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.

/pdf/fitting-linear-mixed-effects-models-using-lme4-4zzrpoy9lh.pdf

Fitting Linear Mixed-Effects Models Using lme4

https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.2517-6161.1977.tb01600.x

Maximum likelihood from incomplete data via the EM algorithm

Deep learning is a form of machine learning that enables computers to learn from experience and understand the world in terms of a hierarchy of concepts. Because the computer gathers knowledge from experience, there is no need for a human computer operator to formally specify all the knowledge that the computer needs. The hierarchy of concepts allows the computer to learn complicated concepts by building them out of simpler ones; a graph of these hierarchies would be many layers deep. This book introduces a broad range of topics in deep learning. The text offers mathematical and conceptual background, covering relevant concepts in linear algebra, probability theory and information theory, numerical computation, and machine learning. It describes deep learning techniques used by practitioners in industry, including deep feedforward networks, regularization, optimization algorithms, convolutional networks, sequence modeling, and practical methodology; and it surveys such applications as natural language processing, speech recognition, computer vision, online recommendation systems, bioinformatics, and videogames. Finally, the book offers research perspectives, covering such theoretical topics as linear factor models, autoencoders, representation learning, structured probabilistic models, Monte Carlo methods, the partition function, approximate inference, and deep generative models. Deep Learning can be used by undergraduate or graduate students planning careers in either industry or research, and by software engineers who want to begin using deep learning in their products or platforms. A website offers supplementary material for both readers and instructors.

Deep Learning

Meta-Analysis in Clinical Trials*

We describe latent Dirichlet allocation (LDA), a generative probabilistic model for collections of discrete data such as text corpora. LDA is a three-level hierarchical Bayesian model, in which each item of a collection is modeled as a finite mixture over an underlying set of topics. Each topic is, in turn, modeled as an infinite mixture over an underlying set of topic probabilities. In the context of text modeling, the topic probabilities provide an explicit representation of a document. We present efficient approximate inference techniques based on variational methods and an EM algorithm for empirical Bayes parameter estimation. We report results in document modeling, text classification, and collaborative filtering, comparing to a mixture of unigrams model and the probabilistic LSI model.

/pdf/latent-dirichlet-allocation-4gmscb06en.pdf

Latent dirichlet allocation

Algorithms for multiple imputation and posterior simulation from incomplete multivariate data with ignorable nonresponse

Bayesian casual inference : Appraches to Estimating the Effect of Treating Hospital Type on Cancer Survival in Sweden Using Principal Stratification

Bayesian causal inference: Approaches to estimating the effect of treating hospital type on cancer survival in Sweden using principal stratification

Random effects and longitudinal models are becoming increasingly popular in the analysis of many types of data, including medical and biopharmaceutical, because of their richness and flexibility. They can be, however, difficult to fit using traditional statistical tools. Fortunately, there now exists a burgeoning collection of newer computational methods that can be applied to draw inferences with such models. This review attempts to provide an introduction to some of these techniques by describing them as extensions of the EM algorithm, currently a standard tool for the analysis of longitudinal and random effects models. For clarity of exposition, the extensions are classified into three types: large-sample iterative; large-sample simulation, and small-sample simulation.

Computational aspects of analysing random effects/longitudinal models

In a p-variate weighted least squares multiple regression based on n points, each of the p regression coefficients can be calculated from a univariate weighted regression through the origin using N = ( n p ) composite points. This result can be applied to (1) display the p-dimensional multiple regression as p univariate regressions each based on N composite points, (2) isolate collections of points with large leverage, and (3) characterize the region of regression coefficients that can be obtained from fixed data with varying weights. Consequently, composite points are diagnostically useful in the context of robust regression.

Donald B. Rubin

Papers

Algorithms for multiple imputation and posterior simulation from incomplete multivariate data with ignorable nonresponse

Bayesian causal inference: Approaches to estimating the effect of treating hospital type on cancer survival in Sweden using principal stratification

Computational aspects of analysing random effects/longitudinal models

Composite Points in Weighted Least Squares Regressions

Underlying Bayesian Theory