The present status of confidence interval estimation on variance components in balanced and unbalanced random models

Much research has been conducted to develop confidence Intervals on linear combinations and ratios of variance components in balanced and unbalanced random models.This paper first presents confidence intervals on functions of variance components in balanced designs.These results assume that classical analysis of variance sums of squares are independent and have exact scaled chi-squared distributions.In unbalanced designs, either one or both of these assumptions are violated, and modifications to the balanced model intervals are required.We report results of some recent work that examines various modifications for some particular unbalanced designs.     

Posterior bimodality in the balanced one-way random-effects model

Summary. Although some researchers have examined posterior multimodality for specific richly parameterized models, multimodality is not well characterized for any such model. The paper characterizes bimodality of the joint and marginal posteriors for a conjugate analysis of the balanced one-way random-effects model with a flat prior on the mean. This apparently simple model has surprisingly complex and even bizarre mode behaviour. Bimodality usually arises when the data indicate a much larger between-groups variance than does the prior. We examine an example in detail, present a graphical display for describing bimodality and use real data sets from a statistical practice to shed light on the practical relevance of bimodality for these models.     

Limiting admissible estimators for variance components

Simultaneous estimation of the vector of the variance components for mixed and random models under the quadratic loss function is considered. For a large class of such models there are identified classes of admissible biased invariant quadratic estimators that are better than some admissible unbiased estimators. Numerous numerical results presented in the paper show that for many of the commonly used balanced models the improvements in the quadratic risk may be considerable over a large set of the parameter space.     

Kronecker Products in ANOVA—A First Step

In the complete balanced model for the analysis of variance, the equivalence of sums of squares and quadratic forms is seen to imply well-fitting patterns involving Kronecker products of identity matrices and scalar multiples of matrices with all elements equal to 1. The questions of symmetry, idempotency, and orthogonality so central to this topic are answered by simple multiplications; ranks are determined from simple traces. The associations between the forms of the two-factor model are presented here in a way that is accessible to first-year students and makes generalizations to higher order models transparent. The lack of patterns in incomplete or unbalanced models is noted. Additional steps in design and analysis are suggested in the references.     

Simultaneous inferences for variance ratios of some mixed linear models

Earlier investigations used a one-sided inequality to consltuct confidence regions for the variance ratios or balanced randoiu models. In this study, confidence regions are based on a two-sided generalisation of this inequality and the results are illustrated by estimating the parameters of some elementary random models.     

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Estimation of covariance components in the multivariate random-effect model with nested covariance structure is discussed. There are two covariance matrices to be estimated, namely, the between-group and the within-group covariance matrices. These two covariance matrices are most often estimated by forming a multivariate analysis of variance and equating mean square matrices to their expectations. Such a procedure involves taking the difference between the between-group mean square and the within-group mean square matrices, and often produces an estimated between-group covariance matrix that is not nonnegative definite. We present estimators of the two covariance matrices that are always proper covariance matrices. The estimators are the restricted maximum likelihood estimators if the random effects are normally distributed. The estimation procedure is extended to more complicated models, including the twofold nested and the mixed-effect models. A numerical example is presented to illustrate the use of the estimation procedure.     

Nonparametric Survival Estimation when Death is Reported with Delay

In disease registries there can be a delay between death of asubject and the reporting of this death to the data analyst.If researchers use the Kaplan-Meier estimator and implicitlyassumed that subjects who have yet to have death reported arestill alive, i.e. are censored at the time of analysis, the Kaplan-Meierestimator is typically inconsistent. Assuming censoring is independentof failure, we provide a simple estimator that is consistentand asymptotically efficient. We also provide estimates of theasymptotic variance of our estimator and simulations that demonstratethe favorable performance of these estimators. Finally, we demonstrateour methods by analyzing AIDS survival data. This analysis underscoresthe pitfalls of not accounting for delay when estimating thesurvival distribution and suggests a significant reduction inbias by using our estimator.     

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.     

Accent on Teaching Materials

This department includes the two sections New Developments in Statistical Computing and Statistical Computing Software Reviews; suitable contents for each of these sections are described under the respective section heading. Articles submitted for the department, outside the two sections, should not be highly technical and should be relevant to the teaching or practice of statistical computing.

An unbiased estimator of e is used to motivate a simple simulation exercise that requires only observations from the distribution uniform on (0, 1). Antithetic variables are introduced and applied to the simulation problem to give a second unbiased estimator of e with reduced variance.     

Incomplete covariates data in generalized linear models

We consider regression analysis when part of covariates are incomplete in generalized linear models. The incomplete covariates could be due to measurement error or missing for some study subjects. We assume there exists a validation sample in which the data is complete and is a simple random subsample from the whole sample. Based on the idea of projection-solution method in Heyde (1997, Quasi-Likelihood and its Applications: A General Approach to Optimal Parameter Estimation. Springer, New York), a class of estimating functions is proposed to estimate the regression coefficients through the whole data. This method does not need to specify a correct parametric model for the incomplete covariates to yield a consistent estimate, and avoids the ‘curse of dimensionality’ encountered in the existing semiparametric method. Simulation results shows that the finite sample performance and efficiency property of the proposed estimates are satisfactory. Also this approach is computationally convenient hence can be applied to daily data analysis.     

Adaptive Inference for Multi-Stage Survey Data

Multi-level models can be used to account for clustering in data from multi-stage surveys. In some cases, the intraclass correlation may be close to zero, so that it may seem reasonable to ignore clustering and fit a single-level model. This article proposes several adaptive strategies for allowing for clustering in regression analysis of multi-stage survey data. The approach is based on testing whether the PSU-level variance component is zero. If this hypothesis is retained, then variance estimates are calculated ignoring clustering; otherwise, clustering is reflected in variance estimation. A simple simulation study is used to evaluate the various procedures.     

Cocaine Dependence Treatment Data: Methods for Measurement Error Problems With Predictors Derived From Stationary Stochastic Processes

In a cocaine dependence treatment study, we use linear and nonlinear regression models to model posttreatment cocaine craving scores and first cocaine relapse time. A subset of the covariates are summary statistics derived from baseline daily cocaine use trajectories, such as baseline cocaine use frequency and average daily use amount. These summary statistics are subject to estimation error and can therefore cause biased estimators for the regression coefficients. Unlike classical measurement error problems, the error we encounter here is heteroscedastic with an unknown distribution, and there are no replicates for the error-prone variables or instrumental variables. We propose two robust methods to correct for the bias: a computationally efficient method-of-moments-based method for linear regression models and a subsampling extrapolation method that is generally applicable to both linear and nonlinear regression models. Simulations and an application to the cocaine dependence treatment data are used to illustrate the efficacy of the proposed methods. Asymptotic theory and variance estimation for the proposed subsampling extrapolation method and some additional simulation results are described in the online supplementary material.     

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.     

On a theorem of graybill

Graybill (1976) gives a theorem for testing variance components in balanced random models. Unfortunately, the theorem does not hold true. We pin down the reason that causes the theorem to fail and give a correct version of it.     

Estimation of linear models for field experiments

General mixed linear models for experiments conducted over a series of sltes and/or years are described. The ordinary least squares (OLS) estlmator is simple to compute, but is not the best unbiased estimator. Also, the usuaL formula for the varlance of the OLS estimator is not correct and seriously underestimates the true variance. The best linear unbiased estimator is the generalized least squares (GLS) estimator. However, t requires an inversion of the variance-covariance matrix V, whlch is usually of large dimension. Also, in practice, V is unknown.

We presented an estlmator [Vcirc] of the matrix V using the estimators of variance components [for sites, blocks (sites), etc.]. We also presented a simple transformation of the data, such that an ordinary least squares regression of the transformed data gives the estimated generalized least squares (EGLS) estimator. The standard errors obtained from the transformed regression serve as asymptotic standard errors of the EGLS estimators. We also established that the EGLS estlmator is unbiased.

An example of fitting a linear model to data for 18 sites (environments) located in Brazil is given. One of the site variables (soil test phosphorus) was measured by plot rather than by site and this established the need for a covariance model such as the one used rather than the usual analysis of variance model. It is for this variable that the resulting parameter estimates did not correspond well between the OLS and EGLS estimators. Regression statistics and the analysis of variance for the example are presented and summarized.     

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.     

The Use of Meteorological Data in the Prediction of Tropical Cyclones

Recent studies have shown that the ability to forecast tropical cyclone motion and intensity has not improved significantly over the past decade, in spite of apparent advances in the statistical and dynamical modeling of the motion- and intensity-change processes. This static situation has focused attention on the meteorological data and analysis techniques that provide input to these models. The quality and quantity of data have apparently not kept pace with the modeling efforts; in fact, increased sophistication in modeling alone may be counterproductive. Steps are being taken in the meteorological community to correct this problem. This study examines the entire data problem, discusses the remedial measures being taken, and points out some of the pitfalls in the statistical use of meteorological data.     

Umvu estimators for the population mean and variance based on random effects models for lognormal data

Cross-classified data are often obtained in controlled experimental situations and in epidemiologic studies. As an example of the latter, occupational health studies sometimes require personal exposure measurements on a random sample of workers from one or more job groups, in one or more plant locations, on several different sampling dates. Because the marginal distributions of exposure data from such studies are generally right-skewed and well-approximated as lognormal, researchers in this area often consider the use of ANOVA models after a logarithmic transformation. While it is then of interest to estimate original-scale population parameters (e.g., the overall mean and variance), standard candidates such as maximum likelihood estimators (MLEs) can be unstable and highly biased. Uniformly minimum variance unbiased (UMVU) cstiniators offer a viable alternative, and are adaptable to sampling schemes that are typiral of experimental or epidemiologic studies. In this paper, we provide UMVU estimators for the mean and variance under two random effects ANOVA models for logtransformed data. We illustrate substantial mean squared error gains relative to the MLE when estimating the mean under a one-way classification. We illustrate that the results can readily be extended to encompass a useful class of purely random effects models, provided that the study data are balanced.     

On the role of minque in testing of hypotheses under hiked linear models

Testing of hypotheses under balanced ANOVA models is fairly simple and generally based on the usual ANOVA sums of squares. Difficulties may arise in special cases when these sums of squares do not form a complete sufficient statistic. There is a huge literature on this subject which was recently surveyed in Seifert's contribution to the book of Mumak (1904). But there are only a few results about unbalanced models. In such models the consideration of likelihood ratios leads to more complex sums of squares known from MINQUE theory.

Uniform optimality of testsusually reduces to local optimality. Here we prespnt a small review of methods proposed for testing of hypotheses in unbalanced models. where MINQUEI playb a major role. We discuss the use of iterated MINQUE for the construction of asymptotically optimal tests described in Humak (1984) and approximate tests based on locally uncorrelated linear combinations of MINQUE estimators by Seifert (1985), We show that the latter tests coincide with robust locally optimal invariant tests proposeci by Kariya and Sinha and Das and Sinha, if the number of variance components is two. Explicit expressions for corresponding tests are given for the unbalanced two-way cross classification random model, which covers some other models as special cases. A simulation study under lines the relevance of MINQUE for testing of hypotheses problems.     

Bayesian approach to estimation of intraclass correlation using reference prior

For the balanced variance component model when the intraclass correlation coefficient is of interest, Bayesian analysis is often appropriate. Berger and Bernardo’s (1992a) grouped ordering reference prior approach is used to analyze this model. The reference priors are developed and compared for the posterior inference with real and simulated data. We examine whether the reference priors satisfy the probability-matching criterion. Further, the reference prior is shown to be good in the sense of correct frequentist coverage probability of the posterior quantile.