Joint frailty models for zero-inflated recurrent events in the presence of a terminal event

Comorbidities affecting re-admission and survival in COVID-19: Application of joint frailty model

BACKGROUND

One of the common concerns of healthcare systems is the potential for re-admission of COVID-19 patients. In addition to adding costs to the healthcare system, re-admissions also endanger patient safety. Recognizing the factors that influence re-admission, can help provide appropriate and optimal health care. The aim of this study was to assess comorbidities that affect re-admission and survival in COVID-19 patients using a joint frailty model.

METHODS

This historical cohort study was done using data of patients with COVID-19 who were re-hospitalized more than twice in a referral hospital in North of Iran. We used the joint frailty model to investigate prognostic factors of survival and recurrence, simultaneously using R version 3.5.1 (library "frailtypack"). P-values less than 0.05 were considered as statistically significant.

RESULTS

A total of 112 patients with mean (SD) age of 63.76 (14.58) years old were recruited into the study. Forty-eight (42.9%) patients died in which 53.83% of them were re-admitted for a second time. Using adjusted joint model, the hazard of re-admission increased with cancer (Hazard ratio (HR) = 1.92) and hyperlipidemia (HR = 1.22). Furthermore, the hazard of death increased with hyperlipidemia (HR = 4.05) followed by age (HR = 1.76) and cancer (HR = 1.64). It Also decreased with lung disease (HR = 0.11), hypothyroidism (HR = 0.32), and hypertension (HR = 0.97).

CONCLUSION

Considering the correlation between re-admission and mortality in the joint frailty model, malignancy and hyperlipidemia increased the risk of both re-admission and mortality. Moreover, lung disease probably due to the use of corticosteroids, was a protective factor against both mortality and re-admission.

Regression analysis of general mixed recurrent event data

In modern biomedical datasets, it is common for recurrent outcomes data to be collected in an incomplete manner. More specifically, information on recurrent events is routinely recorded as a mixture of recurrent event data, panel count data, and panel binary data; we refer to this structure as general mixed recurrent event data. Although the aforementioned data types are individually well-studied, there does not appear to exist an established approach for regression analysis of the three component combination. Often, ad-hoc measures such as imputation or discarding of data are used to homogenize records prior to the analysis, but such measures lead to obvious concerns regarding robustness, loss of efficiency, and other issues. This work proposes a maximum likelihood regression estimation procedure for the combination of general mixed recurrent event data and establishes the asymptotic properties of the proposed estimators. In addition, we generalize the approach to allow for the existence of terminal events, a common complicating feature in recurrent event analysis. Numerical studies and application to the Childhood Cancer Survivor Study suggest that the proposed procedures work well in practical situations.

A parametric model to jointly characterize rate, duration, and severity of exacerbations in episodic diseases

BACKGROUND

The natural history of many chronic diseases is characterized by periods of increased disease activity, commonly referred to as flare-ups or exacerbations. Accurate characterization of the burden of these exacerbations is an important research objective.

METHODS

The purpose of this work was to develop a statistical framework for nuanced characterization of the three main features of exacerbations: their rate, duration, and severity, with interrelationships among these features being a particular focus. We jointly specified a zero-inflated accelerated failure time regression model for the rate, an accelerated failure time regression model for the duration, and a logistic regression model for the severity of exacerbations. Random effects were incorporated into each component to capture heterogeneity beyond the variability attributable to observed characteristics, and to describe the interrelationships among these components.

RESULTS

We used pooled data from two clinical trials in asthma as an exemplary application to illustrate the utility of the joint modeling approach. The model fit clearly indicated the presence of heterogeneity in all three components. A novel finding was that the new therapy reduced not just the rate but also the duration of exacerbations, but did not have a significant impact on their severity. After controlling for covariates, exacerbations among more frequent exacerbators tended to be shorter and less likely to be severe.

CONCLUSIONS

We conclude that a joint modeling framework, programmable in available software, can provide novel insights about how the rate, duration, and severity of episodic events interrelate, and enables consistent inference on the effect of treatments on different disease outcomes. Trial registration Ethics approval was obtained from the University of British Columbia Human Ethics Board (H17-00938).

Dynamic semiparametric transformation models for recurrent event data with a terminal event

Recurrent event data with a terminal event commonly arise in many longitudinal follow-up studies. This article proposes a class of dynamic semiparametric transformation models for the marginal mean functions of the recurrent events with a terminal event, where some covariate effects may be time-varying. An estimation procedure is developed for the model parameters, and the asymptotic properties of the resulting estimators are established. In addition, relevant significance tests are suggested for examining whether or not covariate effects vary with time, and a model checking procedure is presented for assessing the adequacy of the proposed models. The finite sample performance of the proposed estimators is examined through simulation studies, and an application to a medical cost study of chronic heart failure patients is provided.

Efficient Multiple Imputation for Sensitivity Analysis of Recurrent Events Data with Informative Censoring

Missing data are commonly encountered in clinical trials due to dropout or nonadherence to study procedures. In trials in which recurrent events are of interest, the observed count can be an undercount of the events if a patient drops out before the end of the study. In many applications, the data are not necessarily missing at random and it is often not possible to test the missing at random assumption. Consequently, it is critical to conduct sensitivity analysis. We develop a control-based multiple imputation method for recurrent events data, where patients who drop out of the study are assumed to have a similar response profile to those in the control group after dropping out. Specifically, we consider the copy reference approach and the jump to reference approach. We model the recurrent event data using a semiparametric proportional intensity frailty model with the baseline hazard function completely unspecified. We develop nonparametric maximum likelihood estimation and inference procedures. We then impute the missing data based on the large sample distribution of the resulting estimators. The variance estimation is corrected by a bootstrap procedure. Simulation studies demonstrate the proposed method performs well in practical settings. We provide applications to two clinical trials.

Mixture proportional hazards cure model with latent variables

A mixture proportional hazards cure model with latent variables is proposed. The proposed model assesses the effects of the observed and latent risk factors on the hazards of uncured subjects and the cure rate through a proportional hazards model and a logistic model, respectively. Factor analysis is employed to measure the latent variables through correlated multiple indicators. Maximum likelihood estimation is performed through a Gaussian quadratic technique that approximates the integration over the latent variables. A piecewise constant function is used for the unspecified baseline hazard of uncured subjects. The proposed method can be conveniently implemented by using SAS Proc NLMIXED. Simulation studies are conducted to evaluate the performance of the proposed approach. An application to a study concerning the risk factors of chronic kidney disease for type 2 diabetic patients is provided.

Bayesian inference on the number of recurrent events: A joint model of recurrence and survival

The number of recurrent events before a terminating event is often of interest. For instance, death terminates an individual's process of rehospitalizations and the number of rehospitalizations is an important indicator of economic cost. We propose a model in which the number of recurrences before termination is a random variable of interest, enabling inference and prediction on it. Then, conditionally on this number, we specify a joint distribution for recurrence and survival. This novel conditional approach induces dependence between recurrence and survival, which is often present, for instance, due to frailty that affects both. Additional dependence between recurrence and survival is introduced by the specification of a joint distribution on their respective frailty terms. Moreover, through the introduction of an autoregressive model, our approach is able to capture the temporal dependence in the recurrent events trajectory. A non-parametric random effects distribution for the frailty terms accommodates population heterogeneity and allows for data-driven clustering of the subjects. A tailored Gibbs sampler involving reversible jump and slice sampling steps implements posterior inference. We illustrate our model on colorectal cancer data, compare its performance with existing approaches and provide appropriate inference on the number of recurrent events.

Joint modeling of longitudinal data with informative cluster size adjusted for zero-inflation and a dependent terminal event

Repeated measures are often collected in longitudinal follow-up from clinical trials and observational studies. In many situations, these measures are adherent to some specific event and are only available when it occurs; an example is serum creatinine from laboratory tests for hospitalized acute kidney injuries. The frequency of event recurrences is potentially correlated with overall health condition and hence may influence the distribution of the outcome measure of interest, leading to informative cluster size. In particular, there may be a large portion of subjects without any events, thus no longitudinal measures are available, which may be due to insusceptibility to such events or censoring before any events, and this zero-inflation nature of the data needs to be taken into account. On the other hand, there often exists a terminal event that may be correlated with the recurrent events. Previous work in this area suffered from the limitation that not all these issues were handled simultaneously. To address this deficiency, we propose a novel joint modeling approach for longitudinal data adjusting for zero-inflated and informative cluster size as well as a terminal event. A three-stage semiparametric likelihood-based approach is applied for parameter estimation and inference. Extensive simulations are conducted to evaluate the performance of our proposal. Finally, we utilize the Assessment, Serial Evaluation, and Subsequent Sequelae of Acute Kidney Injury (ASSESS-AKI) study for illustration.

Sample size calculation for recurrent event data with additive rates models

This paper discusses the design of clinical trials where the primary endpoint is a recurrent event with the focus on the sample size calculation. For the problem, a few methods have been proposed but most of them assume a multiplicative treatment effect on the rate or mean number of recurrent events. In practice, sometimes the additive treatment effect may be preferred or more appealing because of its intuitive clinical meaning and straightforward interpretation compared to a multiplicative relationship. In this paper, new methods are presented and investigated for the sample size calculation based on the additive rates model for superiority, non-inferiority, and equivalence trials. They allow for flexible baseline rate function, staggered entry, random dropout, and overdispersion in event numbers, and simulation studies show that the proposed methods perform well in a variety of settings. We also illustrate how to use the proposed methods to design a clinical trial based on real data.

Latent class mediator for multiple indicators of mediation

This paper demonstrates the utility of latent classes in evaluating the effect of an intervention on an outcome through multiple indicators of mediation. These indicators are observed intermediate variables that identify an underlying latent class mediator, with each class representing a different mediating pathway. The use of a latent class mediator allows us to avoid modeling the complex interactions between the multiple indicators and ensures the decomposition of the total mediating effects into additive effects from individual mediating pathways, a desirable feature for evaluating multiple indicators of mediation. This method is suitable when the goal is to estimate the total mediating effects that can be decomposed into the additive effects of distinct mediating pathways. Each indicator may be involved in multiple mediation pathways and at the same time multiple indicators may contribute to a single mediating pathway. The relative importance of each pathway may vary across subjects. We applied this method to the analysis of the first 6 months of data from a 2-year clustered randomized trial for adults in their first episode of schizophrenia. Four indicators of mediation are considered: individual resiliency training; family psychoeducation; supported education and employment; and a structural assessment for medication. The improvement in symptoms was found to be mediated by the latent class mediator derived from these four service indicators. Simulation studies were conducted to assess the performance of the proposed model and showed that the simultaneous estimation through the maximum likelihood yielded little bias when the entropy of the indicators was high.

Semiparametric analysis of zero-inflated recurrent events with a terminal event

Recurrent event data frequently arise in longitudinal studies and observations on recurrent events could be terminated by a major failure event such as death. In many situations, there exist a large fraction of subjects without any recurrent events of interest. Among these subjects, some are unsusceptible to recurrent events, while others are susceptible but have no recurrent events being observed due to censoring. In this article, we propose a zero-inflated generalized joint frailty model and a sieve maximum likelihood approach to analyze zero-inflated recurrent events with a terminal event. The model provides a considerable flexibility in formulating the effects of covariates on both recurrent events and the terminal event by specifying various transformation functions. In addition, Bernstein polynomials are employed to approximate the unknown cumulative baseline hazard (intensity) function. The estimation procedure can be easily implemented and is computationally fast. Extensive simulation studies are conducted and demonstrate that our proposed method works well for practical situations. Finally, we apply the method to analyze myocardial infarction recurrences in the presence of death in a clinical trial with cardiovascular outcomes.

Regression analysis of mixed panel-count data with application to cancer studies

Both panel-count data and panel-binary data are common data types in recurrent event studies. Because of inconsistent questionnaires or missing data during the follow-ups, mixed data types need to be addressed frequently. A recently proposed semiparametric approach uses a proportional means model to facilitate regression analyses of mixed panel-count and panel-binary data. This method can use all available information regardless of the record type and provide unbiased estimates. However, the large number of nuisance parameters in the nonparametric baseline hazard function makes the estimating procedure very complicated and time-consuming. We approximated the baseline hazard function to simplify the estimating procedure. Simulation studies showed that our method performed similarly to that of the previous semiparametric likelihood-based method, but with much faster speed. Approximating the baseline hazard not only reduced the computational burden but also made it possible to implement the estimating procedure in a standard software, such as SAS.

A novel model for the X-chromosome inactivation association on survival data

The X-linked genetic association is overlooked in most of the genetic studies because of the complexity of X-chromosome inactivation process. In fact, the biological process of the gene at the same locus can vary across different subjects. Besides, the skewness of X-chromosome inactivation is inherently subject-specific (even tissue-specific within subjects) and cannot be accurately represented by a population-level parameter. To tackle this issue, a new model is proposed to incorporate the X-linked genetic association into right-censored survival data. The novel model can present that the X-linked genes on different subjects may go through different biological processes via a mixed distribution. Further, a random effect is employed to describe the uncertainty of the biological process for every subject. The proposed method can derive the probability for the escape of X-chromosome inactivation and derive the unbiased estimates of the model parameters. The Legendre-Gauss Quadrature method is used to approximate the integration over the random effect. Finite sample performance of this method is examined via extensive simulation studies. An application is illustrated with the implementation on a cancer genetic study with right-censored survival data.

Variable selection in joint frailty models of recurrent and terminal events

Recurrent event data are commonly encountered in biomedical studies. In many situations, they are subject to an informative terminal event, for example, death. Joint modeling of recurrent and terminal events has attracted substantial recent research interests. On the other hand, there may exist a large number of covariates in such data. How to conduct variable selection for joint frailty proportional hazards models has become a challenge in practical data analysis. We tackle this issue on the basis of the "minimum approximated information criterion" method. The proposed method can be conveniently implemented in SAS Proc NLMIXED for commonly used frailty distributions. Its finite-sample behavior is evaluated through simulation studies. We apply the proposed method to model recurrent opportunistic diseases in the presence of death in an AIDS study.

A bivariate joint frailty model with mixture framework for survival analysis of recurrent events with dependent censoring and cure fraction

In the study of multiple failure time data with recurrent clinical endpoints, the classical independent censoring assumption in survival analysis can be violated when the evolution of the recurrent events is correlated with a censoring mechanism such as death. Moreover, in some situations, a cure fraction appears in the data because a tangible proportion of the study population benefits from treatment and becomes recurrence free and insusceptible to death related to the disease. A bivariate joint frailty mixture cure model is proposed to allow for dependent censoring and cure fraction in recurrent event data. The latency part of the model consists of two intensity functions for the hazard rates of recurrent events and death, wherein a bivariate frailty is introduced by means of the generalized linear mixed model methodology to adjust for dependent censoring. The model allows covariates and frailties in both the incidence and the latency parts, and it further accounts for the possibility of cure after each recurrence. It includes the joint frailty model and other related models as special cases. An expectation-maximization (EM)-type algorithm is developed to provide residual maximum likelihood estimation of model parameters. Through simulation studies, the performance of the model is investigated under different magnitudes of dependent censoring and cure rate. The model is applied to data sets from two colorectal cancer studies to illustrate its practical value.

Semiparametric frailty models for zero-inflated event count data in the presence of informative dropout

Recurrent events data are commonly encountered in medical studies. In many applications, only the number of events during the follow-up period rather than the recurrent event times is available. Two important challenges arise in such studies: (a) a substantial portion of subjects may not experience the event, and (b) we may not observe the event count for the entire study period due to informative dropout. To address the first challenge, we assume that underlying population consists of two subpopulations: a subpopulation nonsusceptible to the event of interest and a subpopulation susceptible to the event of interest. In the susceptible subpopulation, the event count is assumed to follow a Poisson distribution given the follow-up time and the subject-specific characteristics. We then introduce a frailty to account for informative dropout. The proposed semiparametric frailty models consist of three submodels: (a) a logistic regression model for the probability such that a subject belongs to the nonsusceptible subpopulation; (b) a nonhomogeneous Poisson process model with an unspecified baseline rate function; and (c) a Cox model for the informative dropout time. We develop likelihood-based estimation and inference procedures. The maximum likelihood estimators are shown to be consistent. Additionally, the proposed estimators of the finite-dimensional parameters are asymptotically normal and the covariance matrix attains the semiparametric efficiency bound. Simulation studies demonstrate that the proposed methodologies perform well in practical situations. We apply the proposed methods to a clinical trial on patients with myelodysplastic syndromes.

A joint model for recurrent events and a semi-competing risk in the presence of multi-level clustering

Clinical trial designs often include multiple levels of clustering in which patients are nested within clinical sites and recurrent outcomes are nested within patients who may also experience a semi-competing risk. Traditional survival methods that analyze these processes separately may lead to erroneous inferences as they ignore possible dependencies. To account for the association between recurrent events and a semi-competing risk in the presence of two levels of clustering, we developed a semi-parametric joint model. The Gaussian quadrature with a piecewise constant baseline hazard was used to estimate the unspecified baseline hazards and the likelihood. Simulations showed that the proposed joint model has good statistical properties (i.e. <5% bias and 95% coverage) compared to the shared frailty and joint frailty models when informative censoring and multiple levels of clustering were present. The proposed method was applied to data from an AIDS clinical trial to investigate the impact of antiretroviral treatment on recurrent AIDS-defining events in the presence of a semi-competing risk of death and multi-level clustering and showed a significant dependency between AIDS-defining events and death at the patient level but not at the clinic level.

Exploring causality mechanism in the joint analysis of longitudinal and survival data

In many biomedical studies, disease progress is monitored by a biomarker over time, eg, repeated measures of CD4 in AIDS and hemoglobin in end-stage renal disease patients. The endpoint of interest, eg, death or diagnosis of a specific disease, is correlated with the longitudinal biomarker. In this paper, we examine and compare different models of longitudinal and survival data to investigate causal mechanisms, specifically, those related to the role of random effects. We illustrate the methods by data from two clinical trials: an AIDS study and a liver cirrhosis study.

Joint modeling of recurrent events and a terminal event adjusted for zero inflation and a matched design

In longitudinal studies, matched designs are often employed to control the potential confounding effects in the field of biomedical research and public health. Because of clinical interest, recurrent time-to-event data are captured during the follow-up. Meanwhile, the terminal event of death is always encountered, which should be taken into account for valid inference because of informative censoring. In some scenarios, a certain large portion of subjects may not have any recurrent events during the study period due to nonsusceptibility to events or censoring; thus, the zero-inflated nature of data should be considered in analysis. In this paper, a joint frailty model with recurrent events and death is proposed to adjust for zero inflation and matched designs. We incorporate 2 frailties to measure the dependency between subjects within a matched pair and that among recurrent events within each individual. By sharing the random effects, 2 event processes of recurrent events and death are dependent with each other. The maximum likelihood based approach is applied for parameter estimation, where the Monte Carlo expectation-maximization algorithm is adopted, and the corresponding R program is developed and available for public usage. In addition, alternative estimation methods such as Gaussian quadrature (PROC NLMIXED) and a Bayesian approach (PROC MCMC) are also considered for comparison to show our method's superiority. Extensive simulations are conducted, and a real data application on acute ischemic studies is provided in the end.

A flexible family of transformation cure rate models

In this paper, we introduce a flexible family of cure rate models, mainly motivated by the biological derivation of the classical promotion time cure rate model and assuming that a metastasis-competent tumor cell produces a detectable-tumor mass only when a specific number of distinct biological factors affect the cell. Special cases of the new model are, among others, the promotion time (proportional hazards), the geometric (proportional odds), and the negative binomial cure rate model. In addition, our model generalizes specific families of transformation cure rate models and some well-studied destructive cure rate models. Exact likelihood inference is carried out by the aid of the expectationŰmaximization algorithm; a profile likelihood approach is exploited for estimating the parameters of the model while model discrimination problem is analyzed by the aid of the likelihood ratio test. A simulation study demonstrates the accuracy of the proposed inferential method. Finally, as an illustration, we fit the proposed model to a cutaneous melanoma data-set. Copyright © 2017 John Wiley & Sons, Ltd.

Modeling event count data in the presence of informative dropout with application to bleeding and transfusion events in myelodysplastic syndrome

In many biomedical studies, it is often of interest to model event count data over the study period. For some patients, we may not follow up them for the entire study period owing to informative dropout. The dropout time can potentially provide valuable insight on the rate of the events. We propose a joint semiparametric model for event count data and informative dropout time that allows for correlation through a Gamma frailty. We develop efficient likelihood-based estimation and inference procedures. The proposed nonparametric maximum likelihood estimators are shown to be consistent and asymptotically normal. Furthermore, the asymptotic covariances of the finite-dimensional parameter estimates attain the semiparametric efficiency bound. Extensive simulation studies demonstrate that the proposed methods perform well in practice. We illustrate the proposed methods through an application to a clinical trial for bleeding and transfusion events in myelodysplastic syndrome. Copyright © 2017 John Wiley & Sons, Ltd.

A gap time model based on a multiplicative marginal rate function that accounts for zero-recurrence units

In this article, we propose an alternative gap time model based on a multiplicative marginal rate function, which is formulated considering each gap time conditional on the previous recurrence times. In this formulation, the gap times are treated equally and the relation between successive events is no longer a problem. Furthermore, this article considers the inclusion of a proportion of zero-recurrence units (for which the event of interest will not occur) into the model to analyze recurrent event data. Inference aspects of the proposed model are discussed through maximum likelihood approach. A simulation study is carried out to examine the performance of the estimation procedure. The model is applied to hospital readmission data among colorectal cancer patients.