Restricted maximum likelihood estimation of joint mean‐covariance models

The 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 Statistics 40: 225–242; 2012 © 2012 Statistical Society of Canada     

Efficient estimation for the proportional hazards model with competing risks and current status data

The proportional hazards model is the most commonly used model in regression analysis of failure time data and has been discussed by many authors under various situations (Kalbfleisch & Prentice, 2002. The Statistical Analysis of Failure Time Data, Wiley, New York). This paper considers the fitting of the model to current status data when there exist competing risks, which often occurs in, for example, medical studies. The maximum likelihood estimates of the unknown parameters are derived and their consistency and convergence rate are established. Also we show that the estimates of regression coefficients are efficient and have asymptotically normal distributions. Simulation studies are conducted to assess the finite sample properties of the estimates and an illustrative example is provided. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Model‐based clustering of longitudinal data

A new family of mixture models for the model‐based clustering of longitudinal data is introduced. The covariance structures of eight members of this new family of models are given and the associated maximum likelihood estimates for the parameters are derived via expectation–maximization (EM) algorithms. The Bayesian information criterion is used for model selection and a convergence criterion based on the Aitken acceleration is used to determine the convergence of these EM algorithms. This new family of models is applied to yeast sporulation time course data, where the models give good clustering performance. Further constraints are then imposed on the decomposition to allow a deeper investigation of the correlation structure of the yeast data. These constraints greatly extend this new family of models, with the addition of many parsimonious models. The Canadian Journal of Statistics 38:153–168; 2010 © 2010 Statistical Society of Canada     

Improving on the sample covariance matrix for a complex elliptically contoured distribution

In this paper the problem of estimating the scale matrix in a complex elliptically contoured distribution (complex ECD) is addressed. An extended Haff–Stein identity for this model is derived. It is shown that the minimax estimators of the covariance matrix obtained under the complex normal model remain robust under the complex ECD model when the Stein loss function is employed.     

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     

Contaminated normal modeling with application to microarray data analysis

A contaminated beta model $(1-\gamma) B(1,1) + \gamma B(\alpha,\beta)$ is often used to describe the distribution of $P$ ‐values arising from a microarray experiment. The authors propose and examine a different approach: namely, using a contaminated normal model $(1-\gamma) N(0,\sigma^2) + \gamma N(\mu,\sigma^2)$ to describe the distribution of $Z$ statistics or suitably transformed $T$ statistics. The authors then address whether a researcher who has $Z$ statistics should analyze them using the contaminated normal model or whether the $Z$ statistics should be converted to $P$ ‐values to be analyzed using the contaminated beta model. The authors also provide a decision‐theoretic perspective on the analysis of $Z$ statistics. The Canadian Journal of Statistics 38: 315–332; 2010 © 2010 Statistical Society of Canada     

Variability explained by covariates in linear mixed‐effect models for longitudinal data

Variability explained by covariates or explained variance is a well‐known concept in assessing the importance of covariates for dependent outcomes. In this paper we study R² statistics of explained variance pertinent to longitudinal data under linear mixed‐effect models, where the R² statistics are computed at two different levels to measure, respectively, within‐ and between‐subject variabilities explained by the covariates. By deriving the limits of R² statistics, we find that the interpretation of explained variance for the existing R² statistics is clear only in the case where the covariance matrix of the outcome vector is compound symmetric. Two new R² statistics are proposed to address the effect of time‐dependent covariate means. In the general case where the outcome covariance matrix is not compound symmetric, we introduce the concept of compound symmetry projection and use it to define level‐one and level‐two R² statistics. Numerical results are provided to support the theoretical findings and demonstrate the performance of the R² statistics. The Canadian Journal of Statistics 38: 352–368; 2010 © 2010 Statistical Society of Canada     

Nonparametric likelihood and doubly robust estimating equations for marginal and nested structural models

This article considers Robins's marginal and nested structural models in the cross‐sectional setting and develops likelihood and regression estimators. First, a nonparametric likelihood method is proposed by retaining a finite subset of all inherent and modelling constraints on the joint distributions of potential outcomes and covariates under a correctly specified propensity score model. A profile likelihood is derived by maximizing the nonparametric likelihood over these joint distributions subject to the retained constraints. The maximum likelihood estimator is intrinsically efficient based on the retained constraints and weakly locally efficient. Second, two regression estimators, named hat and tilde, are derived as first‐order approximations to the likelihood estimator under the propensity score model. The tilde regression estimator is intrinsically and weakly locally efficient and doubly robust. The methods are illustrated by data analysis for an observational study on right heart catheterization. The Canadian Journal of Statistics 38: 609–632; 2010 © 2010 Statistical Society of Canada     

A longitudinal study of children's aggressive behaviours based on multivariate mixed models with incomplete data

The authors consider children's behavioural and emotional problems and their relationships with possible predictors. They propose a multivariate transitional mixed‐effects model for a longitudinal study and simultaneously address non‐ignorable missing data in responses and covariates, measurement errors in covariates, and multivariate modelling of the responses and covariate processes. A real dataset is analysed in details using the proposed method with some interesting results. The Canadian Journal of Statistics 37: 435–452; 2009 © 2009 Statistical Society of Canada     

Pair‐copula constructions for non‐Gaussian DAG models

We propose a new type of multivariate statistical model that permits non‐Gaussian distributions as well as the inclusion of conditional independence assumptions specified by a directed acyclic graph. These models feature a specific factorisation of the likelihood that is based on pair‐copula constructions and hence involves only univariate distributions and bivariate copulas, of which some may be conditional. We demonstrate maximum‐likelihood estimation of the parameters of such models and compare them to various competing models from the literature. A simulation study investigates the effects of model misspecification and highlights the need for non‐Gaussian conditional independence models. The proposed methods are finally applied to modeling financial return data. The Canadian Journal of Statistics 40: 86–109; 2012 © 2012 Statistical Society of Canada     

Testing for generalized linear mixed models with cluster correlated data under linear inequality constraints

Generalized linear mixed models (GLMMs) are often used for analyzing cluster correlated data, including longitudinal data and repeated measurements. Full unrestricted maximum likelihood (ML) approaches for inference on both fixed‐and random‐effects parameters in GLMMs have been extensively studied in the literature. However, parameter orderings or constraints may occur naturally in practice, and in such cases, the efficiency of a statistical method is improved by incorporating the parameter constraints into the ML estimation and hypothesis testing. In this paper, inference for GLMMs under linear inequality constraints is considered. The asymptotic properties of the constrained ML estimators and constrained likelihood ratio tests for GLMMs have been studied. Simulations investigated the empirical properties of the constrained ML estimators, compared to their unrestricted counterparts. An application to a recent survey on Canadian youth smoking patterns is also presented. As these survey data exhibit natural parameter orderings, a constrained GLMM has been considered for data analysis. The Canadian Journal of Statistics 40: 243–258; 2012 © 2012 Crown in the right of Canada     

Improving variance function estimation in semiparametric longitudinal data analysis

We propose an efficient and robust method for variance function estimation in semiparametric longitudinal data analysis. The method utilizes a local log‐linear approximation for the variance function and adopts a generalized estimating equation approach to account for within subject correlations. We show theoretically and empirically that our method outperforms estimators using working independence that ignores the correlations. The Canadian Journal of Statistics 39: 656–670; 2011. © 2011 Statistical Society of Canada     

Bootstrap tests for variance components in generalized linear mixed models

In many applications of generalized linear mixed models to clustered correlated or longitudinal data, often we are interested in testing whether a random effects variance component is zero. The usual asymptotic mixture of chi‐square distributions of the score statistic for testing constrained variance components does not necessarily hold. In this article, the author proposes and explores a parametric bootstrap test that appears to be valid based on its estimated level of significance under the null hypothesis. Results from a simulation study indicate that the bootstrap test has a level much closer to the nominal one while the asymptotic test is conservative, and is more powerful than the usual asymptotic score test based on a mixture of chi‐squares. The proposed bootstrap test is illustrated using two sets of real‐life data obtained from clinical trials. The Canadian Journal of Statistics © 2009 Statistical Society of Canada