Bayesian Interval Estimates of Variance Components Used in Quality Improvement Studies

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.

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.

In several sciences but especially in animal and plant breeding, the general mixed model with fixed and random effects plays a great role. Statistical inference on variance components means tests of hypotheses about variance components, constructing confidence intervals for them, estimating them, and using the variance components to get best estimates for fixed effects as well as to predict random effects.Many problems in the statistical inference of variance components already arise in even the most simple mixed model, describing nested designs; they are already present in the balanced nested designs, but become more pronounced in the unbalanced case. To find a guideline for solving the problems for the general mixed model, the study of the nested designs is worthwile.In chapter 1 the outline of the author's research on statistical inference of variance components is given.In chapter 2 the historical development of variance components estimation is described. More than 125 years ago the notion of variance components appears explicitly in astronomy. The expansion began with the development of quantitative genetics in 1918 after the first World War, bull the tremendous increase in research in variance components dates from after the second World War.Just as in the fixed effects model, the notions of vectors, vector spaces and projections of vectors on vector spaces, can be fruitfully used for the mixed effects model. This approach is given in chapter 3.For balanced nested designs exact tests about ratios of variance components and the calculation of their power, are well- known. For the unbalanced three-stage nested designs an exact test exists about the variance component belonging to the second stage, but for the first stage variance component no exact test was available in practice before 1974. in chapter 4 the onset of the author's research on the testing problem is described. It was possible to calculate the critical level or P-value for the exact test.The estimation of variance components was directed towards finding, for several designs, the best quadratic unbiased estimators for the variance components. In chapter 5 the onset of author's research on the estimation problem is described. On introduction of the concept of a permissible estimator it becomes clear which approaches to estimate variance components are unsatisfactory. To use permissible estimators for variance components means to use estimators which give non-negative estimates. A solution for a non-negative estimator has been found using the least squares approach as a unified procedure in variance components estimation.To demonstrate the danger of the use of the best quadratic unbiased estimator in the simplest random model, the balanced one-way classification, the probabilities for negative estimates were calculated in chapter 6.In chapter 7 a new exact test about ratios of variance components in the unbalanced three-stage nested design is given.The description of the least squares method as a unified procedure for the estimation of variance components and for the derivation of permissible (non-negative) estimators of variance components is given in chapter 8. The use of the main estimation procedures such as iterated Least Squares (or I-MINQUE), Maximum Likelihood (ML) and Restricted Maximum Likelihood (REML) estimators, is also discussed there.Finally in chapter 9 the use of variance components in predicting the random effects is discussed. Best Linear Unbiased Prediction (BLUP), which is extensively used in animal breeding, has not so far been applied on the same scale in plant breeding. How to use it is the subject of the last section 9.2.

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.

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.

Multilevel models (MLMs) can be used to examine treatment heterogeneity in single-case experimental designs (SCEDs). With small sample sizes, common issues for estimating between-case variance components in MLMs include nonpositive definite matrix, biased estimates, misspecification of covariance structures, and invalid Wald tests for variance components with bounded distributions. To address these issues, unconstrained optimization, model selection procedure based on parametric bootstrap, and restricted likelihood ratio test (RLRT)-based procedure are introduced. Using simulation studies, we compared the performance of two types of optimization methods (constrained vs. unconstrained) when the covariance structures are correctly specified or misspecified. We also examined the performance of a model selection procedure to obtain the optimal covariance structure. The results showed that the unconstrained optimization can avoid nonpositive definite issues to a great extent without a compromise in model convergence. The misspecification of covariance structures would cause biased estimates, especially with small between case variance components. However, the model selection procedure was found to attenuate the magnitude of bias. A practical guideline was generated for empirical researchers in SCEDs, providing conditions under which trustworthy point and interval estimates can be obtained for between-case variance components in MLMs, as well as the conditions under which the RLRT-based procedure can produce acceptable empirical type I error rate and power.

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.

Estimation of Variance Components of Quantitative Traits in Inbred Populations

Multilevel logistic regression models are increasingly being used to analyze clustered data in medical, public health, epidemiological, and educational research. Procedures for estimating the parameters of such models are available in many statistical software packages. There is currently little evidence on the minimum number of clusters necessary to reliably fit multilevel regression models. We conducted a Monte Carlo study to compare the performance of different statistical software procedures for estimating multilevel logistic regression models when the number of clusters was low. We examined procedures available in BUGS, HLM, R, SAS, and Stata. We found that there were qualitative differences in the performance of different software procedures for estimating multilevel logistic models when the number of clusters was low. Among the likelihood-based procedures, estimation methods based on adaptive Gauss-Hermite approximations to the likelihood (glmer in R and xtlogit in Stata) or adaptive Gaussian quadrature (Proc NLMIXED in SAS) tended to have superior performance for estimating variance components when the number of clusters was small, compared to software procedures based on penalized quasi-likelihood. However, only Bayesian estimation with BUGS allowed for accurate estimation of variance components when there were fewer than 10 clusters. For all statistical software procedures, estimation of variance components tended to be poor when there were only five subjects per cluster, regardless of the number of clusters.

Impact of pedigree depth and inclusion of historical data in the estimation of variance components and breeding values in Macrobrachium rosenbergii

BackgroundMixed models are commonly used for the estimation of variance components and genetic evaluation of livestock populations. Some evaluation models include two types of additive genetic effects, direct and maternal. Estimates of variance components obtained with models that account for maternal effects have been the subject of a long-standing controversy about strong negative estimates of the covariance between direct and maternal effects. Genomic imprinting is known to be in some cases statistically confounded with maternal effects. In this study, we analysed the consequences of ignoring paternally inherited effects on the partitioning of genetic variance.ResultsWe showed that the existence of paternal parent-of-origin effects can bias the estimation of variance components when maternal effects are included in the evaluation model. Specifically, we demonstrated that adding a constraint on the genetic parameters of a maternal model resulted in correlations between relatives that were the same as those obtained with a model that fits only paternally inherited effects for most pairs of individuals, as in livestock pedigrees. The main consequence is an upward bias in the estimates of the direct and maternal additive genetic variances and a downward bias in the direct-maternal genetic covariance. This was confirmed by a simulation study that investigated five scenarios, with the trait affected by (1) only additive genetic effects, (2) only paternally inherited effects, (3) additive genetic and paternally inherited effects, (4) direct and maternal additive genetic effects and (5) direct and maternal additive genetic plus paternally inherited effects. For each scenario, the existence of a paternally inherited effect not accounted for by the estimation model resulted in a partitioning of the genetic variance according to the predicted pattern. In addition, a model comparison test confirmed that direct and maternal additive models and paternally inherited models provided an equivalent fit.ConclusionsIgnoring paternally inherited effects in the maternal models for genetic evaluation can lead to a specific pattern of bias in variance component estimates, which may account for the unexpectedly strong negative direct-maternal genetic correlations that are typically reported in the literature.Electronic supplementary materialThe online version of this article (doi:10.1186/s12711-015-0141-5) contains supplementary material, which is available to authorized users.

This chapter describes the facilities for estimation of variance components and analysis of linear mixed models using the method of residual maximum likelihood (REML), sometimes also known as restricted maximum likelihood. The REML algorithm estimates the treatment effects and variance components in a linear mixed model: that is, a linear model with both fixed and random effects. Like regression, REML can be used to analyse unbalanced data sets; but, unlike regression, it can also account for more than one source of variation in the data, providing an estimate of the variance components associated with the random terms in the model.

SUMMARY The effect of non-normality in estimating variance components in a random effects or mixed effects linear model is studied. It is shown that for a wide variety of balanced situations the interval estimation of the upper variance component is likely to be relatively little affected by non-normality in the error component. A method is suggested for constructing a confidence interval for a ratio of variance components in a balanced model, without assuming normality.

A composite likelihood approach to (co)variance components estimation

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.