Pseudo Expectation Approach to Variance Component Estimation

Quadratic forms utilizing solutions to best linear unbiased prediction equations after absorbing all fixed effects into the equations for random effects were computed and pseudo expectations derived. Pseudo expectations are taken as if a priori values are equal to true values rather than being taken as constants. Quadratic forms different from restricted maximum likelihood forms yielded estimators that did not depend on the inverse of mixed model equations, and estimators were always positive. This approach offers computational advantages over most other methods. Theoretically, this method has properties similar to restricted maximum likelihood. Application to models involving additive genetic relationships and models with covariances among levels of two random factors were outlined.     

A General Purpose Approximation to Restricted Maximum Likelihood: The Tilde-Hat Approach

Restricted maximum likelihood estimates of variances and covariances are often preferred by animal breeders but can be expensive or impossible to compute for large data sets. Less expensive, approximate restricted maximum likelihood estimates can be obtained by using quadratic forms that resemble the restricted maximum likelihood quadratics but have expectations easier to compute. Quadratic forms for the tilde-hat approach resemble the restricted maximum likelihood quadratics more closely than previous approximations. The strategy is not difficult computationally even for models containing additive genetic relationships. Of three approximate strategies tested, tilde-hat gave estimates closest to restricted maximum likelihood in an actual data set. All three approximations were strongly biased downward by selection in simulated data. The restricted maximum likelihood quadratic seems to account for selection well, whereas substitute quadratics account for selection poorly.     

Sensitivity of Methods of Variance Component Estimation to Culling Type of Selection

Two pseudoexpectation methods of variance component estimation were examined for selection bias from culling and were compared with Henderson's simple method and restricted maximum likelihood. Milk yield data were simulated for 100 herds, 50 sires, and approximately 2000 cows per replicate. Heritability was .25, and repeatability was .50. Each cow had two records. Twenty replicates were made. Two data sets were created; one was unselected and included both records of the cow, and the second excluded second records of cows with first records below the herd mean. Sire, cow, and error variances were estimated according to a mixed model that included fixed herd, parity, and random sire, cow, and error effects. Estimates of sire, cow, and error variances and heritability and repeatability were unbiased and were similar for all methods for unselected data. For selected data, sire and cow variances were overestimated and error variances were underestimated with Henderson's simple method and the pseudoexpectation methods. Accordingly, heritabilities and repeatabilities were overestimated. Biases were greatest for Henderson's simple method and differences between the two pseudoexpectation methods were small. With restricted maximum likelihood, sire variances were unbiased but error variances were slightly overestimated and cow variances underestimated. Use of restricted maximum likelihood is preferred with selected data.     

Sequential estimation of genetic and phenotypic parameters in multitrait mixed model analysis

Single-trait and multitrait (2-, 3-, 4-, and 5-trait) restricted maximum likelihood methods were applied to the same set of data with complete information on all traits. Results suggest that parameter estimates from a data set vary depending upon the type of analysis (single- or multitrait model) and upon the other traits included in multitrait analysis. The choice of parameter estimation method for a breeding design should be based on the breeding goal. In parameter estimation or sire evaluation, traits included in a multitrait analysis should correspond to the traits of interest in the breeding goal. Multitrait analysis explores all intercorrelations simultaneously in parameter estimation and thus provides a complete picture of all interrelationships among traits. In contrast, single-trait analysis produces pairwise (simple) correlations and ignores the possible contribution of other related traits under study to the pairwise correlation. The 5-trait model analysis through canonical transformation was about 300% more efficient in terms of computer time than single-trait model analysis of the same 5 traits. In this study, parameter estimates converged faster under multitrait analysis through canonical transformation than under single-trait analysis.     

Estimation of Variances and Covariances Within the Allowable Parameter Space

Two computationally simple methods for estimation of variances and covariances, with estimates always within the allowable parameter space, are presented for multiple traits. Both methods involve transformations of data to convert observations into independent sets of traits. A Monte Carlo simulation comparison of eight methods for estimating variances and covariances was conducted. The two proposed methods compared favorably with a general purpose restricted maximum likelihood method. However, one of the methods was slow to converge and requires additional work to make it more useful.     

Correlations of various direct and maternal effects for calving ease

Multitrait restricted maximum likelihood methods were used to estimate heritabilities of direct and maternal effects for calving ease and their genetic correlation for heifers and cows. Heritabilities were 4.9% for direct effects and 4.8% for maternal effects of heifers and 1.1% and .7% for cows. Genetic or direct maternal correlations were--.40 in heifers and .07 in adult cows. Restricted maximum likelihood methods also were used to estimate the genetic correlation (.995) between direct effects for calving ease measured in heifers and in cows.     

Quantitative trait loci mapping for dairy cattle production traits using a maximum likelihood method

A maximum likelihood method was developed for QTL mapping in half-sib designs and compared to the regression method in analyses of both field and simulated data. The field data consisted of milk production evaluations of 433 progeny tested sons of 6 sires and 64 microsatellite markers distributed over 12 chromosomes. Based on permutation tests, 5 significant QTL were detected in the field data by the regression method compared with 10 by the maximum likelihood method (P < 0.05). In field data analysis, the maximum likelihood method detected more significant QTL and had a smaller residual variance than the regression method. The simulation included 9 scenarios differing in number of families, family size, QTL variance, and marker density, each replicated 100 times. The simulation results suggested that, as for the regression method, the precision of estimating QTL from the maximum likelihood method improves with increasing number of sons per sire, increasing the ratio of QTL to phenotypic variance, and decreasing marker interval. The maximum likelihood method had a smaller dispersion of estimated QTL positions than the regression method in 6 of 9 scenarios simulated. Overall, the maximum likelihood method shows potential advantage in QTL detection over the regression method, especially in the situations with less favorable conditions for QTL detection.     

Novel algorithm for tomographic reconstruction of atmospheric chemicals with sparse sampling

Numerical studies were performed to evaluate a new air monitoring method for reconstructing chemical exposures and source emissions, based upon optical remote sensing (ORS) and computed tomography (CT). With an ORS-CT system, two-dimensional maps of chemical concentrations can be created that have good spatial and temporal resolution. The mathematical algorithm used to compute the distribution is critical for accurate and useable reconstructions of the concentrations. In this research, a novel reconstruction method was tested that uses maximum likelihood expectation maximization (MLEM) combined with two techniques called grid-translation and multi-grid (GT-MG). To evaluate this method, computer simulations were performed using 120 test maps of varying complexity and a simulated ORS system with four instruments and a total of 40 path-integrated measurements. The results were quantitatively compared with two previously used reconstruction methods (single-grid and grid-translation). Results using the GT-MG method were dramatically improved over previously used methods. Quantitatively, peak exposure errors were reduced by up to 85% and artifacts were dramatically minimized.     

Estimating Variance Components in a Class of Mixed Models by Restricted Maximum Likelihood

An efficient algorithm is described for computing restricted maximum likelihood estimates of variance components in a class of models. The class of models is characterized by effects to be absorbed, which are nested within herds, other fixed effects, random sire effects, one other group of random effects nested within herds, and a random residual. The model can be generalized to account for relationships among sires. Matrix inversion is not used. A realistic example is presented.     

C. R. Henderson, the statistician; and his contributions to variance components estimation.

C. R. Henderson's 1953 Biometrics paper "Estimation of Variance and Covariance Components" is an outstanding landmark in the discipline of statistics. It sets out the very first ideas of how to estimate variance components from unbalanced (unequal subclass numbers) data in situations more complicated than the one-way classification (completely randomized design). As such it had three important, long-lasting impacts. First, it provided methods for actually using unbalanced data, even in large quantity, for estimating variance components. And this has played a tremendous role in population genetics and in animal breeding where the use of estimated variance components is vital to the application of selection theory and selection index techniques. Second, that 1953 paper stimulated numerous statisticians to become interested in random effects, mixed models, and variance components estimation, with such statistical greats as H. O. Hartley and C. R. Rao making contributions in the late 1960s and early 1970s. By then, improved methods of estimating variance components from unbalanced data had been developed, namely maximum likelihood (ML) and restricted maximum likelihood (REML). Once computing power had expanded to the point where these methods became feasible, Henderson made notable contributions to these methods, allied to his two great interests: animal breeding and feasible computing procedures. For both of these, his mixed model equations were a salient feature. Third, these methods reached a wide audience of geneticists and statisticians.     

Application of a mixed normal mixture model for the estimation of Mastitis-related parameters

The current methodology for estimating genetic parameters for SCC (SCS) does not account for the difference in SCS between healthy cows and cows with an intramammary infection (IMI). We propose a two-component finite mixed normal mixture model to estimate IMI prevalence, separate SCS subpopulation means, individual posterior probabilities of IMI, and SCS variance components. The theory is presented and the expectation-conditional maximization algorithm is utilized to compute maximum likelihood estimates. The methodology is illustrated on two simulated data sets based on the current knowledge of SCS parameters. Maximum likelihood estimates of IMI prevalence and SCS subpopulation means were close to simulated values, except for the estimate of IMI prevalence when both subpopulations were almost confounded. Individual posterior probabilities of IMI were always higher among infected than among healthy cows. Error and additive variance components obtained under the mixture model were closer to simulated values than restricted maximum likelihood estimates obtained assuming a homogeneous SCS distribution, especially when subpopulations were completely separated and when mixing proportion was highest. Convergence was linear and rapid when priors were chosen with caution. The advantages of the methodology are demonstrated, and its feasibility for large data sets is discussed.     

Methods for fitting a parametric probability distribution to most probable number data

Every year hundreds of thousands, if not millions, of samples are collected and analyzed to assess microbial contamination in food and water. The concentration of pathogenic organisms at the end of the production process is low for most commodities, so a highly sensitive screening test is used to determine whether the organism of interest is present in a sample. In some applications, samples that test positive are subjected to quantitation. The most probable number (MPN) technique is a common method to quantify the level of contamination in a sample because it is able to provide estimates at low concentrations. This technique uses a series of dilution count experiments to derive estimates of the concentration of the microorganism of interest. An application for these data is food-safety risk assessment, where the MPN concentration estimates can be fitted to a parametric distribution to summarize the range of potential exposures to the contaminant. Many different methods (e.g., substitution methods, maximum likelihood and regression on order statistics) have been proposed to fit microbial contamination data to a distribution, but the development of these methods rarely considers how the MPN technique influences the choice of distribution function and fitting method. An often overlooked aspect when applying these methods is whether the data represent actual measurements of the average concentration of microorganism per milliliter or the data are real-valued estimates of the average concentration, as is the case with MPN data. In this study, we propose two methods for fitting MPN data to a probability distribution. The first method uses a maximum likelihood estimator that takes average concentration values as the data inputs. The second is a Bayesian latent variable method that uses the counts of the number of positive tubes at each dilution to estimate the parameters of the contamination distribution. The performance of the two fitting methods is compared for two data sets that represent Salmonella and Campylobacter concentrations on chicken carcasses. The results demonstrate a bias in the maximum likelihood estimator that increases with reductions in average concentration. The Bayesian method provided unbiased estimates of the concentration distribution parameters for all data sets. We provide computer code for the Bayesian fitting method.     

Integration of Interbull's multiple across-country evaluation approach breeding values into the multiple-trait single-step random regression test-day genetic evaluation for yield traits of Australian Red breeds

Interbull's multiple across-country evaluaftion provides national breeding organizations with breeding values for internationally used bulls, which integrate performance data obtained in different breeding populations, environments, and production systems. However, breeding value-based selection decisions on domestic individuals born to foreign sires can only benefit from Interbull breeding values if they are integrated such that their information can contribute to the breeding values of all related domestic animals. For that purpose, several methods have been proposed which either model Interbull breeding values as prior information in a Bayesian approach, as additional pseudo data points, or as correlated traits, where these methods also differ in their software and parameterization requirements. Further, the complexity of integration also depends on the traits and genetic evaluation models. Especially random regression models require attention because of the dimensionality discrepancy between the number of Interbull breeding values and the number of modeled genetic effects. This paper presents the results from integrating 16,063 Interbull breeding values into the domestic single-step random regression test-day model for milk, fat, and protein yield for Australian Red dairy cattle breeds. Interbull breeding values were modeled as pseudo data points with data point-specific residual variances derived within animal across traits, ignoring relationships between integrated animals. Results suggest that the integration was successful with regard to alignment of Interbull breeding values with their domestic equivalent as well as with regard to the individual and population-wide increase in reliabilities. Depending on the relationship structure between integration candidates, further work is required to account for those relationships in a computationally feasible manner. Other traits with separate parity effects nationally could use a similar approach, even if not modeled with a test-day model.     

Correlations between calving ease and calf survival

Multitrait restricted maximum likelihood methods were used for estimation of heritabilities and genetic and phenotypic correlations between calving ease and calf survival. These analyses examined both direct and maternal effects. Genetic correlation for direct effects between calving ease and calf survival was approximately -.8 in both heifers and cows, indicating favorable associations, whereas phenotypic correlations were -.32 and -.22.     

Estimating unit costs of nutrients from market prices of feedstuffs

The determinations of nutrient costs and breakeven prices of feedstuffs are important activities in the field of nutritional economics. Current methods are either extensions of the Petersen method, which dates back to 1932, or mathematical programming methods generally set as linear programming problems. Both sets of methods have severe limitations that make them unsuitable or biased for deriving unit costs of nutrients and aggregate breakeven costs of feedstuffs in a given market. We propose a least-squares method that, under verifiable conditions, yields maximum likelihood estimates of unit costs of nutrients and breakeven prices of feedstuffs. Assumptions to the maximum likelihood estimates are verified with studentized residuals, leverage values, and variance inflation factors. The method has been programmed into a Windows application and can be used by those with little formal training in statistics.     

Simulation study on covariance component estimation for two binary traits in an underlying continuous scale

The usefulness of the variance and covariance component estimation methods based on a threshold model was studied in a multiple-trait situation with two binary traits. Estimation equations that yield marginal maximum likelihood estimates of variance components on the underlying continuous variable scale and point estimates of location parameters with empirical Bayesian properties are described. Methods were tested on simulated data sets that were generated to exhibit three different incidences, 25, 15, and 5%. Results were compared with analyses of the same data sets with a REML method based on normal distribution and a linear model. Heritabilities and residual correlations calculated from discrete observations were transformed to underlying parameters. In estimation of heritabilities, all methods performed equally well at all incidence levels and with no detectable bias. As suggested by threshold theory, the genetic correlation was accurately estimated directly from the observations without any need of correction for incidence. Marginal maximum likelihood estimates of genetic correlations were similar to linear model estimates; discrepancies from the true parameters were consistent with both methods. In estimation of residual correlations, the method with the linear model approach yielded satisfactory estimates only at the highest incidence level, 25%. For 5% incidence, the uncorrected estimate of residual correlation was 50% less than the true value, and after correction for incidence, the parameter was overestimated by 90%. The estimates of residual correlation from the threshold model were regarded fair, except at the lowest level of incidence, where the estimate was 27% higher than the true value. Results indicated that when an accurate estimate of residual correlation is needed, the marginal maximum likelihood estimates are superior to the estimates calculated with the linear model. Using correction for the incidence level for residual correlation did not work well except at the highest incidence level.     

Nonlinear Model for Describing Convergence of Iterative Methods of Variance Component Estimation

The common intercept approach of reducing the number of iterations in variance component estimation by restricted maximum likelihood is equivalent to assuming a nonlinear model for the convergence curve.     

Estimation of genetic (co)variances for milk yield in first three lactations using an animal model and restricted maximum likelihood

Genetic relationships among lactation records are of interest because most selection of bulls is on first lactations. Selection also complicates estimation of genetic parameters. Techniques unbiased by selection should be used. Estimation of genetic and environmental (co)variances was done using restricted maximum likelihood with an expectation-maximization algorithm for an animal model. The algorithm involved solving mixed model equations by direct inversion of coefficient matrix that became feasible by neglecting relationships across herds. From data consisting of first to third lactation milk records of New York Holsteins, two computationally manageable subsets were selected of 15 herds each totaling 3070 and 2900 cows. Each cow had a recorded first lactation and a recorded second lactation if she had a recorded third record. Herds were chosen according to frequency of related animals and about 200 cows per herd. After 18 rounds of iteration, changes in estimates between successive rounds were constantly decreasing and small. Estimates averaged from both subsets gave heritabilities of h1(2) = .33, h2(2) = .33, h3(2) = .34, genetic correlations of rg12 = .86, rg13 = .85, rg23 = .87, and phenotypic correlations of rp12 = .57, rp13 = .52, rp23 = .65.     

Maximum Likelihood Estimation of Variance Components in Dairy Cattle Breeding Research

A common problem in animal breeding research is estimation of variance and covariance components. Usual methods of estimation have been described by Henderson in 1953. In 1973 Henderson reported a computing algorithm for maximum likelihood estimation of variance components which may have been overlooked. This note reviews the Henderson computing technique and illustrates application of maximum likelihood estimation to two specific types of models for large, unbalanced data sets.     

Comparison of Iterative Procedures for Solving Equations for Sire Evaluation

Gauss-Seidel, successive overrelaxation, end-of-round relaxation, and block iteration methods of obtaining solutions for sire effects from equations rising from progeny with records in mixed model procedures were compared. Equations transformed to provide direct solutions for genetic group plus sire effects as well as constrained and unconstrained equations were compared also. Equations for milk yield for the Northeast Artificial Insemination Sire Comparison numbered 301 for Ayrshires, 325 for Brown Swiss, 6,010 for Holsteins, and 926 for Jerseys after absorption of herd-year-season effects. Numbers of coefficients were 15 to 20% less for transformed equations, which decreased computing time per round of interation about 15%. Solutions for transformed equations converged more rapidly than solutions for untransformed equations with convergence criterion the ratio of the residual norm to the norm of the right-hand sides. Successive overrelaxation generally was more efficient than Gauss-Seidel iteraton. Solutions for equations constrained to full rank converged more slowly than unconstrained equations. Block iteration was more efficient than single equation iteration.