Variance Components Estimators for the Balanced Two-Way Mixed Model

Lee, Kwan R.; Kapadia, C. H.

doi:10.2307/2531404

Cited by 15 publications

(5 citation statements)

References 8 publications

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“…In the balanced case, REML estimators without restrictions on the parameter space have closed form and are identical to their counterparts from (M)ANOVA (e.g., Corbeil and Searle 1976;Lee and Kapadia 1984). For normally distributed data, restricting the eigenvalues of W À1 B in the one-way classification to a minimum of unity in fact is equivalent to REML estimation of S B and S W constrained to the parameter space (Amemiya 1985, Appendix).…”

Section: Bias Due To Sampling Variancementioning

confidence: 99%

Perils of Parsimony: Properties of Reduced-Rank Estimates of Genetic Covariance Matrices

Meyer

Kirkpatrick

2008

Genetics

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Eigenvalues and eigenvectors of covariance matrices are important statistics for multivariate problems in many applications, including quantitative genetics. Estimates of these quantities are subject to different types of bias. This article reviews and extends the existing theory on these biases, considering a balanced one-way classification and restricted maximum-likelihood estimation. Biases are due to the spread of sample roots and arise from ignoring selected principal components when imposing constraints on the parameter space, to ensure positive semidefinite estimates or to estimate covariance matrices of chosen, reduced rank. In addition, it is shown that reduced-rank estimators that consider only the leading eigenvalues and -vectors of the ''between-group'' covariance matrix may be biased due to selecting the wrong subset of principal components. In a genetic context, with groups representing families, this bias is inverse proportional to the degree of genetic relationship among family members, but is independent of sample size. Theoretical results are supplemented by a simulation study, demonstrating close agreement between predicted and observed bias for large samples. It is emphasized that the rank of the genetic covariance matrix should be chosen sufficiently large to accommodate all important genetic principal components, even though, paradoxically, this may require including a number of components with negligible eigenvalues. A strategy for rank selection in practical analyses is outlined.

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Section: Bias Due To Sampling Variancementioning

confidence: 99%

Perils of Parsimony: Properties of Reduced-Rank Estimates of Genetic Covariance Matrices

Meyer

Kirkpatrick

2008

Genetics

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“…The magnitude ofthese biases depends on the true values of the parameters and on the design. The bias increases rapidly with the number of fixed effects (Corbeil and Searle, 1976;Harville, 1977;Lee and Kapadia, 1984). For this reason, the bias of MLE's is increasingly serious, the more levels of fixed effects (e.g., means for environments, sexes) the experimental design involves.…”

Section: Overview Ofthe Problemmentioning

confidence: 99%

Maximum-Likelihood Approaches Applied to Quantitative Genetics of Natural Populations

Shaw¹

1987

Evolution

213

295

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Growing interest in adaptive evolution in natural populations has spurred efforts to infer genetic components of variance and covariance of quantitative characters. Here, I review difficulties inherent in the usual least-squares methods of estimation. A useful alternative approach is that of maximum likelihood (ML). Its particular advantage over least squares is that estimation and testing procedures are well defined, regardless of the design of the data. A modified version of ML, REML, eliminates the bias of ML estimates of variance components. Expressions for the expected bias and variance of estimates obtained from balanced, fully hierarchical designs are presented for ML and REML. Analyses of data simulated from balanced, hierarchical designs reveal differences in the properties of ML, REML, and F-ratio tests of significance. A second simulation study compares properties of REML estimates obtained from a balanced, fully hierarchical design (within-generation analysis) with those from a sampling design including phenotypic data on parents and multiple progeny. It also illustrates the effects of imposing nonnegativity constraints on the estimates. Finally, it reveals that predictions of the behavior of significance tests based on asymptotic theory are not accurate when sample size is small and that constraining the estimates seriously affects properties of the tests. Because of their great flexibility, likelihood methods can serve as a useful tool for estimation of quantitative-genetic parameters in natural populations. Difficulties involved in hypothesis testing remain to be solved.

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“…Thus , using the joint density function of the observations Yij kS from the model in (6.1.1) (see equation (6.3.1», or the sufficient statistics Y..., SSA,SSB,and SSE (see equations (6.3.2) to (6.3.5)), the likelihood function can be obtained and is given by Equating to zero the partial derivatives of .en(L) with respect to /1-, a;, ag, and a;,we obtain Verdooren, 1980;Lee and Kapadia . 1984).…”

Section: Maximum Likelihood Estimatorsmentioning

confidence: 99%

“…The REML estimators of a} , ai, and a; under 3Corbeil and Searle (1976) have used improper REMLestimators by ignoring the nonnegativity requirements for the variance components (seeVerdooren, 1980;Lee and Kapadia , 1984).…”

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confidence: 99%

Two-Way Nested Classification

Sahai

Ojeda

2004

Analysis of Variance for Random Models

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In the preceding two chapters, we have considered experimental situations where the levels of two factor s are crossed. In this and the following chapter we consider experiments where the levels of one ofthe factors are nested within the levels of the other factor. The data for a two-way nested classification are similar to that of a single factor classification except that now replications are grouped into different sets arising from the levels of the nested factor for a given level of the main factor. Suppose the main factor A has a levels and the nested factor B has ab levels which are grouped into a sets of b levels each , and n observations are made at each level of the factor B giving a total of abn observations. The nested or hierarchical designs of this type are very important in many industrial and genetic investigations. For example, suppose an experiment is designed to investigate the variability of a certain material by randomly selecting a batches , b sample s are made from each batch, and finally n analyses are performed on each sample . The purpo se of the investig ation may be to make inferences about the relative contribution of each source of variation to the total variance or to make inferences about the variance components individually. For another example, suppose in a breeding experiment a random sample of a sires is taken, each sire is mated to a sample of b dams, and finally n offspring are produced from each sire-dam mating. Again, the purpose of the investigation may be to study the relative magnitude of the variance components or to make inferences about them individually. MATHEMATICAL MODElThe random effects model for a two-way nested classification can be written as Yijk = f.-L + a ; + {3j(i) + eijk.

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Variance Components Estimators for the Balanced Two-Way Mixed Model

Cited by 15 publications

References 8 publications

Perils of Parsimony: Properties of Reduced-Rank Estimates of Genetic Covariance Matrices

Perils of Parsimony: Properties of Reduced-Rank Estimates of Genetic Covariance Matrices

Maximum-Likelihood Approaches Applied to Quantitative Genetics of Natural Populations

Two-Way Nested Classification

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