One-way Random Model Research Articles

Effects of non-normality on the power function in a one-way random model

The effects of non-normality on type-I and type-II errors in a one-way random model are investigated for moderate departures from normality. It is found that the probabilities of both errors are more sensitive to the kurtosis of between group effects than that of within group effects.

A note on the satterthwaite confidence interval for a variance

When using a Satterthwaite chi-squared approximation, it is generally thought that the approximation is satisfactory when it is applied to a positive linear combination of mean squares. In this note, we describe how the Williams - Tukey idea for getting a confidence interval for the among groups variance in a random one-way model can be incorporated into Satterthwaite’s procedure for getting a confidence interval for a variance. This adjusted Satterthwaite procedure insures that his chi-squared approximation is always applied to positive linear combinations of mean squares. A small simulation is included which suggests that the adjustment to the Satterthwaite procedure is effective.

Maximum likelihood estimation of variance components of heteroscedastic random anova model

the estimation of variance components of heteroscedastic random model is discussed in this paper. Maximum Likelihood (ML) is described for one-way heteroscedastic random models. The proportionality condition that cell variance is proportional to the cell sample size, is used to eliminate the efffect of heteroscedasticity. The algebraic expressions of the estimators are obtained for the model. It is seen that the algebraic expressions of the estimators depend mainly on the inverse of the variance-covariance matrix of the observation vector. So, the variance-covariance matrix is obtained and the formulae for the inversions are given. A Monte Carlo study is conducted. Five different variance patterns with different numbers of cells are considered in this study. For each variance pattern, 1000 Monte Carlo samples are drawn. Then the Monte Carlo biases and Monte Carlo MSE’s of the estimators of variance components are calculated. In respect of both bias and MSE, the Maximum Likelihood (ML) estimators of varian...

Hypotheses tests for variance components in some multivariate mixed models

Abstract A multivariate mixed model involving exactly one random effect is considered and a locally best invariant (LBI) test is derived for testing the significance of the multivariate variance component corresponding to the random effect. For balanced models, the likelihood ratio test (LRT) is also derived for the same testing problem. In general, the LBI test is different from Pillai's trace test, except for balanced models. The standard multivariate tests based on Wilk's Λ, Pillai's trace and Roy's maximum root are also valid tests for this problem and simulated powers of these tests and the LBI test are reported for several bivariate unbalanced one-way random models. As expected, the LBI test has some advantage over the others for local alternatives. For several bivariate balanced one-way random models, simulation results are also given on the power of the LRT and the tests based on Wilk's Λ, Pillai's trace and Roy's maximum root in order to compare these tests.

On the Effect of Unbalancedness and Heteroscedasticity on the ANOVA Estimator of Group Variance Component in One‐Way Random Model

AbstractThe expression for rth cumulant of ANOVA estimator of group variance component is derived in the One‐way unbalanced random model under heteroscedasticity. The expression is used to study the effect of unbalancedness and heteroscedasticity on the mean and variance of the estimator, numerically. The computed results reveal that the unbalancedness and heteroscedasticity have a combined effect on the mean and variance of the estimator. For certain situations of unequal group sizes and error variances, the mean and variance of the estimator are increased and for certain other situations the values are decreased.

Fixed width interval estimation in a random one way model

Various multistage procedures are introduced to construct fixed width confidence intervals for the mean of the random one-way model. Several results concerning the asymptotic performance of the proposed intervals are given along with simulation results to provide a feel and illustrate the ideas in the moderate sample size situations. We point out that these new sampling techniques provide very practical solution in comparison with the regular classical method of estimation.

Distribution of variance ratio in unbalanced random model under heterogeneity of error variances

The effects of heteroscedasticity have been studied on the mean and variance of F ratio and on the power of F-test in unbalanced one-way random model, numerically. The computed results reveal that the heteroscedasticity and unbalanoedness have combined effects. The mean and variance of F as well as the power of F-test increase with inequality of error variances under balanced and those unbalanced situations where more variable groups have larger size. The effects are of serious nature when more variable groups have smaller size.

On invariant quadratic unbiased estimation of variance components

The present study deals with three different invarint quadratic unbiased estimators (IQUE) for variance components namely quadratic least squares estimators (QLSE), weighted quadratic least squares estimators (WQLSE) and Mitra type estimators (MTE). The variance and covariances of these three different estimators are presented for unbalanced one-way random model. The relative performances of these estimators are assessed based on different optimality criteria like, D-optimality, T-optimality and M-optimality together with variances of these estimators. As a result, it has been shown that MTE has optimal properties.

Quadratic unbiased estimation without invariance and its application in the unbalanced one-way random model

Quadratic unbiased estimation in a general mixed model is considered. It was recently proved in Stępniak (1987, Ann. Inst. Statist. Math. 39, 563–573) that any admissible estimator in the problem is a limit of unique locally best quadratics in a wide sense. We focus on the usual locally best quadratic unbiased estimators (LBQUE's). It is shown, that any locally best invariant quadratic unbiased estimator (or MINQUE with invariance) is a limit of LBQUE's. Explicit formulae for LBQUE's and their variances are given. The unbalanced one-way random model is considered in detail.

On the distributions of components sums of squares in one-way unbalanced random model under heter06ene0us error variances

Exact sampling distributions of sums of squares in the unbalanced one-way random model are obtained under heterogeneous error variances. These distributions are used to investigate the effect of heteroscedasticity and unbalancedness on the probability of negative estimate of the group variance component. The computed results reveal that heteroscedasticity affects the probability of negative estimate in all situations of group sizes. Further, the probability decreases with heterogeneity of error variances for balanced situations and increases with variability among group size for equal error variances case.

Simultaneous tolerance intervals in the random one-way model with covariates

Simultaneous tolerance intervals developed by Limam and Thomas (19881, for the normal regression model, are generalized to the random one-way model with covariates. Simultaneous tolerance intervals for unit means are developed for the balanced model. A simulation study is used to estimate the exact confidence of the tolerance intervals for models with one covariate.

Prediction in a class of mixed models with two variance components

Abstract A best unbiased predictor (BUP) of an arbitrary linear combination of fixed and random effects in mixed linear models is available when the true values of the variance ratios are known. When the true values are unknown, a two-stage predictor, obtained from the BUP by replacing the true values by estimated values, can be used. In this article, exact mean squared errors of two-stage predictors are obtained for a class of mixed models with two variance components that includes the balanced one-way random model and other analysis-of-variance models with proportional frequencies and one balanced random factor.

Robust tests and confidence intervals for error variance in a regression model and for functions of variance components inan unbalanced random one-way model

Tests and confidence intervals for the error variance in a standard regression model are obtained, using weighted jackkife method. The resulting inferences are shown to be robust against nonnormality. A weighted jackknife version of Spjotvoll's (1967) test of against in the unbalanced random one-way model is developed (where {vi} and {eij} are mutually independent random variables with mean zero and variances respectively. Power comparisons are made in a simulation study with the unweighted jackknife version of Arvesen and Layard (1975).

Mean Squared Error of Prediction in Models for Studying Ecological and Agronomic Systems

The mean squared error of prediction (MSEP) as a criterion for evaluating models used for studying ecological and agronomic systems is considered. It is shown that the different sources of error that have been discussed in the literature for such models can be rigorously defined in terms of the MSEP. A situation that occurs frequently for such models is that the model parameters are determined independently of data used to test the model, and the population of interest is structured in subpopulations. It is shown that in this case obtaining estimators of MSEP and of the individual error contributions can be reduced to the classic problem of estimating components of variance in a oneway random model. For comparison, the case where the parameters are estimated from the same data used to test the model is also addressed here.

A Bayesian Approach to the Estimation of Variance Components for the Unbalanced One-Way Random Model

The estimation of variance components in the one-way random model with unequal sample sizes is studied. A simulation study that indicates that modes of posterior distributions have good sampling properties compared with other estimators is presented. The posterior distributions are calculated using a noninformative prior distribution that is uniform on the intraclass correlation. A simulation study for the estimation of the ratio of the variance components is also presented, together with a study of the sampling properties of highest posterior density regions for this ratio, Bayesian estimators appear to be viable competitors to the many classical alternatives in a sampling framework.

A Bayesian Approach to the Estimation of Variance Components for the Unbalanced One-Way Random Model

A Bayesian Approach to the Estimation of Variance Components for the Unbalanced One-Way Random Model

On the Non-Null Distribution of Anova F-Ratio in One Way Unbalanced Random Model

Expressions for the non-null distribution of analysis of variance F-ratio and the power of F-test in the unbalanced one-way random model have been obtained. The exact values of power of the test have been worked out, numerically, and compared with those obtained by using ( i) the Laguerre polynomial and (ii) the two moments approximations. The computed results reveal that the two moments approximation, for the non-null distribution of F-ratio, is quite good and there is no need for the Laguerre polynomial approimation, which is computationally very tedious for practical applications.

Comparisons of Alternative Predictors Under the Balanced One-Way Random Model

Comparisons of Alternative Predictors Under the Balanced One-Way Random Model

Comparisons of Alternative Predictors under the Balanced One-Way Random Model

Abstract Prediction of an arbitrary linear combination of the random effects of a balanced one-way random model is investigated. Alternative two-stage predictors are compared on the basis of their conditional (on the random effects) and unconditional bias and mean squared errors. When the true value of the ratio of expected mean squares is known, there exists a best linear unbiased predictor (BLUP). When the true value is unknown, a two-stage predictor, obtained from the BLUP by replacing the true value with an estimated value, can be used. When the ratio of expected mean squares is estimated by maximum likelihood, Bayesian methods, or various related methods, a two-stage predictor is obtained whose properties compare favorably with, for example, those of the least squares predictor and the positive-part James—Stein predictor.

Robustness of the Restricted Maximum Likelihood Estimator Derived Under Normality as Applied to Data with Skewed Distributions

A one-way random model was used to simulate observations corresponding to sire and error variance components that were either skewed or normally distributed. The objectives were to investigate the dispersion and asymptotic biasedness properties of variance components estimated by an Expectation Maximization algorithm of the Restricted Maximum Likelihood estimator derived under normality when applied to these distributions. Results indicate that if the sire effects are distributed as a continuous normal variable, the dispersion of estimated sire variances is approximately normal regardless of the distribution of the error component. Similarly, the sire distribution had little effect on the estimated error variances when normal errors were generated. However, when either distribution was skewed, the variances of corresponding variance estimates increased. The skewed sire distributions increased the dispersion of heritability estimates. The mean variance estimates across replications indicate variance components estimated by Restricted Maximum Likelihood estimator are robust against skewness in terms of unbiasedness.Skewness may also be caused by the arbitrary scoring of a trait in subjective categories. To examine this, observations were classified in two through six classes coded as zero through five with respect to the number of classes. A separate analysis was performed with each set of classes. The greatest deviation of estimated variances from the true parameter occurred in the binomially distributed case. As the number of classes increased, the mean estimate asymptotically approached the true parameter. Estimates of heritability decreased as the occurrence of a score became rare within a class, regardless of the total number of classes.