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

It is sometimes argued that Bayesian inference is unaffected by data-dependent stopping rules. Although this can be true in a formal sense, it is likely that there will be heightened sensitivity to prior assumptions when data-dapendent rules are used rather than stopping rules that do not depend on the data. That is, there is an interaction between violations of prior assumptions and data-dependent stopping rules such that the violations have more severe consequences in repeated practice when data-dependent rules are used. We illustrate this fact in a simple example where 95% intervals are created using a flat prior when in fact the correct prior is normal with positive prior precision ρ. The coverage probabilities of the nominal 95% intervals are less tightly concentrated around .95 when data-dependent stopping rules are used, and the effect becomes stronger as ρ becomes larger.

Sensitivity of Bayes Inference with Data-Dependent Stopping Rules

The precarious state of the educational system existing in the inner-cities of the U.S., including its potential causes and solutions, has been a popular topic of debate in recent years. Part of the difficulty in resolving this debate is the lack of solid empirical evidence regarding the true impact of educational initiatives. For example, educational researchers rarely are able to engage in controlled, randomized experiments. The efficacy of so-called “school choice” programs has been a particularly contentious issue. A current multi-million dollar evaluation of the New York School Choice Scholarship Program (NYSCSP) endeavors to shed some light on this issue. This study can be favorably contrasted with other school choice evaluations in terms of the consideration that went into the randomized experimental design (a completely new design, the Propensity Matched Pairs Design, is being implemented) and the rigorous data collection and compliance-encouraging efforts. In fact, this study benefits from the authors’ previous experiences with the analysis of data from the Milwaukee Parental Choice Program, which, although randomized, was relatively poorly implemented as an experiment.

/pdf/school-choice-in-ny-city-a-bayesian-analysis-of-an-imperfect-2cpc1cbpsw.pdf

School Choice in NY City: A Bayesian Analysis of an Imperfect Randomized Experiment

The estimation of causal effects in nonrandomized studies should comprise two distinct phases: design, with no outcome data available; and analysis of the outcome data according to a specified protocol. Here, we review and compare point and interval estimates of common statistical procedures for estimating causal effects (i.e. matching, subclassification, weighting, and model-based adjustment) with a scalar continuous covariate and a scalar continuous outcome. We show, using an extensive simulation, that some highly advocated methods have poor operating characteristics. In many conditions, matching for the point estimate combined with within-group matching for sampling variance estimation, with or without covariance adjustment, appears to be the most efficient valid method of those evaluated. These results provide new conclusions and advice regarding the merits of currently used procedures.

/pdf/estimation-of-causal-effects-of-binary-treatments-in-2c3wssmzz0.pdf

Estimation of causal effects of binary treatments in unconfounded studies with one continuous covariate

/pdf/a-research-agenda-for-assessment-and-propagation-of-model-2snwfff7ox.pdf

A Research Agenda for Assessment and Propagation of Model Uncertainty

One type of pharmacokinetic/pharmacodynamic (PK/PD) relationship that is used to characterize the therapeutic action of a drug is the relationship between some univariate summary of the plasma-concentration-versus-time profile and the drug effect on a response outcome. Operationally, such a relationship may be observed in a large clinical trial where randomly sampled patients are randomized to different values of the concentration summary. If, under such conditions, the relationship between concentration and effect does not depend on the dose needed to attain the target concentration, such a relationship will be called a true PK/PD relationship. When the true PK/PD relationship is assessed as an object of estimation in a dose-controlled clinical trial (i.e. when dose is randomized), observed drug concentration is an outcome variable. The estimated PK/PD relationship between observed outcome and observed concentration, which we then refer to as the conventional PK/PD relationship, may be biased for the true PK/PD relationship. Because of this bias, the conventional relationship is called confounded for the true one. We show that diagnostics for confounding can be devised under reasonable assumptions. We then apply these diagnostics to PK/PD assessments of adults and children on oxcarbazepine adjunctive therapy. It was necessary to demonstrate the similarity of the true PK/PD relationships of adults and children on adjunctive therapy in order to support the approval of oxcarbazepine monotherapy in children by a bridging argument.

Donald B. Rubin

Papers

Sensitivity of Bayes Inference with Data-Dependent Stopping Rules

School Choice in NY City: A Bayesian Analysis of an Imperfect Randomized Experiment

Estimation of causal effects of binary treatments in unconfounded studies with one continuous covariate

A Research Agenda for Assessment and Propagation of Model Uncertainty

Diagnostics for confounding in PK/PD models for oxcarbazepine