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

INTRODUCTION In the previous chapter we analyzed stratified randomized experiments, where a sample of size N was partitioned into J strata, and within each stratum a completely randomized experiment was conducted. In this chapter we consider a special case of the stratified randomized experiment. Each stratum contains exactly two units, with one randomly selected to be assigned to the treatment group, and the other one assigned to the control group. Such a design is known as a pairwise randomized experiment or paired comparison . Although this can be viewed simply as a special case of a stratified randomized experiment, there are two features of this design that warrant special attention. First, the fact that there is only a single unit in each treatment group in each stratum (or pair in this case) implies that the Neyman sampling variance estimator that we discussed in the chapters on completely randomized experiments (Chapter 6) and stratified randomized experiments (Chapter 9) cannot be used; that estimator requires the presence of at least two units assigned to each treatment in each stratum. Second, each stratum has the same proportion of treated units, which allows us to analyze the within-stratum estimates symmetrically; the natural estimator for the average treatment effect weights each stratum equally. As in the case of stratified randomized experiments, the motivation for eliminating some of the possible assignments in pairwise randomized experiments is that a priori those values of the assignment vectors that are eliminated are expected to lead to less informative inferences. This argument relies on the within-pair variation in potential outcomes being small relative to the between-pair variation. Often the assignment to pairs is based on covariates. Units are matched to other units based on their similarity in these covariates, with the expectation that this similarity corresponds to similarity in the potential outcomes under each treatment. Suppose, for example, that the treatment is an expensive surgical procedure for a relatively common medical condition. It may not be financially feasible to apply the treatment to many individuals.

Pairwise Randomized Experiments

Causality in Experiments and Observational Studies

By 'partially post-hoc' subgroup analyses, we mean analyses that compare existing data from a randomized experiment-from which a subgroup specification is derived-to new, subgroup-only experimental data. We describe a motivating example in which partially post hoc subgroup analyses instigated statistical debate about a medical device's efficacy. We clarify the source of such analyses' invalidity and then propose a randomization-based approach for generating valid posterior predictive p-values for such partially post hoc subgroups. Lastly, we investigate the approach's operating characteristics in a simple illustrative setting through a series of simulations, showing that it can have desirable properties under both null and alternative hypotheses.

/pdf/valid-randomization-based-p-values-for-partially-post-hoc-3f17tn73ea.pdf

Valid randomization-based p-values for partially post hoc subgroup analyses.

Management of chronic diseases often involves monitoring one or more outcomes over time. Modeling long term disease progression in untreated patients can be useful for drug development as well as patient monitoring. The outcomes of interest in chronic diseases often have expected monotone or approximately monotone progression over time, possibly preceded by a period during which the outcome measurement is relatively constant. Many diseases have some treatment available for current patients, if only to relieve some symptoms or to slow progression, making disease modeling in untreated current patients difficult. Then historical data can often be useful for developing models for disease progression without treatment. 
The first part of this thesis presents a model for disease progression in untreated patients suffering from Fabry disease, an X-linked recessive disorder. We develop a Bayesian hierarchical random effects regression model for serum creatinine, a critically important blood measurement for Fabry patients, using a historical data base compiled from medical records and patient registries. We then use this model and results from the historical patients' data to aid in multiply imputing missing placebo data from a clinical trial for a new drug specifically developed to treat Fabry disease; data are missing when placebo patients drop out or switch to the new drug, and thus imputation using only the current untreated patients would rely solely on extrapolation. 
The second and third parts of this thesis expand upon two topics encountered in developing the disease progression models considered in Part One. Part Two outlines a general strategy for testing the correctness of software for fitting Bayesian models, capitalizing on properties of Bayesian posterior distributions. Software testing becomes a critically important enterprise with models fit using Markov Chain Monte Carlo methods. Part Three presents a method for constructing vague prior distributions for complex Bayesian models, based on the asymptotic normality of the likelihood function. Throughout, we illustrate concepts and techniques using the disease progression model developed in Part One.

Modeling monotone nonlinear disease progression and checking the correctness of the associated software

Randomized encouragement designs, terminology established in the seminal article by Holland (1988) although used earlier (eg, Swinton, 1975), are the norm when dealing with human populations At least in much of the world today, thankfully, we cannot force anyone to take a randomly assigned treatment; rather, we can only encourage them to do so, typically after describing some details of what to expect under each of the treatment conditions prior to participation, so-called informed consent As a consequence, human experiments often face the complication of noncompliance with assigned treatment For example, the treatment group may be randomly assigned to be encouraged to study more, whereas the control group receives no extra encouragement In this example, hours of studying will be measured in both groups of the study – treatment and control – as will the primary outcome variable, final achievement on a test Not only is it of interest to study the effects of the encouragement on the amount of studying and on achievement, but it is also of interest to investigate the relationship between the amount of studying, an intermediate outcome variable, on the primary outcome variable in the treatment and control groups

Donald B. Rubin

Papers

Pairwise Randomized Experiments

Causality in Experiments and Observational Studies

Valid randomization-based p-values for partially post hoc subgroup analyses.

Modeling monotone nonlinear disease progression and checking the correctness of the associated software

Bayesian Analysis of a Two-Group Randomized Encouragement Design