Genomic prediction in biparental tropical maize populations in water-stressed and well-watered environments using low-density and GBS SNPs.

One of the most important applications of genomic selection in maize breeding is to predict and identify the best untested lines from biparental populations, when the training and validation sets are derived from the same cross. Nineteen tropical maize biparental populations evaluated in multienvironment trials were used in this study to assess prediction accuracy of different quantitative traits using low-density (~200 markers) and genotyping-by-sequencing (GBS) single-nucleotide polymorphisms (SNPs), respectively. An extension of the Genomic Best Linear Unbiased Predictor that incorporates genotype × environment (GE) interaction was used to predict genotypic values; cross-validation methods were applied to quantify prediction accuracy. Our results showed that: (1) low-density SNPs (~200 markers) were largely sufficient to get good prediction in biparental maize populations for simple traits with moderate-to-high heritability, but GBS outperformed low-density SNPs for complex traits and simple traits evaluated under stress conditions with low-to-moderate heritability; (2) heritability and genetic architecture of target traits affected prediction performance, prediction accuracy of complex traits (grain yield) were consistently lower than those of simple traits (anthesis date and plant height) and prediction accuracy under stress conditions was consistently lower and more variable than under well-watered conditions for all the target traits because of their poor heritability under stress conditions; and (3) the prediction accuracy of GE models was found to be superior to that of non-GE models for complex traits and marginal for simple traits.

Genomic prediction in CIMMYT maize and wheat breeding programs.

Genomic selection (GS) has been implemented in animal and plant species, and is regarded as a useful tool for accelerating genetic gains. Varying levels of genomic prediction accuracy have been obtained in plants, depending on the prediction problem assessed and on several other factors, such as trait heritability, the relationship between the individuals to be predicted and those used to train the models for prediction, number of markers, sample size and genotype × environment interaction (GE). The main objective of this article is to describe the results of genomic prediction in International Maize and Wheat Improvement Center's (CIMMYT's) maize and wheat breeding programs, from the initial assessment of the predictive ability of different models using pedigree and marker information to the present, when methods for implementing GS in practical global maize and wheat breeding programs are being studied and investigated. Results show that pedigree (population structure) accounts for a sizeable proportion of the prediction accuracy when a global population is the prediction problem to be assessed. However, when the prediction uses unrelated populations to train the prediction equations, prediction accuracy becomes negligible. When genomic prediction includes modeling GE, an increase in prediction accuracy can be achieved by borrowing information from correlated environments. Several questions on how to incorporate GS into CIMMYT's maize and wheat programs remain unanswered and subject to further investigation, for example, prediction within and between related bi-parental crosses. Further research on the quantification of breeding value components for GS in plant breeding populations is required.

Genomic-enabled prediction with classification algorithms.

Pearson's correlation coefficient (ρ) is the most commonly reported metric of the success of prediction in genomic selection (GS). However, in real breeding ρ may not be very useful for assessing the quality of the regression in the tails of the distribution, where individuals are chosen for selection. This research used 14 maize and 16 wheat data sets with different trait-environment combinations. Six different models were evaluated by means of a cross-validation scheme (50 random partitions each, with 90% of the individuals in the training set and 10% in the testing set). The predictive accuracy of these algorithms for selecting individuals belonging to the best α=10, 15, 20, 25, 30, 35, 40% of the distribution was estimated using Cohen's kappa coefficient (κ) and an ad hoc measure, which we call relative efficiency (RE), which indicates the expected genetic gain due to selection when individuals are selected based on GS exclusively. We put special emphasis on the analysis for α=15%, because it is a percentile commonly used in plant breeding programmes (for example, at CIMMYT). We also used ρ as a criterion for overall success. The algorithms used were: Bayesian LASSO (BL), Ridge Regression (RR), Reproducing Kernel Hilbert Spaces (RHKS), Random Forest Regression (RFR), and Support Vector Regression (SVR) with linear (lin) and Gaussian kernels (rbf). The performance of regression methods for selecting the best individuals was compared with that of three supervised classification algorithms: Random Forest Classification (RFC) and Support Vector Classification (SVC) with linear (lin) and Gaussian (rbf) kernels. Classification methods were evaluated using the same cross-validation scheme but with the response vector of the original training sets dichotomised using a given threshold. For α=15%, SVC-lin presented the highest κ coefficients in 13 of the 14 maize data sets, with best values ranging from 0.131 to 0.722 (statistically significant in 9 data sets) and the best RE in the same 13 data sets, with values ranging from 0.393 to 0.948 (statistically significant in 12 data sets). RR produced the best mean for both κ and RE in one data set (0.148 and 0.381, respectively). Regarding the wheat data sets, SVC-lin presented the best κ in 12 of the 16 data sets, with outcomes ranging from 0.280 to 0.580 (statistically significant in 4 data sets) and the best RE in 9 data sets ranging from 0.484 to 0.821 (statistically significant in 5 data sets). SVC-rbf (0.235), RR (0.265) and RHKS (0.422) gave the best κ in one data set each, while RHKS and BL tied for the last one (0.234). Finally, BL presented the best RE in two data sets (0.738 and 0.750), RFR (0.636) and SVC-rbf (0.617) in one and RHKS in the remaining three (0.502, 0.458 and 0.586). The difference between the performance of SVC-lin and that of the rest of the models was not so pronounced at higher percentiles of the distribution. The behaviour of regression and classification algorithms varied markedly when selection was done at different thresholds, that is, κ and RE for each algorithm depended strongly on the selection percentile. Based on the results, we propose classification method as a promising alternative for GS in plant breeding.

Genome-enabled prediction of genetic values using radial basis function neural networks.

The availability of high density panels of molecular markers has prompted the adoption of genomic selection (GS) methods in animal and plant breeding. In GS, parametric, semi-parametric and non-parametric regressions models are used for predicting quantitative traits. This article shows how to use neural networks with radial basis functions (RBFs) for prediction with dense molecular markers. We illustrate the use of the linear Bayesian LASSO regression model and of two non-linear regression models, reproducing kernel Hilbert spaces (RKHS) regression and radial basis function neural networks (RBFNN) on simulated data and real maize lines genotyped with 55,000 markers and evaluated for several trait-environment combinations. The empirical results of this study indicated that the three models showed similar overall prediction accuracy, with a slight and consistent superiority of RKHS and RBFNN over the additive Bayesian LASSO model. Results from the simulated data indicate that RKHS and RBFNN models captured epistatic effects; however, adding non-signal (redundant) predictors (interaction between markers) can adversely affect the predictive accuracy of the non-linear regression models.

Prediction of genetic values of quantitative traits with epistatic effects in plant breeding populations.

Though epistasis has long been postulated to have a critical role in genetic regulation of important pathways as well as provide a major source of variation in the process of speciation, the importance of epistasis for genomic selection in the context of plant breeding is still being debated. In this paper, we report the results on the prediction of genetic values with epistatic effects for 280 accessions in the Nebraska Wheat Breeding Program using adaptive mixed least absolute shrinkage and selection operator (LASSO). The development of adaptive mixed LASSO, originally designed for association mapping, for the context of genomic selection is reported. The results show that adaptive mixed LASSO can be successfully applied to the prediction of genetic values while incorporating both marker main effects and epistatic effects. Especially, the prediction accuracy is substantially improved by the inclusion of two-locus epistatic effects (more than onefold in some cases as measured by cross-validation correlation coefficient), which is observed for multiple traits and planting locations. This points to significant potential in using non-additive genetic effects for genomic selection in crop breeding practices.

[Impact of the nail diameter on consolidation of tibial diaphyseal fractures. Retrospective analysis of 96 patients]. / Impacto del diámetro del clavo endomedular en la consolidación de fracturas diafisarias de tibia. Análisis retrospectivo sobre 96 pacientes.

INTRODUCTION: the gold standard for tibial diaphyseal fracture treatment is represented by the intramedullary nail (IMN). This study aimed to assess the relevance of nail diameter in bone healing of tibial diaphyseal fractures. MATERIAL AND METHODS: a retrospective study was conducted analyzing patients with closed 42 OTA/AO tibial fractures, treated with a reamed and locked IMN between January 2014 and December 2020. The variables assessed were gender, age, comorbidities, number of bolts used, canal/nail index (difference between the diameter of the medullary canal and nail), nail/canal ratio (ratio between nail diameter and medullary canal), related to consolidation and failure rates (delay and non-union). RESULTS: 96 patients were included. The consolidation rate was 91.7% (n = 88). Patients with consolidation had a significantly larger nail diameter than those who failed (p = 0.0014), increasing the chance of consolidation 5.30 (p = 0.04) times for each millimeter that the nail increased its diameter. Using a nail > 10 mm increased the chance of consolidation 13.56 times (p = 0.018). A nail/canal ratio 0.80 increased the chance of consolidation 23.33 times (p = 0.005). CONCLUSION: our findings suggested that reamed and locked IMN in tibial diaphyseal fractures should be implanted with the largest possible diameter (> 10 mm and with a nail-to-canal ratio 0.80) to promote bone healing.

INTRODUCCIÓN: el estándar de oro de tratamiento para la mayoría de las fracturas diafisarias de tibia está representado por el clavo endomedular (CEM). El objetivo de este estudio fue analizar la importancia del diámetro del CEM sobre la consolidación de fracturas diafisarias de tibia. MATERIAL Y MÉTODOS: se realizó un estudio retrospectivo en pacientes con fracturas cerradas de tibia 42 OTA/AO, tratados con un CEM fresado y acerrojado, entre Enero de 2014 y Diciembre de 2020. Las variables analizadas fueron género, edad, comorbilidades, cantidad de cerrojos utilizados, relación clavo/canal (diferencia entre el diámetro del canal medular y clavo), el índice clavo/canal (razón entre diámetro del clavo y el canal medular), en relación con la tasa de consolidación y falla (retardo de consolidación y seudoartrosis). RESULTADOS: la serie final se conformó por 96 pacientes y la tasa de consolidación fue de 91.7% (n = 88). Se observó un diámetro de clavo significativamente mayor en los pacientes que consolidaron respecto a los que fallaron (p = 0.0014), incrementando la posibilidad de consolidación 5.30 (p = 0.04) veces, por cada milímetro que el clavo aumentó su diámetro. Se observó un incremento de probabilidad de consolidación de 13.56 (p = 0.018) veces utilizando un clavo > 10 mm de diámetro. El índice clavo/canal 0.80 aumentó la posibilidad de consolidación 23.33 veces (p = 0.005). CONCLUSIÓN: nuestros hallazgos sugieren que los CEM fresados y acerrojados en fracturas diafisarias de tibia deben colocarse del mayor diámetro posible (> 10 mm y con un índice clavo/canal 0.80) para favorecer la consolidación.

Additive genetic variance and covariance between relatives in synthetic wheat crosses with variable parental ploidy levels.

Cultivated bread wheat (Triticum aestivum L.) is an allohexaploid species resulting from the natural hybridization and chromosome doubling of allotetraploid durum wheat (T. turgidum) and a diploid goatgrass Aegilops tauschii Coss (Ae. tauschii). Synthetic hexaploid wheat (SHW) was developed through the interspecific hybridization of Ae. tauschii and T. turgidum, and then crossed to T. aestivum to produce synthetic hexaploid wheat derivatives (SHWDs). Owing to this founding variability, one may infer that the genetic variances of native wild populations vs improved wheat may vary due to their differential origin and evolutionary history. In this study, we partitioned the additive variance of SHW and SHWD with respect to their breed origin by fitting a hierarchical Bayesian model with heterogeneous covariance structure for breeding values to estimate variance components for each breed category, and segregation variance. Two data sets were used to test the proposed hierarchical Bayesian model, one from a multi-year multi-location field trial of SHWD and the other comprising the two species of SHW. For the SHWD, the Bayesian estimates of additive variances of grain yield from each breed category were similar for T. turgidum and Ae. tauschii, but smaller for T. aestivum. Segregation variances between Ae. tauschii-T. aestivum and T. turgidum-T. aestivum populations explained a sizable proportion of the phenotypic variance. Bayesian additive variance components and the Best Linear Unbiased Predictors (BLUPs) estimated by two well-known software programs were similar for multi-breed origin and for the sum of the breeding values by origin for both data sets. Our results support the suitability of models with heterogeneous additive genetic variances to predict breeding values in wheat crosses with variable ploidy levels.

Genetic Gains for Grain Yield in CIMMYT's Semi-Arid Wheat Yield Trials Grown in Suboptimal Environments.

Wheat (Triticum aestivum L.) is a major staple food crop grown worldwide on >220 million ha. Climate change is regarded to have severe effect on wheat yields, and unpredictable drought stress is one of the most important factors. Breeding can significantly contribute to the mitigation of climate change effects on production by developing drought-tolerant wheat germplasm. The objective of our study was to determine the annual genetic gain for grain yield (GY) of the internationally distributed Semi-Arid Wheat Yield Trials, grown during 2002-2003 to 2013-2014 and developed by the Bread Wheat Breeding program at the CIMMYT. We analyzed data from 740 locations across 66 countries, which were classified in low-yielding (LYE) and medium-yielding (MYE) environments according to a cluster analysis. The rate of GY increase (GYC) was estimated relative to four drought-tolerant wheat lines used as constant checks. Our results estimate that the rate of GYC in LYE was 1.8% (38.13 kg ha-1 yr-1), whereas in MYE, it was 1.41% (57.71 kg ha-1 yr-1). The increase in GYC across environments was 1.6% (48.06 kg ha-1 yr-1). The pedigrees of the highest yielding lines through the coefficient of parentage analysis indicated the utilization of three primary sources-'Pastor', 'Baviacora 92', and synthetic hexaploid derivatives-to develop drought-tolerant, high and stably performing wheat lines. We conclude that CIMMYT's wheat breeding program continues to deliver adapted germplasm for suboptimal conditions of diverse wheat growing regions worldwide.

Use of Genomic Estimated Breeding Values Results in Rapid Genetic Gains for Drought Tolerance in Maize.

More than 80% of the 19 million ha of maize ( L.) in tropical Asia is rainfed and prone to drought. The breeding methods for improving drought tolerance (DT), including genomic selection (GS), are geared to increase the frequency of favorable alleles. Two biparental populations (CIMMYT-Asia Population 1 [CAP1] and CAP2) were generated by crossing elite Asian-adapted yellow inbreds (CML470 and VL1012767) with an African white drought-tolerant line, CML444. Marker effects of polymorphic single-nucleotide polymorphisms (SNPs) were determined from testcross (TC) performance of F families under drought and optimal conditions. Cycle 1 (C1) was formed by recombining the top 10% of the F families based on TC data. Subsequently, (i) C2[PerSe_PS] was derived by recombining those C1 plants that exhibited superior per se phenotypes (phenotype-only selection), and (ii) C2[TC-GS] was derived by recombining a second set of C1 plants with high genomic estimated breeding values (GEBVs) derived from TC phenotypes of F families (marker-only selection). All the generations and their top crosses to testers were evaluated under drought and optimal conditions. Per se grain yields (GYs) of C2[PerSe_PS] and that of C2[TC-GS] were 23 to 39 and 31 to 53% better, respectively, than that of the corresponding F population. The C2[TC-GS] populations showed superiority of 10 to 20% over C2[PerSe-PS] of respective populations. Top crosses of C2[TC-GS] showed 4 to 43% superiority of GY over that of C2[PerSe_PS] of respective populations. Thus, GEBV-enabled selection of superior phenotypes (without the target stress) resulted in rapid genetic gains for DT.

Effect of Leaf Rust on Grain Yield and Yield Traits of Durum Wheats with Race-Specific and Slow-Rusting Resistance to Leaf Rust.

Leaf rust, caused by Puccinia triticina, is an important disease of durum wheat (Triticum turgidum) in many countries. We compared the effectiveness of different types of resistance in International Maize and Wheat Improvement Center-derived durum wheat germ plasm for protecting grain yield and yield traits. In all, 10 durum wheat lines with race-specific resistance, 18 with slow-rusting resistance, and 2 susceptible were included in two yield loss trials sown on different planting dates in Mexico with and without fungicide protection under high disease pressure. Eight genotypes with race-specific resistance were immune to leaf rust. Durum wheat lines with slow-rusting resistance displayed a range of severity responses indicating phenotypic diversity. Mean yield losses for susceptible, race-specific, and slow-rusting genotypes were 51, 5, and 26%, respectively, in the normal sowing date trial and 71, 11, and 44% when sown late. Yield losses were associated mainly with a reduction in biomass, harvest index, and kernels per square meter. Slow-rusting durum wheat lines with low disease levels and low yield losses, as well as genotypes with low yield losses despite moderate disease levels, were identified. Such genotypes can be used for breeding durum wheat genotypes with higher levels of resistance and negligible yield losses by using strategies that previously have been shown to be successful in bread wheat.

Genomic prediction models for grain yield of spring bread wheat in diverse agro-ecological zones.

Genomic and pedigree predictions for grain yield and agronomic traits were carried out using high density molecular data on a set of 803 spring wheat lines that were evaluated in 5 sites characterized by several environmental co-variables. Seven statistical models were tested using two random cross-validations schemes. Two other prediction problems were studied, namely predicting the lines' performance at one site with another (pairwise-site) and at untested sites (leave-one-site-out). Grain yield ranged from 3.7 to 9.0 t ha(-1) across sites. The best predictability was observed when genotypic and pedigree data were included in the models and their interaction with sites and the environmental co-variables. The leave-one-site-out increased average prediction accuracy over pairwise-site for all the traits, specifically from 0.27 to 0.36 for grain yield. Days to anthesis, maturity, and plant height predictions had high heritability and gave the highest accuracy for prediction models. Genomic and pedigree models coupled with environmental co-variables gave high prediction accuracy due to high genetic correlation between sites. This study provides an example of model prediction considering climate data along-with genomic and pedigree information. Such comprehensive models can be used to achieve rapid enhancement of wheat yield enhancement in current and future climate change scenario.

Variability in glutenin subunit composition of Mediterranean durum wheat germplasm and its relationship with gluten strength.

The allelic composition at five glutenin loci was assessed by one-dimensional sodium dodecyl sulphate polyacrylamide gel electrophoresis (1D SDS-PAGE) on a set of 155 landraces (from 21 Mediterranean countries) and 18 representative modern varieties. Gluten strength was determined by SDS-sedimentation on samples grown under rainfed conditions during 3 years in north-eastern Spain. One hundred and fourteen alleles/banding patterns were identified (25 at Glu-1 and 89 at Glu-2/Glu-3 loci); 0·85 of them were in landraces at very low frequency and 0·72 were unreported. Genetic diversity index was 0·71 for landraces and 0·38 for modern varieties. All modern varieties exhibited medium to strong gluten type with none of their 13 banding patterns having a significant effect on gluten-strength type. Ten banding patterns significantly affected gluten strength in landraces. Alleles Glu-B1e (band 20), Glu-A3a (band 6), Glu-A3d (bands 6 + 11), Glu-B3a (bands 2 + 4+15 + 19) and Glu-B2a (band 12) significantly increased the SDS-value, and their effects were associated with their frequency. Two alleles, Glu-A3b (band 5) and Glu-B2b (null), significantly reduced gluten strength, but only the effect of the latter locus could be associated with its frequency. Only three rare banding patterns affected gluten strength significantly: Glu-B1a (band 7), found in six landraces, had a negative effect, whereas banding patterns 2 + 4+14 + 15 + 18 and 2 + 4+15 + 18 + 19 at Glu-B3 had a positive effect. Landraces with outstanding gluten strength were more frequent in eastern than in western Mediterranean countries. The geographical pattern displayed from the frequencies of Glu-A1c is discussed.

Technical note: An R package for fitting Bayesian regularized neural networks with applications in animal breeding.

In recent years, several statistical models have been developed for predicting genetic values for complex traits using information on dense molecular markers, pedigrees, or both. These models include, among others, the Bayesian regularized neural networks (BRNN) that have been widely used in prediction problems in other fields of application and, more recently, for genome-enabled prediction. The R package described here (brnn) implements BRNN models and extends these to include both additive and dominance effects. The implementation takes advantage of multicore architectures via a parallel computing approach using openMP (Open Multiprocessing) for the computations. This note briefly describes the classes of models that can be fitted using the brnn package, and it also illustrates its use through several real examples.

Grouping of accessions of Mexican races of maize revisited with SSR markers.

Mexican races of maize (Zea mays L.) represent a valuable genetic resource for breeding and genetic surveys. We applied simple sequence repeat (SSR) markers to characterize 25 accessions of races of maize from Mexico. Our objectives were to (1) study the molecular genetic diversity within and among these accessions and (2) examine their relationships as assumed previously on the basis of morphological data. A total of 497 individuals were fingerprinted with 25 SSR markers. We observed a high total number of alleles (7.84 alleles per locus) and total gene diversity (0.61), confirming the broad genetic base of the maize races from Mexico. In addition, the accessions were grouped into distinct racial complexes on the basis of a model-based clustering approach. The principal coordinate analyses of the four Modern Incipient hybrids corroborated the proposed parental races of Chalqueño, Cónico Norteño, Celaya, and Bolita on the basis of the morphological data. Consequently, for some of the accessions, hybridizations provide a clue that can further be used to explain the associations among the Mexican races of maize.

Methodologies for estimating the sample size required for genetic conservation of outbreeding crops.

The main purpose of germplasm banks is to preserve the genetic variability existing in crop species. The effectiveness of the regeneration of collections stored in gene banks is affected by factors such as sample size, random genetic drift, and seed viability. The objective of this paper is to review probability models and population genetics theory to determine the choice of sample size used for seed regeneration. A number of conclusions can be drawn from the results. First, the size of the sample depends largely on the frequency of the least common allele or genotype. Genotypes or alleles occurring at frequencies of more than 10% can be preserved with a sample size of 40 individuals. A sample size of 100 individuals will preserve genotypes (alleles) that occur at frequencies of 5%. If the frequency of rare genotypes (alleles) drops below 5%, larger sample sizes are required. A second conclusion is that for two, three, and four alleles per locus the sample size required to include a copy of each allele depends more on the frequency of the rare allele or alleles than on the number. Samples of 300 to 400 are required to preserve alleles that are present at a frequency of 1%. Third, if seed is bulked, the expected number of parents involved in any sample drawn from the bulk will be less than the number of parents included in the bulk. Fourth, to maintain a rate of breeding (F) of 1 %, the effective population size (N e) should be at least 150 for three alleles, and 300 for four alleles. Fifth, equalizing the reproductive output of each family to two progeny doubles the effective size of the population. Based on the results presented here, a practical option is considered for regenerating maize seed in a program constrained by limited funds.

Implications of the variance effective population size on the genetic conservation of monoecious species.

The concept of variance effective population size [Ne(v)] and other expressions are reviewed and described for specific sampling steps in germplasm collection and regeneration of monoecious species. Special attention is given to procedures for computing the variance of the number of contributed gametes [V(k)] to the next generation. Drift, as it occurs between generations, was considered to contain a component due to the sampling of parents and a subsequent component due to the sampling of gametes. This demonstrates that drift, caused by reduction of seed viability, damages the genetic integrity of accessions stored in germplasm banks. The study shows how mating designs, such as plant-to-plant or chain crossings with additional female gametic control, can partially alleviate this problem. Optimal procedures for increasing Ne(v) when collecting germplasm in the field are also discussed. The effect of different female and male gametic control strategies on Ne(v) is considered under several situations. Practical examples illustrating the use of V(k) and Ne(v) expressions are given.

Predicted and realized grain yield responses to full-sib family selection in CIMMYT maize (Zea mays L.) populations.

The maize (Zea mays L.) improvement program of the International Maize and Wheat Improvement Center (CIMMYT) develops broad-based maize populations and, until recently, improved all of them through full-sib family selection with international testing. The purpose of this study was to estimate the genetic and genetic × environment variance components for ten of those populations and to measure expected yield improvement from full-sib selection. Mean yield ranged from 3.35-6.81 t ha(-1). For five populations the average yield in the last cycle was higher than in the initial cycles. Several populations showed no improvement or yielded less in the final cycle of selection, either because selection intensity was low or because strong selection pressure was applied simultaneously for several traits. Variation resulting from differences among family means within cycles and from interaction between families and locations within cycles were significant in all populations and cycles. Results indicate that variability among full-sib families was maintained throughout the cycles for all populations. The large σ ge (2) /σ g (2) ratio shown by most populations suggests that yield response per cycle could be maximized if the environments in which progenies are tested were subdivided and classified into similar subsets. The proportion of the predicted response realized in improved yield varied for each population.

Heterosis among populations of maize (Zea mays L.) with different levels of exotic germplasm.

Thirteen maize (Zea mays L.) populations including five adapted, five adapted x exotic, two composites of adapted and exotic, and one exotic selected for adaptability were crossed in a diallel mating system. The parents and 78 crosses and nine check hybrids were evaluated for grain yield and plant height in five environments. The Gardner-Eberhart model Analysis II indicated that additive and nonadditive gene effects accounted for 60 and 40% of the total variation among populations, respectively, for grain yield and 86% and 14% of the total variation, respectively, for plant height. Components of heterosis were significant in the combined analysis for both traits. Adapted Corn Belt populations tended to have higher performance in crosses and greater values of variety heterosis than 50% adapted populations. 'Nebraska Elite Composite', 'Corn Belt' x 'Mexican', and 'Corn Belt' x 'Brazilian' showed high mean yields in crosses, however, they were not among those with high estimates of variety heterosis. One exotic population ('Tuxpeno' x 'Antigua Grupo 2') and three adapted populations ['307 Composite', 'NB(S1)C-3', and 'NK(S1)C-3'] might be combined together to form a high-yielding population. It may be possible to synthesize two useful populations for reciprocal recurrent selection by grouping 'Tuxpeno' x 'Antiqua Grupo 2', 'NB(S1)C-3', and 'NS(FS)LFW-8' into one population and 'NK(S1)C-3', 'Krug'x'Tabloncillo', and '307 Composite' in the other one.

Using the shifted multiplicative model to search for "separability" in crop cultivar trials.

The shifted multiplicative model (SHMM) is used in an exploratory step-down method for identifying subsets of environments in which genotypic effects are "separable" from environmental effects. Subsets of environments are chosen on the basis of a SHMM analysis of the entire data set. SHMM analyses of the subsets may indicate a need for further subdivision and/or suggest that a different subdivision at the previous stage should be tried. The process continues until SHMM analysis indicates that a SHMM with only one multiplicative term and its "point of concurrence" outside (left or right) of the cluster of data points adequately fits the data in all subsets. The method is first illustrated with a simple example using a small data set from the statistical literature. Then results obtained in an international maize (Zea mays L.) yield trial with 20 sites and nine cultivars is presented and discussed.

The area under the function: an index for selecting desirable genotypes.

The linear regression approach has been widely used for selecting high-yielding and stable genotypes targeted to several environments. The genotype mean yield and the regression coefficient of a genotype's performance on an index of environmental productivity are the two main stability parameters. Using both can often complicate the breeder's decision when comparing high-yielding, less-stable genotypes with low-yielding, stable genotypes. This study proposes to combine the mean yield and regression coefficient into a unified desirability index (D i). Thus, D i is defined as the area under the linear regression function divided by the difference between the two extreme environmental indexes. D i is equal to the mean of the i (th) genotype across all environments plus its slope multiplied by the mean of the environmental indexes of the two extreme environments (symmetry). Desirable genotypes are those with a large D i. For symmetric trials the desirability index depends largely on the mean yield of the genotype and for asymmetric trials the slope has an important influence on the desirability index. The use of D i was illustrated by a 20-environments maize yield trial and a 25-environments wheat yield trial. Three maize genotypes out of nine showed values of D i 's that were significantly larger than a hypothetical, stable genotype. These were considered desirable, even though two of them had slopes significantly greater than 1.0. The results obtained from ranking wheat genotypes on mean yield differ from a ranking based on D i .