Mixture of linear mixed models using multivariate t distribution

Linear mixed models are widely used when multiple correlated measurements are made on each unit of interest. In many applications, the units may form several distinct clusters, and such heterogeneity can be more appropriately modelled by a finite mixture linear mixed model. The classical estimation approach, in which both the random effects and the error parts are assumed to follow normal distribution, is sensitive to outliers, and failure to accommodate outliers may greatly jeopardize the model estimation and inference. We propose a new mixture linear mixed model using multivariate t distribution. For each mixture component, we assume the response and the random effects jointly follow a multivariate t distribution, to conveniently robustify the estimation procedure. An efficient expectation conditional maximization algorithm is developed for conducting maximum likelihood estimation. The degrees of freedom parameters of the t distributions are chosen data adaptively, for achieving flexible trade-off between estimation robustness and efficiency. Simulation studies and an application on analysing lung growth longitudinal data showcase the efficacy of the proposed approach.     

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.     

Bayesian Lasso-mixed quantile regression

In this paper, we discuss the regularization in linear-mixed quantile regression. A hierarchical Bayesian model is used to shrink the fixed and random effects towards the common population values by introducing an l₁ penalty in the mixed quantile regression check function. A Gibbs sampler is developed to simulate the parameters from the posterior distributions. Through simulation studies and analysis of an age-related macular degeneration (ARMD) data, we assess the performance of the proposed method. The simulation studies and the ARMD data analysis indicate that the proposed method performs well in comparison with the other approaches.     

A consistent estimator for logistic mixed effect models

We propose a consistent and locally efficient method of estimating the model parameters of a logistic mixed effect model with random slopes. Our approach relaxes two typical assumptions: the random effects being normally distributed, and the covariates and random effects being independent of each other. Adhering to these assumptions is particularly difficult in health studies where, in many cases, we have limited resources to design experiments and gather data in long‐term studies, while new findings from other fields might emerge, suggesting the violation of such assumptions. So it is crucial to have an estimator that is robust to such violations; then we could make better use of current data harvested using various valuable resources. Our method generalizes the framework presented in Garcia & Ma (2016) which also deals with a logistic mixed effect model but only considers a random intercept. A simulation study reveals that our proposed estimator remains consistent even when the independence and normality assumptions are violated. This contrasts favourably with the traditional maximum likelihood estimator which is likely to be inconsistent when there is dependence between the covariates and random effects. Application of this work to a study of Huntington's disease reveals that disease diagnosis can be enhanced using assessments of cognitive performance. The Canadian Journal of Statistics 47: 140–156; 2019 © 2019 Statistical Society of Canada     

Change-point detection in a linear model by adaptive fused quantile method

A novel approach to quantile estimation in multivariate linear regression models with change-points is proposed: the change-point detection and the model estimation are both performed automatically, by adopting either the quantile-fused penalty or the adaptive version of the quantile-fused penalty. These two methods combine the idea of the check function used for the quantile estimation and the L₁ penalization principle known from the signal processing and, unlike some standard approaches, the presented methods go beyond typical assumptions usually required for the model errors, such as sub-Gaussian or normal distribution. They can effectively handle heavy-tailed random error distributions, and, in general, they offer a more complex view on the data as one can obtain any conditional quantile of the target distribution, not just the conditional mean. The consistency of detection is proved and proper convergence rates for the parameter estimates are derived. The empirical performance is investigated via an extensive comparative simulation study and practical utilization is demonstrated using a real data example.     

Selection of Latent Variables for Multiple Mixed‐outcome Models

Latent variable models have been widely used for modelling the dependence structure of multiple outcomes data. However, the formulation of a latent variable model is often unknown a priori, the misspecification will distort the dependence structure and lead to unreliable model inference. Moreover, multiple outcomes with varying types present enormous analytical challenges. In this paper, we present a class of general latent variable models that can accommodate mixed types of outcomes. We propose a novel selection approach that simultaneously selects latent variables and estimates parameters. We show that the proposed estimator is consistent, asymptotically normal and has the oracle property. The practical utility of the methods is confirmed via simulations as well as an application to the analysis of the World Values Survey, a global research project that explores peoples’ values and beliefs and the social and personal characteristics that might influence them.     

Robust mixture modelling using the t distribution

Normal mixture models are being increasingly used to model the distributions of a wide variety of random phenomena and to cluster sets of continuous multivariate data. However, for a set of data containing a group or groups of observations with longer than normal tails or atypical observations, the use of normal components may unduly affect the fit of the mixture model. In this paper, we consider a more robust approach by modelling the data by a mixture of t distributions. The use of the ECM algorithm to fit this t mixture model is described and examples of its use are given in the context of clustering multivariate data in the presence of atypical observations in the form of background noise.     

A semiparametric nonlinear mixed model approach to phase I profile monitoring

When process data follow a particular curve in quality control, profile monitoring is suitable and appropriate for assessing process stability. Previous research in profile monitoring focusing on nonlinear parametric (P) modeling, involving both fixed and random-effects, was made under the assumption of an accurate nonlinear model specification. Lately, nonparametric (NP) methods have been used in the profile monitoring context in the absence of an obvious linear P model. This study introduces a novel technique in profile monitoring for any nonlinear and auto-correlated data. Referred to as the nonlinear mixed robust profile monitoring (NMRPM) method, it proposes a semiparametric (SP) approach that combines nonlinear P and NP profile fits for scenarios in which a nonlinear P model is adequate over part of the data but inadequate of the rest. These three methods (P, NP, and NMRPM) account for the auto-correlation within profiles and treats the collection of profiles as a random sample with a common population. During Phase I analysis, a version of Hotelling’s T² statistic is proposed for each approach to identify abnormal profiles based on the estimated random effects and obtain the corresponding control limits. The performance of the NMRPM method is then evaluated using a real data set. Results reveal that the NMRPM method is robust to model misspecification and performs adequately against a correctly specified nonlinear P model. Control charts with the NMRPM method have excellent capability of detecting changes in Phase I data with control limits that are easily computable.     

A mixed effects model for analyzing area under the curve of longitudinally measured biomarkers with missing data

A simple approach for analyzing longitudinally measured biomarkers is to calculate summary measures such as the area under the curve (AUC) for each individual and then compare the mean AUC between treatment groups using methods such as t test. This two-step approach is difficult to implement when there are missing data since the AUC cannot be directly calculated for individuals with missing measurements. Simple methods for dealing with missing data include the complete case analysis and imputation. A recent study showed that the estimated mean AUC difference between treatment groups based on the linear mixed model (LMM), rather than on individually calculated AUCs by simple imputation, has negligible bias under random missing assumptions and only small bias when missing is not at random. However, this model assumes the outcome to be normally distributed, which is often violated in biomarker data. In this paper, we propose to use a LMM on log-transformed biomarkers, based on which statistical inference for the ratio, rather than difference, of AUC between treatment groups is provided. The proposed method can not only handle the potential baseline imbalance in a randomized trail but also circumvent the estimation of the nuisance variance parameters in the log-normal model. The proposed model is applied to a recently completed large randomized trial studying the effect of nicotine reduction on biomarker exposure of smokers.     

Robust transformation mixed‐effects models for longitudinal continuous proportional data

The authors propose a robust transformation linear mixed‐effects model for longitudinal continuous proportional data when some of the subjects exhibit outlying trajectories over time. It becomes troublesome when including or excluding such subjects in the data analysis results in different statistical conclusions. To robustify the longitudinal analysis using the mixed‐effects model, they utilize the multivariate t distribution for random effects or/and error terms. Estimation and inference in the proposed model are established and illustrated by a real data example from an ophthalmology study. Simulation studies show a substantial robustness gain by the proposed model in comparison to the mixed‐effects model based on Aitchison's logit‐normal approach. As a result, the data analysis benefits from the robustness of making consistent conclusions in the presence of influential outliers. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

A heteroscedastic measurement error model based on skew and heavy-tailed distributions with known error variances

In this paper, we study inference in a heteroscedastic measurement error model with known error variances. Instead of the normal distribution for the random components, we develop a model that assumes a skew-t distribution for the true covariate and a centred Student's t distribution for the error terms. The proposed model enables to accommodate skewness and heavy-tailedness in the data, while the degrees of freedom of the distributions can be different. Maximum likelihood estimates are computed via an EM-type algorithm. The behaviour of the estimators is also assessed in a simulation study. Finally, the approach is illustrated with a real data set from a methods comparison study in Analytical Chemistry.     

A linear mixed model for analyzing longitudinal skew-normal responses with random dropout

In this paper, a linear mixed effects model is used to fit skewed longitudinal data in the presence of dropout. Two distributional assumptions are considered to produce background for heavy tailed models. One is the linear mixed model with skew-normal random effects and normal errors and the other one is the linear mixed model with skew-normal errors and normal random effects. An ECM algorithm is developed to obtain the parameter estimates. Also an empirical Bayes approach is used for estimating random effects. A simulation study is implemented to investigate the performance of the presented algorithm. Results of an application are also reported where standard errors of estimates are calculated using the Bootstrap approach.     

Semiparametric mixed-effects models for clustered doubly censored data

The Cox proportional frailty model with a random effect has been proposed for the analysis of right-censored data which consist of a large number of small clusters of correlated failure time observations. For right-censored data, Cai et al. [3] proposed a class of semiparametric mixed-effects models which provides useful alternatives to the Cox model. We demonstrate that the approach of Cai et al. [3] can be used to analyze clustered doubly censored data when both left- and right-censoring variables are always observed. The asymptotic properties of the proposed estimator are derived. A simulation study is conducted to investigate the performance of the proposed estimator.     

A Simulation Comparison of Approximate Tests for Fixed Effects in Random Coefficients Growth Curve Models

Often, the response variables on sampling units are observed repeatedly over time. The sampling units may come from different populations, such as treatment groups. This setting is routinely modeled by a random coefficients growth curve model, and the techniques of general linear mixed models are applied to address the primary research aim. An alternative approach is to reduce each subject’s data to summary measures, such as within-subject averages or regression coefficients. One may then test for equality of means of the summary measures (or functions of them) among treatment groups. Here, we compare by simulation the performance characteristics of three approximate tests based on summary measures and one based on the full data, focusing mainly on accuracy of p-values. We find that performances of these procedures can be quite different for small samples in several different configurations of parameter values. The summary-measures approach performed at least as well as the full-data mixed models approach.     

Joint (k,l)-hurdle random effects models for mixed longitudinal power series and normal outcomes

A random effects model for analyzing mixed longitudinal normal and count outcomes with and without the possibility of non ignorable missing outcomes is presented. The count response is inflated in two points (k and l) and the (k, l)-Hurdle power series is used as its distribution. The new distribution contains, as special submodels, several important distributions which are discussed, such as (k, l)-Hurdle Poisson and (k, l)-Hurdle negative binomial and (k, l)-Hurdle binomial distributions among others. Random effects are used to take into account the correlation between longitudinal outcomes and inflation parameters. A full likelihood-based approach is used to yield maximum likelihood estimates of the model parameters. A simulation study is performed in which for count outcome (k, l)-Hurdle Poisson, (k, l)-Hurdle negative binomial and (k, l)-Hurdle binomial distributions are considered. To illustrate the application of such modelling the longitudinal data of body mass index and the number of joint damage are analyzed.     

Augmented mixed beta regression models with skew-normal independent distributions: Bayesian analysis of labor force data

Abstract

Augmented mixed beta regression models are suitable choices for modeling continuous response variables on the closed interval [0, 1]. The random eeceeects in these models are typically assumed to be normally distributed, but this assumption is frequently violated in some applied studies. In this paper, an augmented mixed beta regression model with skew-normal independent distribution for random effects are used. Next, we adopt a Bayesian approach for parameter estimation using the MCMC algorithm. The methods are then evaluated using some intensive simulation studies. Finally, the proposed models have applied to analyze a dataset from an Iranian Labor Force Survey.     

Parametric bootstrap tests for unbalanced nested designs under heteroscedasticity

In this article, we consider the two-factor unbalanced nested design model without the assumption of equal error variance. For the problem of testing ‘main effects’ of both factors, we propose a parametric bootstrap (PB) approach and compare it with the existing generalized F (GF) test. The Type I error rates of the tests are evaluated using Monte Carlo simulation. Our studies show that the PB test performs better than the GF test. The PB test performs very satisfactorily even for small samples while the GF test exhibit poor Type I error properties when the number of factorial combinations or treatments goes up. It is also noted that the same tests can be used to test the significance of the random effect variance component in a two-factor mixed effects nested model under unequal error variances.     

A taxonomy of mixing and outcome distributions based on conjugacy and bridging

Abstract

The generalized linear mixed model (GLMM) is commonly used for the analysis of hierarchical non Gaussian data. It combines an exponential family model formulation with normally distributed random effects. A drawback is the difficulty of deriving convenient marginal mean functions with straightforward parametric interpretations. Several solutions have been proposed, including the marginalized multilevel model (directly formulating the marginal mean, together with a hierarchical association structure) and the bridging approach (choosing the random-effects distribution such that marginal and hierarchical mean functions share functional forms). Another approach, useful in both a Bayesian and a maximum-likelihood setting, is to choose a random-effects distribution that is conjugate to the outcome distribution. In this paper, we contrast the bridging and conjugate approaches. For binary outcomes, using characteristic functions and cumulant generating functions, it is shown that the bridge distribution is unique. Self-bridging is introduced as the situation in which the outcome and random-effects distributions are the same. It is shown that only the Gaussian and degenerate distributions have well-defined cumulant generating functions for which self-bridging holds.     

Mixtures of general location model with factor analyzer covariance structure for clustering mixed type data

Cluster analysis is one of the most widely used method in statistical analyses, in which homogeneous subgroups are identified in a heterogeneous population. Due to the existence of the continuous and discrete mixed data in many applications, so far, some ordinary clustering methods such as, hierarchical methods, k-means and model-based methods have been extended for analysis of mixed data. However, in the available model-based clustering methods, by increasing the number of continuous variables, the number of parameters increases and identifying as well as fitting an appropriate model may be difficult. In this paper, to reduce the number of the parameters, for the model-based clustering mixed data of continuous (normal) and nominal data, a set of parsimonious models is introduced. Models in this set are extended, using the general location model approach, for modeling distribution of mixed variables and applying factor analyzer structure for covariance matrices. The ECM algorithm is used for estimating the parameters of these models. In order to show the performance of the proposed models for clustering, results from some simulation studies and analyzing two real data sets are presented.     

Missing values: sparse inverse covariance estimation and?an?extension to sparse regression

We propose an ℓ ₁-regularized likelihood method for estimating the inverse covariance matrix in the high-dimensional multivariate normal model in presence of missing data. Our method is based on the assumption that the data are missing at random (MAR) which entails also the completely missing at random case. The implementation of the method is non-trivial as the observed negative log-likelihood generally is a complicated and non-convex function. We propose an efficient EM algorithm for optimization with provable numerical convergence properties. Furthermore, we extend the methodology to handle missing values in a sparse regression context. We demonstrate both methods on simulated and real data.