Inferences for joint modelling of repeated ordinal scores and time to event data
16 Aug 2010-Computational and Mathematical Methods in Medicine (Hindawi Publishing Corporation)-Vol. 11, Iss: 3, pp 281-295
TL;DR: This article has attempted to analyse clinical trials and other follow-up studies through a latent variable model to account for the dependence between ordered categorical responses and survival time for different causes due to unobserved factors.
Abstract: In clinical trials and other follow-up studies, it is natural that a response variable is repeatedly measured during follow-up and the occurrence of some key event is also monitored. There has been a considerable study on the joint modelling these measures together with information on covariates. But most of the studies are related to continuous outcomes. In many situations instead of observing continuous outcomes, repeated ordinal outcomes are recorded over time. The joint modelling of such serial outcomes and the time to event data then becomes a bit complicated. In this article we have attempted to analyse such models through a latent variable model. In view of the longitudinal variation on the ordinal outcome measure, it is desirable to account for the dependence between ordered categorical responses and survival time for different causes due to unobserved factors. A flexible Monte Carlo EM (MCEM) method based on exact likelihood is proposed that can simultaneously handle the longitudinal ordinal data and also the censored time to event data. A computationally more efficient MCEM method based on approximation of the likelihood is also proposed. The method is applied to a number of ordinal scores and survival data from trials of a treatment for children suffering from Duchenne Muscular Dystrophy. Finally, a simulation study is conducted to examine the finite sample properties of the proposed estimators in the joint model under two different methods.
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TL;DR: Joint model analysis was more parsimonious as compared to separate analysis, as it reduces type I error and subject-specific analysis improved its model fit, and the observed correlation between the outcomes that have emerged from the association of intercepts is validated.
Abstract: Adherence and CD4 cell count change measure the progression of the disease in HIV patients after the commencement of HAART. Lack of information about associated factors on adherence to HAART and CD4 cell count reduction is a challenge for the improvement of cells in HIV positive adults. The main objective of adopting joint modeling was to compare separate and joint models of longitudinal repeated measures in identifying long-term predictors of the two longitudinal outcomes: CD4 cell count and adherence to HAART. A longitudinal retrospective cohort study was conducted to examine the joint predictors of CD4 cell count change and adherence to HAART among HIV adult patients enrolled in the first 10 months of the year 2008 and followed-up to June 2012. Joint model was employed to determine joint predictors of two longitudinal response variables over time. Furthermore, the generalized linear mixed effect model had been used for specification of the marginal distribution, conditional to correlated random effect. A total of 792 adult HIV patients were studied to analyze the longitudinal joint model study. The result from this investigation revealed that age, weight, baseline CD4 cell count, ownership of cell phone, visiting times, marital status, residence area and level of disclosure of the disease to family members had significantly affected both outcomes. From the two-way interactions, time * owner of cell phone, time * sex, age * sex, age * level of education as well as time * level of education were significant for CD4 cell count change in the longitudinal data analysis. The multivariate joint model with linear predictor indicates that CD4 cell count change was positively correlated (p ≤ 0.0001) with adherence to HAART. Hence, as adherence to HAART increased, CD4 cell count also increased; and those patients who had significant CD4 cell count change at each visiting time had been encouraged to be good adherents. Joint model analysis was more parsimonious as compared to separate analysis, as it reduces type I error and subject-specific analysis improved its model fit. The joint model operates multivariate analysis simultaneously; and it has great power in parameter estimation. Developing joint model helps validate the observed correlation between the outcomes that have emerged from the association of intercepts. There should be a special attention and intervention for HIV positive adults, especially for those who had poor adherence and with low CD4 cell count change. The intervention may be important for pre-treatment counseling and awareness creation. The study also identified a group of patients who were with maximum risk of CD4 cell count change. It is suggested that this group of patients needs high intervention for counseling.
27 citations
TL;DR: A joint model to analyze the structure and intensity of the association between longitudinal measurements of an ordinal marker and time to a relevant event is proposed and is applied to the assessment of breast cancer risk in women attending a population‐based screening program.
Abstract: We propose a joint model to analyze the structure and intensity of the association between longitudinal measurements of an ordinal marker and time to a relevant event. The longitudinal process is defined in terms of a proportional-odds cumulative logit model. Time-to-event is modeled through a left-truncated proportional-hazards model, which incorporates information of the longitudinal marker as well as baseline covariates. Both longitudinal and survival processes are connected by means of a common vector of random effects. General inferences are discussed under the Bayesian approach and include the posterior distribution of the probabilities associated to each longitudinal category and the assessment of the impact of the baseline covariates and the longitudinal marker on the hazard function. The flexibility provided by the joint model makes possible to dynamically estimate individual event-free probabilities and predict future longitudinal marker values. The model is applied to the assessment of breast cancer risk in women attending a population-based screening program. The longitudinal ordinal marker is mammographic breast density measured with the Breast Imaging Reporting and Data System (BI-RADS) scale in biennial screening exams. © 2016 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.
22 citations
TL;DR: This work proposes to simultaneously model the survival time with a stratified Cox proportional hazards model and the longitudinal categorical outcomes with a generalized linear mixed model to address the dependence between survival time and longitudinal outcomes due to unobserved factors.
Abstract: In biomedical or public health research, it is common for both survival time and longitudinal categorical outcomes to be collected for a subject, along with the subject’s characteristics or risk factors. Investigators are often interested in finding important variables for predicting both survival time and longitudinal outcomes which could be correlated within the same subject. Existing approaches for such joint analyses deal with continuous longitudinal outcomes. New statistical methods need to be developed for categorical longitudinal outcomes. We propose to simultaneously model the survival time with a stratified Cox proportional hazards model and the longitudinal categorical outcomes with a generalized linear mixed model. Random effects are introduced to account for the dependence between survival time and longitudinal outcomes due to unobserved factors. The Expectation–Maximization (EM) algorithm is used to derive the point estimates for the model parameters, and the observed information matrix is adopted to estimate their asymptotic variances. Asymptotic properties for our proposed maximum likelihood estimators are established using the theory of empirical processes. The method is demonstrated to perform well in finite samples via simulation studies. We illustrate our approach with data from the Carolina Head and Neck Cancer Study (CHANCE) and compare the results based on our simultaneous analysis and the separately conducted analyses using the generalized linear mixed model and the Cox proportional hazards model. Our proposed method identifies more predictors than by separate analyses.
6 citations
TL;DR: PA had the smallest AB-PE and ASE-PE for the longitudinal submodel among the three approaches for the small and moderate sample sizes and JLCM was desirable for the none-association and the large sample size.
Abstract: In recent years, the joint models have been widely used for modeling the longitudinal and time-to-event data simultaneously. In this study, we proposed an approach (PA) to study the longitudinal and survival outcomes simultaneously in heterogeneous populations. PA relaxes the assumption of conditional independence (CI). We also compared PA with joint latent class model (JLCM) and separate approach (SA) for various sample sizes (150, 300, and 600) and different association parameters (0, 0.2, and 0.5). The average bias of parameters estimation (AB-PE), average SE of parameters estimation (ASE-PE), and coverage probability of the 95% confidence interval (CP) among the three approaches were compared. In most cases, when the sample sizes increased, AB-PE and ASE-PE decreased for the three approaches, and CP got closer to the nominal level of 0.95. When there was a considerable association, PA in comparison with SA and JLCM performed better in the sense that PA had the smallest AB-PE and ASE-PE for the longitudinal submodel among the three approaches for the small and moderate sample sizes. Moreover, JLCM was desirable for the none-association and the large sample size. Finally, the evaluated approaches were applied on a real HIV/AIDS dataset for validation, and the results were compared.
5 citations
TL;DR: A large number of clinical applications and methodological development in the area of joint models of longitudinal and time-to-event outcomes have come up in the last 20 or more years.
Abstract: Over the last 20 or more years a lot of clinical applications and methodological development in the area of joint models of longitudinal and time-to-event outcomes have come up. In these studies, p...
4 citations
References
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TL;DR: This work argues that the Cox proportional hazards regression model method is superior to naive methods where one maximizes the partial likelihood of the Cox model using the observed covariate values and improves on two-stage methods where empirical Bayes estimates of the covariate process are computed and then used as time-dependent covariates.
Abstract: The relationship between a longitudinal covariate and a failure time process can be assessed using the Cox proportional hazards regression model We consider the problem of estimating the parameters in the Cox model when the longitudinal covariate is measured infrequently and with measurement error We assume a repeated measures random effects model for the covariate process Estimates of the parameters are obtained by maximizing the joint likelihood for the covariate process and the failure time process This approach uses the available information optimally because we use both the covariate and survival data simultaneously Parameters are estimated using the expectation-maximization algorithm We argue that such a method is superior to naive methods where one maximizes the partial likelihood of the Cox model using the observed covariate values It also improves on two-stage methods where, in the first stage, empirical Bayes estimates of the covariate process are computed and then used as time-dependent covariates in a second stage to find the parameters in the Cox model that maximize the partial likelihood
911 citations
"Inferences for joint modelling of r..." refers background in this paper
...Considerable literature [5,8,9,19] is available on models called selection model, where the joint density of repeated measures vector and failure time is obtained as the product of the conditional density of the failure time due to any event given the longitudinal outcomes and the marginal density of these outcomes....
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TL;DR: This paper formulates a class of models for the joint behaviour of a sequence of longitudinal measurements and an associated sequence of event times, including single-event survival data, using results from a clinical trial into the treatment of schizophrenia.
Abstract: This paper formulates a class of models for the joint behaviour of a sequence of longitudinal measurements and an associated sequence of event times, including single-event survival data. This class includes and extends a number of specific models which have been proposed recently, and, in the absence of association, reduces to separate models for the measurements and events based, respectively, on a normal linear model with correlated errors and a semi-parametric proportional hazards or intensity model with frailty. Special cases of the model class are discussed in detail and an estimation procedure which allows the two components to be linked through a latent stochastic process is described. Methods are illustrated using results from a clinical trial into the treatment of schizophrenia.
883 citations
"Inferences for joint modelling of r..." refers background in this paper
...Considerable literature [5,8,9,19] is available on models called selection model, where the joint density of repeated measures vector and failure time is obtained as the product of the conditional density of the failure time due to any event given the longitudinal outcomes and the marginal density of these outcomes....
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TL;DR: Methods for fitting a broad class of models of this type, in which both the repeated CD4-lymphocyte counts and the survival time are modelled using random effects are proposed, are proposed and applied to results of AIDS clinical trials.
Abstract: The purpose of this article is to model the progression of CD4-lymphocyte count and the relationship between different features of this progression and survival time. The complicating factors in this analysis are that the CD4-lymphocyte count is observed only at certain fixed times and with a high degree of measurement error, and that the length of the vector of observations is determined, in part, by the length of survival. If probability of death depends on the true, unobserved CD4-lymphocyte count, then the survival process must be modelled. Wu and Carroll (1988, Biometrics 44, 175-188) proposed a random effects model for two-sample longitudinal data in the presence of informative censoring, in which the individual effects included only slopes and intercepts. We propose methods for fitting a broad class of models of this type, in which both the repeated CD4-lymphocyte counts and the survival time are modelled using random effects. These methods permit us to estimate parameters describing the progression of CD4-lymphocyte count as well as the effect of differences in the CD4 trajectory on survival. We apply these methods to results of AIDS clinical trials.
400 citations
"Inferences for joint modelling of r..." refers background in this paper
...Considerable literature [5,8,9,19] is available on models called selection model, where the joint density of repeated measures vector and failure time is obtained as the product of the conditional density of the failure time due to any event given the longitudinal outcomes and the marginal density of these outcomes....
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TL;DR: In this paper, an approximation to Laplace's method for integrals is applied to marginal distributions of data arising from models in which both fixed and random effects enter nonlinearly.
Abstract: SUMMARY An approximation to Laplace's method for integrals is applied to marginal distributions of data arising from models in which both fixed and random effects enter nonlinearly. The approach provides alternative derivations of some recent algorithms for fitting such models, and it has direct ties with Gaussian restricted maximum likelihood and the accompanying mixed model equations.
358 citations
"Inferences for joint modelling of r..." refers background in this paper
...There has been several studies on nonlinear mixed effect (NLME) models [15] where the exact likelihood is approximated on the basis of Taylor expansions or Laplace approximations [3,17]....
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TL;DR: It is illustrated that the LMVUB and LMMSE estimators, derived under a simple linear regression model, are quite competitive compared to the pseudo maximum likelihood estimator (PMLE) derived by modeling the censoring probabilities.
Abstract: A general linear regression model for the usual least squares estimated rate of change (slope) on censoring time is described as an approximation to account for informative right censoring in estimating and comparing changes of a continuous variable in two groups. Two noniterative estimators for the group slope means, the linear minimum variance unbiased (LMVUB) estimator and the linear minimum mean squared error (LMMSE) estimator, are proposed under this conditional model. In realistic situations, we illustrate that the LMVUB and LMMSE estimators, derived under a simple linear regression model, are quite competitive compared to the pseudo maximum likelihood estimator (PMLE) derived by modeling the censoring probabilities. Generalizations to polynomial response curves and general linear models are also described.
340 citations
"Inferences for joint modelling of r..." refers background in this paper
...The truncation of a patient’s response can raise similar problems to those that have been considered in the event of patient dropout [12,14,18]....
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