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

Abstract A common complication that can arise with analyses of high-dimensional data is the repeated use of hypothesis tests. A second complication, especially with small samples, is the reliance on asymptotic p -values. Our proposed approach for addressing both complications uses a scientifically motivated scalar summary statistic, and although not entirely novel, seems rarely used. The method is illustrated using a crossover study of seventeen participants examining the effect of exposure to ozone versus clean air on the DNA methylome, where the multivariate outcome involved 484,531 genomic locations. Our proposed test yields a single null randomization distribution, and thus a single Fisher-exact p -value that is statistically valid whatever the structure of the data. However, the relevance and power of the resultant test requires the careful a priori selection of a single test statistic. The common practice using asymptotic p -values or meaningless thresholds for “significance” is inapposite in general. 

/pdf/high-dimensional-randomization-based-inference-capitalizing-lljh9vid.pdf

High-dimensional randomization-based inference capitalizing on classical design and modern computing

Randomization‐Based Evaluations

: Incomplete data occur commonly in the application of statistics to real data, and there exists a substantial literature on the problem. This article provides an overview of issues for the statistically knowledgeable but not necessarily statistically sophisticated reader. It is intended to appear as an entry in The Encyclopedia of Statistical Sciences. (Author)

Incomplete Data - Encyclopedia Entry.

Consider a completely randomized experiment with two groups, say treatment and control, and a covariate X that has been observed on a subset of the units. The objective is to estimate τ, the treatment effect on the variable Y which is recorded for all units. Although the average Y difference in the two groups, , is unbiased for τ, it is common to try to use X to increase the precision of the estimate of τ by forming a covariance adjustment, especially if the correlation between X and Y, ρ, is large. If X is not fully observed it is not clear whether to and/or how to form the covariance adjusted estimate. In addition to , we consider two covariance adjusted estimates of τ, which ignores the units without X, and which fills in group means for the missing values. We compare the three estimators' variances and find the conditions under which each estimator is best.

Donald B. Rubin

Papers

High-dimensional randomization-based inference capitalizing on classical design and modern computing

Randomization‐Based Evaluations

Incomplete Data - Encyclopedia Entry.

The assessment of a treatment effect when a covariate is observed on a subset of the units: i. simple estimators

Asymptotics and connections to non-Bayesian approaches