Maximum Likelihood Estimation of Multivariate Covariance Components for the Balanced One-Way Layout

Klotz, Jerome; Putter, Joseph

doi:10.1214/aoms/1177697618

Cited by 29 publications

(29 citation statements)

References 7 publications

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“…Using the canonical transformation yieldŝ S B ¼ ðTðL À IÞT9Þ=m, and it is readily seen thatŜ B is guaranteed to be nonnegative definite by truncating the elements of L at a minimum of unity, i.e., by replacing L with L w T ¼ Diagfminð1; l i Þg. The resulting estimator is the REML estimator (Klotz and Putter 1969;Amemiya 1985;Anderson et al 1986). In a genetic context where we estimate the matrix of environmental covariances aŝ S E ¼Ŝ W À ða À 1ÞŜ B (with a À1 the degree of relationship among group members), additional constraints may be required to ensure thatŜ E is within the parameter space (Meyer and Kirkpatrick 2008).…”

Section: Review: Principles Of Penalized Estimationmentioning

confidence: 99%

Better Estimates of Genetic Covariance Matrices by “Bending” Using Penalized Maximum Likelihood

Meyer

Kirkpatrick

2010

Genetics

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Obtaining accurate estimates of the genetic covariance matrix S G for multivariate data is a fundamental task in quantitative genetics and important for both evolutionary biologists and plant or animal breeders. Classical methods for estimating S G are well known to suffer from substantial sampling errors; importantly, its leading eigenvalues are systematically overestimated. This article proposes a framework that exploits information in the phenotypic covariance matrix S P in a new way to obtain more accurate estimates of S G . The approach focuses on the ''canonical heritabilities'' (the eigenvalues of S À1 P S G ), which may be estimated with more precision than those of S G because S P is estimated more accurately. Our method uses penalized maximum likelihood and shrinkage to reduce bias in estimates of the canonical heritabilities. This in turn can be exploited to get substantial reductions in bias for estimates of the eigenvalues of S G and a reduction in sampling errors for estimates of S G . Simulations show that improvements are greatest when sample sizes are small and the canonical heritabilities are closely spaced. An application to data from beef cattle demonstrates the efficacy this approach and the effect on estimates of heritabilities and correlations. Penalized estimation is recommended for multivariate analyses involving more than a few traits or problems with limited data.

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Section: Review: Principles Of Penalized Estimationmentioning

confidence: 99%

Better Estimates of Genetic Covariance Matrices by “Bending” Using Penalized Maximum Likelihood

Meyer

Kirkpatrick

2010

Genetics

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“…Earlier, Klotz and Putter (1969) derived maximumlikelihood (ML) and REML estimators of S B for the balanced one-way classification, which were constrained to be p.s.d. Anderson et al (1986) extended this to a p.s.d.…”

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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“…In the literature, beginning with Anderson [3], Morris and Olkin [11], Klotz and Putter [9], and Amemyia and Fuller [1] had studied these problems as well. Anderson et al [2] first successfully obtained the MLEs of covariance matrices for the completely balanced multivariate one-way random effect model in which only two matrices are involved, namely for the case (i) above, K = { : 1 2 }.…”

Section: Introductionmentioning

confidence: 99%

Maximum likelihood estimation of Wishart mean matrices under Löwner order restrictions

Tsai

2007

Journal of Multivariate Analysis

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For Wishart density functions, there remains a long-time question unsolved. That is whether there exists the closed-form MLEs of mean matrices over the partially Löwner ordering sets. In this note, we provide an affirmative answer by demonstrating a unified procedure on exactly how the closed-form MLEs are obtained for the simple ordering case. Under the Kullback-Leibler loss function, a property of obtained MLEs is further studied. Some applications of the obtained closed-form MLEs, including the comparison between our ML estimates and Calvin and Dykstra's [Maximum likelihood estimation of a set of covariance matrices under Löwner order restrictions with applications to balanced multivariate variance components models, Ann. Statist. 19 (1991) 850-869.] which obtained by iterative algorithm, are also made.

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Maximum Likelihood Estimation of Multivariate Covariance Components for the Balanced One-Way Layout

Cited by 29 publications

References 7 publications

Better Estimates of Genetic Covariance Matrices by “Bending” Using Penalized Maximum Likelihood

Better Estimates of Genetic Covariance Matrices by “Bending” Using Penalized Maximum Likelihood

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

Maximum likelihood estimation of Wishart mean matrices under Löwner order restrictions

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