The Residual Likelihood Ratio Test for the Variance Ratio in a Linear Model with Two Variance Components

Residual maximum likelihood has proved to be a successful approach to the estimation of variance components. In this paper, its counterpart in testing, the residual likelihood ratio test, is applied to testing the ratio of two variance components. The test is compared with the Wald test and the locally most powerful invariant test.     

Variance Component Estimation for Mixed Model Analysis of cDNA Microarray Data

Microarrays provide a valuable tool for the quantification of gene expression. Usually, however, there is a limited number of replicates leading to unsatisfying variance estimates in a gene‐wise mixed model analysis. As thousands of genes are available, it is desirable to combine information across genes. When more than two tissue types or treatments are to be compared it might be advisable to consider the array effect as random. Then information between arrays may be recovered, which can increase accuracy in estimation. We propose a method of variance component estimation across genes for a linear mixed model with two random effects. The method may be extended to models with more than two random effects. We assume that the variance components follow a log‐normal distribution. Assuming that the sums of squares from the gene‐wise analysis, given the true variance components, follow a scaled χ²‐distribution, we adopt an empirical Bayes approach. The variance components are estimated by the expectation of their posterior distribution. The new method is evaluated in a simulation study. Differentially expressed genes are more likely to be detected by tests based on these variance estimates than by tests based on gene‐wise variance estimates. This effect is most visible in studies with small array numbers. Analyzing a real data set on maize endosperm the method is shown to work well. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Tests of Bivariate Symmetry with a One-sided Alternative in the Analysis of Variance

We consider a two-factor experiment in which the factors have the same number of levels with a natural ordering among levels. We test the hypothesis that the effects of the two treatments are symmetric against a one-sided alternative using the likelihood ratio criteria. Test of the one-sided alternative as a null hypothesis against no restriction has also been studied. Exact distribution theory under the null hypothesis is derived and is shown to be a weighted mixture of chi-square distributions. An example is used to illustrate the procedure.     

华南虎现状及谱系分析

华南虎目前处于极危状态。野生个体已极为少见,只有中国一些动物园内饲养着 53只。按其亲缘关系分为上海系和贵阳系,上海系的华南虎携带优秀的多子基因,但其子代的生活力不强;而贵阳系的虎虽然产子率不高,但子代的生活力很强,现存的6只贵阳系华南虎具有极高的基因保存与基因研究价值,应建立基因库。这些虎分散饲养于22个单位,大大降低了适龄虎的交配繁殖机会,浪费了本已有限的珍贵资源,是抑制其种群发展的最主要因素。目前其种群的年龄结构非常合理,育龄虎占种群的62.3％,幼龄虎为28.3％,高龄虎很少,仅为9.4％,今后种群的发展乐观。     

On the Optimality of F-test on Variance Components in Mixed Linear Models

The analysis-of-variance tests for hypotheses on random effects in regular linear models are considered. Conditions are given for these tests to be uniformly most powerful unbiased or uniformly most powerful invariant unbiased. An example shows that the difference between these conditions can be serious.     

用方差分析法对葡萄分类性状的评价

本文根据方差分析原理,提出了用F值作为葡萄植物分类性状的评价方法．该法客现可靠,能够区分性状在不同分类水平下的作用大小．文中对156个葡萄分类性状进行了评价,我们认为这种方法同样适合其他植物分类性状的评价．     

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.     

Finite Sample Properties of the Maximum Likelihood Estimator and of Likelihood Ratio Tests in Hidden Markov Models

Hidden Markov models were successfully applied in various fields of time series analysis, especially for analyzing ion channel recordings. The maximum likelihood estimator (MLE) has recently been proven to be asymptotically normally distributed. Here, we investigate finite sample properties of the MLE and of different types of likelihood ratio tests (LRTs) by means of simulation studies. The MLE is shown to reach the asymptotic behavior within sample sizes that are common for various applications. Thus, reliable estimates and confidence intervals can be obtained. We give an approximative scaling function for the estimation error for finite samples, and investigate the power of different LRTs suitable for applications to ion channels, including tests for superimposed hidden Markov processes. Our results are applied to physiological sodium channel data.     

零配比正交试验设计与结果分析

发现了正交配比试验的一种特殊情况-零配比正交试验,极差与平方和特征分析表明:常规正交(配比)试验极差与方差分析法均不能直接用于零配比正交试验结果分析．推导了3因素、3水平、无互作零配比正交试验极差估计与平方和分解校正方法,可有效提高试验结果分析的可靠性．     

The Method of Unweighted Means in Unbalanced Analysis of Variance

For factorial experiments with unbalanced data the method of unweighted means is an alternate method of analysis which is computationally simpler than likelihood ratio based F-tests. The quality of the null approximation to the F-distribution of the statistic for the method of unweighted means and the power of this test relative to the likelihood ratio test are discussed. Recommendations are made of when in the course of data analysis the superior operating characteristics of the likelihood ratio test may not outweigh the computational simplicity of the method of unweighted means test.     

Detection of Cyclic Trends in Incidence Data Using the Efficiency Score Method

In studies involving a cyclic regularity, researchers usually have a good working knowledge regarding the peak time in the cycle. Capitalizing on this information, we derive the asymptotically uniformly most powerful unbiased test for detecting a cyclic trend using the likelihood score, and present the asymptotic power function of the test and the approximate formula for sample size. Numerical studies demonstrate great advantages of the proposed test over the standard test in terms of power and sample size. Asymptotic power of the score test is satisfactorily close to actual power. We also generalize this method so that it is applicable for incidence data from unequally spaced intervals or risk populations of unequal size.     

On power and sample size calculations for likelihood ratio tests in generalized linear models

A direct extension of the approach described in Self, Mauritsen, and Ohara (1992, Biometrics 48, 31-39) for power and sample size calculations in generalized linear models is presented. The major feature of the proposed approach is that the modification accommodates both a finite and an infinite number of covariate configurations. Furthermore, for the approximation of the noncentrality of the noncentral chi-square distribution for the likelihood ratio statistic, a simplification is provided that not only reduces substantial computation but also maintains the accuracy. Simulation studies are conducted to assess the accuracy for various model configurations and covariate distributions.     

Testing the Intraclass Version of Kappa Coeffcient of Agreement with Binary Scale and Sample Size Determination

The intraclass version of kappa coefficient has been commonly applied as a measure of agreement for two ratings per subject with binary outcome in reliability studies. We present an efficient statistic for testing the strength of kappa agreement using likelihood scores, and derive asymptotic power and sample size formula. Exact evaluation shows that the score test is generally conservative and more powerful than a method based on a chi‐square goodness‐of‐fit statistic (Donner and Eliasziw , 1992, Statistics in Medicine 11 , 1511–1519). In particular, when the research question is one directional, the one‐sided score test is substantially more powerful and the reduction in sample size is appreciable.     

Estimation of Design and Genetic Components of Variance in Double Cross Mating Designs

The present paper deals with the estimation of design and genetic components of variance based on full double crosses and partial tetra-allel crosses. We have presented the explicit experissions for the symmetric sums estimators of design components of variance based on Hinkelmann's model. Consequently we have also presented explicitly the genetic components of variance based on full double crosses and partial tetra-allel crosses. The procedure is also illustrated with the help of a hypothetical data.     

A New Algorithm for Explicit Determination of Variance Component Estimations in Mixed Models and Mixed Classifications of Balanced ANOVA

This paper deals with the balanced case of the analysis of variance. The use of a classification function leads to an easy determination of all possible sources of variation of any mixed classification. For mixed models a new method is derived, which allows to represent explicit the ANOVA-estimations of the variance components respectively the estimation of the mean sum of squares of the fixed effects for all sources of variation. Thereby the corresponding F-quotients and the approximate confidence intervals of variance components are received in a simple way.     

A Bootstrap Procedure for Adaptive Selection of the Test Statistic in Flexible Two‐Stage Designs

Adaptive two‐stage designs allow a data‐driven change of design characteristics during the ongoing trial. One of the available options is an adaptive choice of the test statistic for the second stage of the trial based on the results of the interim analysis. Since there is often only a vague knowledge of the distribution shape of the primary endpoint in the planning phase of a study, a change of the test statistic may then be considered if the data indicate that the assumptions underlying the initial choice of the test are not correct. Collings and Hamilton proposed a bootstrap method for the estimation of the power of the two‐sample Wilcoxon test for shift alternatives. We use this approach for the selection of the test statistic. By means of a simulation study, we show that the gain in terms of power may be considerable when the initial assumption about the underlying distribution was wrong, whereas the loss is relatively small when in the first instance the optimal test statistic was chosen. The results also hold true for comparison with a one‐stage design. Application of the method is illustrated by a clinical trial example.     

Numerical comparison between powers of maximum likelihood and analysis of variance methods for QTL detection in progeny test designs: the case of monogenic inheritance

Simulations are used to compare four statistics for the detection of a quantitative trait locus (QTL) in daughter and grand-daughter designs as defined by Soller and Genizi (1978) and Weller et al. (1990): (1) the Fisher test of a linear model including a marker effect within sire or grand-sire effect; (2) the likelihood ratio test of a segregation analysis without the information given by the marker; (3) the likelihood ratio test of a segregation analysis considering the information from the marker; and (4) the lod score which is the likelihood ratio test of absence of linkage between the marker and the QTL. In all cases the two segregation analyses are more powerful for QTL detection than are either the linear method or the lod score. The differences in power are generally limited but may be significant (in a ratio of 1 to 3 or 4) when the QTL has a small effect (0.2 standard deviations) and is not closely linked to the marker (recombination rate of 20% or more).