Adaptive interval estimation in one-way random effects models

Two-stage procedures are introduced to control the width and coverage (validity) of confidence intervals for the estimation of the mean, the between groups variance component and certain ratios of the variance components in one-way random effects models. The procedures use the pilot sample data to estimate an “optimal” group size and then proceed to determine the number of groups by a stopping rule. Such sampling plans give rise to unbalanced data, which are consequently analyzed by the harmonic mean method. Several asymptotic results concerning the proposed procedures are given along with simulation results to assess their performance in moderate sample size situations. The proposed procedures were found to effectively control the width and probability of coverage of the resulting confidence intervals in all cases and were also found to be robust in the presence of missing observations. From a practical point of view, the procedures are illustrated using a real data set and it is shown that the resulting unbalanced designs tend to require smaller sample sizes than is needed in a corresponding balanced design where the group size is arbitrarily pre-specified.     

Interval estimation of the mean in a two-stage nested model

The problem of constructing confidence intervals to estimate the mean in a two-stage nested model is considered. Several approximate intervals, which are based on both linear and nonlinear estimators of the mean are investigated. In particular, the method of bootstrap is used to correct the bias in the ‘usual’ variance of the nonlinear estimators. It is found that the intervals based on the nonlinear estimators did not achieve the nominal confidence coefficient for designs involving a small number of groups. Further, it turns out that the intervals are generally conservative, especially at small values of the intraclass correlation coefficient, and that the intervals based on the nonlinear estimators are more conservative than those based on the linear estimators. Compared with the others, the intervals based on the unweighted mean of the group means performed well in terms of coverage and length. For small values of the intraclass correlation coefficient, the ANOVA estimators of the variance components are recommended, otherwise the unweighted means estimator of the between groups variance component should be used. If one is fortunate enough to have control over the design, he is advised to increase the number of groups, as opposed to increasing group sizes, while avoiding groups of size one or two.     

Performance of Confidence Intervals in Regression Models with Unbalanced One-Fold Nested Error Structures

Abstract

In this article we consider the problem of constructing confidence intervals for a linear regression model with unbalanced nested error structure. A popular approach is the likelihood-based method employed by PROC MIXED of SAS. In this article, we examine the ability of MIXED to produce confidence intervals that maintain the stated confidence coefficient. Our results suggest that intervals for the regression coefficients work well, but intervals for the variance component associated with the primary level cannot be recommended. Accordingly, we propose alternative methods for constructing confidence intervals on the primary level variance component. Computer simulation is used to compare the proposed methods. A numerical example and SAS code are provided to demonstrate the methods.     

The present status of confidence interval estimation on variance components in balanced and unbalanced random models

Much research has been conducted to develop confidence Intervals on linear combinations and ratios of variance components in balanced and unbalanced random models.This paper first presents confidence intervals on functions of variance components in balanced designs.These results assume that classical analysis of variance sums of squares are independent and have exact scaled chi-squared distributions.In unbalanced designs, either one or both of these assumptions are violated, and modifications to the balanced model intervals are required.We report results of some recent work that examines various modifications for some particular unbalanced designs.     

Approximate confidence intervals for a ratio of two variance components

New approximate confidence intervals for the ratio of two variance components in an unbalanced mixed .linear model with a single set of random effects are proposed. Contrary to the confidence intervals known in the literature the new intervals preserve the confidence coefficient and cover the exact confidence interval which, however, is not easy to establish as it requires the solution of complicated nonlinear equations.     

Inferences on the Among-Group Variance Component in Unbalanced Heteroscedastic One-Fold Nested Design

This article studies the hypothesis testing and interval estimation for the among-group variance component in unbalanced heteroscedastic one-fold nested design. Based on the concepts of generalized p-value and generalized confidence interval, tests and confidence intervals for the among-group variance component are developed. Furthermore, some simulation results are presented to compare the performance of the proposed approach with those of existing approaches. It is found that the proposed approach and one of the existing approaches can maintain the nominal confidence level across a wide array of scenarios, and therefore are recommended to use in practical problems. Finally, a real example is illustrated.     

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.     

A comparison of inference procedures in unbalanced split-plot designs

Several procedures for constructing confidence intervals and testing hypotheses about fixed effects in unbalanced split-plot experiments are described in this paper. These procedures can also be used for unbalanced repeated measures experiments when the repeated measures satisfy the Huyhn-Feldt (1970) conditions. A number of these procedures require that the whole plot error mean square has a distribution proportional to a chi-square distribution and that it be independent of estimators of the parameter functions. Often, neither of these conditions are met in unbalanced split-plot experiments. Simulation studies of a small design of eight observations and larger designs with 34 to 48 observations are used to investigate the performance of the different procedures.     

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.     

Confidence intervals and tests on the variance ratio in random models with two variance components

The LM test is modified to test any value of the ratio of two variance components in a mixed effects linear model with two variance components. The test is exact, so it can be used to construct exact confidence intervals on this ratio.Exact Neyman-Pearson (NP) tests on the variance ratio are described.Their powers provide attainable upper bounds on powers of tests on the variance ratio.Efficiencies of LM tests, which include ANOVA tests, and NP tests are compared for unbalanced, random, one-way ANOVA models.Confidence intervals corresponding to LM tests and NP tests are described.     

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 for variance component ratios in unbalanced linear mixed models

Methods for constructing confidence intervals for variance component ratios in general unbalanced mixed models are developed. The methods are based on inverting the distribution of the signed root of the log-likelihood ratio statistic constructed from either the restricted maximum likelihood or the full likelihood. As this distribution is intractable, the inversion is rather based on using a saddlepoint approximation to its distribution. Apart from Wald's exact method, the resulting intervals are unrivalled in terms of achieving accuracy in overall coverage, underage, and overage. Issues related to the proper “reference set” with which to judge the coverage as well as issues connected to variance ratios being nonnegative with lower bound 0 are addressed. Applications include an unbalanced nested design and an unbalanced crossed design.     

Maximum likelihood estimation of variance components in a completely random bib design

ABSTRACT

The purposes of this paper are to abstract from a number of articles variance component estimation procedures which can be used for completely random balanced incomplete block designs, to develop an iterated least squares (ITLS) computing algorithm for calculating maximum likelihood estimates, and to compare these procedures by use of simulated experiments. Based on the simulated experiments, the estimated mean square errors of the ITLS estimates are generally less than*those for previously proposed analysis of variance and symmetric sums estimators.     

Using adaptive weighted least squares to reduce the lengths of confidence intervals

The author proposes an adaptive method which produces confidence intervals that are often narrower than those obtained by the traditional procedures. The proposed methods use both a weighted least squares approach to reduce the length of the confidence interval and a permutation technique to insure that its coverage probability is near the nominal level. The author reports simulations comparing the adaptive intervals to the traditional ones for the difference between two population means, for the slope in a simple linear regression, and for the slope in a multiple linear regression having two correlated exogenous variables. He is led to recommend adaptive intervals for sample sizes superior to 40 when the error distribution is not known to be Gaussian.     

A modified harmonic mean test procedure for variance components

A complete class of tests of variance components is characterized within the class of tests statistics of the form of a ratio of a linear combination of chi-squared random variables to an independent chi-squared random variable. This result is used in the context of general unbalanced mixed models to show that the harmonic mean method results in an inadmissible test of the random treatment effects. The harmonic mean procedure is then modified in such a way that the modified test uniformly dominates the original test. Two competitive tests are the LMP (locally most powerful) and Wald's tests, which have optimal power properties against small and large alternatives, respectively. A Monte Carlo simulation study reveals that the modified test outperforms both the LMP and Wald's tests in badly unbalanced designs and that it is a viable alternative in less unbalanced designs.     

Bayesian confidence intervals for means and variances of lognormal and bivariate lognormal distributions

The lognormal distribution is currently used extensively to describe the distribution of positive random variables. This is especially the case with data pertaining to occupational health and other biological data. One particular application of the data is statistical inference with regards to the mean of the data. Other authors, namely Zou et al. (2009), have proposed procedures involving the so-called “method of variance estimates recovery” (MOVER), while an alternative approach based on simulation is the so-called generalized confidence interval, discussed by Krishnamoorthy and Mathew (2003). In this paper we compare the performance of the MOVER-based confidence interval estimates and the generalized confidence interval procedure to coverage of credibility intervals obtained using Bayesian methodology using a variety of different prior distributions to estimate the appropriateness of each. An extensive simulation study is conducted to evaluate the coverage accuracy and interval width of the proposed methods. For the Bayesian approach both the equal-tail and highest posterior density (HPD) credibility intervals are presented. Various prior distributions (Independence Jeffreys' prior, Jeffreys'-Rule prior, namely, the square root of the determinant of the Fisher Information matrix, reference and probability-matching priors) are evaluated and compared to determine which give the best coverage with the most efficient interval width. The simulation studies show that the constructed Bayesian confidence intervals have satisfying coverage probabilities and in some cases outperform the MOVER and generalized confidence interval results. The Bayesian inference procedures (hypothesis tests and confidence intervals) are also extended to the difference between two lognormal means as well as to the case of zero-valued observations and confidence intervals for the lognormal variance. In the last section of this paper the bivariate lognormal distribution is discussed and Bayesian confidence intervals are obtained for the difference between two correlated lognormal means as well as for the ratio of lognormal variances, using nine different priors.     

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.     

Harmonic mean approach to unbalanced mixed effects models under heteroscedastic variances

In this paper we consider unbalanced mixed models (Scheffe's model) under heteroscedastic variances. By using the harmonic mean approach, It is shown that the problems appear to be anologous to those problems from balanced mixed models under homoscedastic variance. Thus, by using harmonic mean approach, statistical inferences about fixed effects and variance components are derived by using those from balanced models under homoscedastic variance. Laguerre polynomial expansion is used Lo approximate sampling distributions of relevant statistics.     

EMPIRICAL EVIDENCE FOR ADAPTIVE CONFIDENCE INTERVALS AND IDENTIFICATION OF OUTLDSRS USING METHODS OF TRIMMING

In an attempt to apply robust procedures, conventional t-tables are used to approximate critical values of a Studentized t-statistic which is formed from the ratio of a trimmed mean to the square root of a suitably normed Winsorized sum of squared deviations. It is shown here that the approximation is poor if the proportion of trimming is chosen to depend on the data. Instead a data dependent alternative is given which uses adaptive trimming proportions and confidence intervals based on trimmed likelihood statistics. Resulting statistics have high efficiency at the normal model, proper coverage for confidence intervals, yet retain breakdown point one half. Average lengths of confidence intervals are competitive with those of recent Studentized confidence intervals based on the biweight over a range of underlying distributions. In addition, the adaptive trimming is used to identify potential outliers. Evidence in the form of simulations and data analysis support the new adaptive trimming approach.     

Inferences on the lognormal mean for complete samples

In many engineering problems it is necessary to draw statistical inferences on the mean of a lognormal distribution based on a complete sample of observations. Statistical demonstration of mean time to repair (MTTR) is one example. Although optimum confidence intervals and hypothesis tests for the lognormal mean have been developed, they are difficult to use, requiring extensive tables and/or a computer. In this paper, simplified conservative methods for calculating confidence intervals or hypothesis tests for the lognormal mean are presented. In this paper, “conservative” refers to confidence intervals (hypothesis tests) whose infimum coverage probability (supremum probability of rejecting the null hypothesis taken over parameter values under the null hypothesis) equals the nominal level. The term “conservative” has obvious implications to confidence intervals (they are “wider” in some sense than their optimum or exact counterparts). Applying the term “conservative” to hypothesis tests should not be confusing if it is remembered that this implies that their equivalent confidence intervals are conservative. No implication of optimality is intended for these conservative procedures. It is emphasized that these are direct statistical inference methods for the lognormal mean, as opposed to the already well-known methods for the parameters of the underlying normal distribution. The method currently employed in MIL-STD-471A for statistical demonstration of MTTR is analyzed and compared to the new method in terms of asymptotic relative efficiency. The new methods are also compared to the optimum methods derived by Land (1971, 1973).