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

Several studies have shown that spatial working memory is impaired in schizophrenia patients. In our study, schizophrenia patients and normal controls participated in a memory test designed to measure both spatial and object working memory. Standard analyses were inappropriate because the test items had differing levels of difficulty. To account for this problem, the data were analyzed using a Bayesian Bivariate Item Response Theory (BBIRT) model. Item response theory is a method for analyzing test scores in which the test items themselves are parameterized in addition to the test-takers’ abilities. Analyzing the data in this way accounts for the fact that the questions were not all equally difficult, and also produces results which are more generalizable and less test-dependent. The analysis was carried out using Gibbs sampling, a Markov Chain Monte Carlo technique which improves upon standard EM methods for IRT models by producing standard error estimates which more accurately represent uncertainty about the parameters.

Working Memory Impairments in Schizophrenia Patients: A Bayesian Bivariate IRT Analysis

Large-sample inference and frequency properties of Bayesian inference

Rubin, D. B. (2005), Causal Inference Using Potential Outcomes : Design, Modeling, Decisions, Journal Of The American Statistical Association, 100, 322-331: Comment by Wong and Reply

A method is given for estimating, in a subjective sense, the effect of non-respondents in a sample survey. The method is based on Bayesian techniques and yields a “confidence interval” for the value of the average response in the survey if all non-respondents had responded. Background information which is recorded for both respondents and non-respondents plays an important role although is not needed. The technique is illustrated with real survey data of 660 schools; 488 had 80 dependent variables recorded, 172 had no dependent variables recorded, and all 660 had 35 background variables recorded. On the basis of this example, it appears as if the method can be useful in practical problems. The general idea, of which the method presented here is only a specific example, can be applied to any problem with non-respondents and/or missing data.

A method for formalizing subjective notions about the effect of non‐respondents in sample surveys

Although education abroad in the US offers participants demonstrable benefits, direct and opportunity costs are cited as primary barriers to broader participation. Yet the degree to which low-income status deters studying abroad and whether additional need-based aid beyond Pell Grants encourages participation remain uncertain. Moreover, not all education abroad programs are equivalent in terms of costs. This study is the first to examine whether need-based aid recipients differentially choose programs of varying duration or programs offered by various provider types. The sample consisted of 221,981 students from 36 institutions of the Consortium for Analysis of Student Success through International Education (CASSIE). Within that sample, 60,477 received Pell grants. Of those recipients, 39% received additional need-based aid. Regression models controlling for student background and context indicated that Pell grant recipients were 3% less likely to study abroad than peers receiving no such aid, and receipt of additional aid increased likelihood by 1% relative to Pell-only recipients. While aid was unrelated to study abroad duration, low-income students were less likely to study with third-party providers. The findings invite financial aid officers to determine thresholds of additional aid necessary to increase participation and to collaborate more systematically with counterparts in international education. 

/pdf/need-based-aid-participation-in-education-abroad-and-program-1qpdk8tl.pdf

Donald B. Rubin

Papers

Working Memory Impairments in Schizophrenia Patients: A Bayesian Bivariate IRT Analysis

Large-sample inference and frequency properties of Bayesian inference

Rubin, D. B. (2005), Causal Inference Using Potential Outcomes : Design, Modeling, Decisions, Journal Of The American Statistical Association, 100, 322-331: Comment by Wong and Reply

A method for formalizing subjective notions about the effect of non‐respondents in sample surveys

Need-Based Aid, Participation in Education Abroad, and Program Type Choice