the slope steeper in patients with normal CEA than those with elevated CEA. Worse RFS was observed for elevated preoperative CA19-9 (> 37 U/mL) (n = 738) versus normal preoperative CA19-9 (≤ 37 U/mL) (n = 4310) (3-year RFS rate: 59.4% versus 78.0%; unadjusted hazard ratio [HR]: 2.02; 95% confidence interval [CI]:1.79 to 2.28), and significant interaction was found between CA19-9 and CEA (P for interaction = 0.001). Increased risk and interaction with CEA were also observed for OS. In the Cox multivariable analysis, elevated CA19-9 was associated with shorter RFS and OS regardless of preoperative CEA level, even after adjustment for other prognostic factors (HR: 2.08, 95% CI:1.75 to 2.47; HR: 2.25, 95% CI:1.80 to 2.81). Subgroup analyses and sensitivity analyses yielded largely similar results. These associations were maintained in patients with stage II disease (n = 2724).

Conclusions

Preoperative CA19-9 is an independent prognostic factor in CRC patients. Preoperative CA19-9 can be clinically used as a routine biomarker for CRC patients, especially with preoperative normal serum CEA.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12885-022-10051-2.

Multi-parameter regression survival modelling with random effects

We consider a parametric modelling approach for survival data where covariates are allowed to enter the model through multiple distributional parameters (i.e., scale and shape). This is in contrast with the standard convention of having a single covariate-dependent parameter, typically the scale. Taking what is referred to as a multi-parameter regression (MPR) approach to modelling has been shown to produce flexible and robust models with relatively low model complexity cost. However, it is very common to have clustered data arising from survival analysis studies, and this is something that is under developed in the MPR context. The purpose of this article is to extend MPR models to handle multivariate survival data by introducing random effects in both the scale and the shape regression components. We consider a variety of possible dependence structures for these random effects (independent, shared and correlated), and estimation proceeds using a h-likelihood approach. The performance of our estimation procedure is investigated by a way of an extensive simulation study, and the merits of our modelling approach are illustrated through applications to two real data examples, a lung cancer dataset and a bladder cancer dataset.

Penalized variable selection for cause-specific hazard frailty models with clustered competing-risks data

Competing risks data usually arise when an occurrence of an event precludes other types of events from being observed. Such data are often encountered in a clustered clinical study such as a multi-center clinical trial. For the clustered competing-risks data which are correlated within a cluster, competing-risks models allowing for frailty terms have been recently studied. To the best of our knowledge, however, there is no literature on variable selection methods for cause-specific hazard frailty models. In this article, we propose a variable selection procedure for fixed effects in cause-specific competing risks frailty models using a penalized h-likelihood (HL). Here, we study three penalty functions, LASSO, SCAD, and HL. Simulation studies demonstrate that the proposed procedure using the HL penalty works well, providing a higher probability of choosing the true model than LASSO and SCAD methods without losing prediction accuracy. The proposed method is illustrated by using two kinds of clustered competing-risks cancer data sets.

Effect of Kidney Function on Relationships between Lifestyle Behaviors and Mortality or Cardiovascular Outcomes: A Pooled Cohort Analysis

BACKGROUND

Adherence to healthy behaviors reduces risks of cardiovascular disease and death in the general population. However, among people with kidney disease, a group at higher risk for cardiovascular disease, such benefits have not been established.

METHODS

We pooled data from three cohort studies with a total of 27,271 participants. Kidney function was categorized on the basis of eGFR (≥60, 45 to <60, and <45 ml/min per 1.73 m²). We used proportional hazard frailty models to estimate associations between healthy behaviors (not smoking, at recommended body mass index [BMI], physical activity, healthy diet, and moderate to no alcohol intake) and outcomes (all-cause death, major coronary events, ischemic stroke, and heart failure events).

RESULTS

All recommended lifestyle behaviors were significantly associated with lower risks of death, regardless of eGFR. Not smoking (versus current) and any moderate to vigorous physical activity (versus none) was significantly associated with reduced risks of major coronary and heart failure events, regardless of eGFR. Any (versus no) moderate or vigorous physical activity significantly associated with decreased risk of ischemic stroke events only among those with eGFR ≥60. Moderate to no daily alcohol intake (versus excessive) was significantly associated with an increased risk of major coronary events, regardless of eGFR. For heart failure events, a BMI of 18.5 to 30 associated with decreased risk, regardless of eGFR. Across all eGFR categories, the magnitude of risk reduction for death and all cardiovascular outcomes increased with greater numbers of recommended lifestyle behaviors.

CONCLUSIONS

Recommended lifestyle behaviors are associated with lower risk of death and cardiovascular disease events among individuals with or without reduced kidney function, supporting lifestyle behaviors as potentially modifiable risk factors for people with kidney disease.

Nurse staffing practices and adverse events in acute care hospitals: The research protocol of a multisite patient-level longitudinal study

Aims

We describe an innovative research protocol to: (a) examine patient‐level longitudinal associations between nurse staffing practices and the risk of adverse events in acute care hospitals and; (b) determine possible thresholds for safe nurse staffing.

Design

A dynamic cohort of adult medical, surgical and intensive care unit patients admitted to 16 hospitals in Quebec (Canada) between January 2015–December 2019.

Methods

Patients in the cohort will be followed from admission until 30‐day postdischarge to assess exposure to selected nurse staffing practices in relation to the subsequent occurrence of adverse events. Five staffing practices will be measured for each shift of an hospitalization episode, using electronic payroll data, with the following time‐varying indicators: (a) nursing worked hours per patient; (b) skill mix; (c) overtime use; (d) education mix and; and (e) experience. Four high‐impact adverse events, presumably associated with nurse staffing practices, will be measured from electronic health record data retrieved at the participating sites: (a) failure‐to‐rescue; (b) in‐hospital falls; (c) hospital‐acquired pneumonia and; and (d) venous thromboembolism. To examine the associations between the selected nurse staffing exposures and the risk of each adverse event, separate multivariable Cox proportional hazards frailty regression models will be fitted, while adjusting for patient, nursing unit and hospital characteristics, and for clustering. To assess for possible staffing thresholds, flexible non‐linear spline functions will be fitted. Funding for the study began in October 2019 and research ethics/institutional approval was granted in February 2020.

Discussion

To our knowledge, this study is the first multisite patient‐level longitudinal investigation of the associations between common nurse staffing practices and the risk of adverse events. It is hoped that our results will assist hospital managers in making the most effective use of the scarce nursing resources and in identifying staffing practices that minimize the occurrence of adverse events.

Fast Algorithms for Conducting Large-Scale GWAS of Age-at-Onset Traits Using Cox Mixed-Effects Models

Age-at-onset is one of the critical traits in cohort studies of age-related diseases. Large-scale genome-wide association studies (GWAS) of age-at-onset traits can provide more insights into genetic effects on disease progression and transitions between stages. Moreover, proportional hazards (or Cox) regression models can achieve higher statistical power in a cohort study than a case-control trait using logistic regression. Although mixed-effects models are widely used in GWAS to correct for sample dependence, application of Cox mixed-effects models (CMEMs) to large-scale GWAS is so far hindered by intractable computational cost. In this work, we propose COXMEG, an efficient R package for conducting GWAS of age-at-onset traits using CMEMs. COXMEG introduces fast estimation algorithms for general sparse relatedness matrices including, but not limited to, block-diagonal pedigree-based matrices. COXMEG also introduces a fast and powerful score test for dense relatedness matrices, accounting for both population stratification and family structure. In addition, COXMEG generalizes existing algorithms to support positive semidefinite relatedness matrices, which are common in twin and family studies. Our simulation studies suggest that COXMEG, depending on the structure of the relatedness matrix, is orders of magnitude computationally more efficient than coxme and coxph with frailty for GWAS. We found that using sparse approximation of relatedness matrices yielded highly comparable results in controlling false-positive rate and retaining statistical power for an ethnically homogeneous family-based sample. By applying COXMEG to a study of Alzheimer's disease (AD) with a Late-Onset Alzheimer's Disease Family Study from the National Institute on Aging sample comprising 3456 non-Hispanic whites and 287 African Americans, we identified the APOE ε4 variant with strong statistical power (P = 1e-101), far more significant than that reported in a previous study using a transformed variable and a marginal Cox model. Furthermore, we identified novel SNP rs36051450 (P = 2e-9) near GRAMD1B, the minor allele of which significantly reduced the hazards of AD in both genders. These results demonstrated that COXMEG greatly facilitates the application of CMEMs in GWAS of age-at-onset traits.

Frailty modelling approaches for semi-competing risks data

In the semi-competing risks situation where only a terminal event censors a non-terminal event, observed event times can be correlated. Recently, frailty models with an arbitrary baseline hazard have been studied for the analysis of such semi-competing risks data. However, their maximum likelihood estimator can be substantially biased in the finite samples. In this paper, we propose effective modifications to reduce such bias using the hierarchical likelihood. We also investigate the relationship between marginal and hierarchical likelihood approaches. Simulation results are provided to validate performance of the proposed method. The proposed method is illustrated through analysis of semi-competing risks data from a breast cancer study.

Impact of model misspecification in shared frailty survival models

Survival models incorporating random effects to account for unmeasured heterogeneity are being increasingly used in biostatistical and applied research. Specifically, unmeasured covariates whose lack of inclusion in the model would lead to biased, inefficient results are commonly modeled by including a subject-specific (or cluster-specific) frailty term that follows a given distribution (eg, gamma or lognormal). Despite that, in the context of parametric frailty models, little is known about the impact of misspecifying the baseline hazard or the frailty distribution or both. Therefore, our aim is to quantify the impact of such misspecification in a wide variety of clinically plausible scenarios via Monte Carlo simulation, using open-source software readily available to applied researchers. We generate clustered survival data assuming various baseline hazard functions, including mixture distributions with turning points, and assess the impact of sample size, variance of the frailty, baseline hazard function, and frailty distribution. Models compared include standard parametric distributions and more flexible spline-based approaches; we also included semiparametric Cox models. The resulting bias can be clinically relevant. In conclusion, we highlight the importance of fitting models that are flexible enough and the importance of assessing model fit. We illustrate our conclusions with two applications using data on diabetic retinopathy and bladder cancer. Our results show the importance of assessing model fit with respect to the baseline hazard function and the distribution of the frailty: misspecifying the former leads to biased relative and absolute risk estimates, whereas misspecifying the latter affects absolute risk estimates and measures of heterogeneity.

Penalized variable selection for accelerated failure time models with random effects

Accelerated failure time (AFT) models allowing for random effects are linear mixed models under the log-transformation of survival time with censoring and describe dependence in correlated survival data. It is well known that the AFT models are useful alternatives to frailty models. To the best of our knowledge, however, there is no literature on variable selection methods for such AFT models. In this paper, we propose a simple but unified variable-selection procedure of fixed effects in the AFT random-effect models using penalized h-likelihood (HL). We consider four penalty functions (ie, least absolute shrinkage and selection operator (LASSO), adaptive LASSO, smoothly clipped absolute deviation (SCAD), and HL). We show that the proposed method can be easily implemented via a slight modification to existing h-likelihood estimation procedures. We thus demonstrate that the proposed method can also be easily extended to AFT models with multilevel (or nested) structures. Simulation studies also show that the procedure using the adaptive LASSO, SCAD, or HL penalty performs well. In particular, we find via the simulation results that the variable selection method with HL penalty provides a higher probability of choosing the true model than other three methods. The usefulness of the new method is illustrated using two actual datasets from multicenter clinical trials.

Multilevel model with random effects for clustered survival data with multiple failure outcomes

We present a multilevel frailty model for handling serial dependence and simultaneous heterogeneity in survival data with a multilevel structure attributed to clustering of subjects and the presence of multiple failure outcomes. One commonly observes such data, for example, in multi-institutional, randomized placebo-controlled trials in which patients suffer repeated episodes (eg, recurrent migraines) of the disease outcome being measured. The model extends the proportional hazards model by incorporating a random covariate and unobservable random institution effect to respectively account for treatment-by-institution interaction and institutional variation in the baseline risk. Moreover, a random effect term with correlation structure driven by a first-order autoregressive process is attached to the model to facilitate estimation of between patient heterogeneity and serial dependence. By means of the generalized linear mixed model methodology, the random effects distribution is assumed normal and the residual maximum likelihood and the maximum likelihood methods are extended for estimation of model parameters. Simulation studies are carried out to evaluate the performance of the residual maximum likelihood and the maximum likelihood estimators and to assess the impact of misspecifying random effects distribution on the proposed inference. We demonstrate the practical feasibility of the modeling methodology by analyzing real data from a double-blind randomized multi-institutional clinical trial, designed to examine the effect of rhDNase on the occurrence of respiratory exacerbations among patients with cystic fibrosis.

H-likelihood approach for joint modeling of longitudinal outcomes and time-to-event data

In longitudinal studies, a subject may have different types of outcomes that could be correlated. For example, a response variable of interest would be measured repeatedly over time on the same subject and at the same time, an event time representing a single event or competing-risks event is also observed. In this paper, we propose a joint modeling framework that accounts for the inherent association between such multiple outcomes via frailties (unobserved random effects). Among outcomes, at least one outcome is an event time that has a type of a single event or competing-risks event. For inference we use the hierarchical likelihood (h-likelihood) that provides an unified efficient fitting procedure for the joint models. Numerical studies are provided to show the performance of the proposed method and two data examples are shown.

A retrospective cohort study to quantify the contribution of health systems to child survival in Kenya: 1996-2014

Globally, the majority of childhood deaths in the post-neonatal period are caused by infections that can be effectively treated or prevented with inexpensive interventions delivered through even very basic health facilities. To understand the role of inadequate health systems on childhood mortality in Kenya, we assemble a large, retrospective cohort of children (born 1996–2013) and describe the health systems context of each child using health facility survey data representative of the province at the time of a child’s birth. We examine the relationship between survival beyond 59 months of age and geographic distribution of health facilities, quality of services, and cost of services. We find significant geographic heterogeneity in survival that can be partially explained by differences in distribution of health facilities and user fees. Higher per capita density of health facilities resulted in a 25% reduction in the risk of death (HRR = 0.73, 95% CI:0.58 to 0.91) and accounted for 30% of the between-province heterogeneity in survival. User fees for sick-child visits increased risk by 30% (HRR = 1.30, 95% CI:1.11 to 1.53). These results implicate health systems constraints in child mortality, quantify the contribution of specific domains of health services, and suggest priority areas for improvement to accelerate reductions in child mortality.

Analysis of clustered competing risks data using subdistribution hazard models with multivariate frailties

Competing risks data often exist within a center in multi-center randomized clinical trials where the treatment effects or baseline risks may vary among centers. In this paper, we propose a subdistribution hazard regression model with multivariate frailty to investigate heterogeneity in treatment effects among centers from multi-center clinical trials. For inference, we develop a hierarchical likelihood (or h-likelihood) method, which obviates the need for an intractable integration over the frailty terms. We show that the profile likelihood function derived from the h-likelihood is identical to the partial likelihood, and hence it can be extended to the weighted partial likelihood for the subdistribution hazard frailty models. The proposed method is illustrated with a dataset from a multi-center clinical trial on breast cancer as well as with a simulation study. We also demonstrate how to present heterogeneity in treatment effects among centers by using a confidence interval for the frailty for each individual center and how to perform a statistical test for such heterogeneity using a restricted h-likelihood.

Frailty modeling for clustered competing risks data with missing cause of failure

Competing risks data often occur within a center in multi-center clinical trials where the event times within a center may be correlated due to unobserved factors across individuals. In this paper, we consider the cause-specific proportional hazards model with a shared frailty to model the association between the event times within a center in the framework of competing risks. We use a hierarchical likelihood approach, which does not require any intractable integration over the frailty terms. In a clinical trial, cause of death information may not be observed for some patients. In such a case, analyses through exclusion of cases with missing cause of death may lead to biased inferences. We propose a hierarchical likelihood approach for fitting the cause-specific proportional hazards model with a shared frailty in the presence of missing cause of failure. We use multiple imputation methods to address missing cause of death information under the assumption of missing at random. Simulation studies show that the proposed procedures perform well, even if the imputation model is misspecified. The proposed methods are illustrated with data from EORTC trial 30791 conducted by European Organization for Research and Treatment of Cancer (EORTC).

Investigating covariate-by-centre interaction in survival data

In survival analysis, assessing the existence of potential centre effects on the baseline hazard or on the effect of fixed covariates on the baseline hazard, such as treatment-by-centre interaction, is a frequent clinical concern in multicentre studies. Survival models with random effects on the baseline hazard and/or on the effect of the covariates of interest have been largely applied, for instance, to investigate potential centre effects. We aimed to develop a procedure to routinely test for multiple random effects in survival analyses. We propose a statistic and a permutation approach to test whether all or a subset of components of the variance-covariance matrix of random effects are non-zero in a mixed-effects Cox model framework. Performances of the proposed permutation tests are examined under different null hypotheses corresponding to the different components of the variance-covariance matrix, i.e ., to the different random effects considered on the baseline hazard and/or on the covariates effects. Several alternative hypotheses are evaluated using simulations. The results indicate that the permutation tests have valid type I error rates under the null and achieve satisfactory power under all alternatives. The procedure is applied to two European cohorts of haematological stem cell transplants in acute leukaemia to investigate the heterogeneity across centres in leukaemia-free survival and the potential heterogeneity in prognostic factors effects across centres.

Age-dependent improvements in survival after hospitalisation with acute myocardial infarction: an analysis of the Myocardial Ischemia National Audit Project (MINAP)

BACKGROUND

recent studies report an age-dependent decline in mortality after acute myocardial infarction (AMI).

OBJECTIVE

to investigate age-dependent improvements in survival after hospitalisation with AMI.

DESIGN

population-based cohort study using data from the Myocardial Ischaemia National Audit Project.

SUBJECTS

a total of 583,466 patients with AMI admitted to 247 hospitals between 1 January 2003 and 31 December 2010.

METHODS

six-month relative survival (RS) was calculated from the ratio of observed to expected survival using an age-, sex- and biennial year-matched population from the Office for National Statistics. Risk-adjusted mortality rates (RMAR) were estimated using shared frailty regression. Data were stratified by age group, AMI phenotype [(ST-elevation myocardial infarction, (STEMI) and non-STEMI, (NSTEMI)] and period of admission to hospital.

RESULTS

for STEMI, there was an increase in RS for patients aged 65-80 years (84.8 versus 89.2%) and those over 80 years (68.0 versus 71.8%), but not for patients aged 18 to <65 years (96.4 versus 96.9%). For NSTEMI patients aged 18 to <65 years RS was higher, but stable (95.5 versus 96.8%) and improved for patients aged 65-80 years (83.2 versus 88.5%) and patients aged >80 years (68.3% versus 75.5%). Likewise, RMAR improved for patients aged ≥65 years, were stable and higher for patients <65 years.

CONCLUSIONS

there were significant improvements in survival after hospitalisation with AMI in the older but not younger patients. The scope for further reductions in mortality is likely to be much greater for older than younger patients with AMI.

Variable selection in subdistribution hazard frailty models with competing risks data

The proportional subdistribution hazards model (i.e. Fine-Gray model) has been widely used for analyzing univariate competing risks data. Recently, this model has been extended to clustered competing risks data via frailty. To the best of our knowledge, however, there has been no literature on variable selection method for such competing risks frailty models. In this paper, we propose a simple but unified procedure via a penalized h-likelihood (HL) for variable selection of fixed effects in a general class of subdistribution hazard frailty models, in which random effects may be shared 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 using a slight modification to existing h-likelihood estimation approaches. Numerical studies demonstrate that the proposed procedure using the HL penalty performs well, providing a higher probability of choosing the true model than LASSO and SCAD methods without losing prediction accuracy. The usefulness of the new method is illustrated using two actual datasets from multi-center clinical trials.

Permutation tests for centre effect on survival endpoints with application in an acute myeloid leukaemia multicentre study

When analysing multicentre data, it may be of interest to test whether the distribution of the endpoint varies among centres. In a mixed-effect model, testing for such a centre effect consists in testing to zero a random centre effect variance component. It has been shown that the usual asymptotic χ(2) distribution of the likelihood ratio and score statistics under the null does not necessarily hold. In the case of censored data, mixed-effects Cox models have been used to account for random effects, but few works have concentrated on testing to zero the variance component of the random effects. We propose a permutation test, using random permutation of the cluster indices, to test for a centre effect in multilevel censored data. Results from a simulation study indicate that the permutation tests have correct type I error rates, contrary to standard likelihood ratio tests, and are more powerful. The proposed tests are illustrated using data of a multicentre clinical trial of induction therapy in acute myeloid leukaemia patients.

Interval estimation of random effects in proportional hazards models with frailties

Semi-parametric frailty models are widely used to analyze clustered survival data. In this article, we propose the use of the hierarchical likelihood interval for individual frailties. We study the relationship between hierarchical likelihood, empirical Bayesian, and fully Bayesian intervals for frailties. We show that our proposed interval can be interpreted as a frequentist confidence interval and Bayesian credible interval under a uniform prior. We also propose an adjustment of the proposed interval to avoid null intervals. Simulation studies show that the proposed interval preserves the nominal confidence level. The procedure is illustrated using data from a multicenter lung cancer clinical trial.

Sample size in cluster-randomized trials with time to event as the primary endpoint

In cluster-randomized trials, groups of individuals (clusters) are randomized to the treatments or interventions to be compared. In many of those trials, the primary objective is to compare the time for an event to occur between randomized groups, and the shared frailty model well fits clustered time-to-event data. Members of the same cluster tend to be more similar than members of different clusters, causing correlations. As correlations affect the power of a trial to detect intervention effects, the clustered design has to be considered in planning the sample size. In this publication, we derive a sample size formula for clustered time-to-event data with constant marginal baseline hazards and correlation within clusters induced by a shared frailty term. The sample size formula is easy to apply and can be interpreted as an extension of the widely used Schoenfeld's formula, accounting for the clustered design of the trial. Simulations confirm the validity of the formula and its use also for non-constant marginal baseline hazards. Findings are illustrated on a cluster-randomized trial investigating methods of disseminating quality improvement to addiction treatment centers in the USA.

Bias correction in the hierarchical likelihood approach to the analysis of multivariate survival data

Frailty models are useful for measuring unobserved heterogeneity in risk of failures across clusters, providing cluster-specific risk prediction. In a frailty model, the latent frailties shared by members within a cluster are assumed to act multiplicatively on the hazard function. In order to obtain parameter and frailty variate estimates, we consider the hierarchical likelihood (H-likelihood) approach (Ha, Lee and Song, 2001. Hierarchical-likelihood approach for frailty models. Biometrika 88, 233-243) in which the latent frailties are treated as "parameters" and estimated jointly with other parameters of interest. We find that the H-likelihood estimators perform well when the censoring rate is low, however, they are substantially biased when the censoring rate is moderate to high. In this paper, we propose a simple and easy-to-implement bias correction method for the H-likelihood estimators under a shared frailty model. We also extend the method to a multivariate frailty model, which incorporates complex dependence structure within clusters. We conduct an extensive simulation study and show that the proposed approach performs very well for censoring rates as high as 80%. We also illustrate the method with a breast cancer data set. Since the H-likelihood is the same as the penalized likelihood function, the proposed bias correction method is also applicable to the penalized likelihood estimators.