Statistical inference on variance components

Estimation of Variance Components of Quantitative Traits in Inbred Populations

Computer simulation was used to compare minimum variance quadratic estimation (MIVQUE), minimum norm quadratic unbiased estimation (MINQUE), restricted maximum likelihood (REML), maximum likelihood (ML), and Henderson's Method 3 (HM3) on the basis of variance among estimates, mean square error (MSE), bias and probability of nearness for estimation of both individual variance components and three ratios of variance components. The investigation also compared three procedures for dealing with negative estimates and included the use of both individual observations and plot means as the experimental unit of the analysis. The structure of data simulated (field design, mating designs, genetic architecture and imbalance) represented typical analysis problems in quantitative forest genetics. Results of comparing the estimation techniques demonstrated that: estimates of probability of nearness did not discriminate among techniques; bias was discriminatory among procedures for dealing with negative estimates but not among estimation techniques (except ML); sampling variance among estimates was discriminatory among procedures for dealing with negative estimates, estimation techniques and unit of observation; and MSE provided no additional information to variance of the estimates. HM3 and REML were the closest competitors under these criteria; however, REML demonstrated greater robustness to imbalance. Of the three negative estimate procedures, two are of practical significance and guidelines for their application are presented. Estimates from individual observations were always preferable to those from plot means over the experimental levels of this study.

BackgroundGenomic prediction is now an essential technology for genetic improvement in animal and plant breeding. Whereas emphasis has been placed on predicting the breeding values, the prediction of non-additive genetic effects has also been of interest. In this study, we assessed the potential of genomic prediction using non-additive effects for phenotypic prediction in Japanese Black, a beef cattle breed. In addition, we examined the stability of variance component and genetic effect estimates against population size by subsampling with different sample sizes.ResultsRecords of six carcass traits, namely, carcass weight, rib eye area, rib thickness, subcutaneous fat thickness, yield rate and beef marbling score, for 9850 animals were used for analyses. As the non-additive genetic effects, dominance, additive-by-additive, additive-by-dominance and dominance-by-dominance effects were considered. The covariance structures of these genetic effects were defined using genome-wide SNPs. Using single-trait animal models with different combinations of genetic effects, it was found that 12.6–19.5 % of phenotypic variance were occupied by the additive-by-additive variance, whereas little dominance variance was observed. In cross-validation, adding the additive-by-additive effects had little influence on predictive accuracy and bias. Subsampling analyses showed that estimation of the additive-by-additive effects was highly variable when phenotypes were not available. On the other hand, the estimates of the additive-by-additive variance components were less affected by reduction of the population size.ConclusionsThe six carcass traits of Japanese Black cattle showed moderate or relatively high levels of additive-by-additive variance components, although incorporating the additive-by-additive effects did not improve the predictive accuracy. Subsampling analysis suggested that estimation of the additive-by-additive effects was highly reliant on the phenotypic values of the animals to be estimated, as supported by low off-diagonal values of the relationship matrix. On the other hand, estimates of the additive-by-additive variance components were relatively stable against reduction of the population size compared with the estimates of the corresponding genetic effects.

This study evaluated the influence of variance component (VC) estimates, obtained from different models and two relationship matrices, pedigree-based (BLUP) and genomic information-based (ssGBLUP), on genomic predictions of carcass traits in Montana composite cattle. Phenotypic records from 14,422 animals were analysed for ribeye area, rump fat thickness, backfat thickness and marbling, along with pedigree information from 193,129 animals and genomic data from 3911 animals genotyped with 49,457 SNPs. Variance components and heritability estimates were calculated using restricted maximum likelihood under single-trait linear models. Across five models (M1-M5), fixed effects included contemporary group, embryo transfer, age at ultrasound and cow age at calving, while random effects included direct genetic effect and residual. From model M2 onwards, biological type, heterosis and both combined and specific recombination effects were also considered. The Akaike information criterion (AIC) was used to identify the best-fitting model. Different VC estimates were applied in ssGBLUP predictions to evaluate predictive ability based on accuracy, bias and dispersion. Variance component and heritability estimates were similar between methods, although ssGBLUP yielded higher direct additive genetic variances and heritabilities. More parameterised models using ssGBLUP provided a better fit according to AIC. However, less parameterised models showed superior predictive ability, regardless of whether VCs were estimated by BLUP or ssGBLUP. When comparing predictive ability across sources, pedigree-based VC estimates resulted in more accurate predictions. Thus, the choice of model complexity should be guided by the analysis objective and the available data structure.

A composite likelihood approach to (co)variance components estimation

In this study, variance components and breeding value estimations of Anatolian water buffaloes were estimated from the data obtained within the scope of community based water buffalo improvement program of Anatolian water buffalo population of İstanbul. The "BUGA" named software, which was started to be developed with the support of Harran University Scientific Research Project Unit (HUBAP) for use in the estimation of (co)variance components and breeding value were used in this study. The BUGA software used in the research can predict genetic and environmental effects together with variance components as Best Linear Unbiased Prediction (BLUP) with the Restricted Maximum Log likelihood (REML) method. Unlike its counterparts of another software of foreign origin, it has a visual interface. The language of use is Turkish. The dataset includes records such as city, district, village, genealogy information, date of birth, gender, calving date, lactation parity, number of test day milk yield, lactation milk yield and previous calving date of 442 Anatolian buffaloes raised in this population. The records were analysed using the repeatability model with BUGA software and the additive genetic and permanent environmental effects of all animals in the dataset were estimated. Random environment variance, genotypic variance and permanent environment variance values were estimated as 107112.60 kg2, 20045.22 kg2 and 1259.50 kg2, respectively, while heritability (h2) and degree of repeatability (r) were determined as 0.1561 and 0.1659, respectively. The results obtained as an example of case study applied for the possibility of usage the BUGA software in the estimation of genetic parameters and breeding values showed that BUGA software can be easily used in farm recording conditions with the advantage of friendly user features.

Bootstrap is a returned re-sampling method used to estimate the variance component and their variability. Adjusted bootstrap method was used by Wiley in p×i design for normal data in 2001. However, Wiley did not compare the difference between adjusted method and unadjusted method when estimating the variability. To expand Wiley’s 2001 study, our study applied Monte Carlo method to simulate four distribution data. The aim of simulation is to explore the effects of four different estimation methods when estimating the variability of estimated variance components for generalizability theory. The four distribution data are normal distribution data, dichotomous distribution data, polytomous distribution data and skewed distribution data. It is common that researchers focus on normal distribution data and neglect non-normal distribution data, yet non-normal distribution data could always be seen in tests such as TOEFL and GRE. There are several methods to estimate the variability of variance components, including traditional, bootstrap, jackknife and Markov Chain Monte Carlo (MCMC). Former research by Li and Zhang (2009) shows that bootstrap method is significantlybetter than traditional, jackknife, and MCMC methods in estimating the variability for four distribution data. Bootstrap method has superior cross-distribution quality when estimating the variability of estimated variance components. Li and Zhang (2009) also suggest that bootstrap method should be adopted with a divide-and-conquer strategy to obtain good estimated standard error and estimated confidence interval and the criteria of such strategy should be set to: boot-p for person, boot-pi for item, and boot-i for person and item. However, it is unclear that which of the bootstrap methods (adjusted and unadjusted) is better for boot-p, boot-pi, and boot-i. Therefore, our study intends to probe into this comparison as well. This aim of the study is to explore whether adjusted bootstrap method is superior to unadjusted method in improving the estimation of variance components and their variability relative for generalizability theory. The simulation is implemented in R statistical programming environment. To simulate skewed data, HyperbolicDist package is used. Some criteria are set to compare the four methods. The bias is considered when variance components and their standard errors are estimated. The smaller the absolute bias is, the more reliable the result is. The criterion of confidence intervals is 80% interval coverage. If the 80% interval coverage is closer to 0.80, the confidence interval is more reliable. The results indicate that for four distribution data, adjusted bootstrap method is superior to unadjusted bootstrap method whether in point estimation of variance components or in variability estimation of variance components. For its improvement of the estimation of variance components and their variability for generalizability theory, adjusted bootstrap should be adopted as soon as possible.

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.

Crossing of Garut (GG) sheep with St. Croix sheep (HH) and Moulton Charollais (MM) resulting HG sheep (♂HH X ♀GG) and MG sheep (♂MM X ♀GG) was done by Indonesian Research Institute for Animal Production. The work continued by crossing MG and HG sheep to obtain HMG (♂HG X ♀MG) and MHG (♂MG X ♀HG) sheep. The study was continued by selection program based on phenotypic performance. Estimation of breeding value and genetic trends were done in order to study the effectiveness of selection program and to study the genetic progress of each breed. The breeding value was estimated by Best Linear Unbiased Prediction (BLUP) method using Variance Component Estimation (VCE4) software to calculate variance component and using Prediction and Estimation (PEST) to calculate Breeding Value. Genetic variations of composite sheep were generally higher than Garut sheep, represented by higher estimated heritability value. Genetic correlation of body weight at diffrent age is high, that makes selection at one age could also increase body weight at any level of age. In general of this population shows positive genetic trend, but fluctuated from year to year. Composite breed tend to have higher estimated breeding value than Garut sheep. Selection method that was practiced in this population showed in line with selection method using breeding value criteria. Key Words: Breeding Value, Genetic, BLUP, Garut Sheep, St. Croix, Moulton Charollais

A common task in the analysis of multi‐environmental trials (MET) by linear mixed models (LMM) is the estimation of variance components (VCs). Most often, MET data are imbalanced (e.g., due to selection). The imbalance mechanism can be missing completely at random (MCAR), missing at random (MAR), or missing not at random. If the missing‐data pattern in MET was caused by selection, it is usually MAR. In this case, likelihood‐based methods are the preferred methods for analysis as they can account for a MAR data pattern. Likelihood‐based methods used to estimate VCs in LMM have the property that all VC estimates are constrained to be non‐negative; thus, the estimators are generally biased. Therefore, there are two potential causes of bias in MET analysis: a data pattern not being MCAR, and the bias of likelihood‐based VC estimators. The current study tries to dissect and quantify both possible sources of bias. A simulation study with MET data typical of cultivar evaluation trials was conducted in which the missing data pattern and the size of VCs were varied. The results showed that for the simulated MET, bias in VC estimates was similar under MCAR and MAR. Thus, the bias is solely due to the likelihood‐based estimation. Bias increases when increasing the ratio of genotype variance to error variance is small. Bias was similar for MAR and MCAR data patterns. Thus, it may be concluded that selection does not increase bias in VC estimation.

An ill-posed problem which involves heterogonous data can yield good results if the weight of observations is properly introduced into the adjustment model. Variance component estimation can be used in this respect to update and improve the weights based on the results of the adjustment. The variance component estimation will not be as simple as that is in an ordinary adjustment problem, because the result of the solution of an ill-posed problem contains a bias due to stabilizing the adjustment model. This paper investigates the variance component estimation in those ill-posed problems solved by the truncation singular value decomposition. The biases of the variance components are analyzed and the biased-corrected and the biased-corrected non-negative estimators of the variance components are developed. The derivations show that in order to estimate unbiased variance components, it suffices to estimate and remove the bias from the estimated residuals.

This study addresses hypothesis testing and interval estimation for fixed effects and variance component functions in the context of a skew-normal unbalanced two-way classification random effects model. First, we develop an exact test for the fixed effect based on the skew F-distribution. Second, we propose test statistics and confidence intervals for the single variance component, the sum of variance components, and the ratio of variance components using both Bootstrap and generalized approaches. The above constructed statistical methods based on variance component functions are significant generalizations of existing results for normal, balanced, and one-way classification scenarios. Through Monte Carlo simulations, we demonstrate that the proposed methods exhibit robust statistical performance, effectively controlling the Type I error probability while achieving high power under various parameter configurations and sample sizes. Finally, we illustrate the practical utility of these methods by applying them to a case study analyzing provincial per capita consumption expenditure data from 2016 to 2022.

Interval estimation of certain functions of variance components is of interest to research workers in all fields of applications in which the variance component model is used. Confidence intervals for linear functions and ratios of variance components have been proposed by several authors. For the most part, these intervals are approximate with unknown exact probabilities associated with their coverage. In this article a technique is given for the construction of simultaneous confidence intervals for the values of all continuous functions of the variance components in a balanced, general random linear model. These confidence intervals are conservative; that is, the actual confidence level cannot be less than any preset value. The proposed technique is easy to apply as it only requires the optimization of a given continuous function of the variance components over a bounded region. Several examples of continuous functions are considered, including linear functions and ratios of variance components. Furthermore, the technique can be applied to unbalanced, random two-way classification models.

Sampling experiments are conducted in QA and process improvement studies to characterize measurement error and classify sources of process variation. Gage R&R studies and nested sampling studies are used in these situations to estimate the variance components. There are several methods available for estimating variance components from data obtained from these studies. However, interval estimates are not available in the output of commonly used software, and only approximate standard errors of estimated variance components have been proposed for unbalanced nested designs, such as the staggered nested plans. Bayesian methods provide a solution to this problem. Bayes methods guarantee nonnegative estimates of variance components, provide exact interval estimates of individual variance components and functions of individual variance components such as gage repeatability plus reproducibility, and can be computed easily with public domain software. This article demonstrates the value of Bayesian interval estimates and illustrates their calculation using the free Excel add-in BugsXLA.

Inference on fixed effects in mixed models depends on standard errors or test statistics which in turn depend on estimates of variance and covariance components. For unbalanced mixed models, even relatively simple models such as two-way cross-classification models with interaction where one factor is fixed and the other is random, dilemmas arise that have received inadequate attention to date. For example, if one uses SAS PROC MIXED, one can estimate variance components using expected means squares from Type I, II, or III sums of squares, or one can use likelihood-based algorithms such as the default restricted maximum likelihood. If there is a negative variance component estimate, one can set the estimate to zero and proceed with fixed effects inference, or one can allow the variance estimate to remain negative. These decisions affect inference on fixed effects in ways that are not generally well-understood. The purposes of this presentation are to 1) clarify what the main issues are and 2) present some guidelines data analysts can use.