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.     

Nonparametric variance estimation in the analysis of microarray data: a measurement error approach

We investigate the effects of measurement error on the estimationof nonparametric variance functions. We show that either ignoringmeasurement error or direct application of the simulation extrapolation,SIMEX, method leads to inconsistent estimators. Nevertheless,the direct SIMEX method can reduce bias relative to a naiveestimator. We further propose a permutation SIMEX method thatleads to consistent estimators in theory. The performance ofboth the SIMEX methods depends on approximations to the exactextrapolants. Simulations show that both the SIMEX methods performbetter than ignoring measurement error. The methodology is illustratedusing microarray data from colon cancer patients.     

Assessing population genetic structure and variability with RAPD data: Application to Vaccinium macrocarpon (American Cranberry)

A method for estimating and comparing population genetic variation using random amplified polymorphic DNA (RAPD) profiling is presented. An analysis of molecular variance (AMOVA) is extended to accomodate phenotypic molecular data in diploid populations in Hardy-Weinberg equilibrium or with an assumed degree of selfing. We present a two step strategy: 1) Estimate RAPD site frequencies without preliminary assumptions on the unknown population structure, then perform significance testing for population substructuring. 2) If population structure is evident from the first step, use this data to calculate better estimates for RAPD site frequencies and sub-population variance components. A nonparametric test for the homogeneity of molecular variance (HOMOVA) is also presented. This test was designed to statistically test for differences in intrapopulational molecular variances (heteroscedasticity among populations). These theoretical developments are applied to a RAPD data set in Vaccinium macrocarpon (American cranberry) using small sample sizes, where a gradient of molecular diversity is found between central and marginal populations. The AMOVA and HOMOVA methods provide flexible population analysis tools when using data from RAPD or other DNA methods that provide many polymorphic markers with or without direct allelic data.     

Bayesian Inference for the Ratio of the Means of Two Normal Populations with Unequal Variances

The problem of making inferences about the ratio of two normal means has been addressed, both from the frequentist and Bayesian perspectives, by several authors. Most of this work is concerned with the homoscedastic case. In contrast, the situation where the variances are not equal has received little attention. Cox (1985) deals, within the frequentist framework, with a model where the variances are related to the means. His results are mainly based on Fieller's theorem whose drawbacks are well known. In this paper we present a Bayesian analysis of this model and discuss some related problems. An agronomical example is used throughout to illustrate the methods.     

Higher order inference for the consensus mean in inter-laboratory studies

In inter-laboratory studies, a fundamental problem of interest is inference concerning the consensus mean, when the measurements are made by several laboratories which may exhibit different within-laboratory variances, apart from the between laboratory variability. A heteroscedastic one-way random model is very often used to model this scenario. Under such a model, a modified signed log-likelihood ratio procedure is developed for the interval estimation of the common mean. Furthermore, simulation results are presented to show the accuracy of the proposed confidence interval, especially for small samples. The results are illustrated using an example on the determination of selenium in non-fat milk powder by combining the results of four methods. Here, the sample size is small, and the confidence limits for the common mean obtained by different methods produce very different results. The confidence interval based on the modified signed log-likelihood ratio procedure appears to be quite satisfactory.