Search

Generalized Linear Models (GLMs) are a popular class of regression models when the responses follow a distribution in the exponential family. In real data the variability often deviates from the relation imposed by the exponential family distribution, which results in over- or underdispersion. Dispersion effects may even vary in the data. Such datasets do not follow the traditional GLM distributional assumptions, leading to unreliable inference. Therefore, the family of double exponential distributions has been proposed, which models both the mean and the dispersion as a function of covariates in the GLM framework. Since standard maximum likelihood inference is highly susceptible to the possible presence of outliers, we propose the robust double exponential (RDE) estimator. Asymptotic properties and robustness of the RDE estimator are discussed. A generalized robust quasi-deviance measure is introduced which constitutes the basis for a stable robust test. Simulations for binomial and Poisson models show the excellent performance of the RDE estimator and corresponding robust tests. Penalized versions of the RDE estimator are developed for sparse estimation with high-dimensional data and for flexible estimation via generalized additive models (GAMs). Real data applications illustrate the relevance of robust inference for dispersion effects in GLMs and GAMs. for this article are available online. [ABSTRACT FROM AUTHOR]

Many scientific and engineering challenges -- ranging from pharmacokineticdrug dosage allocation and personalized medicine to marketing mix (4Ps)recommendations -- require an understanding of the unobserved heterogeneity inorder to develop the best decision making-processes. In this paper, we developa hypothesis test and the corresponding p-value for testing for thesignificance of the homogeneous structure in linear mixed models. A robustmatching moment construction is used for creating a test that adapts to thesize of the model sparsity. When unobserved heterogeneity at a cluster level isconstant, we show that our test is both consistent and unbiased even when thedimension of the model is extremely high. Our theoretical results rely on a newfamily of adaptive sparse estimators of the fixed effects that do not requireconsistent estimation of the random effects. Moreover, our inference results donot require consistent model selection. We showcase that moment matching can beextended to nonlinear mixed effects models and to generalized linear mixedeffects models. In numerical and real data experiments, we find that thedeveloped method is extremely accurate, that it adapts to the size of theunderlying model and is decidedly powerful in the presence of irrelevantcovariates.

In this article we construct simultaneous confidence bands for a smooth curve using penalized spline estimators. We consider three types of estimation methods: (a) as a standard (fixed effect) nonparametric model, (b) using the mixed-model framework with the spline coefficients as random effects, and (c) a full Bayesian approach. The volume-of-tube formula is applied for the first two methods and compared with Bayesian simultaneous confidence bands from a frequentist perspective. We show that the mixed-model formulation of penalized splines can help obtain, at least approximately, confidence bands with either Bayesian or frequentist properties. Simulations and data analysis support the proposed methods. The R package ConfBands accompanies the article. [ABSTRACT FROM AUTHOR]

Two-way functional data consist of a data matrix whose row and column domains are both structured, for example, temporally or spatially, as when the data are time series collected at different locations in space. We extend one-way functional principal component analysis (PCA) to two-way functional data by introducing regularization of both left and right singular vectors in the singular value decomposition (SVD) of the data matrix. We focus on a penalization approach and solve the nontrivial problem of constructing proper two-way penalties from one- way regression penalties. We introduce conditional cross-validated smoothing parameter selection whereby left-singular vectors are cross- validated conditional on right-singular vectors, and vice versa. The concept can be realized as part of an alternating optimization algorithm. In addition to the penalization approach, we briefly consider two-way regularization with basis expansion. The proposed methods are illustrated with one simulated and two real data examples. Supplemental materials available online show that several "natural" approaches to penalized SVDs are flawed and explain why so. [ABSTRACT FROM AUTHOR]

DNA microarrays can provide insight into genetic changes that characterize different stages of a disease process. Accurate identification of these changes has significant therapeutic and diagnostic implications. Statistical analysis for multistage (multigroup) data is challenging, however. ANOVA-based extensions of two-sample Z-tests, a popular method for detecting differentially expressed genes in two groups, do not work well in multigroup settings. False detection rates are high because of variability of the ordinary least squares estimators and because of regression to the mean induced by correlated parameter estimates. We develop a Bayesian rescaled spike and slab hierarchical model specifically designed for the multigroup gene detection problem. Data preprocessing steps are introduced to deal with unique features of inicroarray data and to enhance selection performance, We show theoretically that spike and slab models naturally encourage sparse solutions through a process called selective shrinkage. This translates into oracle-like gene selection risk performance compared with ordinary least squares estimates. The methodology is illustrated on a large microarray repository of samples from different clinical stages of metastatic colon cancer. Through a functional analysis of selected genes, we show that spike and slab models identify important biological signals while minimizing biologically implausible false detections. [ABSTRACT FROM AUTHOR]