Probability matching priors for two-sided tolerance intervals in balanced one-way and two-way nested random effects models

We consider two-sided Bayesian tolerance intervals, with approximate frequentist validity, for a future observation in balanced one-way and two-way nested random effects models. Probability matching conditions, specific to this problem, are derived in either case via a technique that involves inversion of approximate posterior characteristic functions. In addition to yielding probability matching priors for the present problem, these conditions are useful in evaluating certain other priors that have received attention in the literature.     

Approximate one-sided tolerance limits in random effects model and in some mixed models and comparisons

An approximate closed-form one-sided tolerance limit (TL) in a general mixed model is proposed. One-sided TLs for the distribution of observable random variable and for the distribution of unobservable random variable in one-way random model are obtained as special cases from the one for the general mixed model. Applications to a two-way nested random model are also given. The merits of the TLs are evaluated using Monte Carlo simulation and compared with the existing ones. Our comparison studies indicate that the approximate TLs are quite satisfactory for all parameter and sample size configurations, and better than the existing ones in some cases. Approximate confidence intervals for exceedance probabilities in one-way random effects model are also proposed. The procedures are illustrated using three examples.     

Two-sided Bayesian and frequentist tolerance intervals: general asymptotic results with applications

It is well known that that the construction of two-sided tolerance intervals is far more challenging than that of their one-sided counterparts. In a general framework of parametric models, we derive asymptotic results leading to explicit formulae for two-sided Bayesian and frequentist tolerance intervals. In the process, probability matching priors for such intervals are characterized and their role in finding frequentist tolerance intervals via a Bayesian route is indicated. Furthermore, in situations where matching priors are hard to obtain, we develop purely frequentist tolerance intervals as well. The findings are applied to real data. Simulation studies are seen to lend support to the asymptotic results in finite samples.     

Noninformative priors for the between-group variance in the unbalanced one-way random effects model with heterogeneous error variances

For the unbalanced one-way random effects model with heterogeneous error variances, we propose the non-informative priors for the between-group variance and develop the first- and second-order matching priors. It turns out that the second-order matching priors do not exist and the reference prior and Jeffreys prior do not satisfy a first-order matching criterion. We also show that the first-order matching prior meets the frequentist target coverage probabilities much better than the Jeffreys prior and reference prior through simulation study, and the Bayesian credible intervals based on the matching prior and reference prior give shorter intervals than the existing confidence intervals by examples.     

Closed-form approximate tolerance intervals for some general linear models and comparison studies

The problems of constructing tolerance intervals (TIs) in random effects model and in a mixed linear model are considered. The methods based on the generalized variable (GV) approach and the one based on the modified large sample (MLS) procedure are evaluated with respect to coverage probabilities and expected width in various setups using Monte Carlo simulation. Our comparison studies indicate that the TIs based on the MLS procedure are comparable to or better than those based on the GV approach. As the MLS TIs are in closed-form, they are easier to compute than those based on the GV approach. TIs for a two-way nested model are also derived using the MLS method, and their merits are evaluated using simulation. The procedures are illustrated using a practical example.     

Simultaneous tolerance intervals in the random one-way model with covariates

Simultaneous tolerance intervals developed by Limam and Thomas (19881, for the normal regression model, are generalized to the random one-way model with covariates. Simultaneous tolerance intervals for unit means are developed for the balanced model. A simulation study is used to estimate the exact confidence of the tolerance intervals for models with one covariate.     

Graphical evaluation of the adequacy of the method of unweighted means

A graphical technique is introduced to assess the adequacy of the method of unweighted means in providing approximate F -tests for an unbalanced random model. These tests are similar to those obtained under a balanced ANOVA. The proposed technique is simple and can easily be used to determine the effects of imbalance and values of the variance components on the adequacy of the approximation. The one-way and two-way random models are used to illustrate the proposed methodology. Extensions to higher-order models are also mentioned.     

On Ranges of Confidence Coefficients for Confidence Intervals on Variance Components

The among variance component in the balanced one-factor nested components-of-variance model is of interest in many fields of application. Except for an artificial method that uses a set of random numbers which is of no use in practical situations, an exact-size confidence interval on the among variance has not yet been derived. This paper provides a detailed comparison of three approximate confidence intervals which possess certain desired properties and have been shown to be the better methods among many available approximate procedures. Specifically, the minimum and the maximum of the confidence coefficients for the one- and two-sided intervals of each method are obtained. The expected lengths of the intervals are also compared.     

Confidence intervals on intraclass correlation coefficients in a balanced two-factor random design

A modified large-sample (MLS) approach and a generalized confidence interval (GCI) approach are proposed for constructing confidence intervals for intraclass correlation coefficients. Two particular intraclass correlation coefficients are considered in a reliability study. Both subjects and raters are assumed to be random effects in a balanced two-factor design, which includes subject-by-rater interaction. Computer simulation is used to compare the coverage probabilities of the proposed MLS approach (GiTTCH) and GCI approaches with the Leiva and Graybill [1986. Confidence intervals for variance components in the balanced two-way model with interaction. Comm. Statist. Simulation Comput. 15, 301–322] method. The competing approaches are illustrated with data from a gauge repeatability and reproducibility study. The GiTTCH method maintains at least the stated confidence level for interrater reliability. For intrarater reliability, the coverage is accurate in several circumstances but can be liberal in some circumstances. The GCI approach provides reasonable coverage for lower confidence bounds on interrater reliability, but its corresponding upper bounds are too liberal. Regarding intrarater reliability, the GCI approach is not recommended because the lower bound coverage is liberal. Comparing the overall performance of the three methods across a wide array of scenarios, the proposed modified large-sample approach (GiTTCH) provides the most accurate coverage for both interrater and intrarater reliability.     

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.     

Non-informative priors for the common mean in the one-way random effects model with heterogeneous error variances

The paper develops some objective priors for the common mean in the one-way random effects model with heterogeneous error variances. We derive the first and second order matching priors and reference priors. It turns out that the second order matching prior matches the alternative coverage probabilities up to the second order, and is also an HPD matching prior. However, derived reference priors just satisfy a first order matching criterion. Our simulation studies indicate that the second order matching prior performs better than the reference prior and the Jeffreys prior in terms of matching the target coverage probabilities in a frequentist sense. We also illustrate our results using real data.     

A Comparison of Estimation Methods on the Coverage Probability of Satterthwaite Confidence Intervals for Assay Precision with Unbalanced Data

Construction of closed-form confidence intervals on linear combinations of variance components were developed generically for balanced data and studied mainly for one-way and two-way random effects analysis of variance models. The Satterthwaite approach is easily generalized to unbalanced data and modified to increase its coverage probability. They are applied on measures of assay precision in combination with (restricted) maximum likelihood and Henderson III Type 1 and 3 estimation. Simulations of interlaboratory studies with unbalanced data and with small sample sizes do not show superiority of any of the possible combinations of estimation methods and Satterthwaite approaches on three measures of assay precision. However, the modified Satterthwaite approach with Henderson III Type 3 estimation is often preferred above the other combinations.     

Objective priors for hypothesis testing in one-way random effects models

The Bayes factor is a key tool in hypothesis testing. Nevertheless, the important issue of which priors should be used to develop objective Bayes factors remains open. The authors consider this problem in the context of the one-way random effects model. They use concepts such as orthogonality, predictive matching and invariance to justify a specific form of the priors for common parameters and derive the intrinsic and divergence based prior for the new parameter. The authors show that both intrinsic priors or divergence-based priors produce consistent Bayes factors. They illustrate the methods and compare them with other proposals.     

Bayesian shrinkage estimates for regression coefficients in m populations

For a linear regression model over m populations with separate regression coefficients but a common error variance, a Bayesian model is employed to obtain regression coefficient estimates which are shrunk toward an overall value. The formulation uses Normal priors on the coefficients and diffuse priors on the grand mean vectors, the error variance, and the between-to-error variance ratios. The posterior density of the parameters which were given diffuse priors is obtained. From this the posterior means and variances of regression coefficients and the predictive mean and variance of a future observation are obtained directly by numerical integration in the balanced case, and with the aid of series expansions in the approximately balanced case. An example is presented and worked out for the case of one predictor variable. The method is an extension of Box & Tiao's Bayesian estimation of means in the balanced one-way random effects model.     

Approximate tolerance intervals for the discrete Poisson–Lindley distribution

The Poisson–Lindley distribution is a compound discrete distribution that can be used as an alternative to other discrete distributions, like the negative binomial. This paper develops approximate one-sided and equal-tailed two-sided tolerance intervals for the Poisson–Lindley distribution. Practical applications of the Poisson–Lindley distribution frequently involve large samples, thus we utilize large-sample Wald confidence intervals in the construction of our tolerance intervals. A coverage study is presented to demonstrate the efficacy of the proposed tolerance intervals. The tolerance intervals are also demonstrated using two real data sets. The R code developed for our discussion is briefly highlighted and included in the tolerance package.     

Inferences on the intraclass correlation coefficients in the unbalanced two-way random effects model with interaction

In this paper, the hypothesis testing and interval estimation for the intraclass correlation coefficients are considered in a two-way random effects model with interaction. Two particular intraclass correlation coefficients are described in a reliability study. The tests and confidence intervals for the intraclass correlation coefficients are developed when the data are unbalanced. One approach is based on the generalized p-value and generalized confidence interval, the other is based on the modified large-sample idea. These two approaches simplify to the ones in Gilder et al. [2007. Confidence intervals on intraclass correlation coefficients in a balanced two-factor random design. J. Statist. Plann. Inference 137, 1199–1212] when the data are balanced. Furthermore, some statistical properties of the generalized confidence intervals are investigated. Finally, some simulation results to compare the performance of the modified large-sample approach with that of the generalized approach are reported. The simulation results indicate that the modified large-sample approach performs better than the generalized approach in the coverage probability and expected length of the confidence interval.     

Bayesian inference for a flexible class of bivariate beta distributions

Several bivariate beta distributions have been proposed in the literature. In particular, Olkin and Liu [A bivariate beta distribution. Statist Probab Lett. 2003;62(4):407–412] proposed a 3 parameter bivariate beta model which Arnold and Ng [Flexible bivariate beta distributions. J Multivariate Anal. 2011;102(8):1194–1202] extend to 5 and 8 parameter models. The 3 parameter model allows for only positive correlation, while the latter models can accommodate both positive and negative correlation. However, these come at the expense of a density that is mathematically intractable. The focus of this research is on Bayesian estimation for the 5 and 8 parameter models. Since the likelihood does not exist in closed form, we apply approximate Bayesian computation, a likelihood free approach. Simulation studies have been carried out for the 5 and 8 parameter cases under various priors and tolerance levels. We apply the 5 parameter model to a real data set by allowing the model to serve as a prior to correlated proportions of a bivariate beta binomial model. Results and comparisons are then discussed.     

Asymptotic posterior distributions for balanced nested multi-way models with a large number of main effect levels

We derive the explicit form for the asymptotic posterior distribution of the balanced nested multi-way variance components model with the assumption that the number of the main factor levels tends to infinity while the number of any specific effect factor levels remains fixed. Under the multi-way model, we also study two different parameterizations, called the standard and the centering, and the relationship between certain quadratic forms of random effects and the variance component parameters. The asymptotic results are illustrated by a three-way model and by a simulation study under a two-way case.     

Prediction intervals for general balanced linear random models

The main interest of prediction intervals lies in the results of a future sample from a previously sampled population. In this article, we develop procedures for the prediction intervals which contain all of a fixed number of future observations for general balanced linear random models. Two methods based on the concept of a generalized pivotal quantity (GPQ) and one based on ANOVA estimators are presented. A simulation study using the balanced one-way random model is conducted to evaluate the proposed methods. It is shown that one of the two GPQ-based and the ANOVA-based methods are computationally more efficient and they also successfully maintain the simulated coverage probabilities close to the nominal confidence level. Hence, they are recommended for practical use. In addition, one example is given to illustrate the applicability of the recommended methods.     

Noninformative priors for the nested design

This paper considers noninformative priors for three-stage nested designs. It turns out that the noninformative prior given by Li and Stern (1997) is the one-at-a-time reference prior satisfying a second-order matching criterion when either the variance ratio or linear combinations of the means is of interest. Moreover, it is a joint probability matching prior when both the variance ratio and linear combinations of the means are of interest. These priors are compared with Jeffreys' prior in light of how accurately the coverage probabilities of Bayesian credible intervals match the corresponding frequentist coverage probabilities.