Download or read book An Approximate Likelihood Approach to Nonlinear Mixed Effects Models Via Spline Approximation written by Zhiyu Ge and published by . This book was released on 2018 with total page 50 pages. Available in PDF, EPUB and Kindle. Book excerpt: In dealing with parametric nonlinear mixed effects models, intensive numerical integration often makes exact maximum likelihood estimation impractical given the current computing capacity. Algorithms based on linearization, such as the first order method and the conditional first order method, have the potential of producing highly inconsistent estimates, although numerically they are more efficient. We propose an approximate likelihood approach via spline approximation, which significantly reduces the numerical difficulty associated with the exact maximum likelihood estimation and can give estimates asymptotically equivalent to MLE or up to a controllable asymptotic bias. Theoretical properties of the new algorithm are established for parametric nonlinear mixed effects models with normal additive measurement error. We apply our algorithm to the population pharmacokinetics of phenobarbital and compare results to those obtained with nlme() in S-PLUS. Simulation studies show that our algorithm works equally well as the nlme() for small variability of random effects and outperforms the nlme() for large variability of random effects.

Download or read book Selected Works of Peter J Bickel written by Jianqing Fan and published by Springer Science & Business Media. This book was released on 2012-11-28 with total page 626 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume presents selections of Peter J. Bickel’s major papers, along with comments on their novelty and impact on the subsequent development of statistics as a discipline. Each of the eight parts concerns a particular area of research and provides new commentary by experts in the area. The parts range from Rank-Based Nonparametrics to Function Estimation and Bootstrap Resampling. Peter’s amazing career encompasses the majority of statistical developments in the last half-century or about about half of the entire history of the systematic development of statistics. This volume shares insights on these exciting statistical developments with future generations of statisticians. The compilation of supporting material about Peter’s life and work help readers understand the environment under which his research was conducted. The material will also inspire readers in their own research-based pursuits. This volume includes new photos of Peter Bickel, his biography, publication list, and a list of his students. These give the reader a more complete picture of Peter Bickel as a teacher, a friend, a colleague, and a family man.

Download or read book Statistical Inference For Non linear Mixed Effects Models Involving Ordinary Differential Equations written by and published by . This book was released on 2004 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In the context of nonlinear mixed effect modeling, 'within subject mechanisms' are often represented by a system of nonlinear ordinary differential equations (ODE), whose parameters characterize the different characteristics of the underlying population. These models are useful because they offer a flexible framework where parameters for both individuals and population can be estimated by combining information across all subjects. Estimating parameters for these models becomes challenging in the absence of any analytical solution for the system of ODEs involved in the modeling. In this thesis we proposed two estimation approaches (i) Bayesian Euler's Approximation Method (BEAM) and (ii) Splines Euler's Approximation Method (SEAM). While we proposed SEAM only for the fixed effect models, BEAM is described for fixed as well as mixed effects models. Both of these approaches involve the likelihood approximation based on the naive Euler's numerical approximation method, thereby providing an analytic closed form approximation for the mean function. SEAM combines the Euler's approximation with Spline interpolation to obtain the parameter estimates for each subject separately. On the other hand, BEAM combines the likelihood approximation with the existing Bayesian hierarchical modeling framework to obtain the parameter estimates. For illustration purposes, we presented the real data analyses and simulation studies for both fixed and mixed effects models and compared the results with estimates from the NLS method (fixed effects model) and from the NLME method (mixed effects model). For both type of models, proposed methodologies provide competitive results in terms of estimation accuracy and efficiency. The Bayesian Euler's approximation method was also used to estimate parameters involved in an HIV model, for which an analytical closed form mean function is not available.

Download or read book Generalized Linear Mixed Models written by Charles E. McCulloch and published by IMS. This book was released on 2003 with total page 100 pages. Available in PDF, EPUB and Kindle. Book excerpt: Wiley Series in Probability and Statistics A modern perspective on mixed models The availability of powerful computing methods in recent decades has thrust linear and nonlinear mixed models into the mainstream of statistical application. This volume offers a modern perspective on generalized, linear, and mixed models, presenting a unified and accessible treatment of the newest statistical methods for analyzing correlated, nonnormally distributed data. As a follow-up to Searle's classic, Linear Models, and Variance Components by Searle, Casella, and McCulloch, this new work progresses from the basic one-way classification to generalized linear mixed models. A variety of statistical methods are explained and illustrated, with an emphasis on maximum likelihood and restricted maximum likelihood. An invaluable resource for applied statisticians and industrial practitioners, as well as students interested in the latest results, Generalized, Linear, and Mixed Models features: * A review of the basics of linear models and linear mixed models * Descriptions of models for nonnormal data, including generalized linear and nonlinear models * Analysis and illustration of techniques for a variety of real data sets * Information on the accommodation of longitudinal data using these models * Coverage of the prediction of realized values of random effects * A discussion of the impact of computing issues on mixed models

Download or read book Non Standard Problems in Inference for Additive and Linear Mixed Models written by Sonja Greven and published by Cuvillier Verlag. This book was released on 2008-01-17 with total page 154 pages. Available in PDF, EPUB and Kindle. Book excerpt: Linear mixed models are a powerful inferential tool in modern statistics and have a wide range of applications. Recent advances utilize the connection between penalized spline smoothing and mixed models for efficient implementation of nonparametric and semiparametric regression techniques. These become increasingly important to adequately model the relationship between response variables and covariates. However, despite their common use, some open questions regarding the inference in mixed models still remain. This dissertation is aimed at improving the methodology for inference on random effects. An important special case is testing for polynomial regression against a general smooth alternative modeled by mixed model penalized splines. Our motivating application is the assessment of non-linearity for air pollution dose-response functions in the epidemiological Airgene study. Testing for a zero random effects variance is a non-standard testing problem. First, the tested parameter is on the boundary of the parameter space under the null hypothesis. Second, in linear mixed models observations are generally not independent. While in longitudinal linear mixed models there are at least independent subjects or units, such a subdivision of the data is not possible for mixed model penalized spline smoothing. We first investigate the asymptotic distribution of the restricted likelihood ratio test statistic when testing for polynomial regression using mixed model penalized splines. We show that asymptotic results on boundary testing for independent observations do not hold here. This is due to the asymptotic non-normality of the score statistic. Fundamentally, this is caused by the dependence of observations induced by penalized splines. We find that this dependence cannot be ignored, as it is inherently necessary for the attainment of smooth curves. Different approaches to this testing problem are therefore necessary. Subsequently, we provide finite sample alternatives for testing for zero random effect variances in linear mixed models. The class of models we consider is more general than has previously been covered, including models with moderate numbers of clusters, unbalanced designs, or nonparametric smoothing. We also allow more than one random effect in the model. We propose two approximations to the finite sample null distribution of the restricted likelihood ratio test statistic. Extensive simulations show that both outperform the chi-square mixture approximation and parametric bootstrap currently used, as well as several F-type tests. Finally, we discuss model selection for mixed model penalized splines using the Akaike Information Criterion. The criterion based on the marginal likelihood is found not to be asymptotically unbiased for the expected relative Kullback-Leibler distance. In fact, it is biased towards the simpler model. An alternative is provided using our results on restricted likelihood ratio testing.

Download or read book Mixed Effects Models for Complex Data written by Lang Wu and published by CRC Press. This book was released on 2009-11-11 with total page 431 pages. Available in PDF, EPUB and Kindle. Book excerpt: Although standard mixed effects models are useful in a range of studies, other approaches must often be used in correlation with them when studying complex or incomplete data. Mixed Effects Models for Complex Data discusses commonly used mixed effects models and presents appropriate approaches to address dropouts, missing data, measurement errors, censoring, and outliers. For each class of mixed effects model, the author reviews the corresponding class of regression model for cross-sectional data. An overview of general models and methods, along with motivating examples After presenting real data examples and outlining general approaches to the analysis of longitudinal/clustered data and incomplete data, the book introduces linear mixed effects (LME) models, generalized linear mixed models (GLMMs), nonlinear mixed effects (NLME) models, and semiparametric and nonparametric mixed effects models. It also includes general approaches for the analysis of complex data with missing values, measurement errors, censoring, and outliers. Self-contained coverage of specific topics Subsequent chapters delve more deeply into missing data problems, covariate measurement errors, and censored responses in mixed effects models. Focusing on incomplete data, the book also covers survival and frailty models, joint models of survival and longitudinal data, robust methods for mixed effects models, marginal generalized estimating equation (GEE) models for longitudinal or clustered data, and Bayesian methods for mixed effects models. Background material In the appendix, the author provides background information, such as likelihood theory, the Gibbs sampler, rejection and importance sampling methods, numerical integration methods, optimization methods, bootstrap, and matrix algebra. Failure to properly address missing data, measurement errors, and other issues in statistical analyses can lead to severely biased or misleading results. This book explores the biases that arise when naïve methods are used and shows which approaches should be used to achieve accurate results in longitudinal data analysis.

Download or read book Linear and Nonlinear Mixed effects Models for Repeated Measures Data written by Mary Judith Lindstrom and published by . This book was released on 1987 with total page 368 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Mixed Effects Models in S and S PLUS written by José C. Pinheiro and published by Springer Science & Business Media. This book was released on 2009-04-15 with total page 538 pages. Available in PDF, EPUB and Kindle. Book excerpt: R, linear models, random, fixed, data, analysis, fit.

Download or read book Frontiers In Statistics written by Jianqing Fan and published by World Scientific. This book was released on 2006-07-17 with total page 552 pages. Available in PDF, EPUB and Kindle. Book excerpt: During the last two decades, many areas of statistical inference have experienced phenomenal growth. This book presents a timely analysis and overview of some of these new developments and a contemporary outlook on the various frontiers of statistics.Eminent leaders in the field have contributed 16 review articles and 6 research articles covering areas including semi-parametric models, data analytical nonparametric methods, statistical learning, network tomography, longitudinal data analysis, financial econometrics, time series, bootstrap and other re-sampling methodologies, statistical computing, generalized nonlinear regression and mixed effects models, martingale transform tests for model diagnostics, robust multivariate analysis, single index models and wavelets.This volume is dedicated to Prof. Peter J Bickel in honor of his 65th birthday. The first article of this volume summarizes some of Prof. Bickel's distinguished contributions.

Download or read book Asymptotic Analysis of Mixed Effects Models written by Jiming Jiang and published by CRC Press. This book was released on 2017-09-19 with total page 235 pages. Available in PDF, EPUB and Kindle. Book excerpt: Large sample techniques are fundamental to all fields of statistics. Mixed effects models, including linear mixed models, generalized linear mixed models, non-linear mixed effects models, and non-parametric mixed effects models are complex models, yet, these models are extensively used in practice. This monograph provides a comprehensive account of asymptotic analysis of mixed effects models. The monograph is suitable for researchers and graduate students who wish to learn about asymptotic tools and research problems in mixed effects models. It may also be used as a reference book for a graduate-level course on mixed effects models, or asymptotic analysis.

Download or read book Iterative Estimation Equation Approach for Nonlinear Mixed Effects Models written by Lawrence Lee and published by . This book was released on 2011 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Nonlinear Mixed Effect Model (NLME) is a useful tool for analyzing pharmacokinetic data and other repeated-measures data. Compared to Linear Mixed Effects Models (LME), Nonlinear Mixed Effects Models allow more flexilibility because it allows longitudinal measurement to have a nonlinear relationship with subject-specific parameter and the predictor vector. Maximum likelihood approach can be used to find the estimates of the parameters, which is usually shown to be consistent in linear models. In Nonlinear mixed effects models, because the random effects enter the model nonlinearly, in general, the marginal density of the response does not have a closed-form expression, and hence no closed-form exists for the likelihood. A natural solution to solve this problem is to implement approximation methods for the likelihood function, for example, first order linearization, Laplacian approximation, importance sampling and Gaussian quadrature approximation. However, some of these approaches have a potential of producing inconsistent estimates when the variability in random effects is large. An Iterative Estimation Equations (IEE) approach has been recently studied for a semiparametric regression model for the longitudinal data with unspecified covariance matrix; consistency and asymptotic efficiency have also been demonstrated. We extend this approach to Nonlinear Mixed Effect models. Simulations and case studies are conducted to illustrate the proposed estimation procedures.

Download or read book Robust Mixed Model Analysis written by Jiang Jiming and published by World Scientific. This book was released on 2019-04-10 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mixed-effects models have found broad applications in various fields. As a result, the interest in learning and using these models is rapidly growing. On the other hand, some of these models, such as the linear mixed models and generalized linear mixed models, are highly parametric, involving distributional assumptions that may not be satisfied in real-life problems. Therefore, it is important, from a practical standpoint, that the methods of inference about these models are robust to violation of model assumptions. Fortunately, there is a full scale of methods currently available that are robust in certain aspects. Learning about these methods is essential for the practice of mixed-effects models.This research monograph provides a comprehensive account of methods of mixed model analysis that are robust in various aspects, such as to violation of model assumptions, or to outliers. It is suitable as a reference book for a practitioner who uses the mixed-effects models, and a researcher who studies these models. It can also be treated as a graduate text for a course on mixed-effects models and their applications.

Download or read book Mathematical Reviews written by and published by . This book was released on 2005 with total page 1518 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Mixed Effects Models for the Population Approach written by Marc Lavielle and published by CRC Press. This book was released on 2014-07-14 with total page 385 pages. Available in PDF, EPUB and Kindle. Book excerpt: Wide-Ranging Coverage of Parametric Modeling in Linear and Nonlinear Mixed Effects Models Mixed Effects Models for the Population Approach: Models, Tasks, Methods and Tools presents a rigorous framework for describing, implementing, and using mixed effects models. With these models, readers can perform parameter estimation and modeling across a whole population of individuals at the same time. Easy-to-Use Techniques and Tools for Real-World Data Modeling The book first shows how the framework allows model representation for different data types, including continuous, categorical, count, and time-to-event data. This leads to the use of generic methods, such as the stochastic approximation of the EM algorithm (SAEM), for modeling these diverse data types. The book also covers other essential methods, including Markov chain Monte Carlo (MCMC) and importance sampling techniques. The author uses publicly available software tools to illustrate modeling tasks. Methods are implemented in Monolix, and models are visually explored using Mlxplore and simulated using Simulx. Careful Balance of Mathematical Representation and Practical Implementation This book takes readers through the whole modeling process, from defining/creating a parametric model to performing tasks on the model using various mathematical methods. Statisticians and mathematicians will appreciate the rigorous representation of the models and theoretical properties of the methods while modelers will welcome the practical capabilities of the tools. The book is also useful for training and teaching in any field where population modeling occurs.

Download or read book Computational Approaches for Maximum Likelihood Estimation for Nonlinearmixed Models written by and published by . This book was released on 2000 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The nonlinear mixed model is an important tool for analyzingpharmacokinetic and other repeated-measures data. In particular, these models are used when the measured response for anindividual, has a nonlinear relationship with unknown, random, individual-specificparameters, . Ideally, the method of maximum likelihood is used to find estimates forthe parameters ofthe model after integrating out the random effects in the conditionallikelihood. However, closed form solutions tothe integral are generally not available. As a result, methods have beenpreviously developed to find approximatemaximum likelihood estimates for the parameters in the nonlinear mixedmodel. These approximate methods include FirstOrder linearization, Laplace's approximation, importance sampling, andGaussian quadrature. The methods are availabletoday in several software packages for models of limited sophistication;constant conditional error variance is requiredfor proper utilization of most software. In addition, distributionalassumptions are needed. This work investigates howrobust two of these methods, First Order linearization and Laplace'sapproximation, are to these assumptions. The findingis that Laplace's approximation performs well, resulting in betterestimation than first order linearization when bothmodels converge to a solution. A method must provide good estimates of the likelihood at points inthe parameter space near the solution. This workcompares this ability among the numerical integration techniques, Gaussian quadrature, importance sampling, and Laplace'sapproximation. A new "scaled" and "centered" version of Gaussianquadrature is found to be the most accurate technique. In addition, the technique requires evaluation of the integrand at onlya few abscissas. Laplace's method also performs well; it is more accurate than importance sampling with even 100importance samples over two dimensions. Even so, Laplace's method still does not perform as well as Gaussian quadrature. Overall, Laplace's a.

Download or read book Richly Parameterized Linear Models written by James S. Hodges and published by CRC Press. This book was released on 2016-04-19 with total page 464 pages. Available in PDF, EPUB and Kindle. Book excerpt: A First Step toward a Unified Theory of Richly Parameterized Linear ModelsUsing mixed linear models to analyze data often leads to results that are mysterious, inconvenient, or wrong. Further compounding the problem, statisticians lack a cohesive resource to acquire a systematic, theory-based understanding of models with random effects.Richly Param

Download or read book Topics in Application of Nonparametric Smoothing Splines written by and published by . This book was released on 2004 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: There are two topics in this dissertation. The first topic is 'Smoothing Parameter Selection in Nonparametric Generalized Linear Models via Sixth-order Laplace Approximation' and the second topic is 'Smoothing Spline-based Score Tests for Proportional Hazards Models'. We present a new approach for the automatic selection of the smoothing parameter in nonparametric smoothing spline Generalized Linear Models (GLMs), using the Restricted Maximum Likelihood (REML) method and the sixth-order Laplace approximation of Raudenbush et al. (2000). The proposed approach is compared with Generalized Additive Mixed Model (GAMM, Lin and Zhang 1999) and Generalized Approximate Cross-Validation (GACV, Gu and Xiang 2001) through simulations and is shown to be effective. We propose 'score-type' tests for the proportional hazards assumption and for covariate effects in the Cox model, using the natural smoothing spline representation of the corresponding nonparametric functions of time or covariate. The tests are based on the penalized partial likelihood. By treating the inverse of the smoothing parameter as a variance component, we derive the score tests by testing an equivalent null hypothesis that the corresponding variance component is zero. The tests are shown to have size close to the nominal level and to provide good power against general alternatives in simulations. We apply the proposed tests to data from a cancer clinical trial.