# José C. Pinheiro's research while affiliated with University of Wisconsin–Madison and other places

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## Publications (29)

Linear Mixed-Effects * Theory and Computational Methods for LME Models * Structure of Grouped Data * Fitting LME Models * Extending the Basic LME Model * Nonlinear Mixed-Effects * Theory and Computational Methods for NLME Models * Fitting NLME Models

Clinical trials generally include several outcome measures of interest for assessing treatment efficacy and harm. Traditionally a single measure, the primary outcome, is selected and used as the basis for the design, including sample size and power. Secondary outcomes are then generally ordered with respect to their clinical relevance and importanc...

This chapter presents the theory for the nonlinear mixed-effects model introduced in Chapter 6. A general formulation of NLME models is presented and illustrated with examples. Estimation methods for fitting NLME models, based on approximations to the likelihood function, are described and discussed. The computational methods used in the nlme funct...

In this chapter we present the theory for the linear mixed-effects model introduced in Chapter 1. A general formulation of LME models is presented and illustrated with examples. Estimation methods for LME models, based on the likelihood or the restricted likelihood of the parameters, are described, together with the computational methods used to im...

As illustrated by the examples in Chapter 1, we will be modeling data from experiments or studies in which the observations are grouped according to one or more nested classifications. Often this classification is by “Subject” or some similar experimental unit. Repeated measures data, longitudinal data, and growth curve data are examples of this ge...

This chapter gives an overview of the nonlinear mixed-effects (NLME) model, introducing its main concepts and ideas through the analysis of real-data examples. The emphasis is on presenting the motivation for using NLME models when analyzing grouped data, while introducing some of the key features in the nlme library for fitting and analyzing such...

This chapter describes the capabilities available in the nlme library for fitting and analyzing linear mixed-effects models with uncorrelated, homoscedastic within-group errors. The lme function, for fitting linear mixede ffects models, is described in detail and its various capabilities and associated methods are illustrated through the analyses o...

This chapter presents the theoretical foundations of the nonlinear mixed-effects model for single- and multilevel grouped data, including the general model formulation and its underlying distributional assumptions. Efficient computational methods for maximum likelihood estimation in the NLME model are described and discussed. Different approximatio...

This chapter describes the nonlinear modeling capabilities available in the nlme library. A brief review of the nonlinear least-squares function nls in S is presented and self-starting models for automatically producing starting values for the coefficients in a nonlinear model are introduced and illustrated. The nlsList function for fitting separat...

In this chapter we have shown examples of constructing, summarizing, and graphically displaying grouped Data objects. These objects include the data, stored as a data frame, and a formula that designates different variables as a response, a primary covariate, and as one or more grouping factors. Other variables can be designated as outer or inner f...

The linear mixed-effects model formulation used in Chapters 2 and 4 allows considerable flexibility in the specification of the random-effects structure, but restricts the within-group errors to be independent, identically distributed random variables with mean zero and constant variance. As illustrated in Chapters 2 and 4, this basiclinear mixed-e...

In this chapter the linear mixed-effects model of Chapters 2 and 4 is extended to include heteroscedastic, correlated within-group errors. We show how the estimation and computational methods of Chapter 2 can be extended to this more general linear mixed-effects model. We introduce several classes of variance functions to characterize heteroscedast...

As seen in Chapter 1, mixed-effects models provide a flexible and powerful tool for analyzing balanced and unbalanced grouped data. These models have gained popularity over the last decade, in part because of the development of reliable and efficient software for fitting and analyzing them. The linear and nonlinear mixed-effects (nlme) library in S...

As shown in the examples in Chapter 6, nonlinear mixed-effects models offer a flexible tool for analyzing grouped data with models that depend nonlinearly upon their parameters. As nonlinear models are usually based on a mechanistic model of the relationship between the response and the covariates, their parameters can have a theoretical interpreta...

Many common statistical models can be expressed as linear models that incorporate both fixed effects, which are parameters associated with an entire population or with certain repeatable levels of experimental factors, and random effects, which are associated with individual experimental units drawn at random from a population. A model with both fi...

This document describes the data sets included in the NLMEDATA subdirectory of the nlme distribution. We have adopted naming conventions for the columns in the data frames and for the levels in the factors within the data frames, especially the grouping factor. These naming conventions are not required. We find that they help us remember the roles...

A multilevel mixed-effects model has random effects at each of several nested levels of grouping of the observed responses. We may use these, for example, when modelling observations taken over time on students who are grouped into classes that are grouped into schools that are grouped into districts. If each of the distributions of the random effe...

A concern in the use of sequential testing procedures is the bias associated with the estimates of treatment differences. Clinical trials that stop early because of evidence of therapeutic benefit are prone to exaggerate the magnitude of the treatment effect. We consider methods for estimating and reducing the bias of treatment differences estimato...

This article is about the organization and visualization of data grouped according to several nested classification factors. We describe data structures and display methods we developed in the S-PLUS language for representing and plotting this type of data. In Section 1 we describe a groupedData class of objects used to represent repeated measures...

The estimation of variance-covariance matrices through optimization of an objective function, such as a log-likelihood function, is usually a difficult numerical problem. Since the estimates should be positive semi-definite matrices, we must use constrained optimization, or employ a parametrization that enforces this condition. We describe here fiv...

Contents 1 Introduction 1 2 The lme class and related methods 1 2.1 The lme function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 The print, summary, and anova methods. . . . . . . . . . . . . . . . . . . . . 2 2.3 The plot method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.4 Other methods...

We describe extensions to the nonlinear modeling facilities in release 3 of S and Splus. These extensions provide classesand methods for fitting and analyzing nonlinear mixed effects models with the two-stage estimation method described by Lindstrom and Bates (1990). They are implemented in a combination of S and C code and complement the classes a...

Mixed-effects models provide a powerful and flexible tool for analyzing clustere d data, such as repeated measures data and nested designs. We describe a set of S functions, classes, and methods for the analysis of both linear and nonlinear mixed-effects models. These extend the linear and nonlinear modeling facilities available in release 3 of S a...

Introduction. Several different nonlinear mixed effects models and estimation methods for their parameters have been proposed in recent years (Sheiner and Beal, 1980; Mallet, Mentre, Steimer and Lokiek, 1988; Lindstrom and Bates, 1990; Vonesh and Carter, 1992; Davidian and Gallant, 1992; Wakefield, Smith, Racine-Poon and Gelfand, 1994). We consider...

## Citations

... Gls regression models both the deterministic and the stochastic component, accounting for the structure of the spatial autocorrelation. For each dependent variable, several models with different correlation structures were tested: a null model with no predictors, a model with no correlation structure (in this case, gls and ordinary least squares regression are equivalent), and gls regressions with linear, exponential, gaussian, spherical and rational correlation structures, using the packages lm and nlme in R (R Core Team, 2017) and following the procedure indicated in Pinheiro and Bates (2000) and Crawley (2007). Model selection was performed using AIC (Akaike Information Criterion), an index that compares model performance based on a compromise between parsimony and goodness of fit (Akaike, 1974). ...

... As data to test H2 are naturally hierarchical in structure, with households nested within villages, and villages nested within climatic areas, we will consider the multilevel structure of model variance by fitting mixed-effect models. For the statistical analysis, we will use the package nlme [30] in R statistical software version 4.2.1. ...

... Before estimating the average effect size and heterogeneity among effect sizes, we identified the best random effects structure for our REM by fitting different models using restricted maximum likelihood estimation (function rma.mv, R package metafor, Viechtbauer, 2010), and then comparing these models using likelihood ratio tests (Zuur 2009;Pinheiro & Bates 2000). To do this, first, we ran an intercept-only model without random effects with the following structure: rma.mv (y i~1 , v i ) with y i the observed effect sizes and v i the corresponding sampling variance. ...

... To predict individuals' willingness to adopt protective measures, we conducted mixedeffect regressions, treating the participant as a random effect to account for the interindividual variance in adopting (different) protective measures [41]. Cross-country variance was accounted for by adding country as a dummy predictor (baseline: Germany). ...

... Depending on the therapeutic, clinical questions, and available data, one might need additional mechanistic detail for the pathway model. With sufficient data, we may also be able to fit nonlinear mixed-effect statistical models for the pathway model parameters [20]. When employed with an experimental model focused on cells from human subjects, the approach can start to yield estimates of salient variability that could be informative of pathway-level variability in patients when accompanied with a suitable translational strategy. ...

... Longitudinal data is often analyzed using the linear mixed effects models or generalized estimation equations (5,6). Functional principal component analysis (FPCA) is another popular method that provides a powerful approach for modelling noisy and irregularly measured longitudinal data. ...

... Because plants were of varied sizes since the beginning of the experiment, all growth estimates were assessed as their relative change across time (relative growth rate, RGR) instead of their actual values. Further, we modelled an equal correlation structure within-group observations across time (corCompSymm), a suitable correlation structure for short time series (Pinheiro and Bates, 2000). The significances of the terms in the model were calculated based on the marginality principle to minimize bias estimates (Smith and Cribbie, 2021). ...

... Consequently, the estimated radial velocity is determined by the linear fitting method [30]. The most common method of linear fitting is the least-squares method. ...

... To assess the overall impact of each variable and their interactions on sexual reproductive allocation ratios, we performed an ANOVA on the completed model (Pinheiro and Bates 2000), using the Anova function (car package; Fox and Weisberg 2019) combined with the lmerTest package (Kuznetsova et al. 2017). To evaluate the overall plasticity within each taxon at each site, we determined the nutrient levels with the minimum and maximum allocation values and then calculated the difference by subtracting the lower nutrient level from the higher. ...

... To investigate the environmental variables associated with plant richness and seedling establishment, we used generalised linear mixed-effects models (GLMMs) and LMMs with block nested within transect as random effects to avoid pseudo-replication (Pinheiro & Bates, 2000). Separate analyses were conducted for each of the two dependent variables associated with the plots, species richness and seedling abundance (Table 1). ...