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 Analysis of Transformed Data: Comment

INTRODUCTION As discussed in Chapter 2, Fisher appears to have been the first to grasp fully the importance of physical randomization for credibly assessing causal effects (1925, 1936). A few years earlier, Neyman (1923) had introduced the language and the notation of potential outcomes, using this notation to define causal effects as if the assignments were determined by random draws from an urn, but he did not take the next logical step of appreciating the importance of actually randomizing. It was instead Fisher who made this leap. Given data from a completely randomized experiment, Fisher was intent on assessing the sharp null hypothesis (or exact null hypothesis , Fisher, 1935) of no effect of the active versus control treatment, that is, the null hypothesis under which, for each unit in the experiment, both values of the potential outcomes are identical. In this setting, Fisher developed methods for calculating “p-values.” We refer to them as Fisher Exact P-values (FEPs), although we use them more generally than Fisher originally proposed. Note that Fisher's null hypothesis of no effect of the treatment versus control whatsoever is distinct from the possibly more practical question of whether the typical (e.g., average) treatment effect across all units is zero. The latter is a weaker hypothesis, because the average treatment effect may be zero even when for some units the treatment effect is positive, as long as for some others the effect is negative. We discuss the testing of hypotheses on, and inference for, average treatment effects in Chapter 6. Under Fisher's null hypothesis, and under sharp null hypotheses more generally, for units with either potential outcome observed, the other potential outcome is known; and so, under such a sharp null hypothesis, both potential outcomes are “known” for each unit in the sample – being either directly observed or inferred through the sharp null hypothesis. Consider any test statistic T : a function of the stochastic assignment vector, W ; the observed outcomes, Y obs ; and any pre-treatment variables, X .

Fisher's Exact P-Values for Completely Randomized Experiments

While some stakeholders presume that studying abroad distracts students from efficient pursuit of their programs of study, others regard education abroad as a high impact practice that fosters student engagement and hence college completion. The Consortium for Analysis of Student Success through International Education (CASSIE), compiled semester-by-semester records from 221,981 students across 35 institutions. Of those students, 30,549 had studied abroad. Using nearest-neighbor matching techniques that accounted for a myriad of potentially confounding variables along with matching on institution, the analysis found positive impacts of education abroad on graduation within 4 and 6 years and on cumulative GPA at graduation. A very small increase in credit hours earned emerged, counterbalanced by a small decrease in time-to-degree associated with studying abroad. Overall, the results warrant conclusions that studying abroad does not impede timely graduation. To the contrary, encouraging students to study abroad promotes college completion. These results held similarly for students who had multiple study abroad experiences, and who have studied abroad for varying program lengths. 

/pdf/education-abroad-and-college-completion-155tely9.pdf

Education Abroad and College Completion

A catalytic prior distribution is designed to stabilize a high-dimensional "working model" by shrinking it toward a "simplified model." The shrinkage is achieved by supplementing the observed data with a small amount of "synthetic data" generated from a predictive distribution under the simpler model. We apply this framework to generalized linear models, where we propose various strategies for the specification of a tuning parameter governing the degree of shrinkage and study resultant theoretical properties. In simulations, the resulting posterior estimation using such a catalytic prior outperforms maximum likelihood estimation from the working model and is generally comparable with or superior to existing competitive methods in terms of frequentist prediction accuracy of point estimation and coverage accuracy of interval estimation. The catalytic priors have simple interpretations and are easy to formulate.

https://www.pnas.org/content/pnas/early/2020/05/14/1920913117.full.pdf

Catalytic prior distributions with application to generalized linear models.

Studies employing within-subjects designs may be compared with those employing between-subjects designs in a variety of ways. We discuss and illustrate the comparisons of variabilities, including within-condition variances and precisions as well as the comparisons of means and of mean differences. Our discussion emphasizes the importance of trying to understand the sources of differences.

Donald B. Rubin

Papers

The Analysis of Transformed Data: Comment

Fisher's Exact P-Values for Completely Randomized Experiments

Education Abroad and College Completion

Catalytic prior distributions with application to generalized linear models.

Comparing Within- and Between-Subjects Studies