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This article discusses a general framework for smoothing parameter estimation for models with regular likelihoods constructed in terms of unknown smooth functions of covariates. Gaussian random effects and parametric terms may also be present. By construction the method is numerically stable and convergent, and enables smoothing parameter uncertainty to be quantified. The latter enables us to fix a well known problem with AIC for such models, thereby improving the range of model selection tools available. The smooth functions are represented by reduced rank spline like smoothers, with associated quadratic penalties measuring function smoothness. Model estimation is by penalized likelihood maximization, where the smoothing parameters controlling the extent of penalization are estimated by Laplace approximate marginal likelihood. The methods cover, for example, generalized additive models for nonexponential family responses (e.g., beta, ordered categorical, scaled t distribution, negative binomial and Tweedie distributions), generalized additive models for location scale and shape (e.g., two stage zero inflation models, and Gaussian location-scale models), Cox proportional hazards models and multivariate additive models. The framework reduces the implementation of new model classes to the coding of some standard derivatives of the log-likelihood. Supplementary materials for this article are available online.

The software Selegen-REML/BLUP uses mixed models, and was developed to optimize the routine of plant breeding programs. It addresses the following plants categories: allogamous, automagous, of mixed mating system, and of clonal propagation. It considers several experimental designs, mating designs, genotype x environment interaction, experiments repeated over sites, repeated measures, progenies belonging to several populations, among other factors. The software adjusts effects, estimates variance components, genetic additive, dominance and genotypic values of individuals, genetic gain with selection, effective population size, and other parameters of interest to plant breeding. It allows testing the significance of the effects by means of likelihood ratio test (LRT) and analysis of deviance. It addresses continuous variables (linear models) and categorical variables (generalized linear models). Selegen-REML/ BLUP is friendly, easy to use and interpret, and allows dealing efficiently with most of the situations in plant breeding. It is free and available at http://www.det.ufv.br/ppestbio/corpo_docente.php under the author’s name.

This study evaluated the adaptability, stability, and yield of maize and sorghum hybrids at six different times during the second growing season in Brazill’s Northeast region. Conducted at Fazenda Sol Nascente, located in Balsas, state of Maranhão, the experiment involved seven maize genotypes and seven sorghum genotypes. Sowing was carried out at six distinct times: February 9, 15, 20, 24, 27, and March 3, 2021, with each sowing date considered a separate environment for evaluation. A randomized block design with four replications was used. Genetic parameters for grain productivity were estimated using restricted maximum likelihood (REML), and genotype x environment interactions were analyzed using Best Linear Unbiased Prediction (BLUP). The reaction norm was employed to assess genotype responses to environmental covariates, and the effect of meteorological variables on grain yield was analyzed. Results showed that early sowing promoted optimal performance for both maize and sorghum genotypes. Genotypes P2970 and P3754 (maize) and DKB540 (sorghum) demonstrated superior adaptability, stability, and productivity across different sowing times, emphasizing the critical role of sowing date in crop performance in Brazil’s Northeast region. RESUMO: Este estudo teve como objetivo avaliar a adaptabilidade, a estabilidade e a produtividade de híbridos de milho e sorgo em seis épocas diferentes durante a segunda estação de crescimento na região Nordeste do Brasil. Realizado na Fazenda Sol Nascente, localizada em Balsas, estado do Maranhão, o experimento envolveu sete genótipos de milho e sete genótipos de sorgo. A semeadura foi realizada em seis épocas distintas: 9, 15, 20, 24, 27 de fevereiro e 3 de março de 2021, sendo que cada data de semeadura foi considerada um ambiente separado para avaliação. Foi usado um delineamento de blocos aleatórios com quatro repetições. Os parâmetros genéticos para a produtividade de grãos foram estimados usando a máxima verossimilhança restrita (REML), e as interações genótipo x ambiente foram analisadas usando a melhor previsão linear não tendenciosa (BLUP). A norma de reação foi empregada para avaliar as respostas do genótipo às covariáveis ambientais, e o efeito das variáveis meteorológicas sobre a produtividade de grãos foi analisado. Os resultados mostraram que a semeadura precoce promoveu o desempenho ideal para os genótipos de milho e sorgo. Os genótipos P2970 e P3754 (milho) e DKB540 (sorgo) demonstraram adaptabilidade, estabilidade e produtividade superiores em diferentes épocas de semeadura, enfatizando o papel fundamental da data de semeadura no desempenho da cultura na região Nordeste do Brasil.

This study proposes a new multitrait index based on factor analysis and ideotype‐design (FAI‐BLUP index), and validates its potential on the selection of elephant grass genotypes for energy cogeneration. Factor analysis was carried out, and afterwards, factorial scores of each ideotype were designed according to the desirable and undesirable factors, and the spatial probability was estimated based on genotype‐ideotype distance, enabling genotype ranking. In order to quantify the potential of the FAI‐BLUP index, genetic gains were predicted and compared with the Smith‐Hazel classical index. The FAI‐BLUP index allows ranking the genotypes based on multitrait, free from multicollinearity, and it does not require assigning weights, as in the case of the Smith‐Hazel classical index and its derived indices. Furthermore, the genetic correlation ‐ positive or negative ‐ within each factor was taken into account, preserving their traits relationship, and giving biological meaning to the ideotypes. The FAI‐BLUP index indicated the 15 elephant grass with the highest performance for conversion to bioenergy via combustion, and predicted balanced and desirable genetic gains for all traits. In addition, the FAI‐BLUP index predicted gains of approximately 62% of direct selection, simultaneously for all traits that are desired to be increased, and approximately 33% for traits which are desired to be decreased. The genotypes selected by the FAI‐BLUP index have potential to improve all traits simultaneously, while the Smith‐Hazel classical index predicted gains of 66% for traits that are desired to be increased, and −32% for traits that are desired to be decreased, and it does not have potential to improve all traits simultaneously. The FAI‐BLUP index provides an undoubtable selection process and can be used in any breeding programme aiming at selection based on multitrait. Joining the traditional technique of factor analysis (Exploratory Factor Analysis) with the ideotype‐design (Confirmatory Factor Analysis) we were able to propose a multitrait index (FAI‐BLUP index) that takes into account the relationship between the traits and the final goal of the breeding programme (ideotype). The FAI‐BLUP index allowed ranking the elephant grass genotypes based on multitrait, free from multicollinearity and without assign weights as occur in the Smith‐Hazel classical index and its derived indexes. The FAI‐BLUP index indicated high performance genotypes of elephant grass for energy cogeneration, predicting equilibrate and superior genetic gains. Therefore, The FAI‐BLUP index is a technical advance tool finding interest and application in genetic breeding programmes.

We consider the optimization of smoothing parameters and variance components in models with a regular log likelihood subject to quadratic penalization of the model coefficients, via a generalization of the method of Fellner (1986) and Schall (1991). In particular: (i) we generalize the original method to the case of penalties that are linear in several smoothing parameters, thereby covering the important cases of tensor product and adaptive smoothers; (ii) we show why the method's steps increase the restricted marginal likelihood of the model, that it tends to converge faster than the EM algorithm, or obvious accelerations of this, and investigate its relation to Newton optimization; (iii) we generalize the method to any Fisher regular likelihood. The method represents a considerable simplification over existing methods of estimating smoothing parameters in the context of regular likelihoods, without sacrificing generality: for example, it is only necessary to compute with the same first and second derivatives of the log-likelihood required for coefficient estimation, and not with the third or fourth order derivatives required by alternative approaches. Examples are provided which would have been impossible or impractical with pre-existing Fellner-Schall methods, along with an example of a Tweedie location, scale and shape model which would be a challenge for alternative methods, and a sparse additive modeling example where the method facilitates computational efficiency gains of several orders of magnitude.

The objective was to estimate the repeatability coefficient of quantitative traits in multiple harvests of sweet orange, in order to infer the minimum number of evaluations necessary to identify superior genotypes of orange trees through the methodology of mixed models. The experiment was conducted in randomized blocks containing 55 sweet orange genotypes and three replications. The repeatability coefficients were estimated using the maximum residual likelihood method (REML) and the prediction of genotypic values using the best unbiased linear predictor (BLUP). The Selegen software was used to perform the statistical analysis. The average heritability of genotypes in eight seasons, individual and eight seasons repeatability, selection accuracy in one and eight seasons, repeatability determination coefficient, accuracy of permanent phenotypic values based on M years of assessment and efficiency of M assessments compared to situation where only one assessment is carried out. The predictive accuracy of the selection revealed a significant degree of certainty in the inferences made. Evaluation over six seasons can increase accuracy to 70% in selecting sweet orange genotypes for yield-related traits. For the prediction of average fruit mass, seven harvests are enough to obtain 80% accuracy.

Abstract One of the challenges for genetic and plant breeding is the selection of genotypes that are increasingly adapted, productive, and stable to the cultivation environments, without losing the desired agronomic traits. These evaluations are important for determining the ideal genotypes for each environment. Without these evaluations, the behavior of cultivars or genotypes cannot be predicted, affecting performance, and causing losses for farmers. This work aimed to characterize, by different methodologies, the agronomic performance, adaptability, and stability of soybean genotypes in a preliminary trial, to select the superior genotypes, aiming at the Registration and Protection of Cultivars. The experiment was conducted in the field in a randomized block design with two replications, where 50 soybean genotypes were evaluated, including 3 checks, during two agricultural, two different environments. Agronomic traits were evaluated: number of days to maturity, number of days to flowering, plant height at maturity, the weight of a thousand grains, lodging, oil content, and grain yield. To determine the adaptability and productive stability of the genotypes, GGE biplot multivariate prediction methodologies and REML/BLUP mixed models were used and compared, using the main agronomic traits of production and oil content, which are of industrial importance in human and animal nutrition. The methods of the analysis showed differences regarding the ordering of adaptability and stability, however, they were unanimous in considering the genotypes as superior: 24; 29; 30; 32; 34, and 42 for yield and 4; 17; 37, and 44 for oil content. The predominant pedigree of the selected genotypes consists of bi-parental crosses, except genotype 24, which comes from a quadruple cross. Resumo Um dos desafios do melhoramento genético e vegetal é a seleção de genótipos cada vez mais adaptados, produtivos e estáveis aos ambientes de cultivo, sem perder as características agronômicas desejadas. Essas avaliações são importantes para determinar os genótipos ideais para cada ambiente. Sem essas avaliações, não é possível prever o comportamento de cultivares ou genótipos, afetando o desempenho e causando prejuízos aos agricultores. Este trabalho teve como objetivo caracterizar, por diferentes metodologias, o desempenho agronômico, a adaptabilidade e a estabilidade de genótipos de soja em um ensaio preliminar, para selecionar os genótipos superiores, visando ao Registro e à Proteção de Cultivares. O experimento foi conduzido no campo em um delineamento em blocos casualizados com duas repetições, onde foram avaliados 50 genótipos de soja, incluindo 3 controles, durante duas safras agrícolas, ou seja, dois ambientes diferentes. Foram avaliadas as características agronômicas: número de dias para a maturação, número de dias para a floração, altura da planta na maturação, peso de mil grãos, acamamento, teor de óleo e rendimento de grãos. Para determinar a adaptabilidade e a estabilidade produtiva dos genótipos, foram usadas e comparadas as metodologias de previsão multivariada GGE biplot e os modelos mistos REML/BLUP, usando as principais características agronômicas de produção e teor de óleo, que são de importância industrial na nutrição humana e animal. Os métodos de análise mostraram diferenças quanto à ordenação da adaptabilidade e da estabilidade, mas foram unânimes em considerar os genótipos superiores: 24; 29; 30; 32; 34 e 42 para produção e 4; 17; 37 e 44 para teor de óleo. O pedigree predominante dos genótipos selecionados consiste em cruzamentos biparentais, exceto o genótipo 24, que vem de um cruzamento quádruplo.

We study behavior of the restricted maximum likelihood (REML) estimator under a misspecified linear mixed model (LMM) that has received much attention in recent genome-wide association studies. The asymptotic analysis establishes consistency of the REML estimator of the variance of the errors in the LMM, and convergence in probability of the REML estimator of the variance of the random effects in the LMM to a certain limit, which is equal to the true variance of the random effects multiplied by the limiting proportion of the nonzero random effects present in the LMM. The asymptotic results also establish convergence rate (in probability) of the REML estimators as well as a result regarding convergence of the asymptotic conditional variance of the REML estimator. The asymptotic results are fully supported by the results of empirical studies, which include extensive simulation studies that compare the performance of the REML estimator (under the misspecified LMM) with other existing methods, and real data applications (only one example is presented) that have important genetic implications.

Conventional functional magnetic resonance imaging (FMRI) group analysis makes two key assumptions that are not always justified. First, the data from each subject is condensed into a single number per voxel, under the assumption that within-subject variance for the effect of interest is the same across all subjects or is negligible relative to the cross-subject variance. Second, it is assumed that all data values are drawn from the same Gaussian distribution with no outliers. We propose an approach that does not make such strong assumptions, and present a computationally efficient frequentist approach to FMRI group analysis, which we term mixed-effects multilevel analysis (MEMA), that incorporates both the variability across subjects and the precision estimate of each effect of interest from individual subject analyses. On average, the more accurate tests result in higher statistical power, especially when conventional variance assumptions do not hold, or in the presence of outliers. In addition, various heterogeneity measures are available with MEMA that may assist the investigator in further improving the modeling. Our method allows group effect t-tests and comparisons among conditions and among groups. In addition, it has the capability to incorporate subject-specific covariates such as age, IQ, or behavioral data. Simulations were performed to illustrate power comparisons and the capability of controlling type I errors among various significance testing methods, and the results indicated that the testing statistic we adopted struck a good balance between power gain and type I error control. Our approach is instantiated in an open-source, freely distributed program that may be used on any dataset stored in the universal neuroimaging file transfer (NIfTI) format. To date, the main impediment for more accurate testing that incorporates both within- and cross-subject variability has been the high computational cost. Our efficient implementation makes this approach practical. We recommend its use in lieu of the less accurate approach in the conventional group analysis. ► Our frequentist approach for group analysis incorporates within-subject variability. ► Outliers are modeled with a Laplace distribution in cross-subject variability. ► Significance testing with our t-statistic is more powerful on average. ► Our approach is well balanced in achieving power and in controlling type I errors. ► The approach has relatively low computational cost.

The study of adaptability, stability, and productivity is essential for selecting and recommending superior genotypes. This fact is particularly the case for the introduction of common bean cultivars in Rio de Janeiro, Brazil, whose production is negligible and does not meet the internal demand. Thus, this study estimated genetic parameters and selection gains and undertake a simultaneous selection for adaptability, stability, and productivity in black bean genotypes via mixed models. The investigation was carried out in three municipalities in the state of Rio de Janeiro during three crop years. The trials were set up in a randomized block design with 11 genotypes and three replications. High mean heritability (81%) and selection accuracy (90%), as well as good selection prospects, were observed. Gains between 1.03 and 9.49% were achieved for grain yield. Simultaneous selection was efficient, indicating two black bean lines (CNFP 15290 and CNFP 15361) as the most productive, adaptable, and stable. As such, these lines have the potential to be released as new black bean cultivars for the state of Rio de Janeiro. RESUMO: O estudo de adaptabilidade, estabilidade e produtividade é importante para selecionar e recomendar genótipos superiores. Esse fato é particularmente válido para a introdução de cultivares de feijão comum no Rio de Janeiro, Brasil, cuja produção é ínfima e não atende a demanda interna. Assim, este trabalho teve como objetivo estimar parâmetros genéticos e ganhos de seleção e realizar uma seleção simultânea para adaptabilidade, estabilidade e produtividade de genótipos de feijão preto através de modelos mistos. O estudo foi realizado em três municípios do estado do Rio de Janeiro e em três anos agrícolas. Os ensaios foram instalados em delineamento de blocos casualizados com 11 genótipos e três repetições. Observou-se alta herdabilidade média (81%), acurácia seletiva (90%) e boas perspectivas de seleção. Também foram observados ganhos satisfatórios para a característica entre 1,03 e 9,49%. A seleção simultânea foi eficiente e permitiu selecionar duas linhagens de feijão preto CNFP (15290 e 15361) como sendo as mais produtivas, adaptáveis e estáveis e, portanto, têm potencial para serem lançadas como novas cultivares de feijão preto para o estado do Rio de Janeiro.