Balanced Random Effects Model Research Articles

A comparison of objective priors for Cronbach’s coefficient alpha using a balanced random effects model

. In this article, the reference and probability matching priors for Cronbach’s alpha will be derived. The performance of these two priors will be compared to that of the well-known Jeffreys prior and a divergence prior. Cronbach’s alpha is a measure used to assess the reliability of a set of test items. A simulation study will be considered to compare the performance of the priors, where the coverage rates, average interval lengths, and standard deviations of the interval lengths will be computed. A second simulation study will be considered where the mean relative error will be compared for the various priors using three different loss functions. The following loss functions will be considered, Squared error loss, Absolute error loss, and Linex loss. An illustrative example will also be considered. Throughout the article, the random effects approach will be used.

Theoretical Study for Adjusting ANOVA Procedures For The One-Way Random Effects Model Under Correlated Errors Within Class And Across Classes

In this study, a method was developed to adjust the procedures of analysis of variance for the one-way balanced random effects model when error limits across classes are not independent, while the assumption of independence between the error terms among the different classes is essential in the analysis of variance. The variance-covariance structure and the correlation coefficients for the error limits were defined. The was assumed as the correlation coefficient of error terms within the class and the is the correlation coefficient for error terms in different classes. The work performed by deriving correction factor and calculating the expectation of the mean sum of squares of errors and treatments as well as correcting the f-statistic.

Objective Bayesian Inference for a Generalized Marginal Random Effects Model

An objective Bayesian inference is proposed for the generalized marginal random effects model p(x|μ,σλ)=f((x−μ1)T(V+σλ2I)−1(x−μ1))/det(V+σλ2I). The matrix V is assumed to be known, and the goal is to infer μ given the observations x=(x1,…,xn)T, while σλ is a nuisance parameter. In metrology this model has been applied for the adjustment of inconsistent data x1,…,xn, where the matrix V contains the uncertainties quoted for x1,…,xn. We show that the reference prior for grouping {μ,σλ} is given by π(μ,σλ)∝F22, where F22 denotes the lower right element of the Fisher information matrix F. We give an explicit expression for the reference prior, and we also prove propriety of the resulting posterior as well as the existence of mean and variance of the marginal posterior for μ. Under the additional assumption of normality, we relate the resulting reference analysis to that known for the conventional balanced random effects model in the asymptotic case when the number of repeated within-class observations for that model tends to infinity. We investigate the frequentist properties of the proposed inference for the generalized marginal random effects model through simulations, and we also study its robustness when the underlying distributional assumptions are violated. Finally, we apply the model to the adjustment of current measurements of the Planck constant.

The equivalence between likelihood ratio test and F-test for testing variance component in a balanced one-way random effects model

In mixed linear models, it is frequently of interest to test hypotheses on the variance components. F-test and likelihood ratio test (LRT) are commonly used for such purposes. Current LRTs available in literature are based on limiting distribution theory. With the development of finite sample distribution theory, it becomes possible to derive the exact test for likelihood ratio statistic. In this paper, we consider the problem of testing null hypotheses on the variance component in a one-way balanced random effects model. We use the exact test for the likelihood ratio statistic and compare the performance of F-test and LRT. Simulations provide strong support of the equivalence between these two tests. Furthermore, we prove the equivalence between these two tests mathematically.

A process capability index for averages of observations from new batches in the case of the balanced random effects model

Capability analysis is used in many facets of industrial processes and has been recently introduced into business processes. In this paper a process capability index ( P pl 1 ) is developed for the average of observations from new or unknown batches in the case of a balanced random effects model. Using a Bayesian approach, theoretical and simulation results are derived for the index under two different but related prior distributions. A medical tablet manufacturing example illustrates the flexibility and unique features of the Bayesian simulation method.

Saddlepoint Approximation and Bootstrap Inference for the Satterthwaite Class of Ratios

Saddlepoint approximations are developed for the distributions of Satterthwaite F-type statistics that arise in ANOVA with balanced random-effects models. Such statistics are commonly used in variance component testing and confidence interval construction, because exact F pivotals usually do not exist. The approximations are shown to be uniform in their right tails, and the limiting relative errors for distribution approximations are determined. Based on these approximations, a new method is devised for confidence interval construction of variance component ratios. Simulations show that the method has superior coverage accuracy over existing methods. Prepivoting of the proposed method, using the double-parametric bootstrap, is implemented and shows further coverage improvement. The double bootstrap is most efficiently implemented by using the saddlepoint approximation in lieu of the inner layer of resampling; therefore, only a single outer layer of resampling is required.

Selection of the reference priors for a balanced random effects model

Reference priors for a balanced random effects model with its natural parameter setting were derived by Berger and Bernado (1992a). There are many different priors corresponding to different ordered groupings. The behavior of the posterior inferences for the between variance component in the model is studied for various reference priors. Two frequentist approaches are used to justify appropriate reference priors. It is found that, for a given parameter of interest, different frequentist perspectives may result in selecting different reference priors.

Testing Variance Components in Mixed Models with GeneralizedpValues

Abstract A procedure for testing one-sided hypotheses on variance components in a balanced random effects model is developed. The testing is performed on the basis of a generalized p value and involves a one-dimensional numerical integration. In the special case for which the conventional methods provide an exact test, this test is equivalent to the usual F test. The test is also generalized Bayes in the sense that, under a certain diffuse prior, the p value is numerically the same as the posterior probability of the null hypothesis.

Two-Sided Tolerance Limits for Balanced Random-Effects ANOVA Models

In this article, we derive two-sided approximate β-content tolerance limits for multiway balanced random-effects models. We provide factors, obtained from numerical integration, that can be used to obtain β-content tolerance intervals. We describe methods for extending the results to nested models and discuss the use of the tabled tolerance factors for exact intervals for simple random samples when we have an independent estimate of the variance. We demonstrate the procedure with an experimental design used to evaluate the firing time precision in a high-explosives system.

Harmonic mean approach to uunbalanced random effects models under heteroscedasticity

In this paper we consider unbalanced random effects models under heteroscedastic variances. By using' the harmonic mean approach, it is shown that the problems are analogous to those from balanced random effects models under horaoscedastic variances. Thus, by using the harmonic mean approach, statistical inferences about variance components are derived by using procedures from balanced models under homoscedastic variances. Laguerre polynomial expansion is used to approximate the sampling distributions of relevant statistics.

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.

Counterexamples to a result of broemeling on simultaneous inferences for variance ratios of some mixed linear models

In a recent paper5 Broemeling (1978) extended his earlier work on one-sided confidence regions for the variance ratios of balanced random-effects models to the two-sided case. The extension depends on a probability Inequality which was claimed to be tru We show here that it is false, hence the proof of the main result given in Ms parer is in error W also show Lhat the ntatement of his result remains true in certain special cases.