Additive, Dynamic and Multiplicative Regression

We survey and compare model-based approaches to regression for cross-sectional and longitudinal data which extend the classical parametric linear model for Gaussian responses in several aspects and for a variety of settings. Additive models replace the sum of linear functions of regressors by a sum of smooth functions. In dynamic or state space models, still linear in the regressors, coefficients are allowed to vary smoothly with time according to a Bayesian smoothness prior. We show that this is equivalent to imposing a roughness penalty on time-varying coefficients. Admitting the coefficients to vary with the values of other covariates, one obtains a class of varying-coefficient models (Hastie and Tibshirani, 1993), or in another interpretation, multiplicative models. The roughness penalty approach to non- and semiparametric modelling, together with Bayesian justifications, is used as a unifying and general framework for estimation. The methodological discussion is illustrated by some real data applications

Semiparametric Regression of Multidimensional Genetic Pathway Data: Least-Squares Kernel Machines and Linear Mixed Models

We consider a semiparametric regression model that relates a normal outcome to covariates and a genetic pathway, where the covariate effects are modeled parametrically and the pathway effect of multiple gene expressions is modeled parametrically or nonparametrically using least-squares kernel machines (LSKMs). This unified framework allows a flexible function for the joint effect of multiple genes within a pathway by specifying a kernel function and allows for the possibility that each gene expression effect might be nonlinear and the genes within the same pathway are likely to interact with each other in a complicated way. This semiparametric model also makes it possible to test for the overall genetic pathway effect. We show that the LSKM semiparametric regression can be formulated using a linear mixed model. Estimation and inference hence can proceed within the linear mixed model framework using standard mixed model software. Both the regression coefficients of the covariate effects and the LSKM estimator of the genetic pathway effect can be obtained using the best linear unbiased predictor in the corresponding linear mixed model formulation. The smoothing parameter and the kernel parameter can be estimated as variance components using restricted maximum likelihood. A score test is developed to test for the genetic pathway effect. Model/variable selection within the LSKM framework is discussed. The methods are illustrated using a prostate cancer data set and evaluated using simulations.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/65964/1/j.1541-0420.2007.00799.x.pd

Hierarchical relational models for document networks

We develop the relational topic model (RTM), a hierarchical model of both network structure and node attributes. We focus on document networks, where the attributes of each document are its words, that is, discrete observations taken from a fixed vocabulary. For each pair of documents, the RTM models their link as a binary random variable that is conditioned on their contents. The model can be used to summarize a network of documents, predict links between them, and predict words within them. We derive efficient inference and estimation algorithms based on variational methods that take advantage of sparsity and scale with the number of links. We evaluate the predictive performance of the RTM for large networks of scientific abstracts, web documents, and geographically tagged news.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS309 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Convergence rates of posterior distributions for noniid observations

We consider the asymptotic behavior of posterior distributions and Bayes estimators based on observations which are required to be neither independent nor identically distributed. We give general results on the rate of convergence of the posterior measure relative to distances derived from a testing criterion. We then specialize our results to independent, nonidentically distributed observations, Markov processes, stationary Gaussian time series and the white noise model. We apply our general results to several examples of infinite-dimensional statistical models including nonparametric regression with normal errors, binary regression, Poisson regression, an interval censoring model, Whittle estimation of the spectral density of a time series and a nonlinear autoregressive model.Comment: Published at http://dx.doi.org/10.1214/009053606000001172 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Item Response Theory Equating with the Non-Equivalent Groups with Covariates Design

We use test score equating to be able to compare different test scores from different test forms. Although it is preferable to use non-equivalent groups with anchor test (NEAT) design, it might be impossible to administer an anchor test due to test security or for other reasons. However, we still know that the groups are non-equivalent, which rules out the use of an equivalent groups (EG) design. A possibility, then, is to use non-equivalent groups with covariates (NEC) design. The overall aim of this work was to propose the use of Item Response Theory (IRT) with a NEC design. We propose the use of mixed-measurement IRT with covariates model (Tay, Newman & Vermunt, 2011; 2016) within IRT observed-score equating and IRT true-score equating to model both test scores and covariates. The proposed test equating methods are examined with simulations. The results are compared with IRT observed-score equating and IRT true-score equating methods using the EG and NEAT designs. The results from the simulations show that IRT true-score equating method doesn't work, but support the IRT observed-score equating method for which the standard errors of the equating are lower when covariates are included in the IRT model than if they are excluded. One real test dataset illustrate that the IRT observed-score equating method can be used in practice