Maximum penalized likelihood estimation in a gamma-frailty model

A joint frailty model for recurrent and competing terminal events: Application to delirium in the ICU

Joint models linking longitudinal biomarkers or recurrent event processes with a terminal event, for example, mortality, have been studied extensively. Motivated by studies of recurrent delirium events in patients receiving care in an intensive care unit (ICU), we devise a joint model for a recurrent event process and multiple terminal events. Being discharged alive from the ICU or experiencing mortality may be associated with a patient's hazard of delirium, violating the assumption of independent censoring. Moreover, the direction of the association between the hazards of delirium and mortality may be opposite of the direction of association between the hazards of delirium and ICU discharge. Hence treating either terminal event as independent censoring may bias inferences. We propose a competing joint model that uses a latent frailty to link a patient's recurrent and competing terminal event processes. We fit our model to data from a completed placebo-controlled clinical trial, which studied whether Haloperidol could prevent death and delirium among ICU patients. The clinical trial served as a foundation for a simulation study, in which we evaluate the properties, for example, bias and confidence interval coverage, of the competing joint model. As part of the simulation study, we demonstrate the shortcomings of using a joint model with a recurrent delirium process and a single terminal event to study delirium in the ICU. Lastly, we discuss limitations and possible extensions for the competing joint model. The competing joint model has been added to frailtypack, an R package for fitting an assortment of joint models.

Quantifying variation in home spirometry in people with cystic fibrosis during baseline health, and associations with clinical outcomes

BACKGROUND

Home spirometry is increasingly used to monitor lung function in people with cystic fibrosis (pwCF). Although decreases in lung function in the setting of increased respiratory symptoms are consistent with a pulmonary exacerbation (PEx), the interpretation of home spirometry during asymptomatic periods of baseline health is unclear. The aims of this study were to determine the variation in home spirometry in pwCF during asymptomatic periods of baseline health and to identify associations between this variation and PEx.

METHODS

Near-daily home spirometry measurements were obtained from a cohort of pwCF enrolled in a long-term study of the airway microbiome. Associations between the degree of variation in home spirometry and the time to next PEx were evaluated.

RESULTS

Thirteen subjects (mean age of 29 years and mean percent predicted forced expiratory volume in one second [ppFEV1] of 60) provided a median of 204 spirometry readings taken during 40 periods of baseline health. The mean week-to-week within-subject level of variation in ppFEV1 was 15.2 ± 6.2%. The degree of variation in ppFEV1 during baseline health was not associated with time to PEx.

CONCLUSIONS

Variation in ppFEV1 measured with near-daily home spirometry in pwCF during periods of baseline health exceeded the variation in ppFEV1 expected in clinic spirometry (based on ATS guidelines). The degree of variation in ppFEV1 during baseline health was not associated with time to PEx. These data are relevant for guiding interpretation of home spirometry.

Sample size estimation for recurrent event data using multifrailty and multilevel survival models

In epidemiology and clinical research, recurrent events refer to individuals who are likely to experience transient clinical events repeatedly over an observation period. Examples include hospitalizations in patients with heart failure, fractures in osteoporosis studies and the occurrence of new lesions in oncology. We provided an in-depth analysis of the sample size required for the analysis of recurrent time-to-event data using multifrailty or multilevel survival models. We covered the topic from the simple shared frailty model to models with hierarchical or joint frailties. We relied on a Wald-type test statistic to estimate the sample size assuming either a single or multiple endpoints. Simulations revealed that the sample size increased as heterogeneity increased. We also observed that it was more attractive to include more patients and reduce the duration of follow-up than to include fewer patients and increase the duration of follow-up to obtain the number of events required. Each model investigated can address the question of the number of subjects for recurrent events. However, depending on the research question, one model will be more suitable than another. We illustrated our methodology with the AFFIRM-AHF trial investigating the effect of intravenous ferric carboxymaltose in patients hospitalised for acute heart failure.

A new method for clustered survival data: Estimation of treatment effect heterogeneity and variable selection

We recently developed a new method random-intercept accelerated failure time model with Bayesian additive regression trees (riAFT-BART) to draw causal inferences about population treatment effect on patient survival from clustered and censored survival data while accounting for the multilevel data structure. The practical utility of this method goes beyond the estimation of population average treatment effect. In this work, we exposit how riAFT-BART can be used to solve two important statistical questions with clustered survival data: estimating the treatment effect heterogeneity and variable selection. Leveraging the likelihood-based machine learning, we describe a way in which we can draw posterior samples of the individual survival treatment effect from riAFT-BART model runs, and use the drawn posterior samples to perform an exploratory treatment effect heterogeneity analysis to identify subpopulations who may experience differential treatment effects than population average effects. There is sparse literature on methods for variable selection among clustered and censored survival data, particularly ones using flexible modeling techniques. We propose a permutation-based approach using the predictor's variable inclusion proportion supplied by the riAFT-BART model for variable selection. To address the missing data issue frequently encountered in health databases, we propose a strategy to combine bootstrap imputation and riAFT-BART for variable selection among incomplete clustered survival data. We conduct an expansive simulation study to examine the practical operating characteristics of our proposed methods, and provide empirical evidence that our proposed methods perform better than several existing methods across a wide range of data scenarios. Finally, we demonstrate the methods via a case study of predictors for in-hospital mortality among severe COVID-19 patients and estimating the heterogeneous treatment effects of three COVID-specific medications. The methods developed in this work are readily available in the R ${\textsf {R}}$ package riAFTBART $\textsf {riAFTBART}$ .

A flexible class of generalized joint frailty models for the analysis of survival endpoints

This article focuses on shared frailty models for correlated failure times, as well as joint frailty models for the simultaneous analysis of recurrent events (eg, appearance of new cancerous lesions or hospital readmissions) and a major terminal event (typically, death). As extensions of the Cox model, these joint models usually assume a frailty proportional hazards model for each of the recurrent and terminal event processes. In order to extend these models beyond the proportional hazards assumption, our proposal is to replace these proportional hazards models with generalized survival models, for which the survival function is modeled as a linear predictor through a link function. Depending on the link function considered, these can be reduced to proportional hazards, proportional odds, additive hazards, or probit models. We first consider a fully parametric framework for the time and covariate effects. For proportional and additive hazards models, our approach also allows the use of smooth functions for baseline hazard functions and time-varying coefficients. The dependence between recurrent and terminal event processes is modeled by conditioning on a shared frailty acting differently on the two processes. Parameter estimates are provided using the maximum (penalized) likelihood method, implemented in the R package frailtypack (function GenfrailtyPenal). We perform simulation studies to assess the method, which is also illustrated on real datasets.

Association of Pathologic Complete Response and Long-Term Survival Outcomes Among Patients Treated With Neoadjuvant Chemotherapy or Chemoradiotherapy for NSCLC: A Meta-Analysis

Introduction

Increased efforts to optimize outcomes for early stage NSCLC through the investigation of novel perioperative treatment strategies are ongoing. An emerging question is the role of pathologic response and its association with long-term clinical outcomes after neoadjuvant therapy.

Methods

To investigate the association of pathologic complete response (pCR) and event-free survival (EFS) and overall survival (OS), we performed a systematic review and meta-analysis identifying studies reporting on the prognostic impact of pCR after neoadjuvant chemotherapy or chemoradiotherapy. To evaluate this prognostic value, an aggregated data (AD) meta-analyses was conducted to estimate the pooled hazard ratios (HRs) of EFS and OS for pCR. Using reconstructed individual patient data (IPD), pooled Kaplan-Meier curves were obtained to estimate this association in a more granular fashion. Subgroup analyses were conducted to further explore the impacts of study-level characteristics.

Results

A total of 28 studies comprising 7011 patients were included in the AD meta-analysis, of which, IPD was available for 6274 patients from 24 studies. Results from our AD meta-analysis revealed a pooled pCR rate of 18% (95% confidence interval [CI]: 15%–21%), including significant improvements in OS (HR = 0.50, 95% CI: 0.45–0.56) and EFS (HR = 0.46, 95% CI: 0.37–0.57) on the basis of pCR status. Our IPD analysis revealed a 5-year OS rate of 63% (95% CI: 59.6–67.4) for patients with a pCR compared with 39% (95% CI: 34.5–44.5) for those without a pCR.

Conclusions

pCR after neoadjuvant chemotherapy plus or minus radiotherapy is associated with significant improvements in EFS and survival for patients with resectable NSCLC.

Response-adaptive treatment randomization for multiple comparisons of treatments with recurrentevent responses

Recurrent event responses are frequently encountered during clinical trials of treatments for certain diseases, such as asthma. The recurrence rates of different treatments are often compared by applying the negative binomial model. In addition, a balanced treatment-allocation procedure that assigns the same number of patients to each treatment is often applied. Recently, a response-adaptive treatment-allocation procedure has been developed for trials with recurrent event data, and has been shown to be superior to balanced treatment allocation. However, this response-adaptive treatment allocation procedure is only applicable for the comparison of two treatments. In this paper, we derive response-adaptive treatment-allocation procedures for trials which comprise several treatments. As pairwise comparisons and multiple comparisons with a control are two common multiple-testing scenarios in trials with more than two treatments, corresponding treatment-allocation procedures for these scenarios are also investigated. The redesign of two clinical studies illustrates the clinical benefits that would be obtained from our proposed response-adaptive treatment-allocation procedures.

Left-censored recurrent event analysis in epidemiological studies: a proposal for when the number of previous episodes is unknown

BACKGROUND

When dealing with recurrent events in observational studies it is common to include subjects who became at risk before follow-up. This phenomenon is known as left censoring, and simply ignoring these prior episodes can lead to biased and inefficient estimates. We aimed to propose a statistical method that performs well in this setting.

METHODS

Our proposal was based on the use of models with specific baseline hazards. In this, the number of prior episodes were imputed when unknown and stratified according to whether the subject had been at risk of presenting the event before t = 0. A frailty term was also used. Two formulations were used for this "Specific Hazard Frailty Model Imputed" based on the "counting process" and "gap time." Performance was then examined in different scenarios through a comprehensive simulation study.

RESULTS

The proposed method performed well even when the percentage of subjects at risk before follow-up was very high. Biases were often below 10% and coverages were around 95%, being somewhat conservative. The gap time approach performed better with constant baseline hazards, whereas the counting process performed better with non-constant baseline hazards.

CONCLUSIONS

The use of common baseline methods is not advised when knowledge of prior episodes experienced by a participant is lacking. The approach in this study performed acceptably in most scenarios in which it was evaluated and should be considered an alternative in this context. It has been made freely available to interested researchers as R package miRecSurv.

Functional modeling of recurrent events on time-to-event processes

In clinical practice, it is often the case where the association between the occurrence of events and time-to-event outcomes is of interest; thus, it can be modeled within the framework of recurrent events. The purpose of our study is to enrich the information available for modeling survival with relevant dynamic features, properly taking into account their possibly time-varying nature, as well as to provide a new setting for quantifying the association between time-varying processes and time-to-event outcomes. We propose an innovative methodology to model information carried out by time-varying processes by means of functional data, modeling each time-varying variable as the compensator of marked point process the recurrent events are supposed to derive from. By means of Functional Principal Component Analysis, a suitable dimensional reduction of these objects is carried out in order to plug them into a Cox-type functional regression model for overall survival. We applied our methodology to data retrieved from the administrative databases of Lombardy Region (Italy), related to patients hospitalized for Heart Failure (HF) between 2000 and 2012. We focused on time-varying processes of HF hospitalizations and multiple drugs consumption and we studied how they influence patients' overall survival. This novel way to account for time-varying variables allowed to model self-exciting behaviors, for which the occurrence of events in the past increases the probability of a new event, and to quantify the effect of personal behaviors and therapeutic patterns on survival, giving new insights into the direction of personalized treatment.

Recurrent Events Model Application in Determining the Risk Factors of Bipolar Disorder Recurrence

Objective: Recurrent events data is one of the most important types of survival data whose main feature is correlation between individual’s observations. The aim of this study was to analyze the time to bipolar disorder (BD) relapse and determine the related factors using recurrent events models.

Method: In this retrospective study, records of 104 BD patients with at least one relapse who were admitted for the first time (2001-2015) in Farabi hospital of Kermanshah were gathered to identify the factors influencing the time intervals between the recurrent survivals data using the Cox model with and without frailty (shared frailty), once with frailty gamma distribution and once with log-normal distribution frailty. All calculations were performed using R and SPSS software, versions 3.0.2 and 16 and the level of significance was considered at 0.05.

Results: Among the employed models, Cox model with lognormal shared frailty showed better fit for BD recurrent survival data. According to results of Cox model with lognormal frailty, 2 factors (marital status and history of veteran) were identified to affect the time intervals between relapses.

Conclusion: Because of the better fit of the models with the frailty effect on data, the correlation between the recurrent time intervals of each subject's relapse of BD was confirmed. Also, since the risk of subsequent relapses was less in married and veteran patients, marriage and emotional care supports can be considered as effective factors in reducing the risk of subsequent relapses of this disease.

A joint frailty-copula model for meta-analytic validation of failure time surrogate endpoints in clinical trials

In a meta-analysis framework, the classical approach for the validation of time-to-event surrogate endpoint is based on a two-step analysis. This approach often raises estimation issues. Recently, we proposed a one-step validation approach based on a joint frailty model. This approach was quite time consuming, despite parallel computing, due to individual-level frailties used to take into account heterogeneity in the data at the individual level. We now propose an alternative one-step approach for evaluating surrogacy, using a joint frailty-copula model. The model includes two correlated random effects treatment-by-trial interaction and a shared random effect associated with the baseline risks. At the individual level, the joint survivor functions of time-to-event endpoints are linked using copula functions. We used splines for the baseline hazard functions. We estimated parameters and hazard function using a semiparametric penalized marginal likelihood method, considering various numerical integration methods. Both individual-level and trial-level surrogacy were evaluated using Kendall's tau and coefficient of determination. The performance of the estimators was evaluated using simulation studies. The model was applied to individual patient data meta-analyses in advanced ovarian cancer to assess progression-free survival as a surrogate for overall survival, as part of the evaluation of new therapy. The model showed good performance and was quite robust regarding the integration methods and data variation, regardless of the surrogacy evaluation criteria. Kendall's Tau was better estimated using the Clayton copula model compared to the joint frailty model. The proposed model reduces the convergence and model estimation issues encountered in the two-step approach.

Do Individual and Geographical Deprivation Have the Same Impact on the Risk of Dementia? A 25-Year Follow-up Study

OBJECTIVES

To determine the impact of both individual psychosocioeconomic precariousness and geographical deprivation on risk of dementia in older adults followed-up for 25 years.

METHOD

The sample consisted of 3,431 participants aged 65 years or over from the PAQUID cohort study. Individual psychosocioeconomic precariousness was measured computing eight economic and psychosocial indicators. Geographical deprivation was assessed by the FDep99 index, consisting of four community socioeconomic variables. For both measures, the fourth quartile of the distribution was considered as the more precarious or deprived category, while the first quartile was considered as the less precarious or deprived one. Clinical dementia diagnosis was assessed all along study follow-up. The association between individual psychosocioeconomic precariousness, geographical deprivation and risk of dementia was assessed using illness-death regression models adjusted for age, sex, depression, psychotropic drug consumption, comorbidities, disability, and body mass index, while accounting for death as a competing event.

RESULTS

The risk of dementia was higher for the more psychosocioeconomic precarious participants (HR = 1.51; 95% CI: 1.24-1.84). No increased risk of dementia was found for those living in communities with high index of deprivation.

DISCUSSION

Psychosocioeconomic precariousness, but not geographical deprivation, is associated with a higher risk of dementia.

Bayesian Autoregressive Frailty Models for Inference in Recurrent Events

We propose autoregressive Bayesian semi-parametric models for gap times between recurrent events. The aim is two-fold: inference on the effect of possibly time-varying covariates on the gap times and clustering of individuals based on the time trajectory of the recurrent event. Time-dependency between gap times is taken into account through the specification of an autoregressive component for the frailty parameters influencing the response at different times. The order of the autoregression may be assumed unknown and is an object of inference. We consider two alternative approaches to perform model selection under this scenario. Covariates may be easily included in the regression framework and censoring and missing data are easily accounted for. As the proposed methodologies lie within the class of Dirichlet process mixtures, posterior inference can be performed through efficient MCMC algorithms. We illustrate the approach through simulations and medical applications involving recurrent hospitalizations of cancer patients and successive urinary tract infections.

One-step validation method for surrogate endpoints using data from multiple randomized cancer clinical trials with failure-time endpoints

A surrogate endpoint can be used instead of the most relevant clinical endpoint to assess the efficiency of a new treatment. Before being used, a surrogate endpoint must be validated based on appropriate methods. Numerous validation approaches have been proposed with the most popular used in a context of meta-analysis, based on a two-step analysis strategy. For two failure-time endpoints, two association measurements are usually used, Kendall's τ at the individual level and the adjusted coefficient of determination ( R t r i a l , a d j 2 ) at the trial level. However, R t r i a l , a d j 2 is not always available due to model estimation constraints. We propose a one-step validation approach based on a joint frailty model, including both individual-level and trial-level random effects. Parameters have been estimated using a semiparametric penalized marginal log-likelihood method, and various numerical integration approaches were considered. Both individual- and trial-level surrogacy were evaluated using a new definition of Kendall's τ and the coefficient of determination. Estimators' performances were evaluated using simulation studies and satisfactory results were found. The model was applied to individual patient data meta-analyses in gastric cancer to assess disease-free survival as a surrogate for overall survival, as part of the evaluation of adjuvant therapy.

Adaptive treatment allocation for comparative clinical studies with recurrent events data

In long-term clinical studies, recurrent event data are sometimes collected and used to contrast the efficacies of two different treatments. The event reoccurrence rates can be compared using the popular negative binomial model, which incorporates information related to patient heterogeneity into a data analysis. For treatment allocation, a balanced approach in which equal sample sizes are obtained for both treatments is predominately adopted. However, if one treatment is superior, then it may be desirable to allocate fewer subjects to the less-effective treatment. To accommodate this objective, a sequential response-adaptive treatment allocation procedure is derived based on the doubly adaptive biased coin design. Our proposed treatment allocation schemes have been shown to be capable of reducing the number of subjects receiving the inferior treatment while simultaneously retaining a test power level that is comparable to that of a balanced design. The redesign of a clinical study illustrates the advantages of using our procedure.

Marginal hazard ratio estimates in joint frailty models for heart failure trials

This work is motivated by clinical trials in chronic heart failure disease, where treatment has effects both on morbidity (assessed as recurrent non-fatal hospitalisations) and on mortality (assessed as cardiovascular death, CV death). Recently, a joint frailty proportional hazards model has been proposed for these kind of efficacy outcomes to account for a potential association between the risk rates for hospital admissions and CV death. However, more often clinical trial results are presented by treatment effect estimates that have been derived from marginal proportional hazards models, that is, a Cox model for mortality and an Andersen-Gill model for recurrent hospitalisations. We show how these marginal hazard ratios and their estimates depend on the association between the risk processes, when these are actually linked by shared or dependent frailty terms. First we derive the marginal hazard ratios as a function of time. Then, applying least false parameter theory, we show that the marginal hazard ratio estimate for the hospitalisation rate depends on study duration and on parameters of the underlying joint frailty model. In particular, we identify parameters, for example the treatment effect on mortality, that determine if the marginal hazard ratio estimate for hospitalisations is smaller, equal or larger than the conditional one. How this affects rejection probabilities is further investigated in simulation studies. Our findings can be used to interpret marginal hazard ratio estimates in heart failure trials and are illustrated by the results of the CHARM-Preserved trial (where CHARM is the 'Candesartan in Heart failure Assessment of Reduction in Mortality and morbidity' programme).

A Gamma-frailty proportional hazards model for bivariate interval-censored data

Correlated survival data naturally arise from many clinical and epidemiological studies. For the analysis of such data, the Gamma-frailty proportional hazards (PH) model is a popular choice because the regression parameters have marginal interpretations and the statistical association between the failure times can be explicitly quantified via Kendall's tau. Despite their popularity, Gamma-frailty PH models for correlated interval-censored data have not received as much attention as analogous models for right-censored data. In this work, a Gamma-frailty PH model for bivariate interval-censored data is presented and an easy to implement expectation-maximization (EM) algorithm for model fitting is developed. The proposed model adopts a monotone spline representation for the purposes of approximating the unknown conditional cumulative baseline hazard functions, significantly reducing the number of unknown parameters while retaining modeling flexibility. The EM algorithm was derived from a data augmentation procedure involving latent Poisson random variables. Extensive numerical studies illustrate that the proposed method can provide reliable estimation and valid inference, and is moreover robust to the misspecification of the frailty distribution. To further illustrate its use, the proposed method is used to analyze data from an epidemiological study of sexually transmitted infections.

A Penalized Likelihood Framework for High-Dimensional Phylogenetic Comparative Methods and an Application to New-World Monkeys Brain Evolution

Working with high-dimensional phylogenetic comparative data sets is challenging because likelihood-based multivariate methods suffer from low statistical performances as the number of traits $p $ approaches the number of species $n $ and because some computational complications occur when $p $ exceeds $n$. Alternative phylogenetic comparative methods have recently been proposed to deal with the large $p $ small $n $ scenario but their use and performances are limited. Herein, we develop a penalized likelihood (PL) framework to deal with high-dimensional comparative data sets. We propose various penalizations and methods for selecting the intensity of the penalties. We apply this general framework to the estimation of parameters (the evolutionary trait covariance matrix and parameters of the evolutionary model) and model comparison for the high-dimensional multivariate Brownian motion (BM), Early-burst (EB), Ornstein-Uhlenbeck (OU), and Pagel's lambda models. We show using simulations that our PL approach dramatically improves the estimation of evolutionary trait covariance matrices and model parameters when $p$ approaches $n$, and allows for their accurate estimation when $p$ equals or exceeds $n$. In addition, we show that PL models can be efficiently compared using generalized information criterion (GIC). We implement these methods, as well as the related estimation of ancestral states and the computation of phylogenetic principal component analysis in the R package RPANDA and mvMORPH. Finally, we illustrate the utility of the new proposed framework by evaluating evolutionary models fit, analyzing integration patterns, and reconstructing evolutionary trajectories for a high-dimensional 3D data set of brain shape in the New World monkeys. We find a clear support for an EB model suggesting an early diversification of brain morphology during the ecological radiation of the clade. PL offers an efficient way to deal with high-dimensional multivariate comparative data.

Proportional hazard model estimation under dependent censoring using copulas and penalized likelihood

This paper considers Cox proportional hazard models estimation under informative right censored data using maximum penalized likelihood, where dependence between censoring and event times are modelled by a copula function and a roughness penalty function is used to restrain the baseline hazard as a smooth function. Since the baseline hazard is nonnegative, we propose a special algorithm where each iteration involves updating regression coefficients by the Newton algorithm and baseline hazard by the multiplicative iterative algorithm. The asymptotic properties for both regression coefficients and baseline hazard estimates are developed. The simulation study investigates the performance of our method and also compares it with an existing maximum likelihood method. We apply the proposed method to a dementia patients dataset.

Factors associated with breast cancer recurrences or mortality and dynamic prediction of death using history of cancer recurrences: the French E3N cohort

BACKGROUND

In addition to tumor characteristics and lifestyle factors, cancer relapses are often related to the risk of death but have not been jointly studied. We investigate the prognostic factors of recurrent events and death after a diagnosis of breast cancer and predict individual deaths including a history of recurrences.

METHODS

The E3N (Etude Epidémiologique auprès de Femmes de la Mutuelle Générale de l'Education Nationale) study is a prospective cohort study that was initiated in 1990 to investigate factors associated with the most common types of cancer. Overall survival and three types of recurrent events were considered: locoregional recurrence, metastasis, and second primary breast cancer. Recurrent events and death were analyzed using a joint frailty model.

RESULTS

The analysis included 4926 women from the E3N cohort diagnosed with a first primary invasive breast cancer between June 1990 and June 2008; during the follow-up, 1334 cases had a recurrence (median time of follow-up is 7.2 years) and 469 women died. Cases with high grade, large tumor size, axillary nodal involvement, and negative estrogen and progesterone receptors had a higher risk of recurrence or death. Furthermore, smoking increased the risk of relapse. For cases with a medium risk profile in terms of tumor characteristics and lifestyle factors, the probability of dying between 5 and 10 years after diagnosis was 6, 20 and 36% for 0, 1 or 2 recurrences within the first 5 years after diagnosis, respectively.

CONCLUSIONS

Our study showed the importance of considering baseline lifestyle characteristics and history of relapses to dynamically predict the risk of death in breast cancer cases. Medical experience coupled with an estimate of a patient's survival probability that considers all available information for this patient would enable physicians to make better informed decisions regarding their actions and thus improve clinical output.

High monetary reward rates and caloric rewards decrease temporal persistence

Temporal persistence refers to an individual's capacity to wait for future rewards, while forgoing possible alternatives. This requires a trade-off between the potential value of delayed rewards and opportunity costs, and is relevant to many real-world decisions, such as dieting. Theoretical models have previously suggested that high monetary reward rates, or positive energy balance, may result in decreased temporal persistence. In our study, 50 fasted participants engaged in a temporal persistence task, incentivised with monetary rewards. In alternating blocks of this task, rewards were delivered at delays drawn randomly from distributions with either a lower or higher maximum reward rate. During some blocks participants received either a caloric drink or water. We used survival analysis to estimate participants' probability of quitting conditional on the delay distribution and the consumed liquid. Participants had a higher probability of quitting in blocks with the higher reward rate. Furthermore, participants who consumed the caloric drink had a higher probability of quitting than those who consumed water. Our results support the predictions from the theoretical models, and importantly, suggest that both higher monetary reward rates and physiologically relevant rewards can decrease temporal persistence, which is a crucial determinant for survival in many species.

Cause-Specific Hazard Regression for Competing Risks Data Under Interval Censoring and Left Truncation

Inference for cause-specific hazards from competing risks data under interval censoring and possible left truncation has been understudied. Aiming at this target, a penalized likelihood approach for a Cox-type proportional cause-specific hazards model is developed, and the associated asymptotic theory is discussed. Monte Carlo simulations show that the approach performs very well for moderate sample sizes. An application to a longitudinal study of dementia illustrates the practical utility of the method. In the application, the age-specific hazards of AD, other dementia and death without dementia are estimated, and risk factors of all competing risks are studied.

Analyzing recurrent events when the history of previous episodes is unknown or not taken into account: proceed with caution

OBJECTIVE

Researchers in public health are often interested in examining the effect of several exposures on the incidence of a recurrent event. The aim of the present study is to assess how well the common-baseline hazard models perform to estimate the effect of multiple exposures on the hazard of presenting an episode of a recurrent event, in presence of event dependence and when the history of prior-episodes is unknown or is not taken into account.

METHODS

Through a comprehensive simulation study, using specific-baseline hazard models as the reference, we evaluate the performance of common-baseline hazard models by means of several criteria: bias, mean squared error, coverage, confidence intervals mean length and compliance with the assumption of proportional hazards.

RESULTS

Results indicate that the bias worsen as event dependence increases, leading to a considerable overestimation of the exposure effect; coverage levels and compliance with the proportional hazards assumption are low or extremely low, worsening with increasing event dependence, effects to be estimated, and sample sizes.

CONCLUSIONS

Common-baseline hazard models cannot be recommended when we analyse recurrent events in the presence of event dependence. It is important to have access to the history of prior-episodes per subject, it can permit to obtain better estimations of the effects of the exposures.

Tumor dormancy and frailty models: A novel approach

Frailty models are here proposed in the tumor dormancy framework, in order to account for possible unobservable dependence mechanisms in cancer studies where a non-negligible proportion of cancer patients relapses years or decades after surgical removal of the primary tumor. Relapses do not seem to follow a memory-less process, since their timing distribution leads to multimodal hazards. From a biomedical perspective, this behavior may be explained by tumor dormancy, i.e., for some patients microscopic tumor foci may remain asymptomatic for a prolonged time interval and, when they escape from dormancy, micrometastatic growth results in a clinical disease appearance. The activation of the growth phase at different metastatic states would explain the occurrence of metastatic recurrences and mortality at different times (multimodal hazard). We propose a new frailty model which includes in the risk function a random source of heterogeneity (frailty variable) affecting the components of the hazard function. Thus, the individual hazard rate results as the product of a random frailty variable and the sum of basic hazard rates. In tumor dormancy, the basic hazard rates correspond to micrometastatic developments starting from different initial states. The frailty variable represents the heterogeneity among patients with respect to relapse, which might be related to unknown mechanisms that regulate tumor dormancy. We use our model to estimate the overall survival in a large breast cancer dataset, showing how this improves the understanding of the underlying biological process.

Survival analysis with delayed entry in selected families with application to human longevity

In the field of aging research, family-based sampling study designs are commonly used to study the lifespans of long-lived family members. However, the specific sampling procedure should be carefully taken into account in order to avoid biases. This work is motivated by the Leiden Longevity Study, a family-based cohort of long-lived siblings. Families were invited to participate in the study if at least two siblings were 'long-lived', where 'long-lived' meant being older than 89 years for men or older than 91 years for women. As a result, more than 400 families were included in the study and followed for around 10 years. For estimation of marker-specific survival probabilities and correlations among life times of family members, delayed entry due to outcome-dependent sampling mechanisms has to be taken into account. We consider shared frailty models to model left-truncated correlated survival data. The treatment of left truncation in shared frailty models is still an open issue and the literature on this topic is scarce. We show that the current approaches provide, in general, biased estimates and we propose a new method to tackle this selection problem by applying a correction on the likelihood estimation by means of inverse probability weighting at the family level.

Dynamic prediction of recurrent events data by landmarking with application to a follow-up study of patients after kidney transplant

This paper extends dynamic prediction by landmarking to recurrent event data. The motivating data comprised post-kidney transplantation records of repeated infections and repeated measurements of multiple markers. At each landmark time point ts, a Cox proportional hazards model with a frailty term was fitted using data of individuals who were at risk at landmark s. This model included the time-updated marker values at ts as time-fixed covariates. Based on a stacked data set that merged all landmark data sets, we considered supermodels that allow parameters to depend on the landmarks in a smooth fashion. We described and evaluated four ways to parameterize the supermodels for recurrent event data. With both the study data and simulated data sets, we compared supermodels that were fitted on stacked data sets that consisted of either overlapping or non-overlapping landmark periods. We observed that for recurrent event data, the supermodels may yield biased estimates when overlapping landmark periods are used for stacking. Using the best supermodel amongst the ones considered, we dynamically estimated the probability to remain infection free between ts and a prediction horizon thor, conditional on the information available at ts.

Joint modelling of longitudinal and survival data: incorporating delayed entry and an assessment of model misspecification

A now common goal in medical research is to investigate the inter-relationships between a repeatedly measured biomarker, measured with error, and the time to an event of interest. This form of question can be tackled with a joint longitudinal-survival model, with the most common approach combining a longitudinal mixed effects model with a proportional hazards survival model, where the models are linked through shared random effects. In this article, we look at incorporating delayed entry (left truncation), which has received relatively little attention. The extension to delayed entry requires a second set of numerical integration, beyond that required in a standard joint model. We therefore implement two sets of fully adaptive Gauss-Hermite quadrature with nested Gauss-Kronrod quadrature (to allow time-dependent association structures), conducted simultaneously, to evaluate the likelihood. We evaluate fully adaptive quadrature compared with previously proposed non-adaptive quadrature through a simulation study, showing substantial improvements, both in terms of minimising bias and reducing computation time. We further investigate, through simulation, the consequences of misspecifying the longitudinal trajectory and its impact on estimates of association. Our scenarios showed the current value association structure to be very robust, compared with the rate of change that we found to be highly sensitive showing that assuming a simpler trend when the truth is more complex can lead to substantial bias. With emphasis on flexible parametric approaches, we generalise previous models by proposing the use of polynomials or splines to capture the longitudinal trend and restricted cubic splines to model the baseline log hazard function. The methods are illustrated on a dataset of breast cancer patients, modelling mammographic density jointly with survival, where we show how to incorporate density measurements prior to the at-risk period, to make use of all the available information. User-friendly Stata software is provided.

Dynamic clustering of hazard functions: an application to disease progression in chronic heart failure

We analyse data collected from the administrative datawarehouse of an Italian regional district (Lombardia) concerning patients affected by Chronic Heart Failure. The longitudinal data gathering for each patient hospital readmissions in time, as well as patient-specific covariates, is studied as a realization of non homogeneous Poisson process. Since the aim behind this study is to identify groups of patients behaving similarly in terms of disease progression and then healthcare consumption, we conjectured the time segments between two consecutive hospitalizations to be Weibull distributed in each hidden cluster. Adding a frailty term to take into account the within subjects unknown variability, the corresponding patient-specific hazard functions are reconstructed. Therefore, the comprehensive distribution for each time to event variable is modelled as a Weibull Mixture. We are then able to easily interpret the related hidden groups as healthy, sick, and terminally ill subjects.

A joint frailty-copula model between tumour progression and death for meta-analysis

Dependent censoring often arises in biomedical studies when time to tumour progression (e.g., relapse of cancer) is censored by an informative terminal event (e.g., death). For meta-analysis combining existing studies, a joint survival model between tumour progression and death has been considered under semicompeting risks, which induces dependence through the study-specific frailty. Our paper here utilizes copulas to generalize the joint frailty model by introducing additional source of dependence arising from intra-subject association between tumour progression and death. The practical value of the new model is particularly evident for meta-analyses in which only a few covariates are consistently measured across studies and hence there exist residual dependence. The covariate effects are formulated through the Cox proportional hazards model, and the baseline hazards are nonparametrically modeled on a basis of splines. The estimator is then obtained by maximizing a penalized log-likelihood function. We also show that the present methodologies are easily modified for the competing risks or recurrent event data, and are generalized to accommodate left-truncation. Simulations are performed to examine the performance of the proposed estimator. The method is applied to a meta-analysis for assessing a recently suggested biomarker CXCL12 for survival in ovarian cancer patients. We implement our proposed methods in R joint.Cox package.

Dynamic prediction models for clustered and interval-censored outcomes: Investigating the intra-couple correlation in the risk of dementia

The use of settings such as cohorts or clinical trials with interval-censored data and clustered event times are increasingly popular designs. First, the observed outcomes cannot be considered as independent and random effects survival models were introduced. Second, the failure time is not known exactly but it is only known to have occurred within a certain interval. We propose here an extension of shared frailty models to handle simultaneously the interval censoring, the clustering and also left truncation due to delayed entry in the cohort. A simulation study to evaluate the proposed method was conducted. The estimated results are used to obtain dynamic predictions for clustered patients, with interval-censored failure times and with a given history. We apply our method to the Three-City study, a prospective cohort with periodic follow-up in order to study prognostic factors of dementia. In this application scheme, couples are natural clusters and an intra-couple correlation might be present with a possible increased risk for dementia for subjects whose partner already developed incident dementia. No significant intra-couple correlation for the risk of dementia was observed before and after adjustments for covariates. We also present individual predictions of dementia underlining the usefulness of dynamic prognostic tools that can take into account the clustering. The consideration of frailty models for interval-censoring data and left-truncated data permits useful analysis of very complex clustered data. It could help to improve estimation of the impact of proposed prognostic features in a study with clustering. We proposed here a tractable model and a dynamic prediction tool that can easily be implemented using the R package Frailtypack.

Influence of trial duration on the bias of the estimated treatment effect in clinical trials when individual heterogeneity is ignored

In randomized clinical trials where the times to event of two treatment groups are compared under a proportional hazards assumption, it has been established that omitting prognostic factors from the model entails an underestimation of the hazards ratio. Heterogeneity due to unobserved covariates in cancer patient populations is a concern since genomic investigations have revealed molecular and clinical heterogeneity in these populations. In HIV prevention trials, heterogeneity is unavoidable and has been shown to decrease the treatment effect over time. This article assesses the influence of trial duration on the bias of the estimated hazards ratio resulting from omitting covariates from the Cox analysis. The true model is defined by including an unobserved random frailty term in the individual hazard that reflects the omitted covariate. Three frailty distributions are investigated: gamma, log-normal, and binary, and the asymptotic bias of the hazards ratio estimator is calculated. We show that the attenuation of the treatment effect resulting from unobserved heterogeneity strongly increases with trial duration, especially for continuous frailties that are likely to reflect omitted covariates, as they are often encountered in practice. The possibility of interpreting the long-term decrease in treatment effects as a bias induced by heterogeneity and trial duration is illustrated by a trial in oncology where adjuvant chemotherapy in stage 1B NSCLC was investigated.

Primary squamous cell carcinomas in the thyroid gland: an individual participant data meta-analysis

Primary squamous cell carcinomas arising from the thyroid gland (SCCTh) is extremely rare diseases, which have never been fully studied. Thus, we performed a systematic review and individual participant data meta-analysis of published SCCTh cases, to understand the clinical characteristics and to identify the prognostic factors of primary SCCTh. A literature search was conducted within Medline, EMBASE, Cochrane library databases and KoreaMed using the following Medical Subject Headings (MeSH) keywords: “primary,” “squamous,” “carcinoma,” “cancer,” and “thyroid.” Eighty-four patients' individual data from 39 articles and five patients' data in our institute were selected for analysis (N = 89). The mean age at diagnosis was 63.0 years (range, 24–90) and female preponderance (M:F = 1:2) was noted. The commonest complaint was the anterior neck mass, followed by dyspnea or dysphagia, and extension to the adjacent structure was found in 72%. The median survival was 9.0 months (95% CI, 6.0–23.0) and 3-year survival rate (3YSR) was 37.6% by Kaplan–Meier method, but only 20.1% by a shared frailty model for adjusting heterogeneity. Complete resection (R0) of tumors was the only significant prognostic factor in multivariable analysis, and the benefit of adjuvant treatment was not proved. The prognosis of patients with SCCTh is very poor (20% in 3YSR), but complete resection of disease is correlated with improved survival. To achieve complete surgical eradication of tumors, early detection and accurate diagnosis should be emphasized.

Multilevel mixed effects parametric survival models using adaptive Gauss-Hermite quadrature with application to recurrent events and individual participant data meta-analysis

Multilevel mixed effects survival models are used in the analysis of clustered survival data, such as repeated events, multicenter clinical trials, and individual participant data (IPD) meta-analyses, to investigate heterogeneity in baseline risk and covariate effects. In this paper, we extend parametric frailty models including the exponential, Weibull and Gompertz proportional hazards (PH) models and the log logistic, log normal, and generalized gamma accelerated failure time models to allow any number of normally distributed random effects. Furthermore, we extend the flexible parametric survival model of Royston and Parmar, modeled on the log-cumulative hazard scale using restricted cubic splines, to include random effects while also allowing for non-PH (time-dependent effects). Maximum likelihood is used to estimate the models utilizing adaptive or nonadaptive Gauss-Hermite quadrature. The methods are evaluated through simulation studies representing clinically plausible scenarios of a multicenter trial and IPD meta-analysis, showing good performance of the estimation method. The flexible parametric mixed effects model is illustrated using a dataset of patients with kidney disease and repeated times to infection and an IPD meta-analysis of prognostic factor studies in patients with breast cancer. User-friendly Stata software is provided to implement the methods.

Multilevel competing risk models to evaluate the risk of nosocomial infection

Introduction

Risk factor analyses for nosocomial infections (NIs) are complex. First, due to competing events for NI, the association between risk factors of NI as measured using hazard rates may not coincide with the association using cumulative probability (risk). Second, patients from the same intensive care unit (ICU) who share the same environmental exposure are likely to be more similar with regard to risk factors predisposing to a NI than patients from different ICUs. We aimed to develop an analytical approach to account for both features and to use it to evaluate associations between patient- and ICU-level characteristics with both rates of NI and competing risks and with the cumulative probability of infection.

Methods

We considered a multicenter database of 159 intensive care units containing 109,216 admissions (813,739 admission-days) from the Spanish HELICS-ENVIN ICU network. We analyzed the data using two models: an etiologic model (rate based) and a predictive model (risk based). In both models, random effects (shared frailties) were introduced to assess heterogeneity. Death and discharge without NI are treated as competing events for NI.

Results

There was a large heterogeneity across ICUs in NI hazard rates, which remained after accounting for multilevel risk factors, meaning that there are remaining unobserved ICU-specific factors that influence NI occurrence. Heterogeneity across ICUs in terms of cumulative probability of NI was even more pronounced. Several risk factors had markedly different associations in the rate-based and risk-based models. For some, the associations differed in magnitude. For example, high Acute Physiology and Chronic Health Evaluation II (APACHE II) scores were associated with modest increases in the rate of nosocomial bacteremia, but large increases in the risk. Others differed in sign, for example respiratory vs cardiovascular diagnostic categories were associated with a reduced rate of nosocomial bacteremia, but an increased risk.

Conclusions

A combination of competing risks and multilevel models is required to understand direct and indirect risk factors for NI and distinguish patient-level from ICU-level factors.

A joint frailty model to estimate the recurrence process and the disease-specific mortality process without needing the cause of death

In chronic diseases, such as cancer, recurrent events (such as relapses) are commonly observed; these could be interrupted by death. With such data, a joint analysis of recurrence and mortality processes is usually conducted with a frailty parameter shared by both processes. We examined a joint modeling of these processes considering death under two aspects: 'death due to the disease under study' and 'death due to other causes', which enables estimating the disease-specific mortality hazard. The excess hazard model was used to overcome the difficulties in determining the causes of deaths (unavailability or unreliability); this model allows estimating the disease-specific mortality hazard without needing the cause of death but using the mortality hazards observed in the general population. We propose an approach to model jointly recurrence and disease-specific mortality processes within a parametric framework. A correlation between the two processes is taken into account through a shared frailty parameter. This approach allows estimating unbiased covariate effects on the hazards of recurrence and disease-specific mortality. The performance of the approach was evaluated by simulations with different scenarios. The method is illustrated by an analysis of a population-based dataset on colon cancer with observations of colon cancer recurrences and deaths. The benefits of the new approach are highlighted by comparison with the 'classical' joint model of recurrence and overall mortality. Moreover, we assessed the goodness of fit of the proposed model. Comparisons between the conditional hazard and the marginal hazard of the disease-specific mortality are shown, and differences in interpretation are discussed.

Evaluation of estimation methods and power of tests of discrete covariates in repeated time-to-event parametric models: application to Gaucher patients treated by imiglucerase

Analysis of repeated time-to-event data is increasingly performed in pharmacometrics using parametric frailty models. The aims of this simulation study were (1) to assess estimation performance of Stochastic Approximation Expectation Maximization (SAEM) algorithm in MONOLIX, Adaptive Gaussian Quadrature (AGQ), and Laplace algorithm in PROC NLMIXED of SAS and (2) to evaluate properties of test of a dichotomous covariate on occurrence of events. The simulation setting is inspired from an analysis of occurrence of bone events after the initiation of treatment by imiglucerase in patients with Gaucher Disease (GD). We simulated repeated events with an exponential model and various dropout rates: no, low, or high. Several values of baseline hazard model, variability, number of subject, and effect of covariate were studied. For each scenario, 100 datasets were simulated for estimation performance and 500 for test performance. We evaluated estimation performance through relative bias and relative root mean square error (RRMSE). We studied properties of Wald and likelihood ratio test (LRT). We used these methods to analyze occurrence of bone events in patients with GD after starting an enzyme replacement therapy. SAEM with three chains and AGQ algorithms provided good estimates of parameters much better than SAEM with one chain and Laplace which often provided poor estimates. Despite a small number of repeated events, SAEM with three chains and AGQ gave small biases and RRMSE. Type I errors were closed to 5%, and power varied as expected for SAEM with three chains and AGQ. Probability of having at least one event under treatment was 19.1%.

Concordance measures in shared frailty models: application to clustered data in cancer prognosis

Frailty models are gaining interest in prognostic studies, especially because of the spread of multicenter studies. However, little research has been performed to extend prognostic tools to frailty models, including discrimination measures. As previously performed for the Harrell's c-index, we extended two different discrimination measures (the model-based concordance probability estimation of Gönen and Heller and the nonparametric Uno's c-index) to take into account cluster membership. We calculate measures at three levels: between-group, where only patients with different frailties are compared, within-group, where only patients sharing the same frailty are compared, and overall. We performed simulations to study the impact of group size and the number of groups on these measures. Results showed that the two measures can be extended to frailty models while remaining independent from censoring distribution, provided that the group size is sufficient. We apply the extended measures to two real datasets, a meta-analysis and a large multicenter trial.