Testing methods for the one-way fixed effects ANOVA models of log-normal samples

For one-way fixed effects of log-normal data with unequal variance, the present study proposes a method to deal with heterogeneity. An appropriate hypothesis testing is demonstrated; and one of the approximate tests, such as the Alexander-Govern test, Welch test or James second-order test, is applied to control Type I error rate. Monte Carlo simulation is used to investigate the performance of the F test for log-scale, the F test for original scale, the James second-order test, the Welch test, and the Alexander-Govern test. The simulated results and real data analysis show that the proposed method is valid and powerful.     

Nonparametric estimation in one-way random effects models

We present a rank based method for obtaining point and interval estimates of a scale version of the intraclass correlation coefficient in a one-way random effects model. When compared to the method of Arvesen and Schmitz (1970), the new method is not only applicable to a broader class of situations, but also much easier to implement. Results of a simulation study indicate that the new procedure compares favorably with the Arvesen-Schmitz procedure and the classical normal theory procedure especially If the random components have heavy tailed distributions.     

Best unbiased estimation of fixed effects in mixed ANOVA models

In a linear model with an arbitrary variance–covariance matrix, Zyskind (Ann. Math. Statist. 38 (1967) 1092) provided necessary and sufficient conditions for when a given linear function of the fixed-effect parameters has a best linear unbiased estimator (BLUE). If these conditions hold uniformly for all possible variance–covariance parameters (i.e., there is a UBLUE) and if the data are assumed to be normally distributed, these conditions are also necessary and sufficient for the parametric function to have a uniformly minimum variance unbiased estimator (UMVUE). For mixed-effects ANOVA models, we show how these conditions can be translated in terms of the incidence array, which facilitates verification of the UBLUE and UMVUE properties and facilitates construction of designs having such properties.     

Bayesian estimates in a one-way ANOVA random effects model

Bayesian estimators of variance components are developed, based on posterior mean and posterior mode, respectively, in a one-way ANOVA random effects model with independent prior distributions. The formulas for the proposed estimators are simple. The estimators give sensible results for 'badly-behaved' datasets, where the standard unbiased estimates are negative. They are markedly robust as compared to the existing estimators such as the maximum likelihood estimators and the maximum posterior density estimators.     

Non-negative estimation of variance components in heteroscedastic one-way random-effects ANOVA models

There is a considerable amount of literature dealing with inference about the parameters in a heteroscedastic one-way random-effects ANOVA model. In this paper, we primarily address the problem of improved quadratic estimation of the random-effect variance component. It turns out that such estimators with a smaller mean squared error compared with some standard unbiased quadratic estimators exist under quite general conditions. Improved estimators of the error variance components are also established.     

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.     

Robust Bayesian methods in simple ANOVA models

A robust Bayesian analysis in a conjugate normal framework for the simple ANOVA model is suggested. By fixing the prior mean and varying the prior covariance matrix over a restricted class, we obtain the so-called HiFi and core region, a union and intersection of HPD regions. Based on these robust HPD regions we develop the concept of a ‘robust Bayesian judgement’ procedure. We apply this approach to the simple analysis of variance model with orthogonal designs. The example analyses the costs of an asthma medication obtained by a two-way cross-over study.     

Tests of additivity in mixed and fixed effect two-way ANOVA models with single sub-class numbers

In variety testing as well as in psychological assessment, the situation occurs that in a two-way ANOVA-type model with only one replication per cell, analysis is done under the assumption of no interaction between the two factors. Tests for this situation are known only for fixed factors and normally distributed outcomes. In the following we will present five additivity tests and apply them to fixed and mixed models and to quantitative as well as to Bernoulli distributed data. We consider their performance via simulation studies with respect to the type-I-risk and power. Furthermore, two new approaches will be presented, one being a modification of Tukey’s test and the other being a new experimental design to test for interactions.     

A new computational approach test for one-way ANOVA under heteroscedasticity

This paper proposes a new test statistic based on the computational approach test (CAT) for one-way analysis of variance (ANOVA) under heteroscedasticity. The proposed test was compared with other popular tests according to type I error and power of tests under different combinations of variances, means, number of groups and sample sizes. As a result, it was observed that the proposed test yields better results than other tests in many cases.     

Robust weighted one-way ANOVA: Improved approximation and efficiency

A robust test for the one-way ANOVA model under heteroscedasticity is developed in this paper. The data are assumed to be symmetrically distributed, apart from some outliers, although the assumption of normality may be violated. The test statistic to be used is a weighted sum of squares similar to the Welch [1951. On the comparison of several mean values: an alternative approach. Biometrika 38, 330-336.] test statistic, but any of a variety of robust measures of location and scale for the populations of interest may be used instead of the usual mean and standard deviation. Under the commonly occurring condition that the robust measures of location and scale are asymptotically normal, we derive approximations to the distribution of the test statistic under the null hypothesis and to its distribution under alternative hypotheses. An expression for relative efficiency is derived, thus allowing comparison of the efficiency of the test as a function of the choice of the location and scale estimators used in the test statistic. As an illustration of the theory presented here, we apply it to three commonly used robust location–scale estimator pairs: the trimmed mean with the Winsorized standard deviation; the Huber Proposal 2 estimator pair; and the Hampel robust location estimator with the median absolute deviation.     

A note on Cochran test for homogeneity in one-way ANOVA and meta-analysis

In this paper we provide a formal yet simple and straightforward proof of the asymptotic χ² distribution for Cochran test statistic. Then, we show that the general form of this type of test statistics is invariant for the choice of weights. This fact is important since in practice many such test statistics are constructed with more complicated forms which usually require calculating generalized inverse matrices. Based on our results, we can simplify the construction of the test statistics. More importantly, properties such as anti-conservativeness of this type of test statistics can be drawn from Cochran test statistic. Furthermore, one can improve the performance of the tests by using some modified statistics with correction for small sample size situations.     

ANOVA models for Brownian motion

We investigate longitudinal models having Brownian-motion covariance structure. We show that any such model can be viewed as arising from a related “timeless” classical linear model where sample sizes correspond to longitudinal observation times. This relationship is of practical impact when there are closed-form ANOVA tables for the related classical model. Such tables can be directly transformed into the analogous tables for the original longitudinal model. We in particular provide complete results for one-way fixed and random effects ANOVA on the drift parameter in Brownian motion, and illustrate its use in estimating heterogeneity in tumor growth rates.     

Objective priors for hypothesis testing in one-way random effects models

The Bayes factor is a key tool in hypothesis testing. Nevertheless, the important issue of which priors should be used to develop objective Bayes factors remains open. The authors consider this problem in the context of the one-way random effects model. They use concepts such as orthogonality, predictive matching and invariance to justify a specific form of the priors for common parameters and derive the intrinsic and divergence based prior for the new parameter. The authors show that both intrinsic priors or divergence-based priors produce consistent Bayes factors. They illustrate the methods and compare them with other proposals.     

Nonparametric multiple comparison procedures for unbalanced one-way factorial designs

In this paper, we present several nonparametric multiple comparison (MC) procedures for unbalanced one-way factorial designs. The nonparametric hypotheses are formulated by using normalized distribution functions and the comparisons are carried out on the basis of the relative treatment effects. The proposed test statistics take the form of linear pseudo rank statistics and the asymptotic joint distribution of the pseudo rank statistics for testing treatments versus control satisfies the multivariate totally positive of order two condition irrespective of the correlations among the rank statistics. Therefore, in the context of MCs of treatments versus control, the nonparametric Simes test is validated for the global testing of the intersection hypothesis. For simultaneous testing of individual hypotheses, the nonparametric Hochberg stepup procedure strongly controls the familywise type I error rate asymptotically. With regard to all pairwise comparisons, we generalize various single-step and stagewise procedures to perform comparisons on the relative treatment effects. To further compare with normal theory counterparts, the asymptotic relative efficiencies of the nonparametric MC procedures with respect to the parametric MC procedures are derived under a sequence of Pitman alternatives in a nonparametric location shift model for unbalanced one-way layouts. Monte Carlo simulations are conducted to demonstrate the validity and power of the proposed nonparametric MC procedures.     

Nonparametric repeated significance tests for one-way anova with adoptation to multiple comparisons

Based on two-sample rank order statistics, a repeated significance testing procedure for a multi-sample location problem is considered. The asymptotic distribution theory of the proposed tests is given under the null hypothesis as well as under local alternatives. A Bahadur efficiency result of the repeated significance test relative to the terminal test based solely on the target sample size is presented. In the adaptation of the proposed tests to multiple comparisons, an asymptotically equivalent test statistic in terms of the rank estimators of the location parameters is derived from which the Scheffé method of multiple comparisons can be obtained in a convinient way.     

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.     

The repair alert models with fixed and random effects

This article extends a random preventive maintenance scheme, called repair alert model, when there exist environmental variables that effect on system lifetimes. It can be used for implementing age-dependent maintenance policies on engineering devices. In other words, consider a device that works for a job and is subject to failure at a random time X, and the maintenance crew can avoid the failure by a possible replacement at some random time Z. The new model is flexible to including covariates with both fixed and random effects. The problem of estimating parameters is also investigated in details. Here, the observations are in the form of random signs censoring data (RSCD) with covariates. Therefore, this article generalizes derived statistical inferences on the basis of RSCD albeit without covariates in past literature. To do this, it is assumed that the system lifetime distribution belongs to the log-location-scale family of distributions. A real dataset is also analyzed on basis of the results obtained.     

Model diagnostics for smoothing spline ANOVA models

The author proposes some simple diagnostics for assessing the necessity of selected terms in smoothing spline ANOVA models. The elimination of practically insignificant terms generally enhances the interpretability of the estimates and sometimes may also have inferential implications. The diagnostics are derived from Kullback‐Leibler geometry and are illustrated in the settings of regression, probability density estimation, and hazard rate estimation.