Abbreviations: AMMI, additive main effects and multiplicative interaction; ASV, additive main effects and multiplicative interaction stability value; AVRC, index and the ranks of the mean yields; BLUP, best linear unbiased prediction; EV, averages of the squared eigenvector values; GEI, genotype × environment interaction; HMGV, harmonic mean of genotypic values; HMRPGV, harmonic mean of relative performance of genotypic values; IPCA, interaction principal component axis; LMM, linear mixed-effect model; MET, multi-environment trials; NF, no fungicide; RCBD, randomized complete block design; RMSPD, root mean square prediction difference; SPIC, sums of the absolute value of the IPCA scores; SVD, singular value decomposition; WAASB, weighted average of absolute scores from the singular value decomposition of the matrix of best linear unbiased predictions for the genotype × environment interaction effects generated by an linear mixedeffect model; WAASBY, weighted average of weighted average of absolute scores from the singular value decomposition of the matrix of best linear unbiased predictions for the genotype × environment interaction effects generated by an linear mixed-effect model and response variable; WF, with fungicide; Za, absolute value of the relative contribution of interaction principal component axes to the interaction. bIoMetrY, ModeLInG, And stAtIstIcsPublished in Agron.

ABSTRACT. Methodologies using restricted maximum likelihood/ best linear unbiased prediction (REML/BLUP) in combination with sequential path analysis in maize are still limited in the literature. Therefore, the aims of this study were: i) to use REML/BLUPbased procedures in order to estimate variance components, genetic parameters, and genotypic values of simple maize hybrids, and ii) to fit stepwise regressions considering genotypic values to form a path diagram with multi-order predictors and minimum multicollinearity that explains the relationships of cause and effect among grain yieldrelated traits. Fifteen commercial simple maize hybrids were evaluated in multi-environment trials in a randomized complete block design with four replications. The environmental variance (78.80%) and genotypevs-environment variance (20.83%) accounted for more than 99% of the phenotypic variance of grain yield, which difficult the direct selection of breeders for this trait. The sequential path analysis model allowed the selection of traits with high explanatory power and minimum multicollinearity, resulting in models with elevated fit (R 2 > 0.9 and ε < 0.3). The number of kernels per ear (NKE) and thousand-kernel weight (TKW) are the traits with the largest direct effects on grain yield (r = 0.66 and 0.73, respectively). The high accuracy of selection (0.86 and 0.89) associated with the high heritability of the average (0.732 and 0.794) for NKE and TKW, respectively, indicated good reliability and prospects of success in the indirect selection of hybrids with highyield potential through these traits. The negative direct effect of NKE on TKW (r = -0.856), however, must be considered. The joint use of mixed models and sequential path analysis is effective in the evaluation of maize-breeding trials.

-Diallel analyses are commonly used for the estimate of population genetic effects. Different models can be used, with a direct effect on the inferences. The objective of this study was to determine and compare two diallel analysis models, fixed and random, regarding the combining effects among six wheat genotypes. The experiment was conducted in the county of Capão do Leão/RS in the year 2006. Six wheat genotypes were used that were used for artificial crosses according to a complete diallel model without reciprocals, resulting in 15 hybrid combinations. The data were subjected to diallel analyses according to model 2 of Griffing (fixed) and BLUP (random). The results show that both diallel models indicate similar general combining ability effects.On the other hand, for the specific combining ability, the data must be used with caution, considering the two models simultaneously.

ABSTRACT. The State of Rio Grande do Sul (RS) stands out as the largest wheat producer in Brazil. Wheat is the most emphasized winter cereal in RS, attracting public and private investments directed to wheat genetic breeding. The study of genetic progress should be performed routinely at breeding programs to study the behavior of cultivars developed for homogeneous regions of cultivation. The objectives of this study were: 1) to evaluate the genetic progress of wheat grain yield in RS; 2) to evaluate the influence of cultivar competition trial stratification in homogeneous regions of cultivation on the study of genetic progress. Grain yield data of 122 wheat cultivars evaluated in 137 trials arranged in randomized block design with three or four replications were used. Field trials were carried out in 23 locations in RS divided into two homogeneous regions during the period from 2002 to 2013. Genetic progress for RS and homogeneous regions was studied utilizing the method proposed by Vencovsky. Annual genetic progress for wheat grain yield during the period of 12 years in the State of RS was 2.86%, oscillating between homogeneous regions of cultivation. The difference of annual genetic progress in region 1 (1.82%) in relation to region 2 (4.38%) justifies the study of genetic progress by homogeneous regions of cultivation.

The REML/BLUP statistics are analyses that can be used as selective criteria in the routine of maize breeding programs. The present study aims to determine the genetic potential in crosses of landrace populations applying the REML/BLUP methodology, and to identify populations for the synthesis of new populations and intrapopulation selection for family farming systems, as well as genetic constitutions for use in maize breeding programs. Nine top cross hybrids obtained in the 2012/2013 harvest were evaluated along with their testator, the landraces used as parents, and four commercial hybrids, in a randomized block design, with information taken from the average of each plot. The evaluated traits were: leaf angle, number of ramifications of the tassel, spike insertion height, plant height, spike diameter, number of grains per spike, mass of grains per spike, spike mass, spike length, prolificity, mass of one hundred grains, and grain yield per plot. The data were analyzed using the Selegen-REML/BLUP software. The top cross hybrids Cateto Branco x Planalto, Amarelão x Planalto and the population Cateto Branco are ranked among the ten best crosses, simultaneously, for the traits: leaf angle, number of ramifications of the tassel, spike insertion height, and plant height (Cateto Branco x Planalto), and leaf angle, spike insertion height, and plant height (Amarelão x Planalto and Cateto Branco). The top cross hybrids Criolão x Planalto, Branco 8 Carreiras x Planalto, Caiano Rajado x Planalto, Amarelão x Planalto, Branco Roxo Índio x Planalto stand out for their high genotypic value of the individual BLUP mean components among the ten best genotypes for grain yield, and by combining three or more traits of interest together, being, for effects of selection, the most indicated.