Balanced One-way Random Effects Model Research Articles

Improved Large-Sample Confidence Intervals for Ratios of Variance Components in Nonnormal Distributions

The confidence level of the usual interval for a ratio of variance components is contingent on normally distributed random variables. In this article we focus on confidence intervals for ratios of variance components in balanced one-way random effects models based on the large-sample properties of restricted maximum likelihood (REML) estimators. While this procedure does not require that the random variables be normally distributed, one must estimate a parameter that is a function of the kurtosis of the underlying distributions. Simulation results indicate that REML-based confidence interval methods outperform other well-known methods in the majority of the cases considered.

Bayes Factor Consistency for One-way Random Effects Model

In this article, we consider Bayesian hypothesis testing for the balanced one-way random effects model. A special choice of the prior formulation for the ratio of variance components is shown to yield an explicit closed-form Bayes factor without integral representation. Furthermore, we study the consistency issue of the resulting Bayes factor under three asymptotic scenarios: either the number of units goes to infinity, the number of observations per unit goes to infinity, or both go to infinity. Finally, the behavior of the proposed approach is illustrated by simulation studies.

The Superiorities of Empirical Bayes Estimation of Variance Components in Random Effects Model

In this article, the Bayes estimators of variance components are derived and the parametric empirical Bayes estimators (PEBE) for the balanced one-way classification random effects model are constructed. The superiorities of the PEBE over the analysis of variance (ANOVA) estimators are investigated under the mean square error (MSE) criterion, some simulation results for the PEBE are obtained. Finally, a remark for the main results is given.

A Monte Carlo simulation study of two-sided tolerance intervals in balanced one-way random effects model for non-normal errors

Recently, Ong and Mukerjee [Probability matching priors for two-sided tolerance intervals in balanced one-way and two-way nested random effects models. Statistics. 2011;45:403–411] developed two-sided Bayesian tolerance intervals, with approximate frequentist validity, for a future observation in balanced one-way and two-way nested random effects models. These were obtained using probability matching priors (PMP). On the other hand, Krishnamoorthy and Lian [Closed-form approximate tolerance intervals for some general linear models and comparison studies. J Stat Comput Simul. 2012;82:547–563] studied closed-form approximate tolerance intervals by the modified large-sample (MLS) approach. We compare the performances of these two approaches for normal as well as non-normal error distributions. Monte Carlo simulation methods are used to evaluate the resulting tolerance intervals with regard to achieved confidence levels and expected widths. It turns out that PMP tolerance intervals are less conservative for data with large number of classes and small number of observations per class and the MLS procedure is preferable for smaller sample sizes.

Bias-corrected estimator for intraclass correlation coefficient in the balanced one-way random effects model

BackgroundIntraclass correlation coefficients (ICCs) are used in a wide range of applications. However, most commonly used estimators for the ICC are known to be subject to bias.MethodsUsing second order Taylor series expansion, we propose a new bias-corrected estimator for one type of intraclass correlation coefficient, for the ICC that arises in the context of the balanced one-way random effects model. A simulation study is performed to assess the performance of the proposed estimator. Data have been generated under normal as well as non-normal scenarios.ResultsOur simulation results show that the new estimator has reduced bias compared to the least square estimator which is often referred to as the conventional or analytical estimator. The results also show marked bias reduction both in normal and non-normal data scenarios. In particular, our estimator outperforms the analytical estimator in a non-normal setting producing estimates that are very close to the true ICC values.ConclusionsThe proposed bias-corrected estimator for the ICC from a one-way random effects analysis of variance model appears to perform well in the scenarios we considered in this paper and can be used as a motivation to construct bias-corrected estimators for other types of ICCs that arise in more complex scenarios. It would also be interesting to investigate the bias-variance trade-off.

Nonparametric Bootstrap Confidence Intervals for Variance Components Applied to Interlaboratory Comparisons

Exact confidence intervals for variance components in linear mixed models rely heavily on normal distribution assumptions. If the random effects in the model are not normally distributed, then the true coverage probabilities of these conventional intervals may be erratic. In this paper we examine the performance of nonparametric bootstrap confidence intervals based on restricted maximum likelihood (REML) estimators. Asymptotic theory suggests that these intervals will achieve the nominal coverage value as the sample size increases. Incorporating a small-sample adjustment term in the bootstrap confidence interval construction process improves the performance of these intervals for small to intermediate sample sizes. Simulation studies suggest that the bootstrap standard method (with a transformation) and the bootstrap bias-corrected and accelerated (BCa) method produce confidence intervals that have good coverage probabilities under a variety of distribution assumptions. For an interlaboratory comparison of mercury concentration in oyster tissue, a balanced one-way random effects model is used to quantify the proportion of the variation in mercury concentration that can be attributed to the laboratories. In this application the exact confidence interval using normal distribution theory produces misleading results and inferences based on nonparametric bootstrap procedures are more appropriate.

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.

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.

Comparing Equal-Tail Probability and Unbiased Confidence Intervals for the Intraclass Correlation Coefficient

The conventional confidence interval for the intraclass correlation coefficient assumes equal-tail probabilities. In general, the equal-tail probability interval is biased and other interval procedures should be considered. Unbiased confidence intervals for the intraclass correlation coefficient are readily available. The equal-tail probability and unbiased intervals have exact coverage as they are constructed using the pivotal quantity method. In this article, confidence intervals for the intraclass correlation coefficient are built using balanced and unbalanced one-way random effects models. The expected length of confidence intervals serves as a tool to compare the two procedures. The unbiased confidence interval outperforms the equal-tail probability interval if the intraclass correlation coefficient is small and the equal-tail probability interval outperforms the unbiased interval if the intraclass correlation coefficient is large.

Comparing pivotal and REML-based confidence intervals for heritability

Heritability quantifies the extent to which a physical characteristic is passed from one generation to the next. From a statistical perspective, heritability is the proportion of the phenotypic variance attributable to (additive) genetic effects and is equal to a function of variance components in linear mixed models. Relying on normal distribution assumptions, one can compute exact confidence intervals for heritability using a pivotal quantity procedure. Alternatively, large-sample properties of the restricted maximum likelihood (REML) estimator can be used to construct asymptotic confidence intervals for heritability. Exact and asymptotic intervals are compared loineye muscle area measurements and balanced one-way random effects models having groups of correlated responses. In some cases the two interval methods yield vastly different results and the REML-based confidence interval does not maintain the nomiral coverate value even for seemingly large sample sizes. For finite sample size applications, the validity of the REML-based procedure depends on the correlation structure of the data.

Bootstrap-based tolerance intervals for application to method validation

Recently a new validation procedure was developed using a graphical statistical tool – the so-called accuracy profile – that makes interpretation of results easy and straightforward. Accuracy profiles are estimated by tolerance intervals. Most existing methods for constructing tolerance limits are confined to the restrictive case of normally distributed data. The present study is focused on a nonparametric approach based on bootstrap — in order to get out of this constraint. The Mathematical section recalls some definitions and presents the derivation of the new nonparametric bootstrap approach for setting two-sided mean coverage and guaranteed coverage tolerance limits for a balanced one-way random effects model. The section concludes with a simulation study assessing the performance of the bootstrap methods in comparison to classical methods. Finally, the applicability of the proposed intervals is demonstrated by application to the problem of quantitative analytical method validation based on the accuracy profile. This approach is illustrated by an example consisting in the HPLC determination of the vitamers of vitamin B3 (nicotinamide and nicotinic acid) in milk. The efficiency of the new tolerance intervals is demonstrated as well as the applicability of accuracy profiles in the delicate situation where a correction factor must be applied because there is not a full recovery of the analyte. The comparison of the various tolerance intervals also gives some indication on their interpretation.

Bayesian tolerance intervals for the balanced two-factor nested random effects model

Statistical intervals, properly calculated from sample data, are likely to be substantially more informative to decision makers than obtaining a point estimate alone and are often of paramount interest to practitioners and thus management (and are usually a great deal more meaningful than statistical significance or hypothesis tests). Wolfinger (1998, J Qual Technol 36:162–170) presented a simulation-based approach for determining Bayesian tolerance intervals in a balanced one-way random effects model. In this note the theory and results of Wolfinger are extended to the balanced two-factor nested random effects model. The example illustrates the flexibility and unique features of the Bayesian simulation method for the construction of tolerance intervals.

On Lower Tolerance Limits With Accurate Coverage Probabilities for the Normal Random Effects Model

A method for constructing accurate lower tolerance limits for the balanced one-way normal random-effects model is derived by conditioning on an estimator of the unknown expected mean square ratio. Simulation studies indicate that the present procedure is less conservative than that of several existing methods and gives more accurate coverage rates and smaller standard deviations. Numerical examples are given to illustrate the use of the new procedure. Tables needed to implement the procedure are also included.

Nonparametric estimation of normal ranges given one-way ANOVA random effects assumptions

In this note, we compare three strategies that are commonly used in practice for estimating quantiles and quantile functions when data are assumed generated from a standard balanced one-way ANOVA random effects model, e.g. data from repeated assays, where the goal is to nonparametrically estimate/define a normal range. Strategy 1 consists of averaging the within subject values and then estimating quantiles based upon the averaged values in a standard fashion. Strategy 2 consists of estimating quantiles within each replication and then averaging the marginal quantiles over the replications. Strategy 3 consists of estimating quantiles and ignoring the repeated measures mechanism all together. We show that Strategy 1 is generally a poor choice when the goal is to refine the quantile estimation from a single replication through the application of repeated repetitions of an experiment. Strategy 2 and 3 are shown to be asymptotically equivalent.

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.

Confidence Intervals Based on Rounded Data from the Balanced One-Way Normal Random Effects Model

In standard statistical analyses, data are assumed to be essentially exact. But indeed they are often obtained from a relatively crude gaging method and are thus intrinsically “rounded” to some nearest unit. The discussions in Lee and Vardeman (Lee, Chiang-Sheng, Vardeman, Stephen B. (2001). Interval estimation of a normal process mean from rounded data. Journal of Quality Technology 33:335–348.) and Lee and Vardeman (Lee, Chiang-Sheng, Vardeman, Stephen B. (2002). Interval estimation of a normal process standard deviation from rounded data. Communications in Statistics 31:13–34.) for a rounded sample from a single normal distribution established that nominal confidence levels of standard intervals for μ and σ are much larger than actual coverage probabilities when the rounding is severe. In this article we consider interval estimation in the balanced normal one-way random effects model. We demonstrate the deficiency of standard interval estimators in the presence of rounding and show how likelihood-based results from Lee and Vardeman (Lee, Chiang-Sheng, Vardeman, Stephen B. (2002). Interval estimation of a normal process standard deviation from rounded data. Communications in Statistics 31:13–34.) can be used to produce reliable confidence procedures for the two variance components.

NONNEGATIVE ESTIMATORS FOR THE ONE-WAY RANDOM EFFECTS MODEL

The relative merits of ten estimators for the variance component of the balanced and unbalanced one-way random effects models are compared. Six of the estimators are nonnegative, two of which are obtained by modifying the Minimum Variance Quadratic Unbiased Estimator (MIVQUE) and the Weighted Least Square Estimator (WLS), and two more from the positive parts of these estimators. The Minimum Norm Quadratic Estimator (MINQE), which is nonnegative, is adjusted for reducing its bias. The nonnegative Minimum Mean Square Error Estimator (MIMSQE), the Analysis of Variance (ANOVA) and Unweighted Sums of Squares (USS) estimator are also included.

Pivotal estimation with applications for the intraclass correlation coefficient in the balanced one-way random effects model

In this paper pivotal quantities are used to obtain estimators of parameters. As an application of this technique, a family of estimators is presented for the intraclass correlation coefficient in the balanced one-way random effects model. The family is derived by equating a pivotal quantity to different values of the pivoting distribution, and includes the familiar REML and ML estimators. It is shown that under certain conditions, in terms of mean-squared error, most members of the family of estimators are admissible within the family. The authors provide guidance concerning the choice of an individual member of the family for estimation purposes and indicate how the method can be extended to unbalanced designs.

Bayesian estimators of the intraclass correlation coefficient in the one-way random effects model

In this paper Bayesian methods are used to estimate the intraclass correlation coefficient in the balanced one-way random effects model. An estimator associated with the likelihood function derived from a pivotal quantity along with estimators using reference priors are considered. In addition, an estimator based on a posterior median is examined. These estimators are compared to one aiiothei and to the REML (restricted maximum likelihood) estimator in terms of MSE (mean-squared error) A beta-type approximation to the pivotal likelihood is considered This can be combined with a beta prior to produce closed-form expressions that approximate the posterior mean and mode These approximations generally perform well as judged by Bayes risk. Of the estimators considered the authors recommend the one obtained from the pivotal approach. The authors indicate how the estimation procedures may be extended to the unbalanced one-way random effects model.

USING CONFIDENCE INTERVALS TO OBTAIN A FAMILY OF ESTIMATORS OF THE INTRACLASS CORRELATION COEFFICIENT (OR HERITABILITY)

A family of point estimators is presented for the intraclass correlation coefficient (or heritability) in the balanced one-way random effects model. The family is obtained by equating a pivotal quantity to different values of the pivoting distribution, and includes the familiar ML and REML estimators. In terms of mean-squared error, most members of the family of estimators are admissible within the family. A sire model is used to illustrate the estimation of heritability. The authors provide guidance concerning the choice of an individual member of the family for estimation purposes and indicate how the method can be extended to unbalanced designs.