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

In comparative high-throughput sequencing assays, a fundamental task is the analysis of count data, such as read counts per gene in RNA-seq, for evidence of systematic changes across experimental conditions. Small replicate numbers, discreteness, large dynamic range and the presence of outliers require a suitable statistical approach. We present DESeq2, a method for differential analysis of count data, using shrinkage estimation for dispersions and fold changes to improve stability and interpretability of estimates. This enables a more quantitative analysis focused on the strength rather than the mere presence of differential expression. The DESeq2 package is available at http://www.bioconductor.org/packages/release/bioc/html/DESeq2.html
 .

/pdf/moderated-estimation-of-fold-change-and-dispersion-for-rna-3ywnbkcan0.pdf

Moderated estimation of fold change and dispersion for RNA-seq data with DESeq2

Summary: It is expected that emerging digital gene expression (DGE) technologies will overtake microarray technologies in the near future for many functional genomics applications. One of the fundamental data analysis tasks, especially for gene expression studies, involves determining whether there is evidence that counts for a transcript or exon are significantly different across experimental conditions. edgeR is a Bioconductor software package for examining differential expression of replicated count data. An overdispersed Poisson model is used to account for both biological and technical variability. Empirical Bayes methods are used to moderate the degree of overdispersion across transcripts, improving the reliability of inference. The methodology can be used even with the most minimal levels of replication, provided at least one phenotype or experimental condition is replicated. The software may have other applications beyond sequencing data, such as proteome peptide count data. Availability: The package is freely available under the LGPL licence from the Bioconductor web site (http://bioconductor.org).

http://dmrocke.ucdavis.edu/Class/BST226.2014.Winter/edgeR%20bioinformatics%202010.pdf

edgeR: a Bioconductor package for differential expression analysis of digital gene expression data.

limma is an R/Bioconductor software package that provides an integrated solution for analysing data from gene expression experiments. It contains rich features for handling complex experimental designs and for information borrowing to overcome the problem of small sample sizes. Over the past decade, limma has been a popular choice for gene discovery through differential expression analyses of microarray and high-throughput PCR data. The package contains particularly strong facilities for reading, normalizing and exploring such data. Recently, the capabilities of limma have been significantly expanded in two important directions. First, the package can now perform both differential expression and differential splicing analyses of RNA sequencing (RNA-seq) data. All the downstream analysis tools previously restricted to microarray data are now available for RNA-seq as well. These capabilities allow users to analyse both RNA-seq and microarray data with very similar pipelines. Second, the package is now able to go past the traditional gene-wise expression analyses in a variety of ways, analysing expression profiles in terms of co-regulated sets of genes or in terms of higher-order expression signatures. This provides enhanced possibilities for biological interpretation of gene expression differences. This article reviews the philosophy and design of the limma package, summarizing both new and historical features, with an emphasis on recent enhancements and features that have not been previously described.

/pdf/limma-powers-differential-expression-analyses-for-rna-4yl7ae3arh.pdf

limma powers differential expression analyses for RNA-sequencing and microarray studies

In comparative high-throughput sequencing assays, a fundamental task is the analysis of count data, such as read counts per gene in RNA-Seq data, for evidence of systematic changes across experimental conditions. Small replicate numbers, discreteness, large dynamic range and the presence of outliers require a suitable statistical approach. We present DESeq2, a method for differential analysis of count data. DESeq2 uses shrinkage estimation for dispersions and fold changes to improve stability and interpretability of the estimates. This enables a more quantitative analysis focused on the strength rather than the mere presence of differential expression and facilitates downstream tasks such as gene ranking and visualization. DESeq2 is available as an R/Bioconductor package.

/pdf/moderated-estimation-of-fold-change-and-dispersion-for-rna-59glcgpwlk.pdf

https://www.stat.purdue.edu/~lshu/documents/lme/paper/MixedComp.pdf

Linear mixed models and penalized least squares

The cave of shadows: Addressing the human factor with generalized additive mixed models

Nonlinear Regression: Iterative Estimation and Linear Approximations

The RcppEigen package provides access from R (R Core Team 2012a) to the Eigen (Guennebaud, Jacob, and others 2012) C++ template library for numerical linear algebra. Rcpp (Eddelbuettel and Francois 2011, 2012) classes and specializations of the C++ templated functions as and wrap from Rcpp provide the "glue" for passing objects from R to C++ and back. Several introductory examples are presented. This is followed by an in-depth discussion of various available approaches for solving least-squares problems, including rank-revealing methods, concluding with an empirical run-time comparison. Last but not least, sparse matrix methods are discussed.

/pdf/fast-and-elegant-numerical-linear-algebra-using-the-1ll21iu8ju.pdf

Fast and Elegant Numerical Linear Algebra Using the RcppEigen Package

These Fortran-77 subroutines provide building blocks for Generalized Cross-Validation (GCV) (Craven and Wahba, 1979) calculations in data analysis and data smoothing including ridge regression (Golub, Heath, and Wahba, 1979), thin plate smoothing splines (Wahba and Wendelberger, 1980), deconvolution (Wahba, 1982d), smoothing of generalized linear models (O'sullivan, Yandell and Raynor 1986, Green 1984 and Green and Yandell 1985), and ill-posed problems (Nychka et al, 1984, O'sullivan and Wahba, 1985) We present some of the types of problems for which GCV is a useful method of choosing a smoothing or regularization parameter and we describe the structure of the subroutinesRidge Regression: A familiar example of a smoothing parameter is the ridge parameter X in the ridge regression problem which we write

Douglas M. Bates

Papers

Linear mixed models and penalized least squares

The cave of shadows: Addressing the human factor with generalized additive mixed models

Nonlinear Regression: Iterative Estimation and Linear Approximations

Fast and Elegant Numerical Linear Algebra Using the RcppEigen Package

Gcvpack – routines for generalized cross validation