Joint modeling of multivariate longitudinal mixed measurements and time to event data using a Bayesian approach

In many longitudinal studies multiple characteristics of each individual, along with time to occurrence of an event of interest, are often collected. In such data set, some of the correlated characteristics may be discrete and some of them may be continuous. In this paper, a joint model for analysing multivariate longitudinal data comprising mixed continuous and ordinal responses and a time to event variable is proposed. We model the association structure between longitudinal mixed data and time to event data using a multivariate zero-mean Gaussian process. For modeling discrete ordinal data we assume a continuous latent variable follows the logistic distribution and for continuous data a Gaussian mixed effects model is used. For the event time variable, an accelerated failure time model is considered under different distributional assumptions. For parameter estimation, a Bayesian approach using Markov Chain Monte Carlo is adopted. The performance of the proposed methods is illustrated using some simulation studies. A real data set is also analyzed, where different model structures are used. Model comparison is performed using a variety of statistical criteria.     

Recurrent time-to-event models with ordinal outcomes

A model to accommodate time-to-event ordinal outcomes was proposed by Berridge and Whitehead. Very few studies have adopted this approach, despite its appeal in incorporating several ordered categories of event outcome. More recently, there has been increased interest in utilizing recurrent events to analyze practical endpoints in the study of disease history and to help quantify the changing pattern of disease over time. For example, in studies of heart failure, the analysis of a single fatal event no longer provides sufficient clinical information to manage the disease. Similarly, the grade/frequency/severity of adverse events may be more important than simply prolonged survival in studies of toxic therapies in oncology. We propose an extension of the ordinal time-to-event model to allow for multiple/recurrent events in the case of marginal models (where all subjects are at risk for each recurrence, irrespective of whether they have experienced previous recurrences) and conditional models (subjects are at risk of a recurrence only if they have experienced a previous recurrence). These models rely on marginal and conditional estimates of the instantaneous baseline hazard and provide estimates of the probabilities of an event of each severity for each recurrence over time. We outline how confidence intervals for these probabilities can be constructed and illustrate how to fit these models and provide examples of the methods, together with an interpretation of the results.     

A Discrete-Time Multilevel Mixture Model for Event History Data with Long-Term Survivors,with an Application to an Analysis of Contraceptive Sterilization in Bangladesh

Event history models typically assume that the entire population is at risk of experiencing the event of interest throughout the observation period. However, there will often be individuals, referred to as long-term survivors, who may be considered a priori to have a zero hazard throughout the study period. In this paper, a discrete-time mixture model is proposed in which the probability of long-term survivorship and the timing of event occurrence are modelled jointly. Another feature of event history data that often needs to be considered is that they may come from a population with a hierarchical structure. For example, individuals may be nested within geographical regions and individuals in the same region may have similar risks of experiencing the event of interest due to unobserved regional characteristics. Thus, the discrete-time mixture model is extended to allow for clustering in the likelihood and timing of an event within regions. The model is further extended to allow for unobserved individual heterogeneity in the hazard of event occurrence. The proposed model is applied in an analysis of contraceptive sterilization in Bangladesh. The results show that a woman's religion and education level affect her probability of choosing sterilization, but not when she gets sterilized. There is also evidence of community-level variation in sterilization timing, but not in the probability of sterilization.     

A latent variable model for analyzing mixed longitudinal (k,l)-inflated count and ordinal responses

A random effects model for analyzing mixed longitudinal count and ordinal data is presented where the count response is inflated in two points (k and l) and an (k,l)-Inflated Power series distribution is used as its distribution. A full likelihood-based approach is used to obtain maximum likelihood estimates of parameters of the model. For data with non-ignorable missing values models with probit model for missing mechanism are used.The dependence between longitudinal sequences of responses and inflation parameters are investigated using a random effects approach. Also, to investigate the correlation between mixed ordinal and count responses of each individuals at each time, a shared random effect is used. In order to assess the performance of the model, a simulation study is performed for a case that the count response has (k,l)-Inflated Binomial distribution. Performance comparisons of count-ordinal random effect model, Zero-Inflated ordinal random effects model and (k,l)-Inflated ordinal random effects model are also given. The model is applied to a real social data set from the first two waves of the national longitudinal study of adolescent to adult health (Add Health study). In this data set, the joint responses are the number of days in a month that each individual smoked as the count response and the general health condition of each individual as the ordinal response. For the count response there is incidence of excess values of 0 and 30.     

Latent variable model for mixed correlated power series and ordinal longitudinal responses with non ignorable missing values

We propose a joint model based on a latent variable for analyzing mixed power series and ordinal longitudinal data with and without missing values. A bivariate probit regression model is used for the missing mechanisms. Random effects are used to take into account the correlation between longitudinal responses. A full likelihood-based approach is used to yield maximum-likelihood estimates of the model parameters. Our model is applied to a medical data set, obtained from an observational study on women where the correlated responses are the ordinal response of osteoporosis of the spine and the power series response of the number of joint damages. Sensitivity analysis is also performed to study the influence of small perturbations of the parameters of the missing mechanisms and overdispersion of the model on likelihood displacement.     

Using recurrent time-to-event models with multinomial outcomes to generate toxicity profiles

Most clinical studies, which investigate the impact of therapy simultaneously, record the frequency of adverse events in order to monitor safety of the intervention. Study reports typically summarise adverse event data by tabulating the frequencies of the worst grade experienced but provide no details of the temporal profiles of specific types of adverse events. Such 'toxicity profiles' are potentially important tools in disease management and in the assessment of newer therapies including targeted treatments and immunotherapy where different types of toxicity may be more common at various times during long-term drug exposure. Toxicity profiles of commonly experienced adverse events occurring due to exposure to long-term treatment could assist in evaluating the costs of the health care benefits of therapy. We show how to generate toxicity profiles using an adaptation of the ordinal time-to-event model comprising of a two-step process, involving estimation of the multinomial response probabilities using multinomial logistic regression and combining these with recurrent time to event hazard estimates to produce cumulative event probabilities for each of the multinomial adverse event response categories. Such a model permits the simultaneous assessment of the risk of events over time and provides cumulative risk probabilities for each type of adverse event response. The method can be applied more generally by using different models to estimate outcome/response probabilities. The method is illustrated by developing toxicity profiles for three distinct types of adverse events associated with two treatment regimens for patients with advanced breast cancer.     

Mixed Correlated Bivariate Ordinal and Negative Binomial Longitudinal Responses with Nonignorable Missing Values

Regression models with random effects are proposed for joint analysis of negative binomial and ordinal longitudinal data with nonignorable missing values under fully parametric framework. The presented model simultaneously considers a multivariate probit regression model for the missing mechanisms, which provides the ability of examining the missing data assumptions and a multivariate mixed model for the responses. Random effects are used to take into account the correlation between longitudinal responses of the same individual. A full likelihood-based approach that allows yielding maximum likelihood estimates of the model parameters is used. The model is applied to a medical data, obtained from an observational study on women, where the correlated responses are the ordinal response of osteoporosis of the spine and negative binomial response is the number of joint damage. A sensitivity of the results to the assumptions is also investigated. The effect of some covariates on all responses are investigated simultaneously.     

Non-parametric Tests for Recurrent Events under Competing Risks

Abstract.  We consider a data set on nosocomial infections of patients hospitalized in a French intensive care facility. Patients may suffer from recurrent infections of different types and they also have a high risk of death. To deal with such situations, a model of recurrent events with competing risks and a terminal event is introduced. Our aim was to compare the occurrence rates of two types of events. For this purpose, we propose two tests: one to detect if the occurrence rate of a given type of event increases with time; a second to detect if the instantaneous probability of experiencing an event of a given type is always greater than the one of another type. The asymptotic properties of the test statistics are derived and Monte Carlo methods are used to study the power of the tests. Finally, the procedures developed are applied to the French nosocomial infections data set.     

Mixture regression models for the gap time distributions and illness–death processes

Lifetime Data Analysis - The aim of this study is to provide an analysis of gap event times under the illness–death model, where some subjects experience “illness” before...     

Treatment policy estimands for recurrent event data using data collected after cessation of randomised treatment

In the past, many clinical trials have withdrawn subjects from the study when they prematurely stopped their randomised treatment and have therefore only collected ‘on‐treatment’ data. Thus, analyses addressing a treatment policy estimand have been restricted to imputing missing data under assumptions drawn from these data only. Many confirmatory trials are now continuing to collect data from subjects in a study even after they have prematurely discontinued study treatment as this event is irrelevant for the purposes of a treatment policy estimand. However, despite efforts to keep subjects in a trial, some will still choose to withdraw. Recent publications for sensitivity analyses of recurrent event data have focused on the reference‐based imputation methods commonly applied to continuous outcomes, where imputation for the missing data for one treatment arm is based on the observed outcomes in another arm. However, the existence of data from subjects who have prematurely discontinued treatment but remained in the study has now raised the opportunity to use this ‘off‐treatment’ data to impute the missing data for subjects who withdraw, potentially allowing more plausible assumptions for the missing post‐study‐withdrawal data than reference‐based approaches. In this paper, we introduce a new imputation method for recurrent event data in which the missing post‐study‐withdrawal event rate for a particular subject is assumed to reflect that observed from subjects during the off‐treatment period. The method is illustrated in a trial in chronic obstructive pulmonary disease (COPD) where the primary endpoint was the rate of exacerbations, analysed using a negative binomial model.     

Regression Parameter Estimation from Panel Counts

This paper considers a study where each subject may experience multiple occurrences of an event and the rate of the event occurrences is of primary interest. Specifically, we are concerned with the situations where, for each subject, there are only records of the accumulated counts for the event occurrences at a finite number of time points over the study period. Sets of observation times may vary from subject to subject and differ between groups. We model the mean of the event occurrence number over time semiparametrically, and estimate the regression parameter. The proposed estimation procedures are illustrated with data from a bladder cancer study ( Byar, 1980 ). Both asymptotics and simulation studies on the estimators are presented.     

Principal stratum strategy: Potential role in drug development

A randomized trial allows estimation of the causal effect of an intervention compared to a control in the overall population and in subpopulations defined by baseline characteristics. Often, however, clinical questions also arise regarding the treatment effect in subpopulations of patients, which would experience clinical or disease related events post-randomization. Events that occur after treatment initiation and potentially affect the interpretation or the existence of the measurements are called intercurrent events in the ICH E9(R1) guideline. If the intercurrent event is a consequence of treatment, randomization alone is no longer sufficient to meaningfully estimate the treatment effect. Analyses comparing the subgroups of patients without the intercurrent events for intervention and control will not estimate a causal effect. This is well known, but post-hoc analyses of this kind are commonly performed in drug development. An alternative approach is the principal stratum strategy, which classifies subjects according to their potential occurrence of an intercurrent event on both study arms. We illustrate with examples that questions formulated through principal strata occur naturally in drug development and argue that approaching these questions with the ICH E9(R1) estimand framework has the potential to lead to more transparent assumptions as well as more adequate analyses and conclusions. In addition, we provide an overview of assumptions required for estimation of effects in principal strata. Most of these assumptions are unverifiable and should hence be based on solid scientific understanding. Sensitivity analyses are needed to assess robustness of conclusions.     

A Bivariate Distribution for the Reliability Analysis of Failure Data in Presence of a Forewarning or Primer Event

This article presents a bivariate distribution for analyzing the failure data of mechanical and electrical components in presence of a forewarning or primer event whose occurrence denotes the inception of the failure mechanism that will cause the component failure after an additional random time. The characteristics of the proposed distribution are discussed and several point estimators of parameters are illustrated and compared, in case of complete sampling, via a large Monte Carlo simulation study. Confidence intervals based on asymptotic results are derived, as well as procedures are given for testing the independence between the occurrence time of the forewarning event and the additional time to failure. Numerical applications based on failure data of cable insulation specimens and of two-component parallel systems are illustrated.     

Modeling a mixture of ordinal and continuous repeated measures

We study the correlation structure for a mixture of ordinal and continuous repeated measures using a Bayesian approach. We assume a multivariate probit model for the ordinal variables and a normal linear regression for the continuous variables, where latent normal variables underlying the ordinal data are correlated with continuous variables in the model. Due to the probit model assumption, we are required to sample a covariance matrix with some of the diagonal elements equal to one. The key computational idea is to use parameter-extended data augmentation, which involves applying the Metropolis-Hastings algorithm to get a sample from the posterior distribution of the covariance matrix incorporating the relevant restrictions. The methodology is illustrated through a simulated example and through an application to data from the UCLA Brain Injury Research Center.     

A Bayesian test of homogeneity of association parameter using transition modelling of longitudinal mixed responses

In this paper, a Bayesian framework using a joint transition model for analysing longitudinal mixed ordinal and continuous responses is considered. The joint model considers a multivariate mixed model for the responses in which a transitive cumulative logistic regression model and an autoregressive regression model are used to model ordinal and continuous responses, respectively. Also, to take into account the association between longitudinal ordinal and continuous responses, a dynamic association parameter is used. A test is conducted to see whether this parameter is time-invariant and another test is presented to see whether this parameter is equal to zero or significantly far from zero. Our approach is applied to longitudinal PIAT (Peabody Individual Achievement Test) data where the Bayesian estimates of parameters are obtained.     

Logistic and neural network models for predicting a hospital admission

Feedforward neural networks are often used in a similar manner as logistic regression models; that is, to estimate the probability of the occurrence of an event. In this paper, a probabilistic model is developed for the purpose of estimating the probability that a patient who has been admitted to the hospital with a medical back diagnosis will be released after only a short stay or will remain hospitalized for a longer period of time. As the purpose of the analysis is to determine if hospital characteristics influence the decision to retain a patient, the inputs to this model are a set of demographic variables that describe the various hospitals. The output is the probability of either a short or long term hospital stay. In order to compare the ability of each method to model the data, a hypothesis test is performed to test for an improvement resulting from the use of the neural network model.     

A comparison of ordinal logistic regression models using Classical and Bayesian approaches in an analysis of factors associated with diabetic retinopathy

Estimating the risk factors of a disease such as diabetic retinopathy (DR) is one of the important research problems among bio-medical and statistical practitioners as well as epidemiologists. Incidentally many studies have focused in building models with binary outcomes, that may not exploit the available information. This article has investigated the importance of retaining the ordinal nature of the response variable (e.g. severity level of a disease) while determining the risk factors associated with DR. A generalized linear model approach with appropriate link functions has been studied using both Classical and Bayesian frameworks. From the result of this study, it can be observed that the ordinal logistic regression with probit link function could be more appropriate approach in determining the risk factors of DR. The study has emphasized the ways to handle the ordinal nature of the response variable with better model fit compared to other link functions.     

A dynamic Mover–Stayer model for recurrent event processes subject to resolution

In studies of affective disorder, individuals are often observed to experience recurrent symptomatic exacerbations warranting hospitalization. Interest may lie in modeling the occurrence of such exacerbations over time and identifying associated risk factors. In some patients, recurrent exacerbations are temporally clustered following disease onset, but cease to occur after a period of time. We develop a dynamic Mover–Stayer model in which a canonical binary variable associated with each event indicates whether the underlying disease has resolved. An individual whose disease process has not resolved will experience events following a standard point process model governed by a latent intensity. When the disease process resolves, the complete data intensity becomes zero and no further event will occur. An expectation–maximization algorithm is described for parametric and semiparametric model fitting based on a discrete time dynamic Mover–Stayer model and a latent intensity-based model of the underlying point process.     

On bootstrap testing inference in cure rate models

Survival models deal with the time until the occurrence of an event of interest. However, in some situations the event may not occur in part of the studied population. The fraction of the population that will never experience the event of interest is generally called cure rate. Models that consider this fact (cure rate models) have been extensively studied in the literature. Hypothesis testing on the parameters of these models can be performed based on likelihood ratio, gradient, score or Wald statistics. Critical values of these tests are obtained through approximations that are valid in large samples and may result in size distortion in small or moderate sample sizes. In this sense, this paper proposes bootstrap corrections to the four mentioned tests and bootstrap Bartlett correction for the likelihood ratio statistic in the Weibull promotion time model. Besides, we present an algorithm for bootstrap resampling when the data presents cure fraction and right censoring time (random and non-informative). Simulation studies are conducted to compare the finite sample performances of the corrected tests. The numerical evidence favours the corrected tests we propose. We also present an application in an actual data set.