Modelling Survival Events with Longitudinal Covariates Measured with Error

In survival analysis, time-dependent covariates are usually present as longitudinal data collected periodically and measured with error. The longitudinal data can be assumed to follow a linear mixed effect model and Cox regression models may be used for modelling of survival events. The hazard rate of survival times depends on the underlying time-dependent covariate measured with error, which may be described by random effects. Most existing methods proposed for such models assume a parametric distribution assumption on the random effects and specify a normally distributed error term for the linear mixed effect model. These assumptions may not be always valid in practice. In this article, we propose a new likelihood method for Cox regression models with error-contaminated time-dependent covariates. The proposed method does not require any parametric distribution assumption on random effects and random errors. Asymptotic properties for parameter estimators are provided. Simulation results show that under certain situations the proposed methods are more efficient than the existing methods.     

Robust confidence regions for the semi-parametric regression model with responses missing at random

In this paper, a regression semi-parametric model is considered where responses are assumed to be missing at random. From the empirical likelihood function defined based on the rank-based estimating equation, robust confidence intervals/regions of the true regression coefficient are derived. Monte Carlo simulation experiments show that the proposed approach provides more accurate confidence intervals/regions compared to its normal approximation counterpart under different model error structure. The approach is also compared with the least squares approach, and its superiority is shown whenever the error distribution in the simulation study is heavy tailed or contaminated. Finally, a real data example is given to illustrate our proposed method.     

Errors-in-variables beta regression models

Beta regression models provide an adequate approach for modeling continuous outcomes limited to the interval (0, 1). This paper deals with an extension of beta regression models that allow for explanatory variables to be measured with error. The structural approach, in which the covariates measured with error are assumed to be random variables, is employed. Three estimation methods are presented, namely maximum likelihood, maximum pseudo-likelihood and regression calibration. Monte Carlo simulations are used to evaluate the performance of the proposed estimators and the naïve estimator. Also, a residual analysis for beta regression models with measurement errors is proposed. The results are illustrated in a real data set.     

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.     

Semiparametric Approach to a Random Effects Quantile Regression Model

We consider a random effects quantile regression analysis of clustered data and propose a semiparametric approach using empirical likelihood. The random regression coefficients are assumed independent with a common mean, following parametrically specified distributions. The common mean corresponds to the population-average effects of explanatory variables on the conditional quantile of interest, while the random coefficients represent cluster specific deviations in the covariate effects. We formulate the estimation of the random coefficients as an estimating equations problem and use empirical likelihood to incorporate the parametric likelihood of the random coefficients. A likelihood-like statistical criterion function is yield, which we show is asymptotically concave in a neighborhood of the true parameter value and motivates its maximizer as a natural estimator. We use Markov Chain Monte Carlo (MCMC) samplers in the Bayesian framework, and propose the resulting quasi-posterior mean as an estimator. We show that the proposed estimator of the population-level parameter is asymptotically normal and the estimators of the random coefficients are shrunk toward the population-level parameter in the first order asymptotic sense. These asymptotic results do not require Gaussian random effects, and the empirical likelihood based likelihood-like criterion function is free of parameters related to the error densities. This makes the proposed approach both flexible and computationally simple. We illustrate the methodology with two real data examples.     

SIMULATED MAXIMUM LIKELIHOOD APPLIED TO NON-GAUSSIAN AND NONLINEAR MIXED EFFECTS AND STATE–SPACE MODELS

The paper presents an overview of maximum likelihood estimation using simulated likelihood, including the use of antithetic variables and evaluation of the simulation error of the resulting estimates. It gives a general purpose implementation of simulated maximum likelihood and uses it to re‐visit four models that have previously appeared in the published literature: a state–space model for count data; a nested random effects model for binomial data; a nonlinear growth model with crossed random effects; and a crossed random effects model for binary salamander‐mating data. In the case of the last three examples, this appears to be the first time that maximum likelihood fits of these models have been presented.     

Response dimension reduction: model-based approach

In this paper, a model-based approach to reduce the dimension of response variables in multivariate regression is newly proposed, following the existing context of the response dimension reduction developed by Yoo and Cook [Response dimension reduction for the conditional mean in multivariate regression. Comput Statist Data Anal. 2008;53:334–343]. The related dimension reduction subspace is estimated by maximum likelihood, assuming an additive error. In the new approach, the linearity condition, which is assumed for the methodological development in Yoo and Cook (2008), is understood through the covariance matrix of the random error. Numerical studies show potential advantages of the proposed approach over Yoo and Cook (2008). A real data example is presented for illustration.     

Focused Information Criterion and Model Averaging in Quantile Regression

In this article, we study model selection and model averaging in quantile regression. Under general conditions, we develop a focused information criterion and a frequentist model average estimator for the parameters in quantile regression model, and examine their theoretical properties. The new procedures provide a robust alternative to the least squares method or likelihood method, and a major advantage of the proposed procedures is that when the variance of random error is infinite, the proposed procedure works beautifully while the least squares method breaks down. A simulation study and a real data example are presented to show that the proposed method performs well with a finite sample and is easy to use in practice.     

Conditional Akaike Information Criteria for a Class of Poisson Mixture Models with Random Effects

Focusing on the model selection problems in the family of Poisson mixture models (including the Poisson mixture regression model with random effects and zero‐inflated Poisson regression model with random effects), the current paper derives two conditional Akaike information criteria. The criteria are the unbiased estimators of the conditional Akaike information based on the conditional log‐likelihood and the conditional Akaike information based on the joint log‐likelihood, respectively. The derivation is free from the specific parametric assumptions about the conditional mean of the true data‐generating model and applies to different types of estimation methods. Additionally, the derivation is not based on the asymptotic argument. Simulations show that the proposed criteria have promising estimation accuracy. In addition, it is found that the criterion based on the conditional log‐likelihood demonstrates good model selection performance under different scenarios. Two sets of real data are used to illustrate the proposed method.     

A random‐effects model for clustered circular data

This article considers a circular regression model for clustered data, where both the cluster effects and the regression errors have von Mises distributions. It involves β, a vector of parameters for the fixed effects, and two concentration parameters for the error distribution. A measure of intra‐cluster circular correlation and a predictor for an unobserved cluster random effect are studied. Preliminary estimators for the vector β and the two concentration parameters are proposed, and their performance is compared with that of the maximum likelihood estimators in a simulation study. A numerical example investigating the factors impacting the orientation taken by a sand hopper when released is presented. The Canadian Journal of Statistics 47: 712–728; 2019 © 2019 Statistical Society of Canada     

Cluster analysis in a randomized complete block design

The mixture maximum likelihood approach to clustering is used to allocate treatments from a randomized complete block de-sign into relatively homogeneous groups. The implementation of this approach is straightforward for fixed but not random block effects. The density function in each underlying group is assumed to be normal and clustering is performed on the basis of the estimated posterior probabilities of group membership. A test based on the log likelihood under the mixture model can be used to assess the actual number of groups present. The tech-nique is demonstrated by using it to cluster data from a random-ized complete block experiment.     

Finite mixtures of quantile and M-quantile regression models

In this paper we define a finite mixture of quantile and M-quantile regression models for heterogeneous and /or for dependent/clustered data. Components of the finite mixture represent clusters of individuals with homogeneous values of model parameters. For its flexibility and ease of estimation, the proposed approaches can be extended to random coefficients with a higher dimension than the simple random intercept case. Estimation of model parameters is obtained through maximum likelihood, by implementing an EM-type algorithm. The standard error estimates for model parameters are obtained using the inverse of the observed information matrix, derived through the Oakes (J R Stat Soc Ser B 61:479–482, 1999) formula in the M-quantile setting, and through nonparametric bootstrap in the quantile case. We present a large scale simulation study to analyse the practical behaviour of the proposed model and to evaluate the empirical performance of the proposed standard error estimates for model parameters. We considered a variety of empirical settings in both the random intercept and the random coefficient case. The proposed modelling approaches are also applied to two well-known datasets which give further insights on their empirical behaviour.     

Mixtures of Linear Regression with Measurement Errors

Existing research on mixtures of regression models are limited to directly observed predictors. The estimation of mixtures of regression for measurement error data imposes challenges for statisticians. For linear regression models with measurement error data, the naive ordinary least squares method, which directly substitutes the observed surrogates for the unobserved error-prone variables, yields an inconsistent estimate for the regression coefficients. The same inconsistency also happens to the naive mixtures of regression estimate, which is based on the traditional maximum likelihood estimator and simply ignores the measurement error. To solve this inconsistency, we propose to use the deconvolution method to estimate the mixture likelihood of the observed surrogates. Then our proposed estimate is found by maximizing the estimated mixture likelihood. In addition, a generalized EM algorithm is also developed to find the estimate. The simulation results demonstrate that the proposed estimation procedures work well and perform much better than the naive estimates.     

A class of finite mixture of quantile regressions with its applications

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.     

A semiparametric approach to hidden Markov models under longitudinal observations

We propose a hidden Markov model for longitudinal count data where sources of unobserved heterogeneity arise, making data overdispersed. The observed process, conditionally on the hidden states, is assumed to follow an inhomogeneous Poisson kernel, where the unobserved heterogeneity is modeled in a generalized linear model (GLM) framework by adding individual-specific random effects in the link function. Due to the complexity of the likelihood within the GLM framework, model parameters may be estimated by numerical maximization of the log-likelihood function or by simulation methods; we propose a more flexible approach based on the Expectation Maximization (EM) algorithm. Parameter estimation is carried out using a non-parametric maximum likelihood (NPML) approach in a finite mixture context. Simulation results and two empirical examples are provided.     

Multivariate linear regression with non-normal errors: a solution based on mixture models

In some situations, the distribution of the error terms of a multivariate linear regression model may depart from normality. This problem has been addressed, for example, by specifying a different parametric distribution family for the error terms, such as multivariate skewed and/or heavy-tailed distributions. A new solution is proposed, which is obtained by modelling the error term distribution through a finite mixture of multi-dimensional Gaussian components. The multivariate linear regression model is studied under this assumption. Identifiability conditions are proved and maximum likelihood estimation of the model parameters is performed using the EM algorithm. The number of mixture components is chosen through model selection criteria; when this number is equal to one, the proposal results in the classical approach. The performances of the proposed approach are evaluated through Monte Carlo experiments and compared to the ones of other approaches. In conclusion, the results obtained from the analysis of a real dataset are presented.     

Estimation in regression models with stationary,dependent errors

Often the unknown covariance structure of a stationary, dependent, Gaussian error sequence can be simply parametrised. The error sequence can either be directly observed or observed only through a random sequence containing a deterministic regression model. The method of scoring is used here, in conjunction with recursive estimation techniques, to effect the maximum likelihood estimation of the covariance parameters. Sequences of recursive residuals, useful in model diagnostics and data analysis, are obtained in the estimation procedure.     

Robustness of Parameter Estimation Procedures in Multilevel Models When Random Effects are MEP Distributed

In this paper we examine maximum likelihood estimation procedures in multilevel models for two level nesting structures. Usually, for fixed effects and variance components estimation, level-one error terms and random effects are assumed to be normally distributed. Nevertheless, in some circumstances this assumption might not be realistic, especially as concerns random effects. Thus we assume for random effects the family of multivariate exponential power distributions (MEP); subsequently, by means of Monte Carlo simulation procedures, we study robustness of maximum likelihood estimators under normal assumption when, actually, random effects are MEP distributed.     

Impact of missing data on type 1 error rates in non‐inferiority trials

In this paper, a simulation study is conducted to systematically investigate the impact of different types of missing data on six different statistical analyses: four different likelihood‐based linear mixed effects models and analysis of covariance (ANCOVA) using two different data sets, in non‐inferiority trial settings for the analysis of longitudinal continuous data. ANCOVA is valid when the missing data are completely at random. Likelihood‐based linear mixed effects model approaches are valid when the missing data are at random. Pattern‐mixture model (PMM) was developed to incorporate non‐random missing mechanism. Our simulations suggest that two linear mixed effects models using unstructured covariance matrix for within‐subject correlation with no random effects or first‐order autoregressive covariance matrix for within‐subject correlation with random coefficient effects provide well control of type 1 error (T1E) rate when the missing data are completely at random or at random. ANCOVA using last observation carried forward imputed data set is the worst method in terms of bias and T1E rate. PMM does not show much improvement on controlling T1E rate compared with other linear mixed effects models when the missing data are not at random but is markedly inferior when the missing data are at random. Copyright © 2009 John Wiley & Sons, Ltd.     

Editorial collaborators

A simulation study of the binomial-logit model with correlated random effects is carried out based on the generalized linear mixed model (GLMM) methodology. Simulated data with various numbers of regression parameters and different values of the variance component are considered. The performance of approximate maximum likelihood (ML) and residual maximum likelihood (REML) estimators is evaluated. For a range of true parameter values, we report the average biases of estimators, the standard error of the average bias and the standard error of estimates over the simulations. In general, in terms of bias, the two methods do not show significant differences in estimating regression parameters. The REML estimation method is slightly better in reducing the bias of variance component estimates.