Unbalanced Random Model Research Articles

Test of Hypothesis: In Two Way Unbalanced Random Model

Test of Hypothesis: In Two Way Unbalanced Random Model

Estimation of Heritability in Heteroscedastic One‐Way Unbalanced Random Model

AbstractA modified estimator of heritability is proposed under heteroscedastic one way unbalanced random model. The distribution, moments and probability of permissible values (PPV) for conventional and modified estimators are derived. The behaviour of two estimators has been investigated, numerically, to devise a suitable estimator of heritability under variance heterogeneity. The numerical results reveal that under balanced case the heteroscedasticity affects the bias, MSE and PPV of conventional estimator, marginally. In case of unbalanced situations, the conventional estimator underestimates the parameter when more variable group has more observations and overestimates when more variable group has less observations, MSE of the conventional estimator decreases when more variable group has more observations and increases when more variable group has less observations and PPV is marginally decreased. The MSE and PPV are comparable for two estimators while the bias of modified estimator is less than the conventional estimator particularly for small and medium values of the parameter. These results suggest the use of modified estimator with equal or more observations for more variable group in presence of variance heterogeneity.

Graphical evaluation of the adequacy of the method of unweighted means

A graphical technique is introduced to assess the adequacy of the method of unweighted means in providing approximate F -tests for an unbalanced random model. These tests are similar to those obtained under a balanced ANOVA. The proposed technique is simple and can easily be used to determine the effects of imbalance and values of the variance components on the adequacy of the approximation. The one-way and two-way random models are used to illustrate the proposed methodology. Extensions to higher-order models are also mentioned.

On the Estimation of Heritability from Unbalanced Data

AbstractThe distribution and moments, of ANOVA estimator of heritability are given under unbalanced random model. These expressions are used to investigate the effect of unbalancedness on the bias and variance/MSE of the estimator and also the validity of certain approximations for its variance, numerically. The computed results reveal that the unbalancedness increases both the bias and variance/MSE of the estimator and the Smith‐approximation for the variance of the estimator provides better accuracy.

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.

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

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.

Exact tests for random models with unequal cell frequencies in the last stage

Abstract Khuri and Littell (1987) derived exact tests for testing hypotheses concerning the variance components in an unbalanced random two-way model. The method used in the development of these tests can be extended to more general unbalanced random models. In this article, such an extension is established for models that are unbalanced only with respect to the last stage of their associated designs, which are assumed to have no missing cells. A numerical example is given to illustrate the implementation of the proposed methodology.

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.

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.

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.

On Expectations, Variances, and Covariances of ANOVA Mean Squares by 'Synthesis'

Hartley [1967] has given a general procedure for finding directly the numerical values in the formulas for expectations, variances and covariances of ANOVA mean squares. His procedure is applicable for any analysis of variance and any unbalanced random model with design matrices having exactly one 1 in each row and the remaining elements zero. This paper extends these results to general design matrices and also to any unbalanced mixed model. Generalization to correlated errors and/or correlated random components is also considered. Simple and direct proofs are provided with the help of the 'trace' of a matrix.