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

The Gibbs sampler is an iterative simulation scheme for generating samples that converge to draws from a target distribution π(X) of random variable X. An iterative simulation scheme is used because direct simulation from the target distribution can not be easily implemented. To define the iterative Gibbs sampler, partition the components of X as X = (x1, . . . , xd) where xi is gi-dimensional, and thus X is a g = ∑d i=1 gi dimensional random vector. The Gibbs sampler is easy to implement when the set of d conditional distributions,

A variational control variable for assessing the convergence of the gibbs sampler

Multiple Imputation Methods

Abstracts of papers presented at the 6th Semiannual meeting of the Phytopathological Society of Israel

A Statistical Model for Positron Emission Tomography: Comment

We describe a new method to combine propensity-score matching with regression adjustment in treatment-control studies when outcomes are binary by multiply imputing potential outcomes under control for the matched treated subjects. This enables the estimation of clinically meaningful measures of effect such as the risk difference. We used Monte Carlo simulation to explore the effect of the number of imputed potential outcomes under control for the matched treated subjects on inferences about the risk difference. We found that imputing potential outcomes under control (either single imputation or multiple imputation) resulted in a substantial reduction in bias compared with what was achieved using conventional nearest neighbor matching alone. Increasing the number of imputed potential outcomes under control resulted in more efficient estimation, with more efficient estimation of the estimated risk difference when increasing the number of the imputed potential outcomes. The greatest relative increase in efficiency was achieved by imputing five potential outcomes; once 20 outcomes under control were imputed for each matched treated subject, further improvements in efficiency were negligible. We also examined the effect of the number of these imputed potential outcomes on: (i) estimated standard errors; (ii) mean squared error; (iii) coverage of estimated confidence intervals. We illustrate the application of the method by estimating the effect on the risk of death within 1 year of prescribing beta-blockers to patients discharged from hospital with a diagnosis of heart failure.

Donald B. Rubin

Papers

A variational control variable for assessing the convergence of the gibbs sampler

Multiple Imputation Methods

Abstracts of papers presented at the 6th Semiannual meeting of the Phytopathological Society of Israel

A Statistical Model for Positron Emission Tomography: Comment

Estimating adjusted risk differences by multiply-imputing missing control binary potential outcomes following propensity score-matching