The lme4 R package can be used to fit generalized linear mixed models (GLMMs), which extend the class of linear mixed models (LMMs). The two main extensions provided by GLMMs are (1) allowing for the conditional distribution of the response given the random effects to be non-Gaussian (e.g. binomial, Poisson) and (2) allowing the conditional mean to be a nonlinear function of a linear combination of the fixed and random effect coefficients, via an inverse link function. The conditional mode of the random effects given the observed data, the variance-covariance matrix of the random effects, and the fixed effect parameters are determined using penalized iteratively reweighted least squares. We compute an approximation of the integral over the distributions of the conditional modes to compute the maximum likelihood estimate for a given set of parameters (by default we use the Laplace approximation or, alternatively, the more computationally expensive adaptive Gauss-Hermite quadrature). The package provides all the standard features available for GLMs in base R, including the standard set of accessor functions as well as the possibility of user-specified distributions (within the exponential dispersion family) and link functions.

Grace Wahba (nee Goldsmith, born August 3, 1934), I. J. Schoenberg-Hilldale Professor of Statistics at the University of Wisconsin-Madison (Emerita), is a pioneer in methods for smoothing noisy data. Her research combines theoretical analysis, computation and methodology motivated by innovative scientific applications. Best known for the development of generalized cross-validation (GCV), the connection between splines and Bayesian posterior estimates, and "Wahba's problem," she has developed methods with applications in demographic studies, machine learning, DNA microarrays, risk modeling, medical imaging and climate prediction. Grace grew up in the Washington, DC area and New Jersey, and graduated from Montclair High School. She was educated at Cornell (B.A. 1956), University of Maryland, College Park (M.A. 1962) and Stanford (Ph.D. 1966), and worked in industry for several years before receiving her doctorate in 1966 and settling in Madison in 1967. Although holding several visiting appointments, she has made Madison her home for over 50 years. She is the author of Spline Models for Observational Data which has garnered more than 8000 citations. Grace is treasured as an academic advisor and has mentored 39 Ph.D. students that have resulted in more than 330 academic descendants. She was elected to the United States National Academy of Sciences in 2000 and received an honorary degree of Doctor of Science from the University of Chicago in 2007. Wahba is a Fellow of several academic societies including the American Academy of Arts and Sciences, the American Association for the Advancement of Science, the American Statistical Association and the Institute of Mathematical Statistics. Over the years, she has received a selection of notable awards in the statistics community: R. A. Fisher Lectureship (2014), Gottfried E. Noether Senior Researcher Award (2009), Committee of Presidents of Statistical Societies Elizabeth Scott Award (1996) and the first Emanuel and Carol Parzen Prize for Statistical Innovation (1994).

Statistical methods for genome-wide association studies (GWAS) continue to improve. However, the increasing volume and variety of genetic and genomic data make computational speed and ease of data manipulation mandatory in future software. In our view, a collaborative effort of statistical geneticists is required to develop open source software targeted to genetic epidemiology. Our attempt to meet this need is called the OpenMendel project ( https://openmendel.github.io ). It aims to (1) enable interactive and reproducible analyses with informative intermediate results, (2) scale to big data analytics, (3) embrace parallel and distributed computing, (4) adapt to rapid hardware evolution, (5) allow cloud computing, (6) allow integration of varied genetic data types, and (7) foster easy communication between clinicians, geneticists, statisticians, and computer scientists. This article reviews and makes recommendations to the genetic epidemiology community in the context of the OpenMendel project.

This chapter discusses the analysis of data using nonlinear models such as nonlinear regression, general likelihood models, or Bayesian estimation. It presents two fitting functions, one for general minimization models and one for nonlinear regression. In typical use, they both have three arguments, specifying the model, the data, and starting estimates for the parameters. The formula in nonlinear models is an expression in S, involving data, parameters in the model, and any other relevant quantities. In most nonlinear modeling, the relevant data include not only variables similar to those encountered earlier, but also other quantities such as initial estimates for parameters or fixed values occurring in the model formula. Numerical methods for fitting nonlinear models typically can make use of the derivatives of the objective function or of the predictor with respect to the parameters to be estimated. As in linear regression, the sum-of-squares criterion in nonlinear regression can include weights. In linear regression, an optional argument supplies the weights.

Linear mixed-effects models have increasingly replaced mixed-model analyses of variance for statistical inference in factorial psycholinguistic experiments. Although LMMs have many advantages over ANOVA, like ANOVAs, setting them up for data analysis also requires some care. One simple option, when numerically possible, is to fit the full variance-covariance structure of random effects (the maximal model; Barr et al. 2013), presumably to keep Type I error down to the nominal alpha in the presence of random effects. Although it is true that fitting a model with only random intercepts may lead to higher Type I error, fitting a maximal model also has a cost: it can lead to a significant loss of power. We demonstrate this with simulations and suggest that for typical psychological and psycholinguistic data, higher power is achieved without inflating Type I error rate if a model selection criterion is used to select a random effect structure that is supported by the data.

Generalized additive mixed models are introduced as an extension of the generalized linear mixed model which makes it possible to deal with temporal autocorrelational structure in experimental data. This autocorrelational structure is likely to be a consequence of learning, fatigue, or the ebb and flow of attention within an experiment (the `human factor'). Unlike molecules or plots of barley, subjects in psycholinguistic experiments are intelligent beings that depend for their survival on constant adaptation to their environment, including the environment of an experiment. Three data sets illustrate that the human factor may interact with predictors of interest, both factorial and metric. We also show that, especially within the framework of the generalized additive model, in the nonlinear world, fitting maximally complex models that take every possible contingency into account is ill-advised as a modeling strategy. Alternative modeling strategies are discussed for both confirmatory and exploratory data analysis.

This chapter examines the relationship between the number of reproducing fish and recruitment, number of reproducing females and egg production, and indices of year-class or cohort strength. Stock-recruit data consists of estimates of the number of reproducing fish and the number of subsequent recruits. The problem for biologists is to try to understand the relationship between stock and recruitment at least well enough to know how much the stock can be reduced before recruitment starts to drop. Starting values for stock-recruitment functions may be obtained from parameter estimates from linearized versions of the functions or from a visual fit of the function to data. The chapter shows that a visual of the fitted function with a 95% confidence band is constructed largely. Several authors have demonstrated or discussed how to add additional explanatory variables to a stock-recruitment function to explain additional …

Maximum likelihood or restricted maximum likelihood (REML) estimates of the parameters in linear mixed-effects models can be determined using the lmer function in the lme4 package for R. As for most model-fitting functions in R, the model is described in an lmer call by a formula, in this case including both fixed- and random-effects terms. The formula and data together determine a numerical representation of the model from which the profiled deviance or the profiled REML criterion can be evaluated as a function of some of the model parameters. The appropriate criterion is optimized, using one of the constrained optimization functions in R, to provide the parameter estimates. We describe the structure of the model, the steps in evaluating the profiled deviance or REML criterion, and the structure of classes or types that represents such a model. Sufficient detail is included to allow specialization of these structures by users who wish to write functions to fit specialized linear mixed models, such as models incorporating pedigrees or smoothing splines, that are not easily expressible in the formula language used by lmer.

Description Fit linear and generalized linear mixed-effects models. The models and their components are represented using S4 classes and methods. The core computational algorithms are implemented using the Eigen C++ library for numerical linear algebra and RcppEigen``glue''.

The analysis of experimental data with mixed-effects models requires decisions about the specification of the appropriate random-effects structure. Recently, Barr, Levy, Scheepers, and Tily, 2013 recommended fitting `maximal' models with all possible random effect components included. Estimation of maximal models, however, may not converge. We show that failure to converge typically is not due to a suboptimal estimation algorithm, but is a consequence of attempting to fit a model that is too complex to be properly supported by the data, irrespective of whether estimation is based on maximum likelihood or on Bayesian hierarchical modeling with uninformative or weakly informative priors. Importantly, even under convergence, overparameterization may lead to uninterpretable models. We provide diagnostic tools for detecting overparameterization and guiding model simplification.

Nonlinear Regression-Introduction† D. G. Watts, D. G. WattsSearch for more papers by this authorD. M. Bates, D. M. BatesSearch for more papers by this author D. G. Watts, D. G. WattsSearch for more papers by this authorD. M. Bates, D. M. BatesSearch for more papers by this author First published: 29 September 2014 https://doi.org/10.1002/9781118445112.stat03220Citations: 1 †This article was originally published online in 2006 in Encyclopedia of Statistical Sciences, © John Wiley & Sons, Inc. and republished in Wiley StatsRef: Statistics Reference Online, 2014. Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article.Citing Literature Wiley StatsRef: Statistics Reference OnlineBrowse other articles of this reference work:BROWSE BY TOPICBROWSE A-Z RelatedInformation

The methods in the lme4 package for R for fitting linear mixed models are based on sparse matrix methods, especially the Cholesky decomposition of sparse positive-semidefinite matrices, in a penalized least squares representation of the conditional model for the response given the random effects. The representation is similar to that in Henderson’s mixed-model equations. An alternative representation of the calculations is as a generalized least squares problem. We describe the two representations, show the equivalence of the two representations and explain why we feel that the penalized least squares approach is more versatile and more computationally efficient. 1 Definition of the model We consider linear mixed models in which the random effects are represented by a q-dimensional random vector, B, and the response is represented by an n-dimensional random vector, Y . We observe a value, y, of the response. The random effects are unobserved. For our purposes, we will assume a “spherical”multivariate normal conditional distribution of Y , given B. That is, we assume the variance-covariance matrix of Y |B is simply σ2In, where In denotes the identity matrix of order n. (The term “spherical” refers to the fact that contours of the conditional density are concentric spheres.)