We introduce General Multilevel Models and discuss the estimation procedures that may be used to fit multilevel models. We apply the proposed procedures to three-level binary data generated in a simulation study. We compare the procedures by two criteria, Bias and efficiency. We find that the estimates of the fixed effects and variance components are substantially and significantly biased using Longford's Approximation and Goldstein's Generalized Least Squares approaches by two software packages VARCL and ML3. These estimates are not significantly biased and are very close to real values when we use Markov Chain Monte Carlo (MCMC) using Gibbs sampling or Nonparametric Maximum Likelihood (NPML) approach. The Gaussian Quadrature (GQ) approach, even with small number of mass points results in consistent estimates but computationally problematic. We conclude that the MCMC and the NPML approaches are the recommended procedures to fit multilevel models.

Graphic processing units (GPUs) are rapidly gaining maturity as powerful general parallel computing devices. A key feature in the development of modern GPUs has been the advancement of the programming model and programming tools. Compute Unified Device Architecture (CUDA) is a software platform for massively parallel high-performance computing on Nvidia many-core GPUs. In functional magnetic resonance imaging (fMRI), the volume of the data to be processed, and the type of statistical analysis to perform call for high-performance computing strategies. In this work, we present the main features of the R-CUDA package cudaBayesreg which implements in CUDA the core of a Bayesian multilevel model for the analysis of brain fMRI data. The statistical model implements a Gibbs sampler for multilevel/hierarchical linear models with a normal prior. The main contribution for the increased performance comes from the use of separate threads for fitting the linear regression model at each voxel in parallel. The R-CUDA implementation of the Bayesian model proposed here has been able to reduce significantly the run-time processing of Markov chain Monte Carlo (MCMC) simulations used in Bayesian fMRI data analyses. Presently, cudaBayesreg is only configured for Linux systems with Nvidia CUDA support.

Traditional Rasch estimation of the item and student parameters via marginal maximum likelihood, joint maximum likelihood or conditional maximum likelihood, assume individuals in clustered settings are uncorrelated and items within a test that share a grouping structure are also uncorrelated. These assumptions are often violated, particularly in educational testing situations, in which students are grouped into classrooms and many test items share a common grouping structure, such as a content strand or a reading passage. Consequently, one possible approach is to explicitly recognize the clustered nature of the data and directly incorporate random effects to account for the various dependencies. This article demonstrates how the multilevel Rasch model can be estimated using the functions in R for mixed-effects models with crossed or partially crossed random effects. We demonstrate how to model the following hierarchical data structures: a) individuals clustered in similar settings (e.g., classrooms, schools), b) items nested within a particular group (such as a content strand or a reading passage), and c) how to estimate a teacher x content strand interaction.

The R package HGLMMM has been developed to fit generalized linear models with random effects using the h-likelihood approach. The response variable is allowed to follow a binomial, Poisson, Gaussian or gamma distribution. The distribution of random effects can be specified as Gaussian, gamma, inverse-gamma or beta. Complex structures as multi-membership design or multilevel designs can be handled. Further, dispersion parameters of random components and the residual dispersion (overdispersion) can be modeled as a function of covariates. Overdispersion parameter can be fixed or estimated. Fixed effects in the mean structure can be estimated using extended likelihood or a first order Laplace approximation to the marginal likelihood. Dispersion parameters are estimated using first order adjusted profile likelihood.

The R package mice imputes incomplete multivariate data by chained equations. The software mice 1.0 appeared in the year 2000 as an S-PLUS library, and in 2001 as an R package. mice 1.0 introduced predictor selection, passive imputation and automatic pooling. This article documents mice, which extends the functionality of mice 1.0 in several ways. In mice, the analysis of imputed data is made completely general, whereas the range of models under which pooling works is substantially extended. mice adds new functionality for imputing multilevel data, automatic predictor selection, data handling, post-processing imputed values, specialized pooling routines, model selection tools, and diagnostic graphs. Imputation of categorical data is improved in order to bypass problems caused by perfect prediction. Special attention is paid to transformations, sum scores, indices and interactions using passive imputation, and to the proper setup of the predictor matrix. mice can be downloaded from the Comprehensive R Archive Network. This article provides a hands-on, stepwise approach to solve applied incomplete data problems.

Multilevel acceptance sampling for attributes is used to decide whether a lot from an incoming shipment or outgoing production is accepted or rejected when the product has multiple levels of product quality or multiple types of (mutually exclusive) possible defects. This paper describes a package which provides the tools to create, evaluate, plot, and display the acceptance sampling plans for such lots for both fixed and sequential sampling. The functions for calculating cumulative probabilities for several common multivariate distributions (which are needed in the package) are provided as well.

Variance component models are generally accepted for the analysis of hierarchical structured data. A shortcoming is that outcome variables are still treated as measured without an error. Unreliable variables produce biases in the estimates of the other model parameters. The variability of the relationships across groups and the group-effects on individuals' outcomes differ substantially when taking the measurement error in the dependent variable of the model into account. The multilevel model can be extended to handle measurement error using an item response theory (IRT) model, leading to a multilevelIRT model. This extended multilevel model is in particular suitable for the analysis of educational response data where students are nested in schools and schools are nested within cities/countries.

Multiple imputation is becoming increasingly established as the leading practical approach to modelling partially observed data, under the assumption that the data are missing at random. However, many medical and social datasets are multilevel, and this structure should be reflected not only in the model of interest, but also in the imputation model. In particular, the imputation model should reflect the differences between level 1 variables and level 2 variables (which are constant across level 1 units). This led us to develop the REALCOM-IMPUTE software, which we describe in this article. This software performs multilevel multiple imputation, and handles ordinal and unordered categorical data appropriately. It is freely available on-line, and may be used either as a standalone package, or in conjunction with the multilevel software MLwiN or Stata.

We illustrate how to fit multilevel models in the MLwiN package seamlessly from within Stata using the Stata program runmlwin. We argue that using MLwiN and Stata in combination allows researchers to capitalize on the best features of both packages. We provide examples of how to use runmlwin to fit continuous, binary, ordinal, nominal and mixed response multilevel models by both maximum likelihood and Markov chain Monte Carlo estimation.

Genotyping platforms such as Affymetrix can be used to assess genotype-phenotype as well as copy number-phenotype associations at millions of markers. While genotyping algorithms are largely concordant when assessed on HapMap samples, tools to assess copy number changes are more variable and often discordant. One explanation for the discordance is that copy number estimates are susceptible to systematic differences between groups of samples that were processed at different times or by different labs. Analysis algorithms that do not adjust for batch effects are prone to spurious measures of association. The R package crlmm implements a multilevel model that adjusts for batch effects and provides allele-specific estimates of copy number. This paper illustrates a workflow for the estimation of allele-specific copy number and integration of the marker-level estimates with complimentary Bioconductor software for inferring regions of copy number gain or loss. All analyses are performed in the statistical environment R.