Summary A survey is given of developments in the area of variance components during the last three decades. The survey covers mainly point estimation, interval estimation, and hypothesis testing concerning the variance components for both balanced and unbalanced models.

Variance Components Analysis: A Selective Literature Survey

The variability among data may have quite complex structure and the splitting of variation into its individual components is important in many areas of applications. In particular, the study of variation through a class of linear models known as random and mixed models is a central topic in statistics with wide ramifications in both theory and applications. Sometimes these models are called components of variance models since the interest in the model lies in investigating the variances of the effects of factor levels which are assumed to be randomly selected from a large population of levels. These variances are also known as variance components since the total variance can be expressed as the sum of these variances and the error variance. Knowledge of the variance components, i.e., the information on the relative importance and magnitude of the sources of variability in the data is of critical importance in many problems since the isolation of the total variance into the individual components of variance would result in the physical elimination of the sources of variability in question.

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A bibliography on variance components an introduction and an update: 1984-2002

Reconstructing a gene network from high-throughput molecular data is an important but challenging task, as the number of parameters to estimate easily is much larger than the sample size. A conventional remedy is to regularize or penalize the model likelihood. In network models, this is often done locally in the neighborhood of each node or gene. However, estimation of the many regularization parameters is often difficult and can result in large statistical uncertainties. In this paper we propose to combine local regularization with global shrinkage of the regularization parameters to borrow strength between genes and improve inference. We employ a simple Bayesian model with nonsparse, conjugate priors to facilitate the use of fast variational approximations to posteriors. We discuss empirical Bayes estimation of hyperparameters of the priors, and propose a novel approach to rank-based posterior thresholding. Using extensive model- and data-based simulations, we demonstrate that the proposed inference strategy outperforms popular (sparse) methods, yields more stable edges, and is more reproducible. The proposed method, termed ShrinkNet, is then applied to Glioblastoma to investigate the interactions between genes associated with patient survival.

https://projecteuclid.org/download/pdfview_1/euclid.aoas/1491616871

Gene network reconstruction using global-local shrinkage priors

This paper extends some of the work presented in Redner and Walker [I9841 on the maximum likelihood estimate of parameters in a mixture model to a Bayesian modal estimate. The problem of determining the mode of the joint posterior distribution is discussed. Necessary conditions are given for a choice of parameters to be the mode and a numerical scheme based on the EM algorithm is presented. Some theoretical remarks on the resulting iterative scheme and simulation results are also given.

Estimating the parameters of mixture models with modal estimators

Reconstructing a gene network from high-throughput molecular data is often a challenging task, as the number of parameters to estimate easily is much larger than the sample size. A conventional remedy is to regularize or penalize the model likelihood. In network models, this is often done locally in the neighbourhood of each node or gene. However, estimation of the many regularization parameters is often difficult and can result in large statistical uncertainties. In this paper we propose to combine local regularization with global shrinkage of the regularization parameters to borrow strength between genes and improve inference. We employ a simple Bayesian model with non-sparse, conjugate priors to facilitate the use of fast variational approximations to posteriors. We discuss empirical Bayes estimation of hyper-parameters of the priors, and propose a novel approach to rank-based posterior thresholding. Using extensive model- and data-based simulations, we demonstrate that the proposed inference strategy outperforms popular (sparse) methods, yields more stable edges, and is more reproducible.

The general mixed linear model, containing both the fixed and random effects, is considered. Using gamma priors for the variance components, the conditional posterior distributions of the fixed effects and the variance components, conditional on the random effects, are obtained. Using the normal approximation for the multiple t distribution, approximations are obtained for the posterior distributions of the variance components in infinite series form. The same approximation Is used to obtain closed expressions for the moments of the variance components. An example is considered to illustrate the procedure and a numerical study examines the closeness of the approximations.

Bayesian inference for the variance components in general mixed linear models

This paper presents a procedure to estimate the variance components and fixed effects of mixed linear models. The mode of the joint posterior distribution of all the parameters is obtained by an iterative technique. The proposed method is illustrated with one-way and two-fold nested random models. Two numerical examples demonstrate the iterative solution.

Muthiya Rajagopalan

Papers

Bayesian inference for the variance components in general mixed linear models

Estimating the parameters of mixed linear models with modal estimators