Fixed and Random Effects Selection in Mixed Effects Models

Summary We consider selecting both fixed and random effects in a general class of mixed effects models using maximum penalized likelihood (MPL) estimation along with the smoothly clipped absolute deviation (SCAD) and adaptive least absolute shrinkage and selection operator (ALASSO) penalty functions. The MPL estimates are shown to possess consistency and sparsity properties and asymptotic normality. A model selection criterion, called the IC_Q statistic, is proposed for selecting the penalty parameters ( Ibrahim, Zhu, and Tang, 2008 , Journal of the American Statistical Association 103, 1648–1658). The variable selection procedure based on IC_Q is shown to consistently select important fixed and random effects. The methodology is very general and can be applied to numerous situations involving random effects, including generalized linear mixed models. Simulation studies and a real data set from a Yale infant growth study are used to illustrate the proposed methodology.     

A note on MAR, identifying restrictions, model comparison, and sensitivity analysis in pattern mixture models with and without covariates for incomplete data

Summary Pattern mixture modeling is a popular approach for handling incomplete longitudinal data. Such models are not identifiable by construction. Identifying restrictions is one approach to mixture model identification ( Little, 1995 , Journal of the American Statistical Association 90 , 1112–1121; Little and Wang, 1996 , Biometrics 52 , 98–111; Thijs et al., 2002 , Biostatistics 3 , 245–265; Kenward, Molenberghs, and Thijs, 2003 , Biometrika 90 , 53–71; Daniels and Hogan, 2008 , in Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis) and is a natural starting point for missing not at random sensitivity analysis ( Thijs et al., 2002 , Biostatistics 3 , 245–265; Daniels and Hogan, 2008 , in Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis). However, when the pattern specific models are multivariate normal, identifying restrictions corresponding to missing at random (MAR) may not exist. Furthermore, identification strategies can be problematic in models with covariates (e.g., baseline covariates with time‐invariant coefficients). In this article, we explore conditions necessary for identifying restrictions that result in MAR to exist under a multivariate normality assumption and strategies for identifying sensitivity parameters for sensitivity analysis or for a fully Bayesian analysis with informative priors. In addition, we propose alternative modeling and sensitivity analysis strategies under a less restrictive assumption for the distribution of the observed response data. We adopt the deviance information criterion for model comparison and perform a simulation study to evaluate the performances of the different modeling approaches. We also apply the methods to a longitudinal clinical trial. Problems caused by baseline covariates with time‐invariant coefficients are investigated and an alternative identifying restriction based on residuals is proposed as a solution.     

Statistical designs for two-color spotted microarray experiments

Two-color cDNA or oligonucleotide-based spotted microarrays have been commonly used in measuring the expression levels of thousands of genes simultaneously. To realize the immense potential of this powerful new technology, budgeted within limited resources or other constraints, practical designs with high efficiencies are in demand. In this study, we address the design issue concerning the arrangement of the mRNA samples labeled with fluorescent dyes and hybridized on the slides. A normalization model is proposed to characterize major sources of systematic variation in a two-color microarray experiment. This normalization model establishes a connection between designs for two-color microarray experiments with a particular class of classical row-column designs. A heuristic algorithm for constructing A-optimal or highly efficient designs is provided. Statistical optimality results are found for some of the designs generated from the algorithm. It is believed that the constructed designs are the best or very close to the best possible for estimating the relative gene expression levels among the mRNA samples of interest.     

A modified Bayes information criterion with applications to the analysis of comparative genomic hybridization data

In the analysis of data generated by change-point processes, one critical challenge is to determine the number of change-points. The classic Bayes information criterion (BIC) statistic does not work well here because of irregularities in the likelihood function. By asymptotic approximation of the Bayes factor, we derive a modified BIC for the model of Brownian motion with changing drift. The modified BIC is similar to the classic BIC in the sense that the first term consists of the log likelihood, but it differs in the terms that penalize for model dimension. As an example of application, this new statistic is used to analyze array-based comparative genomic hybridization (array-CGH) data. Array-CGH measures the number of chromosome copies at each genome location of a cell sample, and is useful for finding the regions of genome deletion and amplification in tumor cells. The modified BIC performs well compared to existing methods in accurately choosing the number of regions of changed copy number. Unlike existing methods, it does not rely on tuning parameters or intensive computing. Thus it is impartial and easier to understand and to use.     

Empirical‐likelihood‐based criteria for model selection on marginal analysis of longitudinal data with dropout missingness

Longitudinal data are common in clinical trials and observational studies, where missing outcomes due to dropouts are always encountered. Under such context with the assumption of missing at random, the weighted generalized estimating equation (WGEE) approach is widely adopted for marginal analysis. Model selection on marginal mean regression is a crucial aspect of data analysis, and identifying an appropriate correlation structure for model fitting may also be of interest and importance. However, the existing information criteria for model selection in WGEE have limitations, such as separate criteria for the selection of marginal mean and correlation structures, unsatisfactory selection performance in small‐sample setups, and so forth. In particular, there are few studies to develop joint information criteria for selection of both marginal mean and correlation structures. In this work, by embedding empirical likelihood into the WGEE framework, we propose two innovative information criteria named a joint empirical Akaike information criterion and a joint empirical Bayesian information criterion, which can simultaneously select the variables for marginal mean regression and also correlation structure. Through extensive simulation studies, these empirical‐likelihood‐based criteria exhibit robustness, flexibility, and outperformance compared to the other criteria including the weighted quasi‐likelihood under the independence model criterion, the missing longitudinal information criterion, and the joint longitudinal information criterion. In addition, we provide a theoretical justification of our proposed criteria, and present two real data examples in practice for further illustration.