Bayesian generalized varying coefficient models for longitudinal proportional data with errors-in-covariates

This paper is motivated from a neurophysiological study of muscle fatigue, in which biomedical researchers are interested in understanding the time-dependent relationships of handgrip force and electromyography measures. A varying coefficient model is appealing here to investigate the dynamic pattern in the longitudinal data. The response variable in the study is continuous but bounded on the standard unit interval (0, 1) over time, while the longitudinal covariates are contaminated with measurement errors. We propose a generalization of varying coefficient models for the longitudinal proportional data with errors-in-covariates. We describe two estimation methods with penalized splines, which are formalized under a Bayesian inferential perspective. The first method is an adaptation of the popular regression calibration approach. The second method is based on a joint likelihood under the hierarchical Bayesian model. A simulation study is conducted to evaluate the efficacy of the proposed methods under different scenarios. The analysis of the neurophysiological data is presented to demonstrate the use of the methods.     

Unified Inference for Sparse and Dense Longitudinal Data in Time‐varying Coefficient Models

Time‐varying coefficient models are widely used in longitudinal data analysis. These models allow the effects of predictors on response to vary over time. In this article, we consider a mixed‐effects time‐varying coefficient model to account for the within subject correlation for longitudinal data. We show that when kernel smoothing is used to estimate the smooth functions in time‐varying coefficient models for sparse or dense longitudinal data, the asymptotic results of these two situations are essentially different. Therefore, a subjective choice between the sparse and dense cases might lead to erroneous conclusions for statistical inference. In order to solve this problem, we establish a unified self‐normalized central limit theorem, based on which a unified inference is proposed without deciding whether the data are sparse or dense. The effectiveness of the proposed unified inference is demonstrated through a simulation study and an analysis of Baltimore MACS data.     

Penalized spline joint models for longitudinal and time-to-event data

The joint models for longitudinal data and time-to-event data have recently received numerous attention in clinical and epidemiologic studies. Our interest is in modeling the relationship between event time outcomes and internal time-dependent covariates. In practice, the longitudinal responses often show non linear and fluctuated curves. Therefore, the main aim of this paper is to use penalized splines with a truncated polynomial basis to parameterize the non linear longitudinal process. Then, the linear mixed-effects model is applied to subject-specific curves and to control the smoothing. The association between the dropout process and longitudinal outcomes is modeled through a proportional hazard model. Two types of baseline risk functions are considered, namely a Gompertz distribution and a piecewise constant model. The resulting models are referred to as penalized spline joint models; an extension of the standard joint models. The expectation conditional maximization (ECM) algorithm is applied to estimate the parameters in the proposed models. To validate the proposed algorithm, extensive simulation studies were implemented followed by a case study. In summary, the penalized spline joint models provide a new approach for joint models that have improved the existing standard joint models.     

Two-step and likelihood methods for joint models of longitudinal and survival data

We compare the commonly used two-step methods and joint likelihood method for joint models of longitudinal and survival data via extensive simulations. The longitudinal models include LME, GLMM, and NLME models, and the survival models include Cox models and AFT models. We find that the full likelihood method outperforms the two-step methods for various joint models, but it can be computationally challenging when the dimension of the random effects in the longitudinal model is not small. We thus propose an approximate joint likelihood method which is computationally efficient. We find that the proposed approximation method performs well in the joint model context, and it performs better for more “continuous” longitudinal data. Finally, a real AIDS data example shows that patients with higher initial viral load or lower initial CD4 are more likely to drop out earlier during an anti-HIV treatment.     

Estimation and inference of the joint conditional distribution for multivariate longitudinal data using nonparametric copulas

In this paper we study estimating the joint conditional distributions of multivariate longitudinal outcomes using regression models and copulas. For the estimation of marginal models, we consider a class of time-varying transformation models and combine the two marginal models using nonparametric empirical copulas. Our models and estimation method can be applied in many situations where the conditional mean-based models are not good enough. Empirical copulas combined with time-varying transformation models may allow quite flexible modelling for the joint conditional distributions for multivariate longitudinal data. We derive the asymptotic properties for the copula-based estimators of the joint conditional distribution functions. For illustration we apply our estimation method to an epidemiological study of childhood growth and blood pressure.     

Joint analysis of nonlinear heterogeneous longitudinal data and binary outcome: an application to AIDS clinical studies

Finite mixture models are currently used to analyze heterogeneous longitudinal data. By releasing the homogeneity restriction of nonlinear mixed-effects (NLME) models, finite mixture models not only can estimate model parameters but also cluster individuals into one of the pre-specified classes with class membership probabilities. This clustering may have clinical significance, which might be associated with a clinically important binary outcome. This article develops a joint modeling of a finite mixture of NLME models for longitudinal data in the presence of covariate measurement errors and a logistic regression for a binary outcome, linked by individual latent class indicators, under a Bayesian framework. Simulation studies are conducted to assess the performance of the proposed joint model and a naive two-step model, in which finite mixture model and logistic regression are fitted separately, followed by an application to a real data set from an AIDS clinical trial, in which the viral dynamics and dichotomized time to the first decline of CD4/CD8 ratio are analyzed jointly.     

COVARIATE-ADJUSTED REGRESSION FOR LONGITUDINAL DATA INCORPORATING CORRELATION BETWEEN REPEATED MEASUREMENTS

We propose an estimation method that incorporates the correlation/covariance structure between repeated measurements in covariate-adjusted regression models for distorted longitudinal data. In this distorted data setting, neither the longitudinal response nor (possibly time-varying) predictors are directly observable. The unobserved response and predictors are assumed to be distorted/contaminated by unknown functions of a common observable confounder. The proposed estimation methodology adjusts for the distortion effects both in estimation of the covariance structure and in the regression parameters using generalized least squares. The finite-sample performance of the proposed estimators is studied numerically by means of simulations. The consistency and convergence rates of the proposed estimators are also established. The proposed method is illustrated with an application to data from a longitudinal study of cognitive and social development in children.     

A modified two-stage approach for joint modelling of longitudinal and time-to-event data

Joint models for longitudinal and time-to-event data have been applied in many different fields of statistics and clinical studies. However, the main difficulty these models have to face with is the computational problem. The requirement for numerical integration becomes severe when the dimension of random effects increases. In this paper, a modified two-stage approach has been proposed to estimate the parameters in joint models. In particular, in the first stage, the linear mixed-effects models and best linear unbiased predictorsare applied to estimate parameters in the longitudinal submodel. In the second stage, an approximation of the fully joint log-likelihood is proposed using the estimated the values of these parameters from the longitudinal submodel. Survival parameters are estimated bymaximizing the approximation of the fully joint log-likelihood. Simulation studies show that the approach performs well, especially when the dimension of random effects increases. Finally, we implement this approach on AIDS data.     

Bivariate modelling of longitudinal measurements of two human immunodeficiency type 1 disease progression markers in the presence of informative drop-outs

Summary.  The main statistical problem in many epidemiological studies which involve repeated measurements of surrogate markers is the frequent occurrence of missing data. Standard likelihood-based approaches like the linear random-effects model fail to give unbiased estimates when data are non-ignorably missing. In human immunodeficiency virus (HIV) type 1 infection, two markers which have been widely used to track progression of the disease are CD4 cell counts and HIV–ribonucleic acid (RNA) viral load levels. Repeated measurements of these markers tend to be informatively censored, which is a special case of non-ignorable missingness. In such cases, we need to apply methods that jointly model the observed data and the missingness process. Despite their high correlation, longitudinal data of these markers have been analysed independently by using mainly random-effects models. Touloumi and co-workers have proposed a model termed the joint multivariate random-effects model which combines a linear random-effects model for the underlying pattern of the marker with a log-normal survival model for the drop-out process. We extend the joint multivariate random-effects model to model simultaneously the CD4 cell and viral load data while adjusting for informative drop-outs due to disease progression or death. Estimates of all the model's parameters are obtained by using the restricted iterative generalized least squares method or a modified version of it using the EM algorithm as a nested algorithm in the case of censored survival data taking also into account non-linearity in the HIV–RNA trend. The method proposed is evaluated and compared with simpler approaches in a simulation study. Finally the method is applied to a subset of the data from the 'Concerted action on seroconversion to AIDS and death in Europe' study.     

Testing for constancy in varying coefficient models

We consider varying coefficient models, which are an extension of the classical linear regression models in the sense that the regression coefficients are replaced by functions in certain variables (for example, time), the covariates are also allowed to depend on other variables. Varying coefficient models are popular in longitudinal data and panel data studies, and have been applied in fields such as finance and health sciences. We consider longitudinal data and estimate the coefficient functions by the flexible B-spline technique. An important question in a varying coefficient model is whether an estimated coefficient function is statistically different from a constant (or zero). We develop testing procedures based on the estimated B-spline coefficients by making use of nice properties of a B-spline basis. Our method allows longitudinal data where repeated measurements for an individual can be correlated. We obtain the asymptotic null distribution of the test statistic. The power of the proposed testing procedures are illustrated on simulated data where we highlight the importance of including the correlation structure of the response variable and on real data.     

General multivariate linear models for longitudional studies

Longitudinal studies occcur frequently in many different disciplines. To fully utilize the potential value of the information contained in a longitudinal data, various multivariate linear models have been proposed. The methodology and analysis are somewhat unique in their own ways and their relationships are not well understood and presented. This article describes a general multivaritate linear model for longitudinal data and attempts to provide a constructive formulation of the components in the mean response profile. The objective is to point out the extension and connections of some well-known models that have been obscured by different areas of application. More imporiantly, the model is expressed in a unified regression form from the subject matter considerations. Such an approach is simpler and more intuitive than other ways to modeling and parameter estimation. As a cmsequeace the analyses of the general class cf models for longitudional data can be casily implemented with standard software.     

Joint Modelling of Survival and Longitudinal Data with Informative Observation Times

In this paper, we consider the joint modelling of survival and longitudinal data with informative observation time points. The survival model and the longitudinal model are linked via random effects, for which no distribution assumption is required under our estimation approach. The estimator is shown to be consistent and asymptotically normal. The proposed estimator and its estimated covariance matrix can be easily calculated. Simulation studies and an application to a primary biliary cirrhosis study are also provided.     

Comparing crossing hazard rate functions by joint modelling of survival and longitudinal data

Abstract

It is one of the important issues in survival analysis to compare two hazard rate functions to evaluate treatment effect. It is quite common that the two hazard rate functions cross each other at one or more unknown time points, representing temporal changes of the treatment effect. In certain applications, besides survival data, we also have related longitudinal data available regarding some time-dependent covariates. In such cases, a joint model that accommodates both types of data can allow us to infer the association between the survival and longitudinal data and to assess the treatment effect better. In this paper, we propose a modelling approach for comparing two crossing hazard rate functions by joint modelling survival and longitudinal data. Maximum likelihood estimation is used in estimating the parameters of the proposed joint model using the EM algorithm. Asymptotic properties of the maximum likelihood estimators are studied. To illustrate the virtues of the proposed method, we compare the performance of the proposed method with several existing methods in a simulation study. Our proposed method is also demonstrated using a real dataset obtained from an HIV clinical trial.     

Weighted empirical likelihood for generalized linear models with longitudinal data

In this paper, we introduce the empirical likelihood (EL) method to longitudinal studies. By considering the dependence within subjects in the auxiliary random vectors, we propose a new weighted empirical likelihood (WEL) inference for generalized linear models with longitudinal data. We show that the weighted empirical likelihood ratio always follows an asymptotically standard chi-squared distribution no matter which working weight matrix that we have chosen, but a well chosen working weight matrix can improve the efficiency of statistical inference. Simulations are conducted to demonstrate the accuracy and efficiency of our proposed WEL method, and a real data set is used to illustrate the proposed method.     

Functional approach of flexibly modelling generalized longitudinal data and survival time

We propose a flexible functional approach for modelling generalized longitudinal data and survival time using principal components. In the proposed model the longitudinal observations can be continuous or categorical data, such as Gaussian, binomial or Poisson outcomes. We generalize the traditional joint models that treat categorical data as continuous data by using some transformations, such as CD4 counts. The proposed model is data-adaptive, which does not require pre-specified functional forms for longitudinal trajectories and automatically detects characteristic patterns. The longitudinal trajectories observed with measurement error or random error are represented by flexible basis functions through a possibly nonlinear link function, combining dimension reduction techniques resulting from functional principal component (FPC) analysis. The relationship between the longitudinal process and event history is assessed using a Cox regression model. Although the proposed model inherits the flexibility of non-parametric methods, the estimation procedure based on the EM algorithm is still parametric in computation, and thus simple and easy to implement. The computation is simplified by dimension reduction for random coefficients or FPC scores. An iterative selection procedure based on Akaike information criterion (AIC) is proposed to choose the tuning parameters, such as the knots of spline basis and the number of FPCs, so that appropriate degree of smoothness and fluctuation can be addressed. The effectiveness of the proposed approach is illustrated through a simulation study, followed by an application to longitudinal CD4 counts and survival data which were collected in a recent clinical trial to compare the efficiency and safety of two antiretroviral drugs.     

Two-step estimation of functional linear models with applications to longitudinal data

Functional linear models are useful in longitudinal data analysis. They include many classical and recently proposed statistical models for longitudinal data and other functional data. Recently, smoothing spline and kernel methods have been proposed for estimating their coefficient functions nonparametrically but these methods are either intensive in computation or inefficient in performance. To overcome these drawbacks, in this paper, a simple and powerful two-step alternative is proposed. In particular, the implementation of the proposed approach via local polynomial smoothing is discussed. Methods for estimating standard deviations of estimated coefficient functions are also proposed. Some asymptotic results for the local polynomial estimators are established. Two longitudinal data sets, one of which involves time-dependent covariates, are used to demonstrate the approach proposed. Simulation studies show that our two-step approach improves the kernel method proposed by Hoover and co-workers in several aspects such as accuracy, computational time and visual appeal of the estimators.     

A Quantile Survival Model for Censored Data

In this paper we propose a quantile survival model to analyze censored data. This approach provides a very effective way to construct a proper model for the survival time conditional on some covariates. Once a quantile survival model for the censored data is established, the survival density, survival or hazard functions of the survival time can be obtained easily. For illustration purposes, we focus on a model that is based on the generalized lambda distribution (GLD). The GLD and many other quantile function models are defined only through their quantile functions, no closed‐form expressions are available for other equivalent functions. We also develop a Bayesian Markov Chain Monte Carlo (MCMC) method for parameter estimation. Extensive simulation studies have been conducted. Both simulation study and application results show that the proposed quantile survival models can be very useful in practice.     

Evaluating fit in functional data analysis using model embeddings

The author proposes a general method for evaluating the fit of a model for functional data. His approach consists of embedding the proposed model into a larger family of models, assuming the true process generating the data is within the larger family, and then computing a posterior distribution for the Kullback‐Leibler distance between the true and the proposed models. The technique is illustrated on biomechanical data reported by Ramsay, Flanagan & Wang (1995). It is developed in detail for hierarchical polynomial models such as those found in Lindley & Smith (1972), and is also generally applicable to longitudinal data analysis where polynomials are fit to many individuals.     

Doubly robust estimation of partially linear models for longitudinal data with dropouts and measurement error in covariates

In longitudinal studies, missing responses and mismeasured covariates are commonly seen due to the data collection process. Without cautiousness in data analysis, inferences from the standard statistical approaches may lead to wrong conclusions. In order to improve the estimation for longitudinal data analysis, a doubly robust estimation method for partially linear models, which can simultaneously account for the missing responses and mismeasured covariates, is proposed. Imprecisions of covariates are corrected by taking advantage of the independence between replicate measurement errors, and missing responses are handled by the doubly robust estimation under the mechanism of missing at random. The asymptotic properties of the proposed estimators are established under regularity conditions, and simulation studies demonstrate desired properties. Finally, the proposed method is applied to data from the Lifestyle Education for Activity and Nutrition study.     

Nonparametric Modeling Auxiliary Covariates in Random Coefficient Models

Random coefficient model (RCM) is a powerful statistical tool in analyzing correlated data collected from studies with different clusters or from longitudinal studies. In practice, there is a need for statistical methods that allow biomedical researchers to adjust for the measured and unmeasured covariates that might affect the regression model. This article studies two nonparametric methods dealing with auxiliary covariate data in linear random coefficient models. We demonstrate how to estimate the coefficients of the models and how to predict the random effects when the covariates are missing or mismeasured. We employ empirical estimator and kernel smoother to handle a discrete and continuous auxiliary, respectively. Simulation results show that the proposed methods perform better than an alternative method that only uses data in the validation data set and ignores the random effects in the random coefficient model.