One-way Random Model Research Articles

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.

Higher order inference for the consensus mean in inter-laboratory studies

In inter-laboratory studies, a fundamental problem of interest is inference concerning the consensus mean, when the measurements are made by several laboratories which may exhibit different within-laboratory variances, apart from the between laboratory variability. A heteroscedastic one-way random model is very often used to model this scenario. Under such a model, a modified signed log-likelihood ratio procedure is developed for the interval estimation of the common mean. Furthermore, simulation results are presented to show the accuracy of the proposed confidence interval, especially for small samples. The results are illustrated using an example on the determination of selenium in non-fat milk powder by combining the results of four methods. Here, the sample size is small, and the confidence limits for the common mean obtained by different methods produce very different results. The confidence interval based on the modified signed log-likelihood ratio procedure appears to be quite satisfactory.

신뢰도 연구에서 급내상관계수와 관련한 표본수 결정 방법 비교

신뢰도 연구는 한 명의 평가자가 연구 대상을 반복 측정하거나 여러 명의 평가자가 한 대상을 평가할 때 평가자 내, 평가자 간 일치도를 알아보는 연구로 임상 의학 분야에서 빈번하게 쓰이고 있다. 이 같은 신뢰도 연구에서 적절한 표본수, 평가자수 및 반복수를 결정하는 것은 비용과 시간 측면에서 보다 더 효율적인 연구를 할 수 있게 해주는 중요한 요인이다. 본 연구의 목적은 신뢰도 연구에서 측정치가 정량적일 때 쓰이는 신뢰도 계수인 급내상관계수(ICC)와 관련한 기존의 표본수 산출 방법들을 비교분석하여 적절한 표본수나 반복수를 결정할 때 그 지침을 제공하는데 있다. 기존 논문에서 제시한 Walter 등 (1998), Giraudeau와 Mary (2001), Saito 등 (2006) 그리고 Bonett (2002)의 방법들을 비교하였다. 임의효과 일원배치 모형일 때 같은 조건에서 가장 적은 양의 정보를 필요로 하는 방법을 찾는 목적으로 요인을 변화시켜 가면서 표본수, 반복수, 신뢰구간 폭을 비교한다. 비교해 본 결과, 가장 작은 수의 표본을 필요로 하는 방법은 Giraudeau의 방법, 가장 작은 수의 반복을 필요로 하는 방법은 Saito의 방법으로 나타났다. 가장 많은 수의 표본과 반복을 필요로 한 방법은 Bonett의 방법이었다. 정도는 Giraudeau의 방법이 가장 높았고 Walter, Saito, Bonett 순으로 정도가 떨어졌다. In medical practice and research, the problem of assessing reliability between two or more quantitative measures is quite common. Intraclass correlation coefficient(ICC) is commonly used to scale of reliability. Some methods were developed to calculate the required number of subjects, raters or replicates in one-way or two-way random ANOVA models. This paper, studies and compares the performance of four methods such as Walter et al. (1998), Giraudeau and Mary (2001), Saito et al. (2006) and Bonett (2002). In order to compare the efficiency of methods we compare the number of subjects, replicates and the width of confidence interval of ICC needed for some specific ICC values. In the case of subject size, Giraudeau's method is the best. In case of the number of replicates, Saito's method was superior to others. The width of confidence interval of ICC was narrower for Giraudeau's method than any others.

A note on the existence of the posteriors for one-way random effect probit models

The existence of the posterior distribution for one-way random effect probit models has been investigated when the uniform prior is applied to the overall mean and a class of noninformative priors are applied to the variance parameter. The sufficient conditions to ensure the propriety of the posterior are given for the cases with replicates at some factor levels. It is shown that the posterior distribution is never proper if there is only one observation at each factor level. For this case, however, a class of proper priors for the variance parameter can provide the necessary and sufficient conditions for the propriety of the posterior.

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.

Identification of the variance components in the general two-variance linear model

Bayesian analyses frequently employ two-stage hierarchical models involving two-variance parameters: one controlling measurement error and the other controlling the degree of smoothing implied by the model's higher level. These analyses can be hampered by poorly identified variances which may lead to difficulty in computing and in choosing reference priors for these parameters. In this paper, we introduce the class of two-variance hierarchical linear models and characterize the aspects of these models that lead to well-identified or poorly identified variances. These ideas are illustrated with a spatial analysis of a periodontal data set and examined in some generality for specific two-variance models including the conditionally autoregressive (CAR) and one-way random effect models. We also connect this theory with other constrained regression methods and suggest a diagnostic that can be used to search for missing spatially varying fixed effects in the CAR model.

Tolerance intervals in a two-way nested model with mixed or random effects

We address the problem of deriving a one-sided tolerance interval in a two-way nested model with mixed or random effects. The generalized confidence interval idea is used in the derivation of our tolerance limit, and the results are obtained by suitably modifying the approach in Krishnamoorthy and Mathew [Krishnamoorthy, K. and Mathew, T., 2004, Generalized confidence limits and one-sided tolerance limits in balanced and unbalanced one-way random models. Technometrics, 46, 44–52], for the one-way random model. Our proposed tolerance limit can be estimated by Monte Carlo simulation. We have also been able to develop closed form approximations in some cases. The performance of our tolerance interval is numerically investigated and found to be satisfactory. The results are illustrated with an example.

On the size of the F-test for the one-way random model with heterogeneous error variances

Traditional analysis of variance tests are based on the assumption of homogeneous error variances, which often fails in real experimental situations. Violation of this assumption affects not only the power of the standard F-test, but also its size. When a design is unbalanced, the effect of unequal error variances is even more complex. In this paper, we study the effect of heterogeneous error variances on the size of the F-test concerning the among-group variance component in an unbalanced random one-way model. We also provide a method for computing the true critical value of the F-test for a given level of significance.

Generalized confidence intervals for process capability indices in the one-way random model

The method of generalized confidence intervals is proposed as an alternative method for constructing confidence intervals for process capability indices under the one-way random model for balanced as well as unbalanced data. The generalized lower confidence limits and the coverage probabilities for three commonly used capability indices were studied via simulation, separately for balanced and unbalanced cases. Simulation results showed that the generalized confidence interval procedure is quite satisfactory both in the balanced and unbalanced cases. Examples are provided to illustrate the results.

A parametric bootstrap approach for ANOVA with unequal variances: Fixed and random models

This article is about testing the equality of several normal means when the variances are unknown and arbitrary, i.e., the set up of the one-way ANOVA. Even though several tests are available in the literature, none of them perform well in terms of Type I error probability under various sample size and parameter combinations. In fact, Type I errors can be highly inflated for some of the commonly used tests; a serious issue that appears to have been overlooked. We propose a parametric bootstrap (PB) approach and compare it with three existing location-scale invariant tests—the Welch test, the James test and the generalized F (GF) test. The Type I error rates and powers of the tests are evaluated using Monte Carlo simulation. Our studies show that the PB test is the best among the four tests with respect to Type I error rates. The PB test performs very satisfactorily even for small samples while the Welch test and the GF test exhibit poor Type I error properties when the sample sizes are small and/or the number of means to be compared is moderate to large. The James test performs better than the Welch test and the GF test. It is also noted that the same tests can be used to test the significance of the random effect variance component in a one-way random model under unequal error variances. Such models are widely used to analyze data from inter-laboratory studies. The methods are illustrated using some examples.

Comparison of One-Sided Tolerance Limits in Unbalanced One-Way Random Models

ABSTRACT The objective of this article is to provide and evaluate a new method for constructing tolerance limits in unbalanced one-way random models via generalized inference. A simulation study is conducted to compare the proposed method with that of Krishnamoorthy and Mathew (2004). The simulation results suggest that the proposed method performs well in all designs considered.

Sequential confidence intervals for variance components in one-way random models

A sequential procedure is introduced to construct controlled width confidence intervals for the among groups variance component, as well as for certain ratios of variance components, in one-way random models. Width is controlled in the sense that it does not exceed a predetermined precision d. Several asymptotic results concerning the proposed procedure are given along with simulation results to assess its performance in moderate sample size situations. The proposed interval is also compared with the confidence interval based on the modified large sample method. In addition, some alternatives are suggested to accelerate the proposed procedure. An expression for the asymptotic expected sample size, which was obtained as a by-product of the sequential procedure, is used to discuss planning considerations in one-way random designs.

Optimizing Occupational Exposure Measurement Strategies When Estimating the Log-Scale Arithmetic Mean Value—An Example from the Reinforced Plastics Industry

When assessing occupational exposures, repeated measurements are in most cases required. Repeated measurements are more resource intensive than a single measurement, so careful planning of the measurement strategy is necessary to assure that resources are spent wisely. The optimal strategy depends on the objectives of the measurements. Here, two different models of random effects analysis of variance (ANOVA) are proposed for the optimization of measurement strategies by the minimization of the variance of the estimated log-transformed arithmetic mean value of a worker group, i.e. the strategies are optimized for precise estimation of that value. The first model is a one-way random effects ANOVA model. For that model it is shown that the best precision in the estimated mean value is always obtained by including as many workers as possible in the sample while restricting the number of replicates to two or at most three regardless of the size of the variance components. The second model introduces the 'shared temporal variation' which accounts for those random temporal fluctuations of the exposure that the workers have in common. It is shown for that model that the optimal sample allocation depends on the relative sizes of the between-worker component and the shared temporal component, so that if the between-worker component is larger than the shared temporal component more workers should be included in the sample and vice versa. The results are illustrated graphically with an example from the reinforced plastics industry. If there exists a shared temporal variation at a workplace, that variability needs to be accounted for in the sampling design and the more complex model is recommended.

One- and Two-Sided Tolerance Intervals for General Balanced Mixed Models and Unbalanced One-Way Random Models

In this article we develop procedures for one- and two-sided tolerance intervals for normal general linear models in which there exists a set of independent scaled chi-squared random variables. The proposed procedures are based on the concept of generalized pivotal quantities and are applicable to general mixed models provided that balanced data are available. However, this study focuses on situations involving unbalanced data. Specific attention is given to the unbalanced one-way random model. It is shown that the use of generalized pivotal quantities allows the construction of the tolerance intervals of interest fairly straightforward. Some practical examples are given to illustrate the proposed procedures. Furthermore, detailed statistical simulation studies are conducted to evaluate their performance, showing that the proposed procedures can be recommended for practical use.

불균형 일원랜덤효과모형에서 EQDGs를 이용한 MIVQUE 추정량 비교

본 논문에서는 불균형 일원랜덤효과모형에서 분산성분에 대한 MIVQUE 추정량들의 효율에 대하여 연구하였다. MIVQUE 추정량에 대한 초기 추정치로 3가지 추정치가 사용되었으며 이들을 이용하여 얻어지는 MIVQUE 추정량의 효율성을 척도화된 추정량의 경험적 분위수를 이용하는 EQDGs 플롯을 이용하여 비교하였다. 모의실험 결과 집단간 분산의 초기치를 0으로 하고 집단내 분산의 초기치를 1로 사용한 MIVQUE 추정량의 효율이 추정량의 안정성 관점에서 다른 초기치를 사용한 MIVQUE 추정량에 비하여 약간 효율적인 것으로 나타났다. In this study, the MIVQUE estimators of variance components for the one-way random model with unbalanced data are investigated. In order to compare the efficiency of MIVQUE estimators obtained by using three priori estimates, the Empirical Quantile Dispersion Graphs (EQDGs) are used. From the results of Monte-Carlo study, the MIVQUE estimator using <TEX>${\sigma}^2_{\alpha}\;=\;0\;and\;{\sigma}^2_{varraho}=1$</TEX> as the priori estimate performs well relative to other estimators.

Modeling the coverage probability of a confidence interval on the among-group variance component in the unbalanced random one-way model

The coverage probability of an approximate confidence interval on the among-group variance component, σ α 2 , in a one-way random model is modeled using generalized linear models techniques. The purpose of the proposed modeling is to derive an empirical relationship between the coverage probability on one hand, andk, n., o, and the model’s variance components on the other hand, wherek is the number of groups, n. is the total number of observations, and o is a measure of imbalance for the design used. The latter quantities serve as control variables in the derived model, and the coverage probability is treated as the response variable. Contour plots generated from this model can be easily used to depict the effects of the control variables on the coverage probability. In particular, the plots are utilized to compare four methods for constructing approximate confidence intervals on σ α 2 . Additional advantages of the derived model include prediction of the coverage probability for a given method using only specified values of the control variables, and the determination of operating conditions that result in improved coverage probability within the region of interest.

Reliability And Precision Of Multiple Expressions Of Hang Power Clean External Force

PURPOSE To assess the stability reliability and precision of various expressions of external force obtained during the concentric phase of hang power cleans (HPC) using an inexpensive dynamometer. METHODS Forty-three male (22.7 ± 4.16 yrs), heavy-resistance lifters (body wt 83.4 ± 15.00 kg) with various training backgrounds participated in the study. Subsequent to 3 practice sessions, each subject's 1RM midthigh HPC was established. The 1RM was used for load standardization on subsequent tests. The minimum 1RM HPC required for inclusion in the study was ≥ 70% body wt. One week following 1RM testing, subjects performed duplicate single-repetition HPC at 90% 1RM. A computer-interfaced dynamometer tethered to one end of an Olympic barbell measured bar velocity (sampling rate = 100 Hz) while force was calculated based on load and acceleration. This procedure was repeated 48 h later to assess stability reliability and precision of the various force variables. Reliability was determined via intraclass correlations (ICC; one-way random model), while precision was determined using standard error of measurement (SEm). RESULTS Average rate of force development (ARFD) is the force at a designated point/the respective elapsed time from the initiation of the concentric phase to the designated point. The start gradient is the force @ 50% of time to start position/respective elapsed time, and acceleration gradient is the force @ start position/respective elapsed time for 2nd50%. The mean, std deviation, ICC, and SEm of external force expressions obtained during 90% 1RM hang power cleans are found in Table 1.Table 1: Mean, standard deviation, reliability, and precision for external force variables acquired while performing mid-thigh hang power cleans at a 90% 1RM load.CONCLUSIONS The 9 force expressions presented herein for heavy relative-load HPC appear to have a high level of stability reliability and precision. Nothing from this data set suggests a lack of validity for these variables; however, concurrent validity remains to be established.

Reliability And Precision Of Multiple Expressions Of Hang Power Clean External Force

PURPOSE To assess the stability reliability and precision of various expressions of external force obtained during the concentric phase of hang power cleans (HPC) using an inexpensive dynamometer. METHODS Forty-three male (22.7 ± 4.16 yrs), heavy-resistance lifters (body wt 83.4 ± 15.00 kg) with various training backgrounds participated in the study. Subsequent to 3 practice sessions, each subject's 1RM midthigh HPC was established. The 1RM was used for load standardization on subsequent tests. The minimum 1RM HPC required for inclusion in the study was ≥ 70% body wt. One week following 1RM testing, subjects performed duplicate single-repetition HPC at 90% 1RM. A computer-interfaced dynamometer tethered to one end of an Olympic barbell measured bar velocity (sampling rate = 100 Hz) while force was calculated based on load and acceleration. This procedure was repeated 48 h later to assess stability reliability and precision of the various force variables. Reliability was determined via intraclass correlations (ICC; one-way random model), while precision was determined using standard error of measurement (SEm). RESULTS Average rate of force development (ARFD) is the force at a designated point/the respective elapsed time from the initiation of the concentric phase to the designated point. The start gradient is the force @ 50% of time to start position/respective elapsed time, and acceleration gradient is the force @ start position/respective elapsed time for 2nd50%. The mean, std deviation, ICC, and SEm of external force expressions obtained during 90% 1RM hang power cleans are found in Table 1.Table 1: Mean, standard deviation, reliability, and precision for external force variables acquired while performing mid-thigh hang power cleans at a 90% 1RM load.CONCLUSIONS The 9 force expressions presented herein for heavy relative-load HPC appear to have a high level of stability reliability and precision. Nothing from this data set suggests a lack of validity for these variables; however, concurrent validity remains to be established.

One-Sided Tolerance Limits in Balanced and Unbalanced One-Way Random Models Based on Generalized Confidence Intervals

We consider the problem of deriving one-sided tolerance intervals in the one-way random model with balanced as well as unbalanced data, under the usual normality assumptions. The problems investigated deal with the computation of such intervals for the observable random variable, as well as the unobservable random effect in the one-way random model. The tolerance limits are derived using the concept of a generalized confidence interval. Some approximations are derived for the tolerance limits, and their performance is investigated by simulation. The simulation results show that the proposed tolerance limits are quite satisfactory for practical use.

On the optimum number of levels in the one-way model with random effects

The optimal number of levels is studied for the one-way random model with normally distributed effects. The optimum criteria used are based on the variances of the traditional analysis of variance estimators of the variance components. Exact solutions are compared to earlier results based on lower bounds of the sampling variances. Comparisons are also made to the large-sample variances of the estimates based on restricted maximum likelihood.