Confidence Intervals For Variance Components 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.

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.

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.

Assessing the performance of normal-based and REML-based confidence intervals for the intraclass correlation coefficient

Using normal distribution assumptions, one can obtain confidence intervals for variance components in a variety of applications. A normal-based interval, which has exact coverage probability under normality, is usually constructed from a pivot so that the endpoints of the interval depend on the data as well as the distribution of the pivotal quantity. Alternatively, one can employ a point estimation technique to form a large-sample (or approximate) confidence interval. A commonly used approach to estimate variance components is the restricted maximum likelihood (REML) method. The endpoints of a REML-based confidence interval depend on the data and the asymptotic distribution of the REML estimator. In this paper, simulation studies are conducted to evaluate the performance of the normal-based and the REML-based intervals for the intraclass correlation coefficient under non-normal distribution assumptions. Simulated coverage probabilities and expected lengths provide guidance as to which interval procedure is favored for a particular scenario. Estimating the kurtosis of the underlying distribution plays a central role in implementing the REML-based procedure. An empirical example is given to illustrate the usefulness of the REML-based confidence intervals under non-normality.

Generalized p-Values and Generalized Confidence Intervals for Variance Components in General Random Effect Model with Balanced Data

Various random models with balanced data that are relevant for analyzing practical test data are described, along with several hypothesis testing and interval estimation problems concerning variance components. In this paper, we mainly consider these problems in general random effect model with balanced data. Exact tests and confidence intervals for a single variance component corresponding to random effect are developed by using generalized p-values and generalized confidence intervals. The resulting procedures are easy to compute and are applicable to small samples. Exact tests and confidence intervals are also established for comparing the random-effects variance components and the sum of random-effects variance components in two independent general random effect models with balanced data. Furthermore, we investigate the statistical properties of the resulting tests. Finally, some simulation results on the type I error probability and power of the proposed test are reported. The simulation results indicate that exact test is extremely satisfactory for controlling type I error probability.

Tests and Confidence Intervals for an Extended Variance Component Using the Modified Likelihood Ratio Statistic

Abstract. The large deviation modified likelihood ratio statistic is studied for testing a variance component equal to a specified value. Formulas are presented in the general balanced case, whereas in the unbalanced case only the one‐way random effects model is studied. Simulation studies are presented, showing that the normal approximation to the large deviation modified likelihood ratio statistic gives confidence intervals for variance components with coverage probabilities very close to the nominal confidence coefficient.

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.

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.

Generalized p Values and Confidence Intervals for Variance Components: Applications to Army Test and Evaluation

Various mixed models that are relevant for analyzing Army test data are described, along with several hypothesis testing and interval estimation problems. The problems come up in the context of investigating gun tube accuracy of an M1 Series tank; in particular, for the study of tube-to-tube dispersion. Factors that affect tube-to-tube variability might include the tanks, ammunition lot, ammunition temperature, firing occasions, and so on. Some of these are fixed factors, and others are random factors. The inference problems that arise in the study of tube-to-tube dispersion are somewhat different from those usually encountered in typical ANOVA situations. A unified approach to solving these problems is presented using the concepts of generalized p values and generalized confidence intervals. The performance of the resulting tests and confidence intervals is numerically investigated and is found to be quite satisfactory. Analysis of some Army test data is presented to illustrate the results.

ENUMERATION OF MIXED LINEAR MODELS AND A SAS MACRO FOR COMPUTATION OF CONFIDENCE INTERVALS FOR VARIANCE COMPONENTS

It is well known that there is a one-to-one correspondence between fixed effects ANOVA models involving both crossed and nested factors, and combinatorial objects called "posets". The enumeration of nonisomorphic posets is an interesting and nontrivial combinatorial problem for which answers are available for posets of order 14 or less, i.e. for fixed effects ANOVA models with 14 or fewer factors. Such an enumeration problem does not appear to have been considered for mixed effects models. In this paper we study this problem and obtain a list of nonisomorphic mixed effects models involving five or fewer factors. For instance, it is shown that there are 576 nonisomorphic mixed effects ANOVA models involving five factors. Confidence intervals for variance components in any of these models, with superior coverage properties than those afforded by the application of the Satterthwaite method, may be computed using the methods discussed in the book "Confidence Intervals for Variance Components" by Burdick and Graybill [2]. To facilitate these calculations, we have written a SAS macro that will compute these confidence intervals for any mixed effects saturated ANOVA model involving five or fewer factors. User input to the SAS MACRO is the actual data set along with a matrix indicating whether each factor is fixed or random and the nesting/ crossing configuration among the factors. The operation of the MACRO is illustrated using an example involving real data. Such a MACRO is expected to be extremely useful to practitioners in view of the fact that SAS or any other commonly available statistical software package does not have built in commands for obtaining confidence intervals for variance components discussed in [2]. A copy of this MACRO is available at www.stat.colostate.edurhess/MixedModels.htm.

Confidence intervals for variance components from gauge capability studies

Two methods of constructing confidence intervals for variance components in gauge capability studies are presented in this article. Comparisons are made between the restricted maximum likelihood (REML) method and the modified large sample (MLS) method for constructing confidence intervals for components of repeatability, reproducibility, and total (gauge) variability. The examples considered involve a factorial design and a nonstandard nested design. © 1997 John Wiley & Sons, Ltd.

Confidence intervals for variance components from gauge capability studies

Confidence intervals for variance components from gauge capability studies

Saddlepoint confidence intervals for variance components

In this paper, saddlepoint methods are proposed for confidence intervals for variance components. The accuracy of the saddlepoint approximation is illustrated in this application. A simulation study is also conducted to compare with existing methods in various situations.

Comparison of confidence intervals for variance components with unbalanced data

We compare the performance of three easily-computed confidence intervals for functions of variance components in the one-way random effect model. The confidence intervals are studied for three underlying generating distributions and a variety of sample sizes and degrees of unbalancedness. Confidence intervals based on the asymptotic covariance matrix for maximum likelihood estimates of the parameters perform poorly for small sample sizes for any function of the between variance component, regardless of the underlying distribution of the random effects.

The modification of confidence intervals for variance components in one-way random model using Stein's approach

This paper concerns the problem of “improving” interval estimators of variance components in one-way random model. The traditional method of construction relies on the establishment of confidence intervals, depending on corresponding sums of squares, from the analysis of variance. It has been shown in the paper how information on the estimator for a population mean can be used to “improve” the estimation of variance components.

Interval estimation in two way nested unbalanced random model

The present paper deals with interval estimation of variance components in two way nested unbalanced random model. Employing a suitable transformation a statistic has been developed and its distribution is well approximated by chi-square. The confidence intervals for variance components and the simultaneous confidence interval for their ratios have been derived

Confidence intervals for variance components in the balanced two way model with interaction

Approximate 1 −α upper, lower and two sided confidence intervals for the respective ratios of the individual variances , to the total variation are found for the balanced two-factor crossed components-of-variance model with interaction. Results of a simulation study, performed to estimate the confidence levels of these approximate confidence intervals, are given.

Confidence intervals for variance components in balanced random models when the errors are independent, and related through an autoregressive series

Il saggio propone limiti di confidenza per le componenti di varianza , ottenuti con tre diversi criteri, per modelli casuali bilanciati con disegni a classificazione incrociata. Le tecniche proposte valgono anche per disegni casuali bilanciati a classificazione incrociata con interazione. Per chiarire e sviluppare l'argomento sono illustrati alcuni studi di simulazione. Vengono anche investigati gli effetti delle serie autoregressive degli errori casuali sulle tecniche proposte.