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

This article considers causal inference for treatment contrasts from a randomized experiment using potential outcomes in a finite population setting. Adopting a Neymanian repeated sampling approach that integrates such causal inference with finite population survey sampling, an inferential framework is developed for general mechanisms of assigning experimental units to multiple treatments. This framework extends classical methods by allowing the possibility of randomization restrictions and unequal replications. Novel conditions that are “milder” than strict additivity of treatment effects, yet permit unbiased estimation of the finite population sampling variance of any treatment contrast estimator, are derived. The consequences of departures from such conditions are also studied under the criterion of minimax bias, and a new justification for using the Neymanian conservative sampling variance estimator in experiments is provided. The proposed approach can readily be extended to the case of treatm...

Using Standard Tools From Finite Population Sampling to Improve Causal Inference for Complex Experiments

A great deal of concern has been expressed in recent years about the fairness of standardized admissions tests. Unfairness can be introduced in test content, test administration procedures, and in the use of the test results. The purpose of this study is to investigate one aspect of the test administration procedures-test-taking time. For tests designed to be measures of "power" rather than "speed," it is expected that most examinees will have time to complete nearly all of the questions within the time allotted. If certain demographic subgroups of examinees are systematically given too little time to complete the test sections, it may be that their scores are depressed because of timing limitations and not because of lower ability. Although several studies have investigated the effect of test-taking time on the scores of minority groups, little evidence showing subgroup bias has been found. In two studies of reading comprehension questions on the Law School Admissions Test, Evans and Reilly (1972a, 1972b) found that increasing the amount of time per question did not differentially increase scores, i.e., although scores might improve with increasing time, the increase did not vary with race (1972a, 1972b) or sex (1972a). A similar study utilizing a special quantitative section of the Admission Test for Graduate Study in Business again reported no significant interaction between ethnic groups and time factors (Evans & Reilly, 1973). Although no "biasing" effects resulting from timing were found in the previously cited studies, the results are not definitive for other admissions tests or for other conditions. First, only two types of questions (reading comprehension and regular mathematics) were investigated. Second, these studies did not consider certain important groups of interest. In addition to ethnic difference in the impact of test timing, it might be hypothesized that time would differentially affect scores of women, especially on quantitative sections, since proportionally fewer women than men are in quantitative fields of study. In addition, scores for students who have been out of school for some time might be differentially affected by the test timing because they are less accustomed to working in timed situations and to taking multiple choice tests. None of the studies reviewed in the literature specifically addressed these two problems. The present study includes a wider range of item types as well as subgroups defined by sex and years since receipt of the baccalaureate degree. The following research question was investigated separately for verbal and quantitative experimental sections of the Graduate Record Examinations (GRE) Aptitude Test: Does increasing the amount of time per question have a differential effect on the scores (after controlling for initial ability) of examinee subgroups as defined by sex, by race (Black and White), and by the number of years that have elapsed since the baccalaureate degree was obtained?

Effect of increased test‐taking time on test scores by ethnic group, years out of school, and sex

Single Imputation Methods

Consider a statistical analysis that draws causal inferences from an observational dataset, inferences that are presented as being valid in the standard frequentist senses; i.e. the analysis produces: (1) consistent point estimates, (2) valid p-values, valid in the sense of rejecting true null hypotheses at the nominal level or less often, and/or (3) confidence intervals, which are presented as having at least their nominal coverage for their estimands. For the hypothetical validity of these statements, the analysis must embed the observational study in a hypothetical randomized experiment that created the observed data, or a subset of that hypothetical randomized data set. This multistage effort with thought-provoking tasks involves: (1) a purely conceptual stage that precisely formulate the causal question in terms of a hypothetical randomized experiment where the exposure is assigned to units; (2) a design stage that approximates a randomized experiment before any outcome data are observed, (3) a statistical analysis stage comparing the outcomes of interest in the exposed and non-exposed units of the hypothetical randomized experiment, and (4) a summary stage providing conclusions about statistical evidence for the sizes of possible causal effects. Stages 2 and 3 may rely on modern computing to implement the effort, whereas Stage 1 demands careful scientific argumentation to make the embedding plausible to scientific readers of the proffered statistical analysis. Otherwise, the resulting analysis is vulnerable to criticism for being simply a presentation of scientifically meaningless arithmetic calculations. The conceptually most demanding tasks are often the most scientifically interesting to the dedicated researcher and readers of the resulting statistical analyses. This perspective is rarely implemented with any rigor, for example, completely eschewing the first stage. We illustrate our approach using an example examining the effect of parental smoking on children's lung function collected in families living in East Boston in the 1970s.

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Bridging observational studies and randomized experiments by embedding the former in the latter

A central problem in epidemiology and medical statistics is how to draw inferences about the causal effects of treatments (ie, interventions) from randomized and nonrandomized data For example, does the new drug really reduce heart disease, or does exposure to that chemical in drinking water increase cancer rates relative to drinking water without that chemical? This chapter provides an overview of the approach to the estimation of such causal effects based on the concept of potential outcomes We discuss randomization-based approaches and the Bayesian posterior predictive approach

Donald B. Rubin

Papers

Using Standard Tools From Finite Population Sampling to Improve Causal Inference for Complex Experiments

Effect of increased test‐taking time on test scores by ethnic group, years out of school, and sex

Single Imputation Methods

Bridging observational studies and randomized experiments by embedding the former in the latter

2 Statistical Inference for Causal Effects, With Emphasis on Applications in Epidemiology and Medical Statistics