A comprehensive multivariate approach for GxE interaction analysis in early maturing rice varieties.

The genotype evaluation process requires analysis of GxE interactions to ascertain the responsiveness of a genotype to various environments, including the development of early maturing rice. However, the concept of interaction is relatively specific to grain yield. In contrast, grain yield is highly polygenic, so assessment should be carried out with multivariate approaches. Therefore, multivariate assessment in evaluating GxE interactions should be developed, especially for early maturing rice genotypes. The study aimed to develop a comprehensive multivariate approach to improve the comprehensiveness and responsiveness of GxE interaction analysis. The study was conducted in Bone and Soppeng districts, South Sulawesi, Indonesia, in two seasons. The study used a randomized complete block design, where replications were nested across two seasons and locations. Two check varieties and five early maturing varieties were replicated three times in each environment. Based on this study, a new approach to GxE interaction analysis based on multiple regression index analysis, BLUP analysis, factor analysis, and path analysis was considered adequate, especially for evaluating early maturing rice. This approach combined days to harvest, biological yield, and grain yield in multiple linear regression with weighting based on the combination of all analyses. The effectiveness of the GxE interaction assessment was reflected by high coefficient of determination (R2) and gradient (b) values above 0.8 and 0.9, respectively. Inpari 13 (R2 = 0.9; b=1.05), Cakrabuana (R2 = 0.98; b=0.99), and Padjajaran (R2 = 0.95; b=1.07) also have good grain yield with days to harvesting consideration, namely 7.83 ton ha-1, 98.12 days; 7.37 ton ha-1, 95.52 days; and 7.29 ton ha-1, 97.23 days, respectively. Therefore, this index approach can be recommended in GxE interaction analysis to evaluate early maturing rice genotypes. Furthermore, Inpari 13, Cakrabuana, and Padjajaran are recommended as adaptive early maturing varieties.

Principal component analysis revisited: fast multitrait genetic evaluations with smooth convergence.

A cornerstone in breeding and population genetics is the genetic evaluation procedure, needed to make important decisions on population management. Multivariate mixed model analysis, in which many traits are considered jointly, utilizes genetic and environmental correlations between traits to improve the accuracy. However, the number of parameters in the multitrait model grows exponentially with the number of traits which reduces its scalability. Here, we suggest using principal component analysis to reduce the dimensions of the response variables, and then using the computed principal components as separate responses in the genetic evaluation analysis. As principal components are orthogonal to each other so that phenotypic covariance is abscent between principal components, a full multivariate analysis can be approximated by separate univariate analyses instead which should speed up computations considerably. We compared the approach to both traditional multivariate analysis and factor analytic approach in terms of computational requirement and rank lists according to predicted genetic merit on two forest tree datasets with 22 and 27 measured traits, respectively. Obtained rank lists of the top 50 individuals were in good agreement. Interestingly, the required computational time of the approach only took a few seconds without convergence issues, unlike the traditional approach which required considerably more time to run (7 and 10 h, respectively). The factor analytic approach took approximately 5-10 min. Our approach can easily handle missing data and can be used with all available linear mixed effect model softwares as it does not require any specific implementation. The approach can help to mitigate difficulties with multitrait genetic analysis in both breeding and wild populations.

Of buds and bits: a meta-QTL study identifies stable QTL for berry quality and yield traits in cranberry mapping populations (Vaccinium macrocarpon Ait.).

Introduction: For nearly two centuries, cranberry (Vaccinium macrocarpon Ait.) breeders have improved fruit quality and yield by selecting traits on fruiting stems, termed "reproductive uprights." Crop improvement is accelerating rapidly in contemporary breeding programs due to modern genetic tools and high-throughput phenotyping methods, improving selection efficiency and accuracy. Methods: We conducted genotypic evaluation on 29 primary traits encompassing fruit quality, yield, and chemical composition in two full-sib cranberry breeding populations-CNJ02 (n = 168) and CNJ04 (n = 67)-over 3 years. Genetic characterization was further performed on 11 secondary traits derived from these primary traits. Results: For CNJ02, 170 major quantitative trait loci (QTL; R 2 ≥ 0.10) were found with interval mapping, 150 major QTL were found with model mapping, and 9 QTL were found to be stable across multiple years. In CNJ04, 69 major QTL were found with interval mapping, 81 major QTL were found with model mapping, and 4 QTL were found to be stable across multiple years. Meta-QTL represent stable genomic regions consistent across multiple years, populations, studies, or traits. Seven multi-trait meta-QTL were found in CNJ02, one in CNJ04, and one in the combined analysis of both populations. A total of 22 meta-QTL were identified in cross-study, cross-population analysis using digital traits for berry shape and size (8 meta-QTL), digital images for berry color (2 meta-QTL), and three-study cross-analysis (12 meta-QTL). Discussion: Together, these meta-QTL anchor high-throughput fruit quality phenotyping techniques to traditional phenotyping methods, validating state-of-the-art methods in cranberry phenotyping that will improve breeding accuracy, efficiency, and genetic gain in this globally significant fruit crop.

NARO historical phenotype dataset from rice breeding.

Data from breeding, including phenotypic information, may improve the efficiency of breeding. Historical data from breeding trials accumulated over a long time are also useful. Here, by organizing data accumulated in the National Agriculture and Food Research Organization (NARO) rice breeding program, we developed a historical phenotype dataset, which includes 6052 records obtained for 667 varieties in yield trials in 1991-2018 at six NARO research stations. The best linear unbiased predictions (BLUPs) and principal component analysis (PCA) were used to determine the relationships with various factors, including the year of cultivar release, for 15 traits, including yield. Yield-related traits such as the number of grains per panicle, plant weight, grain yield, and thousand-grain weight increased significantly with time, whereas the number of panicles decreased significantly. Ripening time significantly increased, whereas the lodging degree and protein content of brown rice significantly decreased. These results suggest that panicle-weight-type high-yielding varieties with excellent lodging resistance have been selected. These trends differed slightly among breeding locations, indicating that the main breeding objectives may differ among them. PCA revealed a higher diversity of traits in newer varieties.

Analysis of genotype-by-environment interaction effect in barely genotypes using AMMI and GGE biplot methods.

Genotype-by-environment interaction (GEI) analysis play a key role in any breeding program involving the development of new varieties for cultivation across various environments or in a specific region. The additive main effects and multiplicative interaction (AMMI) method and the GGE biplot are the two main statistical tools that have emerged to analyze GEI in multi-environment trials (METs). The main goal of the present study was to identify the best-performing and stable barley genotypes for the warm regions of Iran. For this purpose, 18 new advanced barley genotypes were investigated in five warm locations in Iran during two cropping seasons (2021-2023). In all experiments, test genotypes were evaluated in a randomized complete block design (RCBD) with three replications. Based on results, grain yield was significantly dependent on environments (E), genotypes (G), and GEI. The GEI effect was further divided into three principal component axes (IPCAs). The AMMI method identified genotypes G3, G9, G10, and G14 as ideal genotypes due to their low IPCA scores and high performances. In the GGE biplot analysis, the initial two PCAs accounted for 49.36 % of the total variation of grain yield, including both G and GEI effects. Based on averaged two-year data, genotypes G3, G4, G10, and G14 showed particular adaptability in the Zabol and Moghan regions. Moreover, the ranking of test environments showed good discriminatory and representative abilities for the Zabol and Moghan regions, so these environments constituted a mega-environment in Iran's warm climate. The genotype ranking indicated G3, G10 and G14 genotypes as the superior genotypes with the highest grain yield and stability in different test environments. Moreover, these results were confirmed by the results obtained by WAASB and WAASBY biplots. In conclusion, genotypes G3, G10 and G14 can be suggested for commercial usage and cultivation in various regions in Iran's warm climate.

BLUPmrMLM: A Fast mrMLM Algorithm in Genome-wide Association Studies.

Multilocus genome-wide association study has become the state-of-the-art tool for dissecting the genetic architecture of complex and multiomic traits. However, most existing multilocus methods require relatively long computational time when analyzing large datasets. To address this issue, in this study, we proposed a fast mrMLM method, namely, best linear unbiased prediction multilocus random-SNP-effect mixed linear model (BLUPmrMLM). First, genome-wide single-marker scanning in mrMLM was replaced by vectorized Wald tests based on the best linear unbiased prediction (BLUP) values of marker effects and their variances in BLUPmrMLM. Then, adaptive best subset selection (ABESS) was used to identify potentially associated markers on each chromosome to reduce computational time when estimating marker effects via empirical Bayes. Finally, shared memory and parallel computing schemes were used to reduce the computational time. In simulation studies, BLUPmrMLM outperformed GEMMA, EMMAX, mrMLM, and FarmCPU as well as the control method (BLUPmrMLM with ABESS removed), in terms of computational time, power, accuracy for estimating quantitative trait nucleotide positions and effects, false positive rate, false discovery rate, false negative rate, and F1 score. In the reanalysis of two large rice datasets, BLUPmrMLM significantly reduced the computational time and identified more previously reported genes, compared with the aforementioned methods. This study provides an excellent multilocus model method for the analysis of large-scale and multiomic datasets. The software mrMLM v5.1 is available at BioCode (https://ngdc.cncb.ac.cn/biocode/tool/BT007388) or GitHub (https://github.com/YuanmingZhang65/mrMLM).

Mapping of dynamic quantitative trait loci for plant height in a RIL population of foxtail millet (Setaria italica L.).

Plant height (PH) is a crucial trait for strengthening lodging resistance and boosting yield in foxtail millet. To identify quantitative trait loci (QTL) and candidate genes associated with PH, we first developed a genetic map using a recombinant inbred line (RIL) population derived from a cross between Aininghuang and Jingu 21. Then, PH phenotyping data and four variations of best linear unbiased prediction (BLUP) were collected from nine environments and three development stages. Next, QTL mapping was conducted using both unconditional and conditional QTL methods. Subsequently, candidate genes were predicted via transcriptome analysis of parental samples at three developmental stages. The results revealed that the genetic map, based on re-sequencing, consisted of 4,360 bin markers spanning 1,016.06 cM with an average genetic distance of 0.23 cM. A total of 19 unconditional QTL, accounting for 5.23%-35.36% of the phenotypic variation explained (PVE), which included 7 major and 4 stable QTL, were identified. Meanwhile, 13 conditional QTL, explaining 5.88%-40.35% of PVE, including 5 major and 3 stable QTL, were discovered. Furthermore, four consistent and stable QTL were identified. Finally, eight candidate genes were predicted through RNA-seq and weighted gene co-expression network analysis (WGCNA). Those findings provide a crucial foundation for understanding the genetic mechanisms underlying PH development and facilitate molecular marker-assisted breeding of ideal plant types in foxtail millet.

Impact of truncating historical data on prediction ability of dairy sheep selection candidates.

Along the last decades, the genetic evaluation methodology has evolved, improving breeding value estimates. Many breeding programmes have historical phenotypic records and large number of generations, but to make use of them could result in more inconveniences than benefits. In this study, the prediction ability of genotyped young animals was assessed by simultaneously evaluating the removal of historical data, two pedigree deepness and two methodologies (traditional BLUP and single-step genomic BLUP or ssGBLUP), using milk yield records of 40 years of three Latxa dairy sheep populations. The linear regression method was used to compare predictions of young rams before and after progeny testing, with six cut-off points, by intervals of 4 years (from 1992 to 2012), and statistics of ratio of accuracies, bias, and dispersion were calculated. The prediction accuracy of selection candidates, when genomic information was included, was the highest in all Latxa populations (between 0.54 and 0.69 with full data set). Nevertheless, the deletion of historical phenotypic data resulted on moderate accuracy gain in the bigger data size populations (mean gain 2.5%), and the smaller population took advantage of a moderate data deletion (2.7% gain by removing data until 2004), reducing accuracy when more records were removed. The bias of validation individuals was lower when the breeding value was predicted based on genomic information (between 2.1 and 13.9), being lower when the biggest amount of data was deleted in the bigger data size populations (5.2% reduction), and the smaller population was benefited from data deletion between 1996 and 2008 (3.8% bias reduction). Meanwhile, the slope of estimated genetic trend was lower when less data were included, and an overestimation of the unknown parent group estimates was observed. The results indicated that ssGBLUP evaluations were outstanding, compared with traditional BLUP evaluations, while the depth of pedigree had a very small influence, and deletion of historical phenotypic data was beneficial. Thus, Latxa routine genetic evaluations would benefit from truncating phenotypic records between 2000 and 2004, the use of two pedigree generations and the implementation of ssGBLUP methodology.

Exploring the Genetic Landscape of Vitiligo in the Pura Raza Español Horse: A Genomic Perspective.

Vitiligo is a depigmentation autoimmune disorder characterized by the progressive loss of melanocytes leading to the appearance of patchy depigmentation of the skin. The presence of vitiligo in horses is greater in those with grey coats. The aim of this study was therefore to perform a genome-wide association study (GWAS) to identify genomic regions and putative candidate loci associated with vitiligo depigmentation and susceptibility in the Pura Raza Español population. For this purpose, we performed a wssGBLUP (weighted single step genomic best linear unbiased prediction) using data from a total of 2359 animals genotyped with Affymetrix Axiom™ Equine 670 K and 1346 with Equine GeneSeek Genomic Profiler™ (GGP) Array V5. A total of 60,136 SNPs (single nucleotide polymorphisms) present on the 32 chromosomes from the consensus dataset after quality control were employed for the analysis. Vitiligo-like depigmentation was phenotyped by visual inspection of the different affected areas (eyes, mouth, nostrils) and was classified into nine categories with three degrees of severity (absent, slight, and severe). We identified one significant genomic region for vitiligo around the eyes, eight significant genomic regions for vitiligo around the mouth, and seven significant genomic regions for vitiligo around the nostrils, which explained the highest percentage of variance. These significant genomic regions contained candidate genes related to melanocytes, skin, immune system, tumour suppression, metastasis, and cutaneous carcinoma. These findings enable us to implement selective breeding strategies to decrease the incidence of vitiligo and to elucidate the genetic architecture underlying vitiligo in horses as well as the molecular mechanisms involved in the disease's development. However, further studies are needed to better understand this skin disorder in horses.

The BLUP method in evaluation of breeding values of Russian spring wheat lines using micro- and macroelements in seeds.

Genomic selection is a technology that allows for the determination of the genetic value of varieties of agricultural plants and animal breeds, based on information about genotypes and phenotypes. The measured breeding value (BV) for varieties and breeds in relation to the target trait allows breeding stages to be thoroughly planned and the parent forms suitable for crossing to be chosen. In this work, the BLUP method was used to assess the breeding value of 149 Russian varieties and introgression lines (4 measurements for each variety or line, 596 phenotypic points) of spring wheat according to the content of seven chemical elements in the grain - K, Ca, Mg, Mn, Fe, Zn, Cu. The quality of the evaluation of breeding values was assessed using cross-validation, when the sample was randomly divided into five parts, one of which was chosen as a test population. The following average values of the Pearson correlation were obtained for predicting the concentration of trace elements: K - 0.67, Ca - 0.61, Mg - 0.4, Mn - 0.5, Fe - 0.38, Zn - 0.46, Cu - 0.48. Out of the 35 models studied, the p-value was below the nominal significant threshold (p-value < 0.05) for 28 models. For 11 models, the p-value was significant after correction for multiple testing (p-value < 0.001). For Ca and K, four out of five models and for Mn two out of five models had a p-value below the threshold adjusted for multiple testing. For 30 varieties that showed the best varietal values for Ca, K and Mn, the average breeding value was 296.43, 785.11 and 4.87 mg/kg higher, respectively, than the average breeding value of the population. The results obtained show the relevance of the application of genomic selection models even in such limited-size samples. The models for K, Ca and Mn are suitable for assessing the breeding value of Russian wheat varieties based on these characteristics.

Sweet potato (Ipomoea batatas L.) genotype selection using advanced indices and statistical models: A multi-year approach.

In Bangladesh, sweet potato holds the fourth position as a crucial carbohydrate source, trailing rice, wheat, and potato. However, locally grown sweet potato varieties often display limited stability and yield. To tackle this challenge, diverse selection methods and statistical models were utilized to pinpoint sweet potato genotypes showcasing both stability and superior yield and quality traits. In the initial two years, multiple selection methods were employed to narrow down the collections based on preferences for yield and its contributing traits. Subsequently, a multi-environment trial (MET) was conducted in the following year to pinpoint superior and stable genotypes with desirable yield and quality characteristics. An integrated approach involving the Multi-Trait Genotype Ideotype Distance Index (MGIDI), Factor Analysis and Ideotype-Design (FAI-BLUP), and Smith-Hazel Index (SH) led to the identification of 71 superior sweet potato genotypes out of a total of 351 in the initial growing season. In the subsequent season, the MGIDI selection index was applied to the 71 genotypes, resulting in the selection of 11 top-performing genotypes. This selection process was complemented by a detailed analysis of the strengths and weaknesses of the selected genotypes. In the MET, the mixed effect model, specifically the linear mixed model (LMM), identified significant genotypic and genotype-environment interaction (GEI) variances. This points to elevated heritability and selection accuracy, ultimately boosting the model's reliability. By combining the strengths of LMM and additive main effects and multiplicative interaction (AMMI), the best linear unbiased prediction (BLUP) index identified H20 as the top-performing genotype for marketable root yield (MRY), H37 for dry weight of root (DW), H8 for beta carotene (BC) and H41 for vitamin c (VC). These genotypes surpassed the overall average in the WAAS index. For simultaneous stability and high performance, the WAASBY index selected H37 for MRY, H6 for DW, H61 for BC, and H3 for VC. Finally, genotypes H3 and H20 were selected using multi-trait stability index (MTSI), as they possessed high performance and stability. Based on the selection sense, the objective has been achieved with regards to the trait MRW, which serves as a major criterion for a superior variety of sweet potato.

Ability of Genomic Prediction to Bi-Parent-Derived Breeding Population Using Public Data for Soybean Oil and Protein Content.

Genomic selection (GS) is a marker-based selection method used to improve the genetic gain of quantitative traits in plant breeding. A large number of breeding datasets are available in the soybean database, and the application of these public datasets in GS will improve breeding efficiency and reduce time and cost. However, the most important problem to be solved is how to improve the ability of across-population prediction. The objectives of this study were to perform genomic prediction (GP) and estimate the prediction ability (PA) for seed oil and protein contents in soybean using available public datasets to predict breeding populations in current, ongoing breeding programs. In this study, six public datasets of USDA GRIN soybean germplasm accessions with available phenotypic data of seed oil and protein contents from different experimental populations and their genotypic data of single-nucleotide polymorphisms (SNPs) were used to perform GP and to predict a bi-parent-derived breeding population in our experiment. The average PA was 0.55 and 0.50 for seed oil and protein contents within the bi-parents population according to the within-population prediction; and 0.45 for oil and 0.39 for protein content when the six USDA populations were combined and employed as training sets to predict the bi-parent-derived population. The results showed that four USDA-cultivated populations can be used as a training set individually or combined to predict oil and protein contents in GS when using 800 or more USDA germplasm accessions as a training set. The smaller the genetic distance between training population and testing population, the higher the PA. The PA increased as the population size increased. In across-population prediction, no significant difference was observed in PA for oil and protein content among different models. The PA increased as the SNP number increased until a marker set consisted of 10,000 SNPs. This study provides reasonable suggestions and methods for breeders to utilize public datasets for GS. It will aid breeders in developing GS-assisted breeding strategies to develop elite soybean cultivars with high oil and protein contents.

Bayesian approach for evaluation of lactation curve in cross bred cattle based on monthly and bimonthly test day milk yield.

In field progeny testing program milk recording at monthly or bimonthly intervals and prediction of first lactation 305-day milk yield (FL305DMY) from these test day yields have been adapted as an alternative to daily milk recording. Wood's incomplete gamma function is the one of the commonly used nonlinear lactation curve model. In recent years Bayesian approach of fitting nonlinear biological models is gaining attention among researchers. In this study Wood's incomplete gamma function was fitted using Bayesian approach using monthly (MTDY) and bimonthly test day (BTDY) yields. The lactation curve parameters thus obtained were used for prediction of FL305DMY. Efficiency of prediction based on monthly and bimonthly test day milk yield were compared using error of prediction. It was found to be 5.78% and 7.59% as root mean square error (RMSE) based on MTDY and BTDY respectively.The Breeding values of 97 Karan Fries sires were estimated using BLUP-AM based on actual and predicted FL305DMY thus obtained. The RMSE was calculated as the difference between estimated breeding values based on actual and predicted yield. It was found that RMSE calculated based on MTDY showed only a marginal superiority of 0.79% over BTDY and showed high degree of correlation with actual yield. Therefore, recording at bimonthly intervals could be an economical alternative without compromising the efficiency.

A method of calculating weighted values of objective traits for BLUP of breeding values to achieve relative desired changes.

A method of calculating weighted values for objective traits from the phenotypic records of all animals in a population was devised as an alternative to the conventional method of calculating weighted values from a family selection index. The genetic improvement of this method was verified by Monte Carlo computer simulation. A base population consisting of 10 males and 50 females, and five separate generations, other than the base population that had been randomly selected, was bred for two traits with different heritabilities. The breeding values of animals in generation five were estimated using the bivariate BLUP method. The three different weighted values obtained from this method and two conventional methods for estimated breeding values of the objective traits were used to estimate aggregate breeding values for selection. The results showed that selection using weighted values calculated from all animals in a population resulted in a greater response to selection, especially when the genetic correlation between the two traits was positive, than selection using other conventional methods. The use of the method devised in this study was expected to result in a greater genetic improvement than the conventional family selection index method for pig breeding programs applied in closed herds in Japan.

Prediction of response to truncated selection based on BLUP of breeding values and its prediction accuracy.

Three methods of predicting the response to truncated selection based on BLUP of breeding values (BVs) were compared under conditions in which the phenotypic values for the progenies of selected animals were not available. The following methods were used to predict the response to selection: (1) based on the mean of estimated breeding values (EBV) in the candidate population for selection ( ∆ g 1 $$ \Delta {\mathrm{g}}_1 $$ ), (2) based on the variance of EBV in the candidate population for selection ( ∆ g 2 $$ \Delta {\mathrm{g}}_2 $$ ), and (3) based on diagonal elements of the inverse matrix on the left-hand side of the mixed model equation ( ∆ g 3 $$ \Delta {\mathrm{g}}_3 $$ ). The deviation of the average BV of the selected animals from the average BV of the candidate population for selection was taken as the true response to selection. The pedigree information and phenotypic values used for comparison were generated by Monte Carlo computer simulation. The results showed that ∆ g 1 $$ \Delta {\mathrm{g}}_1 $$ had the smallest absolute mean error and ∆ g 2 $$ \Delta {\mathrm{g}}_2 $$ had the smallest root-mean-square error. We concluded that it is desirable to use ∆ g 1 $$ \Delta {\mathrm{g}}_1 $$ or ∆ g 2 $$ \Delta {\mathrm{g}}_2 $$ to predict the response to truncated selection based on BLUP of BVs. However, in the population where selection is ongoing, the prediction accuracy of selection response is likely to be affected by the distortion of the distribution and the Bulmer effect for ∆ g 2 $$ \Delta {\mathrm{g}}_2 $$ .

Using machine learning and partial dependence to evaluate robustness of best linear unbiased prediction (BLUP) for phenotypic values.

Best linear unbiased prediction (BLUP) is widely used in plant research to address experimental variation. For phenotypic values, BLUP accuracy is largely dependent on properly controlled experimental repetition and how variable components are outlined in the model. Thus, determining BLUP robustness implies the need to evaluate contributions from each repetition. Here, we assessed the robustness of BLUP values for simulated or empirical phenotypic datasets, where the BLUP value and each experimental repetition served as dependent and independent (feature) variables, respectively. Our technique incorporated machine learning and partial dependence. First, we compared the feature importance estimated with the neural networks. Second, we compared estimated average marginal effects of individual repetitions, calculated with a partial dependence analysis. We showed that contributions of experimental repetitions are unequal in a phenotypic dataset, suggesting that the calculated BLUP value is likely to be influenced by some repetitions more than others (such as failing to detect simulated true positive associations). To resolve disproportionate sources, variable components in the BLUP model must be further outlined.

Short Communication: Reduced GBLUP equations to core animals in the algorithm for proven and young (APY).

The number of animal genotypes is rapidly increasing, and a major challenge for animal models is inverting the genomic relationship matrix (G). Matrix G has a limited dimensionality, and the algorithm for proven and young (APY) makes inverting a large G possible via the inverse of a block diagonal of G with a size equivalent to the dimensionality of G. APY divides genotyped animals into core and non-core groups, and breeding values of non-core animals are conditioned on the breeding values of core animals. Therefore, there is the possibility of opting out equations for non-core animals from the model. A methodology was presented for a reduced APY genomic BLUP (GBLUP) to equations for core animals. Using a small example dataset, the method was validated by the equality of the full and the reduced model analysis results. Absorption of fixed effect equations into random effect equations was successful in reducing the number of equations to solve and producing the same random effect solutions. Extending the method to APY single-step GBLUP (ssGBLUP) was not computationally justifiable. Other reduction techniques exist for ssGBLUP (regardless of APY or non-APY) that work by reducing the number of equations for non-genotyped animals. The number of equations can further be reduced by data pruning.

How trace plots help interpret meta-analysis results.

The trace plot is seldom used in meta-analysis, yet it is a very informative plot. In this article, we define and illustrate what the trace plot is, and discuss why it is important. The Bayesian version of the plot combines the posterior density of τ , the between-study standard deviation, and the shrunken estimates of the study effects as a function of τ . With a small or moderate number of studies, τ is not estimated with much precision, and parameter estimates and shrunken study effect estimates can vary widely depending on the correct value of τ . The trace plot allows visualization of the sensitivity to τ along with a plot that shows which values of τ are plausible and which are implausible. A comparable frequentist or empirical Bayes version provides similar results. The concepts are illustrated using examples in meta-analysis and meta-regression; implementation in R is facilitated in a Bayesian or frequentist framework using the bayesmeta and metafor packages, respectively.

Identification of High-Yielding Genotypes of Barley in the Warm Regions of Iran.

One of the most important effects of climatic changes is increasing temperatures and expanding water deficit stress in tropical and subtropical regions. As the fourth most important cereal crop, barley (Hordeum vulgare L.) is crucial for food and feed security, as well as for a sustainable agricultural system. The present study investigates 56 promising barley genotypes, along with four local varieties (Norooz, Oxin, Golchin, and Negin) in four locations to identify high-yielding and adapted genotypes in the warm climate of Iran. Genotypes were tested in an alpha lattice design with six blocks, which were repeated three times. Traits measured were the number of days to heading and maturity, plant height, thousand kernels weight, and grain yield. A combined analysis of variance showed the significant effects of genotypes (G), environments (E), and their interaction (GEI) on all measured traits. Application of the additive main-effect and multiplicative interaction (AMMI) model to the grain yield data showed that GEI was divided into three significant components (IPCAs), and each accounted for 50.93%, 30.60%, and 18.47%, respectively. Two selection indices [Smith-Hazel (SH) and multiple trait selection index (MTSI)] identified G18, G24, G29, and G57 as desirable genotypes at the four test locations. Using several BLUP-based indices, such as the harmonic mean of genotypic values (HMGV), the relative performance of genotypic values (RPGV), and the harmonic mean of the relative performance of genotypic values (HMRPGV), genotypes G6, G11, G22, G24, G29, G38, G52, and G57 were identified as superior genotypes. The application of GGE analysis identified G6, G24, G29, G52, and G57 as the high-yielding and most stable genotypes. Considering all statistical models, genotypes G24, G29, and G57 can be used, as they are well-adapted to the test locations in warm regions of Iran.

Dynamical BLUP modeling of reaction norm evolution, accommodating changing environments, overlapping generations, and multivariate data.

For theoretical studies, reaction norm evolution in a changing environment can be modeled by means of the multivariate breeder's equation, with the reaction norm parameters treated as traits in their own right. This is, however, not a feasible approach for the use of field data, where the intercept and slope values are not available. An alternative approach is to use infinite-dimensional characters and smooth covariance function estimates found by, e.g., random regression. This is difficult because of the need to find, for example, polynomial basis functions that fit the data reasonably well over time, and because reaction norms in multivariate cases are correlated, such that they cannot be modeled independently. Here, I present an alternative approach based on a multivariate linear mixed model of any order, with dynamical incidence and residual covariance matrices that reflect the changing environment. From such a mixed model follows a dynamical BLUP model for the estimation of the individual reaction norm parameter values at any given parent generation, and for updating of the mean reaction norm parameter values from generation to generation by means of Robertson's secondary theorem of natural selection. This will, for example, make it possible to disentangle the microevolutionary and plasticity components in climate change responses. The BLUP model incorporates the additive genetic relationship matrix in the usual way, and overlapping generations can easily be accommodated. Additive genetic and environmental model parameters are assumed to be known and constant, but it is discussed how they can be estimated by means of a prediction error method. The identifiability by the use of field or laboratory data containing environmental, phenotypic, fitness, and additive genetic relationship data is an important feature of the proposed model.