Estimation of Heritability in Heteroscedastic One‐Way Unbalanced Random Model

A 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.     

The Estimation of Skewness and Kurtosis of Random Effects in the Linear Model

Generalising the ANOVA method of estimating variance components in mixed linear models a simple procedure is presented to estimate skewness and kurtosis of the distributions of the random effects of the model. For the model II of a one-way classification this procedure is demonstrated explicitly.     

Robustness of an S‐Estimator in the One‐Way Random Effects Model

An S‐estimator is defined for the one‐way random effects model, analogous to an S‐estimator in the model of i.i.d. random vectors. The estimator resembles the multivariate S‐estimator with respect to existence and weak continuity. The proof of existence of the estimator yields in addition an upper bound for the breakdown point of the S‐estimator of one of the variance components which is rather low. An improvement of the estimator is proposed which overcomes this deficiency. Nevertheless this estimator is an example that new problems of robustness arise in more structured models.     

On the Estimation of the Nearest Neighbour Distribution,G(t), for Point Processes

A new method is proposed for estimating G(t), the distribution function of the distance from an object to its nearest neighbour in a spatial point process. The new method makes more complete use of the information available and has a smaller mean squared error than that of the existing alternatives. The method appears equally effective with random, clustered and regular patterns.     

Estimation of a Conditional Mean in a Linear Regression Model

Consider the two linear regression models of Y_ij on X_ij, namely Y_ij = β_io + β_il X_ij + ε_ij,j = 1,2,…,n_i, i = 1,2, where ε_ij are assumed to be normally distributed with zero mean and common unknown variance σ². The estimated value of a mean of Y₁ for a given value of X₁ is made to depend on a preliminary test of significance of the hypothesis β₁₁ = β₂₁. The bias and the mean square error of the estimator for the conditional mean of Y₁ are given. The relative efficiency of the estimator to the usual estimator is computed and is used to determine a proper choice of the significance level of the preliminary test.     

On blocking rules for the bootstrap with dependent data

We address the issue of optimal block choice in applicationsof the block bootstrap to dependent data. It is shown that optimalblock size depends significantly on context, being equal ton^1/3, n^1/4 and n^1/5 in the cases of variance or bias estimation,estimation of a onesided distribution function, and estimationof a two-sided distribution function, respectively. A clearintuitive explanation of this phenomenon is given, togetherwith outlines of theoretical arguments in specific cases. Itis shown that these orders of magnitude of block sizes can beused to produce a simple, practical rule for selecting blocksize empirically. That technique is explored numerically.     

On Estimation of Relative Risk in Case Control Studies

An estimator of relative risk in a case control study has been proposed in terms of observed cell frequencies and the probability of disease. The bias of the usual estimator i.e odds ratio as compared to the new estimator has been workedout. The expression of Mean Square Error of proposed estimator has been derived in situations where probability of disease is exactly known and when it is estimated through an independent survey. It has been observed that there is a serious error using odds ratio as an estimate of relative risk when probability of disease is not negligible. In such situations the proposed estimator can be used with advantage.     

The Effect of Design Imbalance on the Power of the F‐test of a Variance Component in the One‐Way Random Model

The ANOVA‐based F‐test used for testing the significance of the random effect variance component is a valid test for an unbalanced one‐way random model. However, it does not have an uniform optimum property. For example, this test is not uniformly most powerful invariant (UMPI). In fact, there is no UMPI test in the unbalanced case (see Khuri , Mathew , and Sinha , 1998). The power of the F‐test depends not only on the design used, but also on the true values of the variance components. As Khuri (1996) noted, we can gain a better insight into the effect of data imbalance on the power of the F‐test using a method for modelling the power in terms of the design parameters and the variance components. In this study, generalized linear modelling (GLM) techniques are used for this purpose. It is shown that GLM, in combination with a method of generating designs with a specified degree of imbalance, is an effective way of studying the behavior of the power of the F‐test in a one‐way random model.     

Prediction of Random Effects in Linear and Generalized Linear Models under Model Misspecification

Summary Statistical models that include random effects are commonly used to analyze longitudinal and correlated data, often with the assumption that the random effects follow a Gaussian distribution. Via theoretical and numerical calculations and simulation, we investigate the impact of misspecification of this distribution on both how well the predicted values recover the true underlying distribution and the accuracy of prediction of the realized values of the random effects. We show that, although the predicted values can vary with the assumed distribution, the prediction accuracy, as measured by mean square error, is little affected for mild‐to‐moderate violations of the assumptions. Thus, standard approaches, readily available in statistical software, will often suffice. The results are illustrated using data from the Heart and Estrogen/Progestin Replacement Study using models to predict future blood pressure values.     

Estimation of heritability using variety trials data from incomplete blocks

An assessment of the heritability of a trait is useful in formulating a breeding strategy for crop improvement. We have considered the estimation of broad-sense heritability from a single-location trial and from multi-locational trials conducted in incomplete blocks. Using residual maximum likelihood estimates of variance components, we estimated the heritability and obtained expressions for the estimate of its bias and its standard error. The estimation procedure is illustrated for 25 barley genotypes evaluated at four locations in West Asia and North Africa during 1992.     

A General Class of Ratio‐Type Estimators Under Super‐Population Model

The problem of constructing classes of estimators for population mean has been widely discussed by various authors under design approach in sample surveys. An attempt by Upadhyaya , Singh , and Vos (1985) has been made to combine the usual mean and ratio estimator with suitable weights in order to define a general class of estimators. This paper is an attempt to study properties of the same estimator under super‐population model. Optimum weights have also been proposed. Results have been supported with some numerical examples.     

On the Estimation of Heritability from Unbalanced Data

The 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.     

Nonparametric Tests in the Unbalanced Multivariate One‐Way Design

A nonparametric model for the multivariate one‐way design is discussed which entails continuous as well as discontinuous distributions and, therefore, allows for ordinal data. Nonparametric hypotheses are formulated by the normalized version of the marginal distribution functions as well as the common distribution functions. The differences between the distribution functions are described by means of the so‐called relative treatment effects, for which unbiased and consistent estimators are derived. The asymptotic distribution of the vector of the effect estimators is derived and under the marignal hypothesis a consistent estimator for the asymptotic covariance matrix is given. Nonparametric versions of the Wald‐type statistic, the ANOVA‐type statistic and the Lawley‐Hotelling statistic are considered and compared by means of a simulation study. Finally, these tests are applied to a psychiatric clinical trial.     

Analysis of the Binary Littermate Data in the One‐Way Layout

In certain experiments with laboratory animals like rodents, it is common to observe the binary littermate data that the fetuses in the same litter provide a set of independent Bernoulli responses conditional on the litter specific abnormal probability. For testing homogeneity of abnormal proportions of several groups, five methods are discussed in the frame of the one‐way layout with litter effects. Simulation results show a superiority of the Rao‐Scott procedure.