Unbiased Estimation Of Variance Components Research Articles

General unbiased estimating equations for variance components in linear mixed models

This paper introduces a general framework for estimating variance components in the linear mixed models via general unbiased estimating equations, which include some well-used estimators such as the restricted maximum likelihood estimator. We derive the asymptotic covariance matrices and second-order biases under general estimating equations without assuming the normality of the underlying distributions and identify a class of second-order unbiased estimators of variance components. It is also shown that the asymptotic covariance matrices and second-order biases do not depend on whether the regression coefficients are estimated by the generalized or ordinary least squares methods. We carry out numerical studies to check the performance of the proposed methods based on typical linear mixed models.

The Augmented Classical Twin Design: Incorporating Genome-Wide Identity by Descent Sharing Into Twin Studies in Order to Model Violations of the Equal Environments Assumption.

The Classical Twin Method (CTM) compares the similarity of monozygotic (MZ) twins with that of dizygotic (DZ) twins to make inferences about the relative importance of genes and environment in the etiology of individual differences. The design has been applied to thousands of traits across the biomedical, behavioral and social sciences and is arguably the most widely used natural experiment known to science. The fundamental assumption of the CTM is that trait relevant environmental covariation within MZ pairs is the same as that found within DZ pairs, so that zygosity differences in within-pair variance must be due to genetic factors uncontaminated by the environment. This equal environments assumption (EEA) has been, and still is hotly contested, and has been mentioned as a possible contributing factor to the missing heritability conundrum. In this manuscript, we introduce a new model for testing the EEA, which we call the Augmented Classical Twin Design which uses identity by descent (IBD) sharing between DZ twin pairs to estimate separate environmental variance components for MZ and DZ twin pairs, and provides a test of whether these are equal. We show through simulation that given large samples of DZ twin pairs, the model provides unbiased estimates of variance components and valid tests of the EEA under strong assumptions (e.g. no epistatic variance, IBD sharing in DZ twins estimated accurately etc.) which may not hold in reality. Sample sizes in excess of 50,000 DZ twin pairs with genome-wide genetic data are likely to be required in order to detect substantial violations of the EEA with moderate power. Consequently, we recommend that the Augmented Classical Twin Design only be applied to datasets with very large numbers of DZ twin pairs (> 50,000 DZ twin pairs), and given the strong assumptions relating to the absence of epistatic variance, appropriate caution be exercised regarding interpretation of the results.

Teams: Heterogeneity, Sorting, and Complementarity

How much do individuals contribute to team output? I propose an econometric framework to quantify individual contributions when only the output of their teams is observed. The identification strategy relies on following individuals who work in different teams over time. I consider two production technologies. For a production function that is additive in worker inputs, I propose a regression estimator and show how to obtain unbiased estimates of variance components that measure the contributions of heterogeneity and sorting. To estimate nonlinear models with complementarity, I propose a mixture approach under the assumption that individual types are discrete, and rely on a mean-field variational approximation for estimation. To illustrate the methods, I estimate the impact of economists on their research output, and the contributions of inventors to the quality of their patents.

Phenotypically Selective Genotyping Realizes More Genetic Gains in a Rainbow Trout Breeding Program in the Presence of Genotype-by-Environment Interactions.

Selective genotyping of phenotypically superior animals may lead to bias and less accurate genomic breeding values (GEBV). Performing selective genotyping based on phenotypes measured in the breeding environment (B) is not necessarily a good strategy when the aim of a breeding program is to improve animals’ performance in the commercial environment (C). Our simulation study compared different genotyping strategies for selection candidates and for fish in C in a breeding program for rainbow trout in the presence of genotype-by-environment interactions when the program had limited genotyping resources and unregistered pedigrees of individuals. For the reference population, selective genotyping of top and bottom individuals in C based on phenotypes measured in C led to the highest genetic gains, followed by random genotyping and then selective genotyping of top individuals in C. For selection candidates, selective genotyping of top individuals in B based on phenotypes measured in B led to the highest genetic gains, followed by selective genotyping of top and bottom individuals and then random genotyping. Selective genotyping led to bias in predicting GEBV. However, in scenarios that used selective genotyping of top fish in B and random genotyping of fish in C, predictions of GEBV were unbiased, with genetic correlations of 0.2 and 0.5 between traits measured in B and C. Estimates of variance components were sensitive to genotyping strategy, with an overestimation of the variance with selective genotyping of top and bottom fish and an underestimation of the variance with selective genotyping of top fish. Unbiased estimates of variance components were obtained when fish in B and C were genotyped at random. In conclusion, we recommend phenotypic genotyping of top and bottom fish in C and top fish in B for the purpose of selecting breeding animals and random genotyping of individuals in B and C for the purpose of estimating variance components when a genomic breeding program for rainbow trout aims to improve animals’ performance in C.

Bias in estimates of variance components in populations undergoing genomic selection: a simulation study

BackgroundAfter the extensive implementation of genomic selection (GS), the choice of the statistical model and data used to estimate variance components (VCs) remains unclear. A primary concern is that VCs estimated from a traditional pedigree-based animal model (P-AM) will be biased due to ignoring the impact of GS. The objectives of this study were to examine the effects of GS on estimates of VC in the analysis of different sets of phenotypes and to investigate VC estimation using different methods. Data were simulated to resemble the Danish Jersey population. The simulation included three phases: (1) a historical phase; (2) 20 years of conventional breeding; and (3) 15 years of GS. The three scenarios based on different sets of phenotypes for VC estimation were as follows: (1) Pheno1: phenotypes from only the conventional phase (1–20 years); (2) Pheno1 + 2: phenotypes from both the conventional phase and GS phase (1–35 years); (3) Pheno2: phenotypes from only the GS phase (21–35 years). Single-step genomic BLUP (ssGBLUP), a single-step Bayesian regression model (ssBR), and P-AM were applied. Two base populations were defined: the first was the founder population referred to by the pedigree-based relationship (P-base); the second was the base population referred to by the current genotyped population (G-base).ResultsIn general, both the ssGBLUP and ssBR models with all the phenotypic and genotypic information (Pheno1 + 2) yielded biased estimates of additive genetic variance compared to the P-base model. When the phenotypes from the conventional breeding phase were excluded (Pheno2), P-AM led to underestimation of the genetic variance of P-base. Compared to the VCs of G-base, when phenotypes from the conventional breeding phase (Pheno2) were ignored, the ssBR model yielded unbiased estimates of the total genetic variance and marker-based genetic variance, whereas the residual variance was overestimated.ConclusionsThe results show that neither of the single-step models (ssGBLUP and ssBR) can precisely estimate the VCs for populations undergoing GS. Overall, the best solution for obtaining unbiased estimates of VCs is to use P-AM with phenotypes from the conventional phase or phenotypes from both the conventional and GS phases.

Estimating Variance Components in Functional Linear Models With Applications to Genetic Heritability

Quantifying heritability is the first step in understanding the contribution of genetic variation to the risk architecture of complex human diseases and traits. Heritability can be estimated for univariate phenotypes from nonfamily data using linear mixed effects models. There is, however, no fully developed methodology for defining or estimating heritability from longitudinal studies. By examining longitudinal studies, researchers have the opportunity to better understand the genetic influence on the temporal development of diseases, which can be vital for populations with rapidly changing phenotypes such as children or the elderly. To define and estimate heritability for longitudinally measured phenotypes, we present a framework based on functional data analysis, FDA. While our procedures have important genetic consequences, they also represent a substantial development for FDA. In particular, we present a very general methodology for constructing optimal, unbiased estimates of variance components in functional linear models. Such a problem is challenging as likelihoods and densities do not readily generalize to infinite-dimensional settings. Our procedure can be viewed as a functional generalization of the minimum norm quadratic unbiased estimation procedure, MINQUE, presented by C. R. Rao, and is equivalent to residual maximum likelihood, REML, in univariate settings. We apply our methodology to the Childhood Asthma Management Program, CAMP, a 4-year longitudinal study examining the long term effects of daily asthma medications on children.

Estimation of genetic parameters and response to selection for a continuous trait subject to culling before testing

The consequences of assuming a zero environmental covariance between a binary trait 'test-status' and a continuous trait on the estimates of genetic parameters by restricted maximum likelihood and Gibbs sampling and on response from genetic selection when the true environmental covariance deviates from zero were studied. Data were simulated for two traits (one that culling was based on and a continuous trait) using the following true parameters, on the underlying scale: h² = 0.4; r(A) = 0.5; r(E) = 0.5, 0.0 or -0.5. The selection on the continuous trait was applied to five subsequent generations where 25 sires and 500 dams produced 1500 offspring per generation. Mass selection was applied in the analysis of the effect on estimation of genetic parameters. Estimated breeding values were used in the study of the effect of genetic selection on response and accuracy. The culling frequency was either 0.5 or 0.8 within each generation. Each of 10 replicates included 7500 records on 'test-status' and 9600 animals in the pedigree file. Results from bivariate analysis showed unbiased estimates of variance components and genetic parameters when true r(E) = 0.0. For r(E) = 0.5, variance components (13-19% bias) and especially (50-80%) were underestimated for the continuous trait, while heritability estimates were unbiased. For r(E) = -0.5, heritability estimates of test-status were unbiased, while genetic variance and heritability of the continuous trait together with were overestimated (25-50%). The bias was larger for the higher culling frequency. Culling always reduced genetic progress from selection, but the genetic progress was found to be robust to the use of wrong parameter values of the true environmental correlation between test-status and the continuous trait. Use of a bivariate linear-linear model reduced bias in genetic evaluations, when data were subject to culling.

Distribution-free estimators of variance components for multivariate linear mixed models

Non-iterative, distribution-free, and unbiased estimators of variance components including minimum norm quadratic unbiased estimators and the method of moments estimators are derived for multivariate linear mixed models. A general inter-cluster variance matrix, a same-member only general inter-response variance matrix, and an uncorrelated intra-cluster error structure for each response are assumed. Some properties of the proposed estimators such as unbiasedness and existence are discussed, and related computational issues are addressed. A simulation study is conducted to compare the proposed estimators with Gaussian (restricted) maximum likelihood estimators in terms of bias and mean square error. An application of gene expression family study is presented to illustrate the proposed estimators.

Unbiased Estimates of Variance Components With Bootstrap Procedures

This article provides general procedures for obtaining unbiased estimates of variance components for any random-model balanced design under any bootstrap sampling plan, with the focus on designs of the type typically used in generalizability theory. The results reported here are particularly helpful when the bootstrap is used to estimate standard errors of estimated variance components. For the p × i design, Wiley (2000) provided formulas for correcting for bias in bootstrap estimates of variance components. This article extends Wiley's results to any design and any bootstrap procedure. There are important differences in approach, however. In particular, in this article unbiased estimates of variance components are obtained directly for any bootstrap sample through the use of modified expected T-term (uncorrected sums of squares) equations.

Exact Optimality of Balanced Designs for Minimum Norm Quadratic Unbiased Estimation of Variance Components in one-way Classified data

Exact Optimality of Balanced Designs for Minimum Norm Quadratic Unbiased Estimation of Variance Components in one-way Classified data

Effects of Selection on Estimates of Variance Components Using Gibbs Sampling and Restricted Maximum Likelihood

The effect of selection on estimates of variance components using Gibbs sampling mean and mode, REML, and minimum variance quadratic unbiased estimation was examined through simulation. One hundred replicates were generated for 27 combinations of three levels each of selection schemes, population structures, and heritabilities. All populations consisted of 400 animals. All methods were empirically unbiased, except for the Gibbs sampling estimate of the mode for small variance components, for which the posterior distribution was skewed. Mean squared errors decreased for Gibbs sampling and REML estimates when data were selected, but mean squared errors increased with selection and were largest for minimum variance quadratic unbiased estimation. No pattern existed for differences in mean squared errors for randomly mated and unselected data, suggesting that the differences were due to the direct effect of selection rather than to changes in population structure. Based on these results, the use of minimum variance quadratic unbiased estimation of variance components may be less accurate than other methods for potentially selected field data. Advantages of Gibbs sampling to estimate variance components include simple programming of the Gibbs sampling algorithm and easy calculation of variance of estimates and confidence intervals.

Advanced Calculus With Applications in Statistics

Preface. Preface to the First Edition. 1. An Introduction to Set Theory. 1.1. The Concept of a Set. 1.2. Set Operations. 1.3. Relations and Functions. 1.4. Finite, Countable, and Uncountable Sets. 1.5. Bounded Sets. 1.6. Some Basic Topological Concepts. 1.7. Examples in Probability and Statistics. 2. Basic Concepts in Linear Algebra. 2.1. Vector Spaces and Subspaces. 2.2. Linear Transformations. 2.3. Matrices and Determinants. 2.4. Applications of Matrices in Statistics. 3. Limits and Continuity of Functions. 3.1. Limits of a Function. 3.2. Some Properties Associated with Limits of Functions. 3.3. The o, O Notation. 3.4. Continuous Functions. 3.5. Inverse Functions. 3.6. Convex Functions. 3.7. Continuous and Convex Functions in Statistics. 4. Differentiation. 4.1. The Derivative of a Function. 4.2. The Mean Value Theorem. 4.3. Taylor's Theorem. 4.4. Maxima and Minima of a Function. 4.5. Applications in Statistics. 5. Infinite Sequences and Series. 5.1. Infinite Sequences. 5.2. Infinite Series. 5.3. Sequences and Series of Functions. 5.4. Power Series. 5.5. Sequences and Series of Matrices. 5.6. Applications in Statistics. 6. Integration. 6.1. Some Basic Definitions. 6.2. The Existence of the Riemann Integral. 6.3. Some Classes of Functions That Are Riemann Integrable. 6.4. Properties of the Riemann Integral. 6.5. Improper Riemann Integrals. 6.6. Convergence of a Sequence of Riemann Integrals. 6.7. Some Fundamental Inequalities. 6.8. RiemannStieltjes Integral. 6.9. Applications in Statistics. 7. Multidimensional Calculus. 7.1. Some Basic Definitions. 7.2. Limits of a Multivariable Function. 7.3. Continuity of a Multivariable Function. 7.4. Derivatives of a Multivariable Function. 7.5. Taylor's Theorem for a Multivariable Function. 7.6. Inverse and Implicit Function Theorems. 7.7. Optima of a Multivariable Function. 7.8. The Method of Lagrange Multipliers. 7.9. The Riemann Integral of a Multivariable Function. 7.10. Differentiation under the Integral Sign. 7.11. Applications in Statistics. 8. Optimization in Statistics. 8.1. The Gradient Methods. 8.2. The Direct Search Methods. 8.3. Optimization Techniques in Response Surface Methodology. 8.4. Response Surface Designs. 8.5. Alphabetic Optimality of Designs. 8.6. Designs for Nonlinear Models. 8.7. Multiresponse Optimization. 8.8. Maximum Likelihood Estimation and the EM Algorithm. 8.9. Minimum Norm Quadratic Unbiased Estimation of Variance Components. 8.10. Scheffe's Confidence Intervals. 9. Approximation of Functions. 9.1. Weierstrass Approximation. 9.2. Approximation by Polynomial Interpolation. 9.3. Approximation by Spline Functions. 9.4. Applications in Statistics. 10. Orthogonal Polynomials. 10.1. Introduction. 10.2. Legendre Polynomials. 10.3. Jacobi Polynomials. 10.4. Chebyshev Polynomials. 10.5. Hermite Polynomials. 10.6. Laguerre Polynomials. 10.7. Least-Squares Approximation with Orthogonal Polynomials. 10.8. Orthogonal Polynomials Defined on a Finite Set. 10.9. Applications in Statistics. 11. Fourier Series. 11.1. Introduction. 11.2. Convergence of Fourier Series. 11.3. Differentiation and Integration of Fourier Series. 11.4. The Fourier Integral. 11.5. Approximation of Functions by Trigonometric Polynomials. 11.6. The Fourier Transform. 11.7. Applications in Statistics. 12. Approximation of Integrals. 12.1. The Trapezoidal Method. 12.2. Simpson's Method. 12.3. NewtonCotes Methods. 12.4. Gaussian Quadrature. 12.5. Approximation over an Infinite Interval. 12.6. The Method of Laplace. 12.7. Multiple Integrals. 12.8. The Monte Carlo Method. 12.9. Applications in Statistics. Appendix. Solutions to Selected Exercises. General Bibliography. Index.

Consideration of Sire Relationships for Estimation of Variance Components with Interaction of Herd and Sire

Effects of sire relationships for sire and interaction of herd and sire were examined using simulation and minimum variance quadratic unbiased estimates of variance components. Data were simulated for 50 herds and 20 sires for five sire relationship matrices, three data structures, and three interaction levels. A total of 1000 replicates were simulated for each combination of relationship matrix, data structure, and interaction level. The minimum variance quadratic unbiased estimates were calculated for the true (simulation) model, for models ignoring relationships for sires and interaction, and for models excluding interaction of herd and sire. Interaction variance was underestimated when relationships were ignored. Underestimation increased with sire relatedness. Sire variance and heritability estimates increased when variance components were estimated using sire models compared with estimates using interaction models. This overestimation increased with interaction level simulated in the data and as the data were more unbalanced. Estimates of sire variance were as much as 2.7 times larger than that expected, and heritability estimates were as much as 2.8 times larger than that expected.

Inadmissibility but near optimality of an estimator of correlated response variance under additive models

Kleffe, Prasad and Rao (J. Amer. Statist. Assoc. 86, 1991) applied the method of minimum norm quadratic unbiased estimation (MINQUE) to obtain locally ‘optimal’ estimators of correlated response variance (CRV) components from interviewer variance experiments involving both interpenetrating and noninterpenetrating interviewer assignments. Evaluation of efficiency of the MINQU estimator relative to the ANOVA estimator of Fellegi (J. Amer. Statist. Assoc. 69, 1974) and the composite estimator of Biemer and Stokes (J. Amer. Statist. Assoc. 80, 1985) did not show considerable gains in efficiency. This raises the question about admissibility of these estimators. We prove that the entire class of admissible estimators in the set of quadratic unbiased estimators coincides with the class of MINQU estimators, and study the relationship of the Biemer–Stokes family of estimators with this class. The additive model considered here is also of independent interest as it provides a simple, but nontrivial, example of a balanced linear model that does not permit uniformly minimum variance unbiased estimation of variance components, but admitting an explicit and simple characterization of all admissible quadratic unbiased estimators.

On least squares estimation of generalized covariance functions

Least squares estimation (LSE) is theoretically related to quadratic unbiased estimation of variance components. It is argued that these methods of estimation of variance components essentially generalize LSE though they are not formally equivalent.

The Unbiased Estimation of Variance Components by a Simple Method.

A set of quadratic forms are proposed, which contain an approximate solution vector and the right-hand side vector in the mixed model equations after absorption of fixed effects on one and another side, respectively. The approximate solutions are based on utilizing information included in the offdiagonal elements as well as in the diagonal elements of the coefficient matrix. The quadratics are translation invariant and whose matrices are not symmetric. Using the quadratics, a method for unbiased estimation of variance components in the mixed linear model is described as a simple approximation to the minimum variance quadratic unbiased estimation. The current approach does not require the calculation of a generalized inverse of the coefficient matrix as do not other approximate approaches. The relative ability of the present method to eliminate bias due to a culling tyqe of selection is examined by Monte Cario simulation.

Optimality of balanced designs for minimum norm quadratic unbiased estimation of variance components

This paper develops an approximate theory for the optimality of balanced designs under minimum norm quadratic unbiased estimation of variance components in one-way classified data.

Application of a two-stage random effects model to longitudinal pulmonary function data from sarcoidosis patients.

We applied a two-stage random effects model to pulmonary function data from 31 sarcoidosis patients to illustrate its usefulness in analysing unbalanced longitudinal data. For the first stage, repeated measurements of percentage of predicted forced vital capacity (FVC%) from an individual were modelled as a function of time since initial clinical assessment. At the second stage, parameters of this function were modelled as a function of certain patient characteristics. We used three methods for estimating the model parameters: maximum likelihood; empirical Bayes; and a two-step least-squares procedure. Similar results were obtained from each, but we recommend the empirical Bayes, since it provides unbiased estimates of variance components. Results indicated that deterioration in FVC% is associated with a higher initial FVC% value and large numbers of both total cells and eosinophils in bronchoalveolar lavage at the initial assessment. Improvement is associated with higher values of pulmonary Gallium uptake at initial assessment and race. Blacks are more likely to improve than whites.

Estimation of parameters in certain hierarchical genetic models

Some hierarchical genetic models are considered. Necessary and sufficient conditions for the existence of best linear unbiased estimators of fixed parameters and best quadratic unbiased estimators of variance components in these models are given. An extended concept of the Gauss-Markov estimator is used to construct best estimators.

Minimum Variance Quadratic Unbiased Estimation (MIVQUE) of Variance Components

Minimum variance quadratic unbiased estimators (MIVQUE's) of variance components from unbalanced data are obtained for the one-way classification random model under normality. Explicit, computable expressions are given for the estimators, their variances, and their covariance. The variance expressions provide readily-calculated lower bounds for the variances of any quadratic unbiased estimators of the variance components. For unbalanced data, the estimators are functions of the data and of constants σ ao 2 and σ e 2, taken as a priori estimates of the variance components σ a 2 and σ e 2. The estimators are, for unbalanced data, only locally minimum variance, i.e., they are only minimum variance when σ ao 2 = σ a 2 and σ eo 2 = σ e 2. However, numerical results suggest that the “MIVQUE” of σ a 2 may have much smaller variance than the usual ANOVA estimator with unbalanced data, even when σ ao 2 and σ eo 2 deviate considerably from σ a 2 and σ e 2 respectively. In contrast, the ANOVA estimator of σ e 2 seem...