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

Comment: Dose-Response Estimands

Propensity score methods were proposed by Rosenbaum and Rubin [Biometrika 70 (1983) 41--55] as central tools to help assess the causal effects of interventions. Since their introduction more than two decades ago, they have found wide application in a variety of areas, including medical research, economics, epidemiology and education, especially in those situations where randomized experiments are either difficult to perform, or raise ethical questions, or would require extensive delays before answers could be obtained. In the past few years, the number of published applications using propensity score methods to evaluate medical and epidemiological interventions has increased dramatically. Nevertheless, thus far, we believe that there have been few applications of propensity score methods to evaluate marketing interventions (e.g., advertising, promotions), where the tradition is to use generally inappropriate techniques, which focus on the prediction of an outcome from background characteristics and an indicator for the intervention using statistical tools such as least-squares regression, data mining, and so on. With these techniques, an estimated parameter in the model is used to estimate some global ``causal'' effect. This practice can generate grossly incorrect answers that can be self-perpetuating: polishing the Ferraris rather than the Jeeps ``causes'' them to continue to win more races than the Jeeps $\Leftrightarrow$ visiting the high-prescribing doctors rather than the low-prescribing doctors ``causes'' them to continue to write more prescriptions. This presentation will take ``causality'' seriously, not just as a casual concept implying some predictive association in a data set, and will illustrate why propensity score methods are generally superior in practice to the standard predictive approaches for estimating causal effects.

Estimating the Causal Effects of Marketing Interventions Using Propensity Score Methodology

Missing data are a common problem in most epidemiological and medical studies, including surveys and clinical trials. Imputation, or filling in the missing values, is an intuitive and flexible way to handle the incomplete data sets that arise because of such missing data. Here, in addition to imputation, including multiple imputation (MI), we discuss several other strategies and their theoretical background, as well as present some examples and advice on computation. Our focus is on MI, which is a statistically valid strategy for handling missing data, although we review other less sound methods, as well as direct maximum likelihood and Bayesian methods for estimating parameters, which are also valid approaches. The analysis of a multiply-imputed data set is now relatively standard using readily available statistical software. The creation of multiply-imputed data sets is more challenging than their analysis but still straightforward relative to other valid methods of handling missing data, and we discuss available software for doing so. Ad hoc methods, including using singly-imputed data sets, almost always lead to invalid inferences and should be eschewed, especially when the focus is on valid interval estimation or testing hypotheses.

19 Incomplete Data in Epidemiology and Medical Statistics

The problem of equating a new standardized test to an old reference test is considered when the samples for equating are not randomly selected from the target population of test takers. Two problems with equating from biased samples are distinguished: (a) bias in the equating function arising from nonrandom selection of the equating sample, and (b) excessive variance in the equating function at scores that are relatively underrepresented in the equating sample relative to the target population. A theorem is presented that suggests that bias may not be a major problem for equating, even when the marginal distributions of scores are distorted by selection. Empirical analysis of data for equating the Armed Services Vocational Aptitude Battery (ASVAB) based on samples of recruits and applicants supports this contention. Analysis of ASVAB data also indicates that excessive variance in the equating function is a more serious issue. Variance-reducing methods, which smooth the test score distributions using extended beta binomial and loglinear polynomial models before equating by the equipercentile method, are presented. Empirical evidence suggests that these smoothing models are successful and yield equating functions that improve on both equipercentile and linear equating of the raw scores.

Test Equating From Biased Samples, With Application to the Armed Services Vocational Aptitude Battery

This article discusses model underlying the use of 1-sample effect size indicators that permit the comparison of effect sizes obtained from different multiple-choice studies by indexing all studies to the results that would have been obtained if there had been only 2 choices. The effect size indicator, the proportion index, (II), is based on a model implying that the probability of knowing a correct response will decrease as more incorrect choices are offered. For specific applications other models may be more appropriate but, for most applications they have encountered, the authors prefer and recommend their model

Donald B. Rubin

Papers

Comment: Dose-Response Estimands

Estimating the Causal Effects of Marketing Interventions Using Propensity Score Methodology

19 Incomplete Data in Epidemiology and Medical Statistics

Test Equating From Biased Samples, With Application to the Armed Services Vocational Aptitude Battery

Further issues in effect size estimation for one-sample multiple-choice-type data