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

A moving average Cholesky factor model in covariance modeling for composite quantile regression with longitudinal data

Mixture of linear regression models provide a popular treatment for modeling nonlinear regression relationship. The traditional estimation of mixture of regression models is based on Gaussian error assumption. It is well known that such assumption is sensitive to outliers and extreme values. To overcome this issue, a new class of finite mixture of quantile regressions (FMQR) is proposed in this article. Compared with the existing Gaussian mixture regression models, the proposed FMQR model can provide a complete specification on the conditional distribution of response variable for each component. From the likelihood point of view, the FMQR model is equivalent to the finite mixture of regression models based on errors following asymmetric Laplace distribution (ALD), which can be regarded as an extension to the traditional mixture of regression models with normal error terms. An EM algorithm is proposed to obtain the parameter estimates of the FMQR model by combining a hierarchical representation of the ALD. Finally, the iterated weighted least square estimation for each mixture component of the FMQR model is derived. Simulation studies are conducted to illustrate the finite sample performance of the estimation procedure. Analysis of an aphid data set is used to illustrate our methodologies.

WINNER OF THE 2007 DEGROOT PRIZE! The prominence of finite mixture modelling is greater than ever. Many important statistical topics like clustering data, outlier treatment, or dealing with unobserved heterogeneity involve finite mixture models in some way or other. The area of potential applications goes beyond simple data analysis and extends to regression analysis and to non-linear time series analysis using Markov switching models. For more than the hundred years since Karl Pearson showed in 1894 how to estimate the five parameters of a mixture of two normal distributions using the method of moments, statistical inference for finite mixture models has been a challenge to everybody who deals with them. In the past ten years, very powerful computational tools emerged for dealing with these models which combine a Bayesian approach with recent Monte simulation techniques based on Markov chains. This book reviews these techniques and covers the most recent advances in the field, among them bridge sampling techniques and reversible jump Markov chain Monte Carlo methods. It is the first time that the Bayesian perspective of finite mixture modelling is systematically presented in book form. It is argued that the Bayesian approach provides much insight in this context and is easily implemented in practice. Although the main focus is on Bayesian inference, the author reviews several frequentist techniques, especially selecting the number of components of a finite mixture model, and discusses some of their shortcomings compared to the Bayesian approach. The aim of this book is to impart the finite mixture and Markov switching approach to statistical modelling to a wide-ranging community. This includes not only statisticians, but also biologists, economists, engineers, financial agents, market researcher, medical researchers or any other frequent user of statistical models. This book should help newcomers to the field to understand how finite mixture and Markov switching models are formulated, what structures they imply on the data, what they could be used for, and how they are estimated. Researchers familiar with the subject also will profit from reading this book. The presentation is rather informal without abandoning mathematical correctness. Previous notions of Bayesian inference and Monte Carlo simulation are useful but not needed.

Attribute-Level Heterogeneity

The finite mixture model is considered as an appropriate instrument for data clustering.Different parsimonious multivariate mixture distributions are introduced for skewed and/or heavy-tailed longitudinal data.The eigenvalue or modified Cholesky decomposition of covariance matrices develops the families of parsimonious mixture models.Thus, the finite mixture of matrix-variate t-distributions for clustering a three-way dataset with heavy-tailed or outlier observations (e.g., multivariate longitudinal data) is more appropriate compared to matrix-variate normal distributions.Accordingly, the present study considered a parsimonious family of the finite mixture of matrix-variate t-distributions using the eigenvalue and modified Cholesky decomposition for within and between covariance matrices, respectively.Finally, parameter estimates were calculated using the expectation-maximization algorithm, and simulations studies and real data analyses were conducted to confirm the obtained results.

We show that the identification results of finite mixture and misclassification models are equivalent in a widely used scenario except for an extra ordering assumption. In the misclassification model, an ordering condition is imposed to pin down the precise values of the latent variable, which are also of interest to researchers and need to be identified. In contrast, finite mixture models are usually identified up to permutations of a latent index, which results in local identification. This local identification is satisfactory because the latent index does not convey any economic meaning. However, reaching global identification is important for estimation, especially when researchers use bootstrap to estimate standard errors. This is because standard errors approximated by bootstrap may be incorrect without a global estimator. We demonstrate that games with multiple equilibria fit in our framework and the global estimator with ordering conditions provides more reliable estimates.

Understanding how variation in reproductive success is related to demography is a critical component in understanding the life history of an organism. Parentage analysis using molecular markers can be used to estimate the reproductive success of different groups of individuals in natural populations. Previous models have been developed for cases where offspring are random samples from the population but these models do not account for the presence of full- and half-sibs commonly found in large clutches of many organisms. Here we develop a model for comparing reproductive success among different groups of individuals that explicitly incorporates within-nest relatedness. Inference for the parameters of the model is done in a Bayesian framework, where we sample from the joint posterior of parental assignments and fertility parameters. We use computer simulations to determine how well our model recovers known parameters and investigate how various data collection scenarios (varying the number of nests or the number of offspring) affect the estimates. We then apply our model to compare reproductive success among different age groups of mottled sculpin, Cottus bairdi, from a natural population. We demonstrate that older adults are more likely to contribute to a nest and that females in the older age groups contribute more eggs to a nest than younger individuals.

Using questionnaires in research relies on the expectation that they measure the same things across different groups of individuals. If this is not true, then interpretations of results can be misleading when researchers compare responses across different groups of individuals or use in it a group that differs from that in which the questionnaire was developed. For the questionnaire we investigated, the Social Communication Questionnaire (SCQ), we found that parents of boys and girls responded to questionnaire items in largely the same way but that the SCQ measured traits and behaviors slightly differently depending on whether the children had autism. Based on these results, we concluded that researchers using this questionnaire should carefully consider these differences when deciding how to interpret findings. SCQ scores as a reflection of "autism-associated traits" in samples that are mostly or entirely made up of individuals without an autism diagnosis may be misleading and we encourage a more precise interpretation of scores as a broader indication of social-communicative and behavioral traits.

Serological cross-reactivity using a SARS-CoV-2 ELISA test in acute Zika virus infection, Colombia

Finite mixture models have been used extensively in clustering applications, where each component of the mixture distribution is assumed to represent an individual cluster. The simplest example describes each cluster in terms of a multivariate Gaussian density with various covariance structures. However, using finite mixture models as a clustering tool is highly flexible and allows for the specification of a wide range of statistical models to describe the data within each cluster. These include modelling each cluster using linear regression models, mixed effects models, generalized linear models, etc. This paper investigates using mixtures of orthogonal regression models to cluster biological data arising from a study of the sugarcane plant.

Pre-analysis of Multi-batch Bioprocesses Data with Finite Mixture Models in the Reduced Feature Subspace

Technological advances led to the generation of large scale complex data. Thus, extraction and retrieval of information to automatically discover latent pattern have been largely studied in the various domains of science and technology. Consequently, machine learning experienced tremendous development and various statistical approaches have been suggested. In particular, data clustering has received a lot of attention. Finite mixture models have been revealed to be one of the flexible and popular approaches in data clustering. Considering mixture models, three crucial aspects should be addressed. The first issue is choosing a distribution which is flexible enough to fit the data. In this paper, a model based on multivariate Beta distributions is proposed. The two other challenges in mixture models are estimation of model's parameters and model complexity. To tackle these challenges, variational inference techniques demonstrated considerable robustness. In this paper, two methods are studied, namely, batch and online variational inferences and the models are evaluated on four medical applications including image segmentation of colorectal cancer, multi‐class colon tissue analysis, digital imaging in skin lesion diagnosis and computer aid detection of Malaria.

The main difficulty in parametric analysis of longitudinal data lies in specifying covariance structure. Several covariance structures, which usually reflect one series of measurements collected over time, have been presented in the literature. However there is a lack of literature on covariance structures designed for repeated measures specified by more than one repeated factor. In this paper a new, general method of modelling covariance structure based on the Kronecker product of underlying factor specific covariance profiles is presented. The method has an attractive interpretation in terms of independent factor specific contribution to overall within subject covariance structure and can be easily adapted to standard software.

Regression analysis is one method in statistics that used to know the pattern of functional relationships between response variables and predictor variables. Combination of parametric and nonparametric regression is semi parametric regression. The most popular estimator for nonparametric or semi parametric regression is spline truncated estimator. Problems in everyday life often using regression modeling with longitudinal data. Longitudinal data is a combination of cross-section data and time-series data. In longitudinal data, between subjects are independent of each other but between observations in the subject are dependent. One of the most important parts of statistical inference is interval estimation. Interval estimation aims to determine predictor variables that have a significant effect on the response variable. This study aims to obtain the form of interval estimation for parameters of semi parametric regression models using spline truncated estimator in longitudinal data. To solve this problem, the Weighted Least Square method and a pivotal quantity method were used for unknown population variance cases. The result of the theoretical study was that pivotal quantity distributed student-t. The shortest parameter interval estimation of semi parametric spline truncated regression model was obtained through the optimization process using the method of Lagrange.Regression analysis is one method in statistics that used to know the pattern of functional relationships between response variables and predictor variables. Combination of parametric and nonparametric regression is semi parametric regression. The most popular estimator for nonparametric or semi parametric regression is spline truncated estimator. Problems in everyday life often using regression modeling with longitudinal data. Longitudinal data is a combination of cross-section data and time-series data. In longitudinal data, between subjects are independent of each other but between observations in the subject are dependent. One of the most important parts of statistical inference is interval estimation. Interval estimation aims to determine predictor variables that have a significant effect on the response variable. This study aims to obtain the form of interval estimation for parameters of semi parametric regression models using spline truncated estimator in longitudinal data. To solve this problem, t...

The data open sharing provides us opportunity to investigate mechanisms underlying autism spectrum disorder (ASD) by employing advanced techniques. In the current study, we employed structural covariance (SC) model to investigate the development of triple networks in ASD. Three hundred and seven ASD and 337 typical controls were collected and further classified into four distinct age cohorts. Night brain seeds belonging to default modal network, salience network, and central executive network were obtained. SC between those seeds, as well as its topological properties, was calculated for each group within each age cohort. Statistical analysis revealed that ASD had dynamic reconfigurations of SC of the triple networks, especially the right fronto-insular cortex. The results might indicate that ASD had specific mechanism within distinct age cohort. Additionally, the big data sharing, together with the SC modal, was able to facilitate understanding of the mechanism underlying ASD.