Unbalanced One-way Random Effects Model Research Articles

Comparison of confidence intervals for variance components in an unbalanced one-way random effects model

Abstract The purpose of this paper is to study and compare the methods for constructing confidence intervals for variance components in an unbalanced one-way random effects model. The methods are based on a classical exact, generalised pivotal quantity, a fiducial inference and a fiducial generalised pivotal quantity. The comparison of criteria involves the empirical coverage probability that maintains at the nominal confidence level of 0.95 and the shortest average length of the confidence interval. The simulation results show that the method based on the generalised pivotal quantity and the fiducial inference perform very well in terms of both the empirical coverage probability and the average length of the confidence interval. The classical exact method performs well in some situations, while the fiducial generalised pivotal quantity performs well in a very unbalanced design. Therefore, the method based on the generalised pivotal quantity is recommended for all situations.

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

ABSTRACTFor 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.

Confidence intervals for intraclass correlation coefficients in variance components models

Confidence intervals for intraclass correlation coefficients in agreement studies with continuous outcomes are model-specific and no generic approach exists. This paper provides two generic approaches for intraclass correlation coefficients of the form [Formula: see text] The first approach uses Satterthwaite's approximation and an F-distribution. The second approach uses the first and second moments of the intraclass correlation coefficient estimate in combination with a Beta distribution. Both approaches are based on the restricted maximum likelihood estimates for the variance components involved. Simulation studies are conducted to examine the coverage probabilities of the confidence intervals for agreement studies with a mix of small sample sizes. Two different three-way variance components models and balanced and unbalanced one-way random effects models are investigated. The proposed approaches are compared with other approaches developed for these specific models. The approach based on the F-distribution provides acceptable coverage probabilities, but the approach based on the Beta distribution results in accurate coverages for most settings in both balanced and unbalanced designs. A real agreement study is provided to illustrate the approaches.

Approximate interval for the between-group variance under heteroscedasticity

The paper presents an approximate confidence interval for the between-group variance in a one-way heteroscedastic random-effects model. Derivation of the interval is inspired by the ideas in Williams [A confidence intervals for variance components, Biometrika 49 (1962), pp. 278–281] for the homoscedastic case. Simulations suggest that the proposed interval performs better than or comparably to the other available procedures as studied in 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], Hartung and Knapp [On confidence intervals for the among-group variance in the one-way random effects model with unequal error variances, J. Statist. Plann. Inference 127 (2005), pp. 157–177], 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]. Moreover, it is relatively easy to compute.

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.

CONFIDENCE INTERVALS FOR VARIANCE COMPONENTS USING NON-NORMAL DISTRIBUTIONS

Simulation studies are conducted to evaluate the performance of confidence intervals for variance components under non-normal distribution assumptions. Confidence intervals based on the pivotal quantity (PQ) method and the large-sample properties of the restricted maximum likelihood (REML) estimator are considered. Of particular interest is the actual coverage value of nominal 95% confidence intervals for a ratio of variance components. In the context of unbalanced one-way random effects models, simulation results and an empirical example involving arsenic concentrations in oyster tissue suggest that the REML-based confidence interval is preferred.

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.

Comparison of Software Algorithms for Calculating REML Wald Type Confidence Limits for the Between-Group Variance Component in a Small Sample One-Way Random Effects Model Example

Confidence limits for variance components in mixed effects models can be readily estimated by procedures such as PROC MIXED (SAS®), MIXED (SPSS®), LME (S-PLUS®), and XTMIXED (Stata®). For a small sample unbalanced one-way random effects model case study, the REML Wald type confidence limits for the between-group variance obtained from PROC MIXED differ substantially from those from MIXED, LME, and XTMIXED. Simulations indicate that the Satterthwaite approximation used by PROC MIXED results in highly inflated confidence interval lengths, while the interval lengths from the other procedures are usually similar but also biased upward. The coverage rates of the confidence limits from all four procedures appear above the nominal value. Extreme upper limits and problems with computing the confidence limits are also possible, especially with PROC MIXED. In general, one should avoid the naive use of statistical software packages when estimating REML confidence limits for variance components with small datasets, and simulations can assist in examining the validity of the results.

Assessing Occupational Exposure via the Unbalanced One-Way Random Effects Model

In this article, an unbalanced one-way random effects model is considered for the log-transformed shift-long exposure measurements. Exact test and confidence interval for the proportion of workers whose mean exposure exceeds the occupational exposure limit are developed based on the concepts of generalized p-value and generalized confidence interval. Some simulation results to compare the performance of the proposed test with that of the existing method are reported. The simulation results indicate that the proposed method appears to have significant gain in the size and power.

One-sided tolerance limits for unbalanced one-way random effects models: a generalized Mee and Owen procedure

This article proposes a new procedure for obtaining one-sided tolerance limits in unbalanced random effects models. The procedure is a generalization of that proposed by Mee and Owen for the balanced situation, and can be easily implemented, because it only needs a non-central-t table. Two simulation studies are carried out to assess the performance of the new procedure and to compare it with one of the other procedures laid out in previous statistical literature. The article findings show that the new procedure is much simpler to compute and performs better than the previous ones, having inferior values of the gamma bias in a wide range of situations, representative of many actual industrial applications, and behaving also reasonably well in more extreme sampling situations. The use of the new limits is illustrated by an application to an actual example from the steel industry.

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.

Tolerance intervals for unbalanced one-way random effects models with covariates and heterogeneous variances

Tolerance intervals are useful in practice to help determine limits for detection monitoring or assessment monitoring of factors that may impact the environment, ecological systems, or other biological processes. This article provides a procedure for construction of one-sided and two-sided tolerance intervals for a normally distributed random variable when the mean and variance of its distribution are estimated using data following an unbalanced one-way random effects model with covariates under heterogeneous error variances. The procedure developed here is based on the concept of a generalized pivotal quantity which has been frequently used to obtain confidence intervals in situations where conventional methods are difficult to apply or fail to provide s satisfactory solutions. For the one-sided case, the generalized pivotal quantity approach yields an exact solution. On the other hand, the method leads to good approximate intervals for the two-sided case. This is confirmed by a detailed simulation study, showing that the method may be recommended for practical use. Two real-data examples are given to illustrate the applicability of the proposed procedures.

Comparison of Confidence Intervals on Between Group Variance in Unbalanced Heteroscedastic One-Way Random Models

In this article, a new method is proposed for constructing confidence intervals on the between group variance in the unbalanced heteroscedastic one-way random effects model via generalized inference. To compare the resulted interval with the interval by Wimmer and Witkovský (2003) and an approximate generalized confidence interval, a simulation study is conducted. The results indicate that the proposed method performs better than the other two methods, especially for very unbalanced designs.

Robust R-Estimation of a Consensus Value from Multi-Center Studies

We consider multi-center experiments (for determining a consensus value) conducted in possibly heterogeneous set-ups leading to unbalanced heteroscedastic one-way random effects models. When normality of both the random components and their homoscedasticity are in doubt, standard statistical methods may not be valid. Two robust R-estimators (for the common location parameter), based on signed-rank statistics, are proposed and their properties studied. When large heteroscedasticity is present or the distribution of random effect is abnormal, the proposed estimators perform better than the classical weighted least squares and selected estimators. This feature is illustrated with an arsenic in oyster tissue problem, along with some other simulation studies.

Bayesian Tolerance Intervals for the Unbalanced One-Way 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). In this note, a simulation-based approach for determining Bayesian tolerance intervals in an unbalanced one-way random effects model is illustrated. Reference and probability matching priors are first derived for a more general mixed linear model from which the priors for β, σ2ε, and ν in the case of the random effects model follow easily, where β is the grand mean, σ2ε is the variance of random errors, and ν = σ2γ/σ2ε is the ratio of the random effect to noise variances. A tensile-strength example illustrates the flexibility and unique features of the Bayesian simulation method for the construction of tolerance intervals. Although this example has only one random effect, the method can be applied similarly to other unbalanced data sets and models with multiple variance components. In the last section, a procedure is discussed to obtain Bayesian tolerance intervals for the three-component balanced hierarchical design model.

Between group variance component interval estimation for the unbalanced heteroscedastic one-way random effects model

The studied topic is motivated by the problem of interlaboratory comparisons. This paper focuses on the confidence interval estimation of the between group variance in the unbalanced heteroscedastic one-way random effects model. Several interval estimators are proposed and compared by means of the simulation study. The most recommended (safest) is the confidence interval based on Bonferroni's inequality.

Generalizing the Welch Test To Non-Zero Hypotheses on the Variance Component in the One-Way Random Effects Model Under Variance Heterogeneity

In an unbalanced and heteroscedastic one-way random effects model of analysis of variance we consider hypotheses concerning the between group variance where in a classical set-up the hypothesis whether this variance component is zero is tested. We consider the still open problem of testing also extended hypotheses concerning this parameter and generalize the prominent Welch test to deal with these hypotheses, too.

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.

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.

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.