MEG source localization under multiple constraints: An extended Bayesian framework

For a balanced two‐way mixed model, the maximum likelihood (ML) and restricted ML (REML) estimators of the variance components were obtained and compared under the non‐negativity requirements of the variance components by Lee and Kapadia (1984). In this note, for a mixed (random blocks) incomplete block model, explicit forms for the REML estimators of variance components are obtained. They are always non‐negative and have smaller mean squared error (MSE) than the analysis of variance (AOV) estimators. The asymptotic sampling variances of the maximum likelihood (ML) estimators and the REML estimators are compared and the balanced incomplete block design (BIBD) is considered as a special case. The ML estimators are shown to have smaller asymptotic variances than the REML estimators, but a numerical result in the randomized complete block design (RCBD) demonstrated that the performances of the REML and ML estimators are not much different in the MSE sense.

Summary - A method is described for the simultaneous estimation of variance components due to several genetic and environmental effects from unbalanced data by restricted maximum likelihood (REML). Estimates are obtained by evaluating the likelihood explicitly and using standard, derivative-free optimization procedures to locate its maximum. The model of analysis considered is the so-called Animal Model which includes the additive genetic merit of animals as a random effect, and incorporates all information on relationships between animals. Furthermore, random effects in addition to animals’ additive genetic effects, such as maternal genetic, dominance or permanent environmental effects are taken into account. Emphasis is placed entirely upon univariate analyses. Simulation is employed to investigate the efficacy of three different maximization techniques and the scope for approximation of sampling errors. Computations are illustrated with a numerical example. variance components - restricted maximum likelihood - animal model - additional random effects - derivative - free approach

Multi-trait (co)variance estimation is an important topic in plant and animal breeding. In this study we compare estimates obtained with restricted maximum likelihood (REML) and Bayesian Gibbs sampling of simulated data and of three traits (diameter, height and branch angle) from a 26-year-old partial diallel progeny test of Scots pine (Pinus sylvestris L.). Based on the results from the simulated data we can conclude that the REML estimates are accurate but the mode of posterior distributions from the Gibbs sampling can be overestimated depending on the level of the heritability. The mean and median of the posteriors were considerably higher than the expected values of the heritabilities. The confidence intervals calculated with the delta method were biased downwardly. The highest probability density (HPD) interval provides a better interval estimate, but could be slightly biased at the lower level. Similar differences between REML and Gibbs sampling estimates were found for the Scots pine data. We conclude that further simulation studies are needed in order to evaluate the effect of different priors on (co)variance components in the genetic individual model.

Additive genetic components of variance and narrow-sense heritabilities were estimated for flowering time and cut-flower yield for generations 8-13 of the Davis population of gerbera, using the least squares (LS) and restricted maximum likelihood (REML)methods. Estimates of heritability for flowering time were 0.54 and 0.50 using REML and LS, respectively, indicating a close agreement between the two methods. However, estimates of heritability for cut-flower yield were 0.30 and 0.46 from REML and LS. This may result from the fact that cut-flower yield was selected in each generation; flowering time was not. Realized heritability for cut-flower yield was estimated to be 0.26 which agreeded more closely with the heritability estimated from REML. The advantages of REML, and its applications in the estimation of components of genetic variance and heritability of plant populations are discussed.

Multilevel modeling or Hierarchical Linear Modeling (HLM) is a statistical approach specifically used to analyze data with a two-level structure. This approach allows an understanding of the contribution of factors at both the individual and group levels to the response variable. One method commonly used in HLM is Restricted Maximum Likelihood (REML). REML is a parameter estimation method that is often applied in statistical models, especially linear models that incorporate random components. This allows more efficient parameter assessment compared to conventional estimation methods. In this research, multilevel model parameter estimation analysis was carried out using the limited maximum likelihood approach. The aim is to determine the multilevel linear regression model on the average UTBK score for health cluster study programs in 2019. This involves selecting the optimal node point based on the minimum Generalized Cross Validation (GCV) and identifying the factors that influence it. Predictor variables considered include interest and capacity of study programs at university level (Level-1), as well as the average UNBK and HDI scores at provincial level (Level-2). The findings of this research indicate that the most appropriate multilevel regression model is formed with three nodes with a minimum GCV value at Level-1 of 864.6593 and at Level-2 of 3.1816. At Level-1, the influencing factor is the interest variable and at Level-2 is the average provincial UNBK score in 2019 and the Human Development Index (HDI).

Group-level variance estimates of zero often arise when fitting multilevel or hierarchical linear models, especially when the number of groups is small. For situations where zero variances are implausible a priori, we propose a maximum penalized likelihood approach to avoid such boundary estimates. This approach is equivalent to estimating variance parameters by their posterior mode, given a weakly informative prior distribution. By choosing the penalty from the log-gamma family with shape parameter greater than 1, we ensure that the estimated variance will be positive. We suggest a default log-gamma(2,λ) penalty with λ → 0, which ensures that the maximum penalized likelihood estimate is approximately one standard error from zero when the maximum likelihood estimate is zero, thus remaining consistent with the data while being nondegenerate. We also show that the maximum penalized likelihood estimator with this default penalty is a good approximation to the posterior median obtained under a noninformative prior.Our default method provides better estimates of model parameters and standard errors than the maximum likelihood or the restricted maximum likelihood estimators. The log-gamma family can also be used to convey substantive prior information. In either case-pure penalization or prior information-our recommended procedure gives nondegenerate estimates and in the limit coincides with maximum likelihood as the number of groups increases.

Data from 1796 animals born between 1978 and 1996 in the Guzera herd of UNESP, Ilha Solteira, SP, were analysed to obtain variance components, heritabilities and genetic correlation for weights at 12, 18 and 24 months of age. The variance components, the heritabilities and the animals breeding values were estimated by the Restricted Maximum Likelihood (REML) and Bayesian methods, with the aim of evaluate properties of the estimates from both methods. The genetic correlations were obtained by CORR procedure of SAS, using the animal breeding values destimated from both methods. The statistic model for the weights at 12, 18 and 24 months, included fixed effects of contemporary groups and the random effects direct additive genetic, maternal genetic, permanent environmental of the cow and residual. The estimates of direct heritabilities were 0.09, 0.08 and 0.12 from REML and 0.07, 0.08 and 0.11 from Bayesian method, the estimates of maternal heritabilities were 0.08, 0.04 and 0.07 from REML and 0.07, 0.04 and 0.06 from Bayesian method, respectively, for weights at 12, 18 and 24 months of age. In both methods the estimates of genetic correlations coefficients were high and similar. The lengths of the confidence intervals from REML were larger than the credibility intervals from Bayesian, suggesting that the Bayesian method gives results with better precision than REML method.

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.

Bayesian methods have been said to solve small sample problems in frequentist methods by reflecting prior knowledge in the prior distribution. However, there are dangers in strongly reflecting prior knowledge or situations where much prior knowledge cannot be used. In order to address the issue, in this article, we considered to apply two Bayesian methods, one with weakly informative prior for random effect parameters (BayesW) and another with non-informative prior (BayesN) and compared their performance of parameter recovery with restricted maximum likelihood (REML) method, a frequentist-based estimation method commonly used in multilevel analysis. Specifically, we conducted two simulation studies using a two-level linear multilevel model and compared the performance of the BayesW, the BayesN, and the REML in various conditions that include scenarios of small sample data with a combination of small sample size both at level-1 and level-2. The results showed that the REML performed better than the BayesW, which showed overall better performance than the BayesN, in terms of convergence, bias, and coverage in some small sample scenarios. Therefore, based on the results of the present study, it is recommended to use the REML when adapting the two-level linear multilevel model and analyzing real-world data with small samples.

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.

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.

Response data in longitudinal studies and group randomized trials are gathered on units that belong to clusters, within which data are usually positively correlated. Therefore, estimates and confidence intervals for intraclass correlation or variance components are helpful when designing a longitudinal study or group randomized trial. Data simulated from both study designs are used to investigate the estimation of variance and covariance parameters from the following procedures: for continuous outcomes, restricted maximum likelihood (REML) and estimating equations (EE); for binary outcomes, restricted pseudo-likelihood (REPL) and estimating equations (EE). We evaluate these procedures to see which provide valid and precise estimates as well as correct standard errors for the intraclass correlation coefficient or variance components. REML seems the better choice for estimating terms related to correlation for models with normal outcomes, especially in group randomized trial situations. Results for REML and EE are mixed when outcomes are continuous and non-normal. With binary outcomes neither REPL nor EE provides satisfactory estimation or inference in longitudinal study situations, while REPL is preferable for group randomized trials.

A mixed-model procedure for analysis of censored data assuming a multivariate normal distribution is described. A Bayesian framework is adopted which allows for estimation of fixed effects and variance components and prediction of random effects when records are left-censored. The procedure can be extended to right- and two-tailed censoring. The model employed is a generalized linear model, and the estimation equations resemble those arising in analysis of multivariate normal or categorical data with threshold models. Estimates of variance components are obtained using expressions similar to those employed in the EM algorithm for restricted maximum likelihood (REML) estimation under normality.

Linear mixed model (LMM) approaches have been widely applied in many areas of research data analysis because they offer great flexibility for different data structures and linear model systems. In this study, emphasis is placed on comparing the properties of two LMM approaches: restricted maximum likelihood (REML) and minimum norm quadratic unbiased estimation (MINQUE) with and without resampling techniques being included. Bias, testing power, Type I error, and computing time were compared between REML and MINQUE approaches with and without Jackknife technique based on 500 simulated data sets. Results showed that MINQUE and REML methods performed equally regarding bias, Type I error, and power. Jackknife-based MINQUE and REML greatly improved power compared to non-Jackknife based linear mixed model approaches. Results also showed that MINQUE is more time-saving compared to REML, especially with the use of resampling techniques and large data set analysis. Results from the actual cotton data analysis were in agreement with our simulated results. Therefore, Jackknife-based MINQUE approaches could be recommended to achieve desirable power with reduced time for a large data analysis and model simulations.

Abstract. A multi-trait repeatability animal model under restricted maximum likelihood (REML) and Bayesian methods was used to estimate genetic parameters of milk, fat, and protein yields in Tunisian Holstein cows. The estimates of heritability for milk, fat, and protein yields from the REML procedure were 0.21 ± 0.05, 0.159 ± 0.04, and 0.158 ± 0.04, respectively. The corresponding results from the Bayesian procedure were 0.273 ± 0.02, 0.198 ± 0.01, and 0.187 ± 0.01. Heritability estimates tended to be larger via the Bayesian than those obtained by the REML method. Genetic and permanent environmental variances estimated by REML were smaller than those obtained by the Bayesian analysis. Inversely, REML estimates of the residual variances were larger than Bayesian estimates. Genetic and permanent correlation estimates were on the other hand comparable by both REML and Bayesian methods with permanent environmental being larger than genetic correlations. Results from this study confirm previous reports on genetic parameters for milk traits in Tunisian Holsteins and suggest that a multi-trait approach can be an alternative for implementing a routine genetic evaluation of the Tunisian dairy cattle population.