Joint Mean-covariance Models Research Articles

Alternative modified Cholesky decomposition of the precision matrix of longitudinal data

The correlation matrix might be of scientific interest for longitudinal data. However, few studies have focused on both robust estimation of the correlation matrix against model misspecification and robustness to outliers in the data, when the precision matrix possesses a typical structure. In this paper, we propose an alternative modified Cholesky decomposition (AMCD) for the precision matrix of longitudinal data, which results in robust estimation of the correlation matrix against model misspecification of the innovation variances. A joint mean-covariance model with multivariate normal distribution and AMCD is established, the quasi-Fisher scoring algorithm is developed, and the maximum likelihood estimators are proven to be consistent and asymptotically normally distributed. Furthermore, a double-robust joint modeling approach with multivariate Laplace distribution and AMCD is established, and the quasi-Newton algorithm for maximum likelihood estimation is developed. The simulation studies and real data analysis demonstrate the effectiveness of the proposed AMCD method.

Bayesian estimation for longitudinal data in a joint model with HPCs

In longitudinal data analysis, linear models are typically utilized. However, deriving the Bayesian estimation with respect to the misspecification of the correlation structure is a challenging task. In this article, we construct a joint mean–covariance model with angles or hyperspherical coordinates (HPCs) for which we then present a Bayesian framework. Based on the connection with the semipartial correlations (SPCs), we focus on the selection (sparsity) priors on these angles. An efficient Markov chain Monte Carlo (MCMC) algorithm is developed for the proposed model, and the positive definiteness of the correlation matrix in posterior computation is automatically guaranteed by our method. Ultimately, we compare the performance of our joint model with some recent methods focusing only on the correlation matrix by using simulations and clinical trial data on smoking.

Robust estimation in multivariate heteroscedastic regression models with autoregressive covariance structures using EM algorithm

In the analysis of repeated or clustered measurements, it is crucial to determine the dynamics that affect the mean, variance, and correlations of the data, which will be possible using appropriate models. One of these models is the joint mean–covariance model, which is a multivariate heteroscedastic regression model with autoregressive covariance structures. In these models, parameter estimation is usually carried on under normality assumption, but the resulting estimators will be very sensitive to the outliers or non-normality of data. In this study, we propose a robust alternative method and an EM algorithm for estimating the parameters of joint mean–covariance models. Robustification is achieved using a multivariate heavy-tailed distribution with the same number of parameters as the multivariate normal distribution. To simplify the estimation procedure, a modified Cholesky decomposition is adopted to factorize the dependence structure in terms of unconstrained autoregressive and scale innovation parameters. Also, the technique for the prediction of future responses is given. An intensive simulation study and a real data example are provided to demonstrate the performance of the proposed method.

Mixture regression for longitudinal data based on joint mean–covariance model

In the process of modeling longitudinal data, we focus on the case that the studied population is comprised of different groups of individuals and individuals within the same group share the similar kind of mean progression trajectories, where finite mixture models (FMM) are often used to address this kind of unobserved heterogeneity in terms of mean. Existing methods, such as parametric and semiparametric mixture regression, usually model the mean in each subpopulation with assumption that observations sharing a common trajectory are independent or their covariance structure is pre-specified, but less research considers modeling of covariance structures while accounting for heterogeneity. In this paper, we introduce a joint model which models the mean and covariance structures simultaneously in a finite normal mixture regression, demonstrating how important the within-subject correlation is in clustering longitudinal data. Model parameters are estimated with an iteratively re-weighted least squares EM (IRLS-EM) algorithm. Our estimators are shown to be consistent and asymptotically normal. We can identify different mean trajectories and covariance structures in all clusters. Simulations show that the proposed method performs well and gives more accurate clustering results by introducing covariance modeling. Real data analysis is also used to illustrate the usefulness of the proposed method, and it presents good performance in clustering COVID-19 deaths for European countries in terms of progression trajectory.

Jmcm: a Python package for analyzing longitudinal data using joint mean-covariance models

The joint mean-covariance models are widely used in longitudinal studies. To draw the mean and covariance patterns of data, joint modeling based on the normality assumptions with the modified Cholesky decompositions (MCD) has been proposed. The method keeps the positive-definiteness and also has advantages in interpreting the statistical roles of the parameters. We proposed a Python package named jmcm to fit the mean-covariance models. It contains the methods for estimating the parameters based on the profile likelihood estimation, plotting the curves with confidence bands based on bootstrap method, doing hypothesis tests based on the Wald tests, etc. This article would review the methods and demonstrate the use of the package with detailed examples. The compatibility of the package would also be included based on simulation study.

Optimal model for weaning-weight of Bunaji Bulls at NAPRI Farm, Shika, Nigeria

Empirical models have over the years been commonly established by animal research centers for the study of weight-age profiles in order to understand the metabolic processes of growth. They provide efficient parameter estimates for mature weight and rate of maturing, but were found to consistently either over-or-under estimate the mature weight. estimate the mature weight. They also perform poorly in predicting weight in early life or beyond the range of input data. At the National Animal Production Research Institute (NAPRI) farm, Shika, Brody was established as the model that provides efficient parameter estimates of weight-age profiles for Bunaji bulls. However, a major drawback of the model is its consistent underestimation of weight prior to six months of age, leading to poor prediction of weaning weight. To address this shortcoming, we propose in this article a joint mean-covariance model that provide optimal parameter estimates for the weaning weight of Bunaji bulls

Robust estimation for the correlation matrix of multivariate longitudinal data

Modelling the covariance structure of multivariate longitudinal data is more challenging than its univariate counterpart, owing to the complex correlated structure among multiple responses. Furthermore, there are little methods focusing on the robustness of estimating the corresponding correlation matrix. In this paper, we propose an alternative Cholesky block decomposition (ACBD) for the covariance matrix of multivariate longitudinal data. The new unconstrained parameterization is capable to automatically eliminate the positive definiteness constraint of the covariance matrix and robustly estimate the correlation matrix with respect to the model misspecifications of the nested prediction error covariance matrices. The entries of the new decomposition are modelled by regression models, and the maximum likelihood estimators of the regression parameters in joint mean–covariance models are computed by a quasi-Fisher iterative algorithm. The resulting estimators are shown to be consistent and asymptotically normal. Simulations and real data analysis illustrate that the new method performs well.

GEE analysis in joint mean-covariance model for longitudinal data

In this paper, we propose generalized estimating equations for the regression parameters in joint mean-covariance model for longitudinal data, motivated by the alternative Cholesky decomposition. This decomposition causes robust estimation of the correlation matrix against model misspecification for innovation variances.

Subject-wise empirical likelihood inference for robust joint mean-covariance model with longitudinal data

Subject-wise empirical likelihood inference for robust joint mean-covariance model with longitudinal data

Joint mean–covariance model in generalized partially linear varying coefficient models for longitudinal data

In longitudinal data analysis, efficient estimation of regression coefficients requires a correct specification of certain covariance structure, and efficient estimation of covariance matrix requires a correct specification of mean regression model. In this article, we propose a general semiparametric model for the mean and the covariance simultaneously using the modified Cholesky decomposition. A regression spline-based approach within the framework of generalized estimating equations is proposed to estimate the parameters in the mean and the covariance. Under regularity conditions, asymptotic properties of the resulting estimators are established. Extensive simulation is conducted to investigate the performance of the proposed estimator and in the end a real data set is analysed using the proposed approach.

Modeling of mean-covariance structures in generalized estimating equations with dropouts

Within the framework of joint mean-covariance models, we study the effects of dropout missing at random (MAR) on the estimation of mean and covariance structures for longitudinal data using generalized estimating equations (GEE).It is evidential that the MAR dropout has more severe influences on the estimation of variance-covariances relative to the mean estimation, as the former involves the estimation of the second moments. We propose to use the inverse probabilityweighted generalized estimating equation (WGEE) method to model the mean and covariance structures, simultaneously, in order to accommodate the effects of MAR dropout. The proposed WGEE approach produces unbiased estimators of parameters in both the mean and covariances for longitudinal data with MAR dropout. Simulation studies are conducted to assess the performance of the proposed approach and a real data analysis for the PANSS data is made to illustrate the effectiveness of the proposed method.

Restricted maximum likelihood estimation of joint mean‐covariance models

AbstractThe class of joint mean‐covariance models uses the modified Cholesky decomposition of the within subject covariance matrix in order to arrive to an unconstrained, statistically meaningful reparameterisation. The new parameterisation of the covariance matrix has two sets of parameters that separately describe the variances and correlations. Thus, with the mean or regression parameters, these models have three sets of distinct parameters. In order to alleviate the problem of inefficient estimation and downward bias in the variance estimates, inherent in the maximum likelihood estimation procedure, the usual REML estimation procedure adjusts for the degrees of freedom lost due to the estimation of the mean parameters. Because of the parameterisation of the joint mean covariance models, it is possible to adapt the usual REML procedure in order to estimate the variance (correlation) parameters by taking into account the degrees of freedom lost by the estimation of both the mean and correlation (variance) parameters. To this end, here we propose adjustments to the estimation procedures based on the modified and adjusted profile likelihoods. The methods are illustrated by an application to a real data set and simulation studies.The Canadian Journal of Statistics40: 225–242; 2012 © 2012 Statistical Society of Canada

Regressograms and Mean-Covariance Models for Incomplete Longitudinal Data

Longitudinal studies are prevalent in biological and social sciences where subjects are measured repeatedly over time. Modeling the correlations and handling missing data are among the most challenging problems in analyzing such data. There are various methods for handling missing data, but data-based and graphical methods for modeling the covariance matrix of longitudinal data are relatively new. We adopt an approach based on the modified Cholesky decomposition of the covariance matrix which handles both challenges. It amounts to formulating parametric models for the regression coefficients of the conditional mean and variance of each measurement given its predecessors. We demonstrate the roles of profile plots and regressograms in formulating joint mean-covariance models for incomplete longitudinal data. Applying these graphical tools to the Fruit Fly Mortality (FFM) data, which has 22% missing values, reveals a logistic curve for the mean function and two different models for the two factors of the modified Cholesky decomposition of the sample covariance matrix. An expectation-maximization algorithm is proposed for estimating the parameters of the mean-covariance models; it performs well for the FFM data and in a simulation study of incomplete longitudinal data.

Joint Mean-Covariance Models with Applications to Longitudinal Data in Partially Linear Model

Semiparametric regression models and estimating covariance functions are very useful in longitudinal study. Unfortunately, challenges arise in estimating the covariance function of longitudinal data collected at irregular time points. In this article, for mean term, a partially linear model is introduced and for covariance structure, a modified Cholesky decomposition approach is proposed to heed the positive-definiteness constraint. We estimate the regression function by using the local linear technique and propose quasi-likelihood estimating equations for both the mean and covariance structures. Moreover, asymptotic normality of the resulting estimators is established. Finally, simulation study and real data analysis are used to illustrate the proposed approach.

Joint estimation of mean-covariance model for longitudinal data with basis function approximations

When the selected parametric model for the covariance structure is far from the true one, the corresponding covariance estimator could have considerable bias. To balance the variability and bias of the covariance estimator, we employ a nonparametric method. In addition, as different mean structures may lead to different estimators of the covariance matrix, we choose a semiparametric model for the mean so as to provide a stable estimate of the covariance matrix. Based on the modified Cholesky decomposition of the covariance matrix, we construct the joint mean-covariance model by modeling the smooth functions using the spline method and estimate the associated parameters using the maximum likelihood approach. A simulation study and a real data analysis are conducted to illustrate the proposed approach and demonstrate the flexibility of the suggested model.

Bayesian modelling of the mean and covariance matrix in normal nonlinear models

An important problem in statistics is the study of longitudinal data taking into account the effect of other explanatory variables such as treatments and time. In this paper, a new Bayesian approach for analysing longitudinal data is proposed. This innovative approach takes into account the possibility of having nonlinear regression structures on the mean and linear regression structures on the variance–covariance matrix of normal observations, and it is based on the modelling strategy suggested by Pourahmadi [M. Pourahmadi, Joint mean-covariance models with applications to longitudinal data: Unconstrained parameterizations, Biometrika, 87 (1999), pp. 667–690.]. We initially extend the classical methodology to accommodate the fitting of nonlinear mean models then we propose our Bayesian approach based on a generalization of the Metropolis–Hastings algorithm of Cepeda [E.C. Cepeda, Variability modeling in generalized linear models, Unpublished Ph.D. Thesis, Mathematics Institute, Universidade Federal do Rio de Janeiro, 2001]. Finally, we illustrate the proposed methodology by analysing one example, the cattle data set, that is used to study cattle growth.

Maximum likelihood estimation for joint mean–covariance models from unbalanced repeated-measures data

This paper develops maximum likelihood estimates for jointly modelling the mean and covariance matrix, for unbalanced repeated measures, using an unconstrained parametrization. Furthermore, the asymptotic distribution of the estimated parameters and the results of a simulation study are presented.

Gaussian estimation and joint modeling of dispersions and correlations in longitudinal data

Analysis of longitudinal, spatial and epidemiological data often requires modelling dispersions and dependence among the measurements. Moreover, data involving counts or proportions usually exhibit greater variation than would be predicted by the Poisson and binomial models. We propose a strategy for the joint modelling of mean, dispersion and correlation matrix of nonnormal multivariate correlated data. The parameter estimation for dispersions and correlations is based on the Whittle’s [P. Whittle, Gaussian estimation in stationary time series, Bull Inst. Statist. Inst. 39 (1962) 105-129.] Gaussian likelihood of the partially standardized data which eliminates the mean parameters. The model formulation for the dispersions and correlations relies on a recent unconstrained parameterization of covariance matrices and a graphical method [M. Pourahmadi, Joint mean–covariance models with applications to longitudinal data: unconstrained parameterization, Biometrika 86 (1999) 677–690] similar to the correlogram in time series analysis. We show that the estimating equations for the regression and dependence parameters derived from a modified Gaussian likelihood (involving two distinct covariance matrices) are broad enough to include generalized estimating equations and its many recent extensions and improvements. The results are illustrated using two datasets.

Covariance matrix estimation using repeated measurements when data are incomplete

Modeling repeated measurements data has been studied extensively lately in the parametric situation. However, it is significant to study the effect of various treatments over a period of time where the repeated measurements on the same subject are expected to be correlated. The correlation among the repeated measurements for all the subjects will be studied via the covariance matrix. The positive definite constraint is one of obstacles that encounters modeling the covariance structure, however the Cholesky decomposition removes this constraint and allows modeling the components of the covariance matrix [M. Pourahmadi, Joint mean-covariance models with applications to longitudinal data: unconstrained parameterization, Biometrika 86 (1999) 677–690]. In this paper, we adopt the estimation procedures introduced by Diggle and Verbyla [Nonparametric estimation of covariance structure in longitudinal data, Biometrics 54 (1998) 401–415] where the variogram cloud as well as the squared residuals are used to estimate the variogram and the variances via the kernel smoothing. Selecting the appropriate bandwidth value is one of the important steps in the estimation process, thus in our data analysis we choose the bandwidth using one of the most simple straight forward methods which is the cross-validation method developed by Rice and Silverman [Estimating the mean and covariance structure nonparametrically when the data are curves, J. Roy. Statist. Soc. B 53 (1991) 233–243] and adapted by others. Finally we apply these nonparametric techniques as well as a graphical method [M. Pourahmadi, Joint mean-covariance models with applications to longitudinal data: unconstrained parameterization, Biometrika 86 (1999) 677–690] to a real life data and use the penalized likelihood criterion like AIC and BIC to compare models of our interest.

Joint mean-covariance models with applications to longitudinal data: unconstrained parameterisation

We provide unconstrained parameterisation for and model a covariance using covariates. The Cholesky decomposition of the inverse of a covariance matrix is used to associate a unique unit lower triangular and a unique diagonal matrix with each covariance matrix. The entries of the lower triangular and the log of the diagonal matrix are unconstrained and have meaning as regression coefficients and prediction variances when regressing a measurement on its predecessors. An extended generalised linear model is introduced for joint modelling of the vectors of predictors for the mean and covariance subsuming the joint modelling strategy for mean and variance heterogeneity, Gabriel's antedependence models, Dempster's covariance selection models and the class of graphical models. The likelihood function and maximum likelihood estimators of the covariance and the mean parameters are studied when the observations are normally distributed. Applications to modelling nonstationary dependence structures and multivariate data are discussed and illustrated using real data. A graphical method, similar to that based on the correlogram in time series, is developed and used to identify parametric models for nonstationary covariances.