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Small area estimation with categorical outcomes often requires intensive computation, as the marginal likelihood does not have a closed form in general. The likelihood analysis is further complicated by deviations in distributional assumptions often arise through outliers in the data. In this paper, the author proposes a robust method for estimating the small area parameters. Finite-sample properties of the estimators are investigated using Monte Carlo simulations. The empirical study shows that the proposed robust method is very useful for bounding the influence of outliers on the small area estimators. To approximate the mean squared errors of the estimators, a parametric bootstrap method is adopted. An application is also provided using actual data from a public health survey.

In this paper, we propose an innovative method for jointly analyzing survival data and longitudinally measured continuous and ordinal data. We use a random effects accelerated failure time model for survival outcomes, a linear mixed model for continuous longitudinal outcomes and a proportional odds mixed model for ordinal longitudinal outcomes, where these outcome processes are linked through a set of association parameters. A primary objective of this study is to examine the effects of association parameters on the estimators of joint models. The model parameters are estimated by the method of maximum likelihood. The finite-sample properties of the estimators are studied using Monte Carlo simulations. The empirical study suggests that the degree of association among the outcome processes influences the bias, efficiency, and coverage probability of the estimators. Our proposed joint model estimators are approximately unbiased and produce smaller mean squared errors as compared to the estimators obtained from separate models. This work is motivated by a large multicenter study, referred to as the Genetic and Inflammatory Markers of Sepsis (GenIMS) study. We apply our proposed method to the GenIMS data analysis.

In many biological experiments, certain values of a biomarker are often nondetectable due to low concentrations of an analyte or the limitations of a chemical analysis device, resulting in left-censored values. There is an increasing demand for the analysis of data subject to detection limits in clinical and environmental studies. In this paper, we develop a novel statistical method for the maximum likelihood estimation in generalized linear models with covariates subject to detection limits. Simulations are carried out to study the relative performance of the proposed estimators, as compared to other existing estimators. The proposed method is also applied to a real dataset from the Maternal-Infant Research on Environmental Chemicals cohort study, where we investigate how different chemical mixtures affect the health outcomes of infants and pregnant women.

The accelerated failure time model is widely used for analyzing censored survival times often observed in clinical studies. It is well-known that the ordinary maximum likelihood estimators of the parameters in the accelerated failure time model are generally sensitive to potential outliers or small deviations from the underlying distributional assumptions. In this paper, we propose and explore a robust method for fitting the accelerated failure time model to survival data by bounding the influence of outliers in both the outcome variable and associated covariates. We also develop a sandwich-type variance-covariance function for approximating the variances of the proposed robust estimators. The finite-sample properties of the estimators are investigated based on empirical results from an extensive simulation study. An application is provided using actual data from a clinical study of primary breast cancer patients.

Frailty models are multiplicative hazard models for studying association between survival time and important clinical covariates. When some values of a clinical covariate are unobserved but known to be below a threshold called the limit of detection (LOD), naive approaches ignoring this problem, such as replacing the undetected value by the LOD or half of the LOD, often produce biased parameter estimate with larger mean squared error of the estimate. To address the LOD problem in a frailty model, we propose a flexible smooth nonparametric density estimator along with Simpson's numerical integration technique. This is an extension of an existing method in the likelihood framework for the estimation and inference of the model parameters. The proposed new method shows the estimators are asymptotically unbiased and gives smaller mean squared error of the estimates. Compared with the existing method, the proposed new method does not require distributional assumptions for the underlying covariates. Simulation studies were conducted to evaluate the performance of the new method in realistic scenarios. We illustrate the use of the proposed method with a data set from Genetic and Inflammatory Markers of Sepsis study in which interlekuin-10 was subject to LOD. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, the authors investigate a robust semi-parametric mixed effects model for analyzing longitudinal data with an unspecified mean response function. The robust method, developed in the framework of the maximum likelihood, is used to bound the influence of potential outliers when estimating the model parameters. The authors also present a robust test procedure for assessing the significance of a variance component in the mixed model. An application is provided using a clinical dataset from a retinopathy of prematurity study in which longitudinal measurements were obtained from premature infants treated with supplemental oxygen. The empirical properties of the proposed estimators are also studied in simulations.

We develop and study an innovative method for jointly modeling longitudinal response and time-to-event data with a covariate subject to a limit of detection. The joint model assumes a latent process based on random effects to describe the association between longitudinal and time-to-event data. We study the role of the association parameter on the regression parameters estimators. We model the longitudinal and survival outcomes using linear mixed-effects and Weibull frailty models, respectively. Because of the limit of detection, missing covariate (explanatory variable, x) values may lead to the non-ignorable missing, resulting in biased parameter estimates with poor coverage probabilities of the confidence interval. We define and estimate the probability of missing due to the limit of detection. Then we develop a novel joint density and hence the likelihood function that incorporates the effect of left-censored covariate. Monte Carlo simulations show that the estimators of the proposed method are approximately unbiased and provide expected coverage probabilities for both longitudinal and survival submodels parameters. We also present an application of the proposed method using a large clinical dataset of pneumonia patients obtained from the Genetic and Inflammatory Markers of Sepsis study.

For the analysis of longitudinal data with nonignorable and nonmonotone missing responses, a full likelihood method often requires intensive computation, especially when there are many follow-up times. The authors propose and explore a Monte Carlo method, based on importance sampling, for approximating the maximum likelihood estimators. The finite-sample properties of the proposed estimators are studied using simulations. An application of the proposed method is also provided using longitudinal data on peptide intensities obtained from a proteomics experiment of trauma patients.

Missing data are common in many experiments, including surveys, clinical trials, epidemiological studies, and environmental studies. Unconstrained likelihood inferences for generalized linear models (GLMs) with nonignorable missing covariates have been studied extensively in the literature. However, parameter orderings or constraints may occur naturally in practice, and thus the efficiency of a statistical method may be improved by incorporating parameter constraints into the likelihood function. In this paper, we consider constrained inference for analysing GLMs with nonignorable missing covariates under linear inequality constraints on the model parameters. Specifically, constrained maximum likelihood (ML) estimation is based on the gradient projection expectation maximization approach. Further, we investigate the asymptotic null distribution of the constrained likelihood ratio test (LRT). Simulations study the empirical properties of the constrained ML estimators and LRTs, which demonstrate improved pr...

In this paper, the author investigates optimal designs for multivariate binary regression models used in many clinical experiments. As the computation of a joint likelihood for multiple binary outcomes is often tedious, the author proposes and explores a pseudo-likelihood approach for choosing an optimal design under minimal parametric assumptions. The proposed design is considered robust in the sense that it provides estimators that are almost as efficient as those obtained from D-optimal designs under correctly specified likelihood functions and it can provide more efficient estimators as compared to D-optimal designs under misspecified likelihood functions. The asymptotic relative efficiencies of the maximum pseudo-likelihood estimators with respect to the exact maximum likelihood estimators are investigated. Monte Carlo simulations are carried out to explore the finite-sample properties of the maximum pseudo-likelihood estimators obtained under the proposed design scheme. The method is also illustrated in an example using actual data from a clinical study.

We encounter missing data in many longitudinal studies. When the missing data are nonignorable, it is important to analyze the data by incorporating the missing data mechanism into the observed data likelihood function. The classical maximum likelihood (ML) method for analyzing longitudinal missing data has been extensively studied in the literature. However, it is well-known that the ordinary ML estimators are sensitive to extreme observations or outliers in the data. In this paper, we propose and explore a robust method, which is developed in the framework of the ML method, and is useful for downweighting any influential observations in the data when estimating the model parameters. We study the empirical properties of the robust estimators in small simulations. We also illustrate the robust method using incomplete longitudinal data on CD4 counts from clinical trials of HIV-infected patients.

We discuss the construction of D-optimal sequential designs for the analysis of longitudinal data or repeated measurements using generalized linear mixed models (GLMMs). We investigate the performance of the design through a simulation study, which indicates that the proposed design can be very successful in improving the efficiency of the ML estimators in GLMMs relative to some common competitors. Our simulations also suggest that the usual normal-theory inference procedures remain valid under the sequential sampling schemes. We also present an example using real data obtained from a clinical study.

For analyzing longitudinal binary data with nonignorable and non-monotone missing responses, a full likelihood method is complicated algebraically, and often requires intensive computation, especially when there are many follow-up times. As an alternative, a pseudo-likelihood approach has been proposed in the literature under minimal parametric assumptions. This formulation only requires specification of the marginal distributions of the responses and missing data mechanism, and uses an independence working assumption. However, this estimator can be inefficient for estimating both time-varying and time-stationary effects under moderate to strong within-subject associations among repeated responses. In this article, we propose an alternative estimator, based on a bivariate pseudo-likelihood, and demonstrate in simulations that the proposed method can be much more efficient than the previous pseudo-likelihood obtained under the assumption of independence. We illustrate the method using longitudinal data on CD4 counts from two clinical trials of HIV-infected patients.

Small area estimation has received considerable attention in recent years because of growing demand for small area statistics. Basic area-level and unit-level models have been studied in the literature to obtain empirical best linear unbiased prediction (EBLUP) estimators of small area means. Although this classical method is useful for estimating the small area means efficiently under normality assumptions, it can be highly influenced by the presence of outliers in the data. In this article, the authors investigate the robustness properties of the classical estimators and propose a resistant method for small area estimation, which is useful for downweighting any influential observations in the data when estimating the model parameters. To estimate the mean squared errors of the robust estimators of small area means, a parametric bootstrap method is adopted here, which is applicable to models with block diagonal covariance structures. Simulations are carried out to study the behaviour of the proposed robust estimators in the presence of outliers, and these estimators are also compared to the EBLUP estimators. Performance of the bootstrap mean squared error estimator is also investigated in the simulation study. The proposed robust method is also applied to some real data to estimate crop areas for counties in Iowa, using farm-interview data on crop areas and LANDSAT satellite data as auxiliary information. The Canadian Journal of Statistics 37: 381–399; 2009 © 2009 Statistical Society of Canada L'estimation de petits domaines a rec cu considerablement d'attention ces dernieres annees en raison de la demande croissante de statistiques regionales. Les modeles au niveau des domaines et des unites ont deja ete etudies dans la litterature et les meilleurs estimateurs lineaires sans biais empiriques (EBLUP) pour les petits domaines ont ete obtenus. Quoique cette methode classique est utile pour estimer les moyennes regionales de fac con efficace sous l'hypothese de normalite, ses resultats sont grandement influences par la presente de donnees aberrantes. Dans cet article, les auteurs etudient les proprietes de robustesse des estimateurs classiques et ils proposent une methode robuste pour l'estimation de petits domaines qui diminue le poids associe aux observations influentes lors de l'estimation des parametres du modele. Afin d'estimer l'erreur quadratique moyenne des estimateurs robustes des moyennes regionales, une methode d'auto-amorc cage parametrique est utilisee. Cette methode peut etre utilisee aux modeles dont la structure de covariance est bloc diagonale. Des simulations sont faites pour etudier le comportement des estimateurs robustes proposes en presence de valeurs aberrantes et aussi pour les comparer aux estimateurs EBLUP. La performance de l'estimateur “boostrap” de l'erreur quadratique moyenne est aussi etudiee dans cette etude de simulations. Cette methode robuste est appliquee a l'estimation de la superficie des cultures pour les comtes de l'Iowa en se basant sur des entrevues au niveau des fermes et en utilisant les donnees provenant du satellite LANDSAT comme information auxiliaire. La revue canadienne de statistique 37: 381–399; 2009 © 2009 Societe statistique du Canada

In this paper, we propose and explore a semiparametric approach to analyzing longitudinal binary data often observed in clinical studies. We applied second-order GEE approach to analyze longitudinal binary responses based on a partially linear single-index model. We use a local polynomial smoothing technique to estimate the single-index. We study the empirical properties of the proposed estimators using simulations. The empirical results demonstrate that if the true underlying model is partially linear, then our proposed method generally provides unbiased and efficient estimators. The proposed method is also applied to some real data sets obtained from longitudinal studies.

Generalized M (or GM) estimation has been extended to the case of a nonlinear regression model with autoregressive and heteroscedastic errors. The robustness properties of the GM estimators have been investigated based on the time-series analog of Hampel's influence function. The asymptotic properties of these estimators have been studied in some detail.

In biomedical and epidemiological studies, often outcomes obtained are of mixed discrete and continuous in nature. Furthermore, due to some technical inconvenience or else, continuous responses are censored and also a few covariates cease to be observed completely. In this paper, we develop a model to tackle these complex situations. Our methodology is developed in a more general framework and provides a full-scale robust analysis of such complex models. The proposed robust maximum likelihood estimators of the model parameters are resistant to potential outliers in the data. We discuss the asymptotic properties of the robust estimators. To avoid computational difficulties involving irreducibly high-dimensional integrals, we propose a Monte Carlo method based on the Metropolis algorithm for approximating the robust maximum likelihood estimators. We study the empirical properties of these estimators in simulations. We also illustrate the proposed robust method using clustered data on blood sugar content from a clinical trial of individuals who were investigated for diabetes.

Problem statement: Modeling survival data with a set of covariates usually assumes that the values of the covariates are fully observed. However, in a variety of applications, some values of a covariate may be left-censored due to inadequate instrument sensitivity to quantify the biospecimen. When data are left-censored, the true values are missing but are known to be smaller than the detection limit. The most commonly used ad-hoc method to deal with nondetect values is to substitute the nondetect values by the detection limit. Such ad-hoc analysis of survival data with an explanatory variable subject to left-censoring may provide biased and inefficient estimators of hazard ratios and survivor functions. Method: We consider a parametric proportional hazards model to analyze time-to-event data. We propose a likelihood method for the estimation and inference of model parameters. In this likelihood approach, instead of replacing the nondetect values by the detection limit, we adopt a numerical integration technique to evaluate the observed data likelihood in the presence of a left-censored covariate. Monte Carlo simulations were used to demonstrate various properties of the proposed regression estimators including the consistency and efficiency. Results: The simulation study shows that the proposed likelihood approach provides approximately unbiased estimators of the model parameters. The proposed method also provides estimators that are more efficient than those obtained under the ad-hoc method. Also, unlike the ad-hoc estimators, the coverage probabilities of the proposed estimators are at their nominal level. Analysis of a large cohort study, genetic and inflammatory marker of sepsis study, shows discernibly different results based on the proposed method. Conclusion: Naive use of detection limit in a parametric survival model may provide biased and inefficient estimators of hazard ratios and survivor functions. The proposed likelihood approach provides approximately unbiased and efficient estimators of hazard ratios and survivor functions.