Robust estimation of variance components

New robust estimates for variance components are introduced. Two simple models are considered: the balanced one-way classification model with a random factor and the balanced mixed model with one random factor and one fixed factor. However, the method of estimation proposed can be extended to more complex models. The new method of estimation we propose is based on the relationship between the variance components and the coefficients of the least-mean-squared-error predictor between two observations of the same group. This relationship enables us to transform the problem of estimating the variance components into the problem of estimating the coefficients of a simple linear regression model. The variance-component estimators derived from the least-squares regression estimates are shown to coincide with the maximum-likelihood estimates. Robust estimates of the variance components can be obtained by replacing the least-squares estimates by robust regression estimates. In particular, a Monte Carlo study shows that for outlier-contaminated normal samples, the estimates of variance components derived from GM regression estimates and the derived test outperform other robust procedures.     

The three-fold nested random effects model

Estimation of the variance components and the mean of the balanced and unbalanced threefold nested design is considered. The relative merits of the following procedures are evaluated: Analysis of variance (ANOVA), maximum likelihood (ML), restricted maximum likelihood (REML), and minimum variance quadratic unbiased estimator (MIVQUE). A new procedure called the weighted analysis of means (WAM) estimator which utilizes prior information on the variance components is proposed. It is found to have optimum properties similar to the REML and MIVQUE, and it is also computationally simpler. For the mean, the overall sample average, grand mean, unweighted mean, and generalized least-squares (GLS) estimator with its weights obtained from the above estimators for the variance components are considered. Comparisons of the above procedures for the variance components and the mean are made from exact expressions for the biases and mean square errors (MSEs) of the estimators and from empirical investigations.     

Estimation of variance components in the mixed effects models: A comparison between analysis of variance and spectral decomposition

The mixed effects models with two variance components are often used to analyze longitudinal data. For these models, we compare two approaches to estimating the variance components, the analysis of variance approach and the spectral decomposition approach. We establish a necessary and sufficient condition for the two approaches to yield identical estimates, and some sufficient conditions for the superiority of one approach over the other, under the mean squared error criterion. Applications of the methods to circular models and longitudinal data are discussed. Furthermore, simulation results indicate that better estimates of variance components do not necessarily imply higher power of the tests or shorter confidence intervals.     

Estimation using penalized quasilikelihood and quasi-pseudo-likelihood in Poisson mixed models

We consider two estimation schemes based on penalized quasilikelihood and quasi-pseudo-likelihood in Poisson mixed models. The asymptotic bias in regression coefficients and variance components estimated by penalized quasilikelihood (PQL) is studied for small values of the variance components. We show the PQL estimators of both regression coefficients and variance components in Poisson mixed models have a smaller order of bias compared to those for binomial data. Unbiased estimating equations based on quasi-pseudo-likelihood are proposed and are shown to yield consistent estimators under some regularity conditions. The finite sample performance of these two methods is compared through a simulation study.     

Confidence intervals and tests on the variance ratio in random models with two variance components

The LM test is modified to test any value of the ratio of two variance components in a mixed effects linear model with two variance components. The test is exact, so it can be used to construct exact confidence intervals on this ratio.Exact Neyman-Pearson (NP) tests on the variance ratio are described.Their powers provide attainable upper bounds on powers of tests on the variance ratio.Efficiencies of LM tests, which include ANOVA tests, and NP tests are compared for unbalanced, random, one-way ANOVA models.Confidence intervals corresponding to LM tests and NP tests are described.     

Expected Mean Squares for Hierarchical Factorial Layouts with Population Imbalance

We introduce an analysis of variance usable for two-factor hierarchical models where observations are incompletely sampled from unbalanced populations of finite effects. Our new approach enables unbiased estimation of the variance components for this type of model and allows hypothesis testing to identify significant effects/sub-class effects. An explanation of how these results can be generalized to factorial layouts with more than two factors is given.     

Unbiasedness of two-stage estimation and prediction procedures for mixed linear models

The traditional method for estimating or predicting linear combinations of the fixed effects and realized values of the random effects in mixed linear models is first to estimate the variance components and then to proceed as if the estimated values of the variance components were the true values. This two-stage procedure gives unbiased estimators or predictors of the linear combinations provided the data vector is symmetrically distributed about its expected value and provided the variance component estimators are translation-invariant and are even functions of the data vector. The standard procedures for estimating the variance components yield even, translation-invariant estimators.     

Variance components estimation for the balanced random effects model under mixed prior distributions

For the balanced random effects models, when the variance components are correlated either naturally or through common prior structures, by assuming a mixed prior distribution for the variance components, we propose some new Bayesian estimators. To contrast and compare the new estimators with the minimum variance unbiased (MVUE) and restricted maximum likelihood estimators (RMLE), some simulation studies are also carried out. It turns out that the proposed estimators have smaller mean squared errors than the MVUE and RMLE.     

A clustering approach to interpretable principal components

A new method for constructing interpretable principal components is proposed. The method first clusters the variables, and then interpretable (sparse) components are constructed from the correlation matrices of the clustered variables. For the first step of the method, a new weighted-variances method for clustering variables is proposed. It reflects the nature of the problem that the interpretable components should maximize the explained variance and thus provide sparse dimension reduction. An important feature of the new clustering procedure is that the optimal number of clusters (and components) can be determined in a non-subjective manner. The new method is illustrated using well-known simulated and real data sets. It clearly outperforms many existing methods for sparse principal component analysis in terms of both explained variance and sparseness.     

QUALITY IMPROVEMENT FROM DISASSEMBLY-REASSEMBLY EXPERIMENTS

The goal of achieving high quality products has led to an emphasis on reducing variation in performance characteristics. It may often happen that one of the product's components is responsible for much of the observed variation. This research is stimulated by the problem of detecting a component that impairs quality by systematically inflating the variance in a product that is assembled from “interchangeable components.” We consider the class of “disassembly-reassembly” experiments, in which components are swapped among assemblies. The specific units used in the experiment are sampled from a large population of units, so it is natural to measure the influence of each factor by its variance component. We present the model for these experiments as a special case of the mixed linear model, compare several estimators for the variance components and consider the problem of testing hypotheses to identify troublesome components.     

Generalized confidence intervals for proportions of total variance in mixed linear models

Exact confidence intervals for a proportion of total variance, based on pivotal quantities, only exist for mixed linear models having two variance components. Generalized confidence intervals (GCIs) introduced by Weerahandi [1993. Generalized confidence intervals (Corr: 94V89 p726). J. Am. Statist. Assoc. 88, 899–905] are based on generalized pivotal quantities (GPQs) and can be constructed for a much wider range of models. In this paper, the author investigates the coverage probabilities, as well as the utility of GCIs, for a proportion of total variance in mixed linear models having more than two variance components. Particular attention is given to the formation of GPQs and GCIs in mixed linear models having three variance components in situations where the data exhibit complete balance, partial balance, and partial imbalance. The GCI procedure is quite general and provides a useful method to construct confidence intervals in a variety of applications.     

Analysis of the reliability of a new scale for housing quality

This article reports the results of a variance components investigation of the reliability of a new scale of measurement of the quality of a dwelling unit. A balanced incomplete block design was utilized. A method for selecting among competing unbiased estimators of the variance components is presented. These components furnish a measure of the stability of the new dwelling quality scale. The validity of this new scale was investigated by comparison with experienced housing inspectors' reports.     

A note on fiducial generalized pivots for in one-way heteroscedastic ANOVA with random effects

The paper deals with generalized confidence intervals for the between-group variance in one-way heteroscedastic (unbalanced) ANOVA with random effects. The approach used mimics the standard one applied in mixed linear models with two variance components, where interval estimators are based on a minimal sufficient statistic derived after an initial reduction by the principle of invariance. A minimal sufficient statistic under heteroscedasticity is found to resemble its homoscedastic counterpart and further analogies between heteroscedastic and homoscedastic cases lead us to two classes of fiducial generalized pivots for the between-group variance. The procedures suggested formerly by Wimmer and Witkovský [Between group variance component interval estimation for the unbalanced heteroscedastic one-way random effects model, J. Stat. Comput. Simul. 73 (2003), pp. 333–346] and Li [Comparison of confidence intervals on between group variance in unbalanced heteroscedastic one-way random models, Comm. Statist. Simulation Comput. 36 (2007), pp. 381–390] are found to belong to these two classes. We comment briefly on some of their properties that were not mentioned in the original papers. In addition, properties of another particular generalized pivot are considered.     

Variance component estimation in multiple regression models having a nested error structure

Multiple regression when regressors are measured on two different sized experimental units involves a nested error structure. This nested error structure consists of two variance components. Sufficient conditions are presented under which UMVU estimators of these variance components exist. When these conditions are not met, two alternative estimators for the two variance components are considered and compared when possible.

This paper considers multiple regression models when regressor variables are associated with different sized experimental units resulting in a nested error structure. Nested error structures occur because of restrictions placed on randomizations. This results in experiments similar to splitplot type experiments which involove two different sizes of experimental units. Data resulting from these type of experiments consists of measurements made on larger sized experimental units as well as measurements made on smaller sized experimental units. Split-plot type experiments occur when certain combinations of the treatment factors are randomly assigned to larger sized experimental units after which these units are split or divided and other combinations of the treatment factors are randomly assigned to the split units.     

Estimation in a Two Variance Components Model When one Component is Known

We study a mixed linear model with two variance components. We suppose that one component is known. The objective of the paper is the estimation of the unknown component. The usual MINQE estimators seem to be unadapted to the problem. So we propose a new family of quadratic estimators, based on a natural class of estimators and the idea upon which the MINQE theory is built. All the estimators are compared on simulated data.     

Confidence intervals for variance components in unbalanced one-way random effects model using non-normal distributions

In scenarios where the variance of a response variable can be attributed to two sources of variation, a confidence interval for a ratio of variance components gives information about the relative importance of the two sources. For example, if measurements taken from different laboratories are nine times more variable than the measurements taken from within the laboratories, then 90% of the variance in the responses is due to the variability amongst the laboratories and 10% of the variance in the responses is due to the variability within the laboratories. Assuming normally distributed sources of variation, confidence intervals for variance components are readily available. In this paper, however, simulation studies are conducted to evaluate the performance of confidence intervals under non-normal distribution assumptions. Confidence intervals based on the pivotal quantity method, fiducial inference, and the large-sample properties of the restricted maximum likelihood (REML) estimator are considered. Simulation results and an empirical example suggest that the REML-based confidence interval is favored over the other two procedures in unbalanced one-way random effects model.     

Approximate confidence intervals for a ratio of two variance components

New approximate confidence intervals for the ratio of two variance components in an unbalanced mixed .linear model with a single set of random effects are proposed. Contrary to the confidence intervals known in the literature the new intervals preserve the confidence coefficient and cover the exact confidence interval which, however, is not easy to establish as it requires the solution of complicated nonlinear equations.     

A revisit to two-way factorial ANOVA with mixed effects and interactions

Abstract

Two mixed models exist in analysis of two-way factorial ANOVA with mixed effects and interactions: the constrained and unconstrained models. The constrained model is unfavored because there is no convincing rationale for the enforced constraints on its random interactions and a lack of clear interpretation about its variance components. The purpose of this study is to further explore the relationship between these two models. We reveal some nice features of the constrained model on partition of the responsive variance. An alternative formulation of ANOVA that follows from this exploration is also presented.     

New Algorithms for Evaluating the Log-Likelihood Function Derivatives in the AI-REML Method

In this study, we propose several improvements of the Average Information Restricted Maximum Likelihood algorithms for estimating the variance components for genetic mapping of quantitative traits. The improved methods are applicable when two variance components are to be estimated. The improvements are related to the algebraic part of the methods and utilize the properties of the underlying matrix structures.

In contrast to previously developed algorithms, the explicit computation of a matrix inverse is replaced by the solution of a linear system of equations with multiple right-hand sides, based on a particular matrix decomposition. The computational costs of the proposed algorithms are analyzed and compared.     

Confidence intervals for the between group variance in the unbalanced one-way random effects model of anaylsis of variance

A confidence interval for the between group variance is proposed which is deduced from Wald'sexact confidence interval for the rtio of the two variance components in the one-way random effects model and the exact confidence interval for the error variance resp.an unbiased estimator of the error variance. In a simulation study the confidence coeffecients for these two intervals are compared with the confidence coefficients of two other commonly used confidence intervals. There the confidence interval derived here yields confidence coefficiends which are always greater than the prescriped level.