ABSTRACT: Genome-wide selection (GWS) is currently a technique of great importance in plant breeding, since it improves efficiency of genetic evaluations by increasing genetic gains. The process is based on genomic estimated breeding values (GEBVs) obtained through phenotypic and dense marker genomic information. In this context, GEBVs of N individuals are calculated through appropriate models, which estimate the effect of each marker on phenotypes, allowing the early identification of genetically superior individuals. However, GWS leads to statistical challenges, due to high dimensionality and multicollinearity problems. These challenges require the use of statistical methods to approach the regularization of the estimation process. Therefore, we aimed to propose a method denominated as triple categorical regression (TCR) and compare it with the genomic best linear unbiased predictor (G-BLUP) and Bayesian least absolute shrinkage and selection operator (BLASSO) methods that have been widely applied to GWS. The methods were evaluated in simulated populations considering four different scenarios. Additionally, a modification of the G-BLUP method was proposed based on the TCR-estimated (TCR/G-BLUP) results. All methods were applied to real data of cassava (Manihot esculenta) with to increase efficiency of a current breeding program. The methods were compared through independent validation and efficiency measures, such as prediction accuracy, bias, and recovered genomic heritability. The TCR method was suitable to estimate variance components and heritability, and the TCR/G-BLUP method provided efficient GEBV predictions. Thus, the proposed methods provide new insights for GWS.

ABSTRACT: Quantitative genetics theory for genomic selection has mainly focused on additive effects. This study presents quantitative genetics theory applied to genomic selection aiming to prove that prediction of genotypic value based on thousands of single nucleotide polymorphisms (SNPs) depends on linkage disequilibrium (LD) between markers and QTLs, assuming dominance and epistasis. Based on simulated data, we provided information on dominance and genotypic value prediction accuracy, assuming mass selection in an open-pollinated population, all quantitative trait loci (QTLs) of lower effect, and reduced sample size. We show that the predictor of dominance value is proportional to the square of the LD value and to the dominance deviation for each QTL that is in LD with each marker. The weighted (by the SNP frequencies) dominance value predictor has greater accuracy than the unweighted predictor. The linear × linear, linear × quadratic, quadratic × linear, and quadratic × quadratic SNP effects are proportional to the corresponding linear combinations of epistatic effects for QTLs and the LD values. LD between two markers with a common QTL causes a bias in the prediction of epistatic values. Compared to phenotypic selection, the efficiency of genomic selection for genotypic value prediction increases as trait heritability decreases. The degree of dominance did not affect the genotypic value prediction accuracy and the approach to maximum accuracy is asymptotic with increases in SNP density. The decrease in the sample size from 500 to 200 did not markedly reduce the genotypic value prediction accuracy.

ABSTRACT To date, the quantitative genetics theory for genomic selection has focused mainly on the relationship between marker and additive variances assuming one marker and one quantitative trait locus (QTL). This study extends the quantitative genetics theory to genomic selection in order to prove that prediction of breeding values based on thousands of single nucleotide polymorphisms (SNPs) depends on linkage disequilibrium (LD) between markers and QTLs, assuming dominance. We also assessed the efficiency of genomic selection in relation to phenotypic selection, assuming mass selection in an open-pollinated population, all QTLs of lower effect, and reduced sample size, based on simulated data. We show that the average effect of a SNP substitution is proportional to LD measure and to average effect of a gene substitution for each QTL that is in LD with the marker. Weighted (by SNP frequencies) and unweighted breeding value predictors have the same accuracy. Efficiency of genomic selection in relation to phenotypic selection is inversely proportional to heritability. Accuracy of breeding value prediction is not affected by the dominance degree and the method of analysis, however, it is influenced by LD extent and magnitude of additive variance. The increase in the number of markers asymptotically improved accuracy of breeding value prediction. The decrease in the sample size from 500 to 200 did not reduce considerably accuracy of breeding value prediction.

Abstract The objectives of this study were to assess linkage disequilibrium (LD) and selection-induced changes in single nucleotide polymorphism (SNP) frequency, and to perform association mapping in popcorn chromosome regions containing quantitative trait loci (QTLs) for quality traits. Seven tropical and two temperate popcorn populations were genotyped for 96 SNPs chosen in chromosome regions containing QTLs for quality traits. The populations were phenotyped for expansion volume, 100-kernel weight, kernel sphericity, and kernel density. The LD statistics were the difference between the observed and expected haplotype frequencies (D), the proportion of D relative to the expected maximum value in the population, and the square of the correlation between the values of alleles at two loci. Association mapping was based on least squares and Bayesian approaches. In the tropical populations, D-values greater than 0.10 were observed for SNPs separated by 100-150 Mb, while most of the D-values in the temperate populations were less than 0.05. Selection for expansion volume indirectly led to increase in LD values, population differentiation, and significant changes in SNP frequency. Some associations were observed for expansion volume and the other quality traits. The candidate genes are involved with starch, storage protein, lipid, and cell wall polysaccharides synthesis.

O Crestamento Bacteriano Comum (CBC), causado por Xanthomonas axonopodis pv. phaseoli (Xap) é a principal doença bacteriana na cultura do feijão-de-vagem e o controle a essa doença usando cultivares resistentes é ainda um desafio. Esse trabalho tem como objetivo estudar a capacidade a capacidade combinatória para a resistência ao CBC em genótipos de Phaseolus vulgaris. Seis genitores (dois genótipos resistentes de feijão comum ao CBC e quatro acessos suscetíveis de feijão-de-vagem) foram cruzados em um esquema de dialelo completo sem recíprocos para estimar a capacidade geral e específica de combinação para a resistência a Xap. A resistência ao CBC foi avaliada por meio da inoculação com dois isolados de Xap e a severidade avaliada a partir de quatro componentes de resistência: área abaixo da curva de progresso da doença, notas de severidade nas folhas, período latente e diâmetro da lesão em vagens. Diferenças entre os dois isoladas foram observados considerando todos os componentes da doença. Além da variabilidade dos patógenos, CGC e CEC foram significativas, indicando que os efeitos aditivos e não-aditivos estão envolvidos no controle da resistência a Xap para os genótipos avaliados, implicando que a resistência a CBC é uma característica com herança complexa. Para fins de melhoramento, os resultados demonstram a necessidade de aplicar métodos de melhoramento que são focados em seleção de gerações avançadas.

Common bacterial blight (CBB), which is caused by Xanthomonas axonopodis pv. phaseoli (Xap), is the main bacterial disease in snap beans and controlling this disease using resistant cultivars is still a challenge. This work aimed to study the combining ability for CBB resistance in Phaseolus vulgaris genotypes. Six parents (two genotypes of CBB-resistant dry bean and four susceptible snap bean accessions) were crossed in a complete diallel scheme without reciprocals to estimate the general and specific ability to Xap resistance. CBB resistance was evaluated by the inoculation with two Xap isolates, and its severity was evaluated based on the four following resistance components: area under the disease progress curve; scores in the leaves; latent period and diameter of pod lesion. Differences between the two isolates were observed considering all the disease components. Besides pathogen variability, significant GCA and SCA indicate that additive and non-additive effects are involved in Xap-resistance control for the evaluated genotypes, implying that CBC resistance is a trait with complex inheritance. For breeding purposes, the result demonstrates the need to apply breeding methods that are focused on advanced generations selection.

O objetivo deste trabalho foi comparar formas de análise de medidas repetidas para o melhoramento da produção de frutos de pinha (Annona squamosa). Vinte progênies de meias-irmãs foram avaliadas por três anos (2003, 2004 e 2005) em delineamento de blocos ao acaso, com cinco repetições, com cada parcela constituída de quatro plantas. A característica avaliada foi o número de frutos por indivíduo. Os modelos de simetria composta, de simetria composta com variâncias heterogêneas, autorregressivo com variâncias heterogêneas, e antedependência estruturada, foram analisados com o programa ASReml. A estimação dos componentes de variância e a predição dos valores genéticos foram feitas com o procedimento REML/BLUP. A comparação dos modelos foi realizada pelo teste de razão de verossimilhança e pelo critério de Akaike. O modelo antedependência estruturada, para os fatores progênie e parcela, e o modelo multivariado, para o fator resíduo, são as melhores abordagens para a análise dos dados, pois propiciam eficiência e parcimônia em relação ao modelo multivariado completo. Com o modelo antedependência estruturada, é possível a identificação de famílias superiores, em cada colheita, e também de famílias com maior número total de frutos.

The objective of this work was to compare the ways of analyzing repeated measures to improve the production of sugar apple (Annona squamosa). Twenty half-sib progenies were evaluated, over three years (2003, 2004 and 2005), in a randomized block design with five replicates, and each plot was constituted of four plants. The evaluated trait was the number of fruit per individual. The models of compound symmetry, autoregressive with heterogeneous variance, the structured ante-dependence, and compound symmetry with heterogeneous variance were analyzed using the ASReml software. The estimation of variance components and the prediction of breeding values were made by the REML/BLUP. The comparison of the models was done by the likelihood ratio test and Akaike's information criterion. The structured ante-dependence model, for the factors progeny and parcel, and the multivariate model, for the residual factor, are the best approaches for data analysis, providing efficiency and parsimony over the full multivariate model. With the structured ante-dependence model, it is possible to identify superior families in each harvest, and also the families with larger total number of fruit.

The aim of the study was to assess inbreeding depression and determine genetic components in landrace populations of popcorn. Six Brazilian popcorn landraces: Amarelo, Beija-Flor, Branco, Rosa-claro, Roxo and Viçosa, were subjected to self-pollination. Experiments with SO populations and their selfing populations (SI) were conducted in a randomized complete block design. The estimates of inbreeding depression varied from 1.2% (popping expansion) to 79% (grain yield). The gene action was predominantly additive for plant height, ear height and hundred-grain-weight. The dominance genetic effects were important for index of prolificacy and grain yield. The genetic effect of popping expansion was bidirectional with nuil dominance and a predominance of additive gene effects. Based on these results, it was possible to identify popcorn populations suitable to extract inbred lines, develop hybrids and use in population improvement. The present study also confirmed that popping expansion is a trait that can be improved efficiently by recurrent selection programs of popcorn.

Los objetivos del estudio fueron evaluar la depresión endogámica y determinar los componentes genéticos en poblaciones de maíz-roseta. Se realizaron autopolinizaciones en seis poblaciones Brasileras: Amarelo, Beija-Flor, Branco, Rosa-claro, Roxo and Viçosa. Las poblaciones SO y las poblaciones de autopolinización SI se evaluaron en un diseño de bloques completos al azar. Las estimaciones de depresión endogámica variaron de 1,2% (capacidad de expansion) a 79% (productividad de granos). Los efectos genéticos fueron predominantemente aditivos para altura de planta, altura de mazorca y peso de cien granos. Los efectos genéticos de dominancia fueron importantes para índice de prolificidad y rendimiento de grano. El efecto genético de la capacidad de expansion fue bidireccional, con nulo efecto de dominancia y predominancia de efectos genéticos aditivos. Basado en los resultados, fue posible identificar poblaciones de maíz-roseta para la obtención de líneas puras, híbridos y mejoramiento poblacional. El presente estudio confirmó, además, que la capacidad de expansion es una característica que puede ser mejorada eficientemente en los programas de selección recurrente de maíz.

Os objetivos deste trabalho foram estimar os parâmetros de adaptabilidade e estabilidade em populações de milho-pipoca relacionadas por ciclos de seleção, inferir sobre a eficiência dos métodos de seleção pelos quais as populações foram obtidas e avaliar os efeitos da seleção sobre os parâmetros de adaptabilidade e estabilidade. Vinte e cinco populações e três testemunhas comerciais foram avaliadas em 14 ensaios realizados nos anos agrícolas de 2003/2004, 2004/2005, 2006/2007, 2008/2009 e 2009/2010, em sete locais. Utilizou-se o delineamento experimental de blocos completos ao acaso, com quatro repetições. Foram analisadas a capacidade de expansão, avaliada em forno de microondas e na pipocadora Metric Weight Volume Tester (MWVT), além da produtividade de grãos. Utilizou-se o método de adaptabilidade e estabilidade de Eberhart & Russell. Em geral, as populações base e melhoradas apresentaram previsibilidade de comportamento em resposta às variações de ambiente. A seleção pode provocar mudanças nos padrões de adaptabilidade e estabilidade, e as diferentes estratégias de seleção empregadas na obtenção das populações apresentaram eficiências semelhantes

The objectives of this work were to estimate the adaptability and stability parameters of popcorn populations related through cycles of selection, to infer about the efficiency of the selection methods by which the populations were obtained and to assess the effects of selection on the parameters of adaptability and stability. Twenty-five populations and three commercial controls were evaluated in 14 trials carried out in the 2003/2004, 2004/2005, 2006/2007, 2008/2009 and 2009/2010 growing seasons, in seven locations. The experimental design used was randomized complete blocks with four replicates. The expansion volume, evaluated in a microwave oven and in the Metric Weight Volume Tester (MWVT), as well as grain yield were analyzed. The Eberhart & Russell adaptability and stability method was used. In general the initial and improved populations showed predictable performance in response to changes in the environment. Selection can cause changes in the adaptability and stability patterns, and the different selection strategies performed in order to obtain the populations showed similar efficiencies

O objetivo deste trabalho foi selecionar famílias S5 de milho-pipoca com base na divergência genética e no desempenho per se. Foram avaliadas 144 famílias S5, obtidas da população Beija-flor. O experimento foi instalado em Viçosa, na safra 2002/2003, no delineamento em blocos aumentados, com as testemunhas IAC 112 e Zélia, intercaladas a cada dez famílias. A divergência genética foi predita pelo método de otimização de Tocher, utilizando a distância generalizada de Mahalanobis. Foram formados 23 grupos. Dentro de cada grupo, estipularam-se pontos de corte para índice de prolificidade (> 1), peso de 100 grãos (< 15 g) e rendimento de grãos (> 1.000 kg ha-1). As famílias restantes dentro de cada grupo foram selecionadas em capacidade de expansão. Assim, foram selecionadas 23 famílias, que podem ser utilizadas no melhoramento populacional na formação de sintéticos, ou em futuros cruzamentos. A seleção de famílias com base na diversidade genética e em desempenho per se foi eficiente, sendo possível selecionar famílias divergentes e, em geral, com excelentes características agronômicas, como capacidade de expansão média de 30,67 mL g-1 e produtividade média de 1.897,69 kg ha-1.

The objective of this work was to select S5 families of popcorn based on their genetic divergence and performance. The study evaluated 144 S5 families, obtained from the Beija-flor population. The experiment was conducted in Viçosa, during the 2002/2003 crop season, in an augmented block design, with the controls IAC 112 and Zélia inserted every 10 families. The genetic divergence was predicted using Tocher's optimization method, based on the Mahalanobis distance. Twenty-three groups were formed. Within each group formed, points were prescribed for cutting prolificity index (> 1), weight of a hundred grains (< 15) and yield (> 1,000 kg ha-1). Twenty-three families were selected for popping expansion, which can be used in population breeding to synthesize synthetic varieties, or advancing the breeding process for subsequent hybrid crosses. The selection of families based on their genetic diversity and performance was quite efficient. Different families with excellent agronomic characteristics, as popping expansion average of 30.67 mL g-1 and yield of 1,897.69 kg ha-1, were selected.

Objetivou-se avaliar a eficiência relativa da seleção massal, entre e dentro e de um novo índice de seleção combinada proposto. O índice, baseado na genealogia, foi obtido por: IC = b1P Sn-1 + b2 <img border=0 width=13 height=13 src="../../../../img/revistas/asagr/v31n2/a03img01.jpg" align=absmiddle>FSn , em que P Sn-1, b1é o valor fenotípico da planta Sn-1; <img border=0 width=15 height=15 src="../../../../img/revistas/asagr/v31n2/a03img01.jpg" align=absmiddle>FSné o valor fenotípico da família Sn descendente da planta Sn-1; b1 é o peso do valor fenotípico individual; e b2 é o peso do valor fenotípico de família Sn. Os pesos foram obtidos por derivação, pressupondo que os mesmos minimizam a variância da diferença entre o índice e o valor genético aditivo da planta Sn - 1. Para comparar os métodos, utilizaram-se famílias e plantas endogâmicas, obtidas da população Viçosa. Fez-se seleção em capacidade de expansão, utilizando os três métodos nas gerações S1, S2, S3 e S4, e avaliou-se a eficiência dos métodos comparando médias em capacidade de expansão de famílias descendentes de selecionadas por cada método em S2, S3, S4 e S5. Para comparação de médias, utilizou-se o teste t ao nível de 5% de probabilidade. Concluiu-se que a seleção combinada proposta foi superior à seleção massal e foi, em geral, superior à seleção entre e dentro. A seleção entre e dentro foi, geralmente, superior à seleção massal.

The study aimed to evaluate the relative efficiency of mass selection, among, within and of proposed new index of combined selection. The index, based on genealogy, was obtained by: IC = b1P Sn-1 + b2 <img border=0 width=13 height=15 src="../../../../img/revistas/asagr/v31n2/a03img01.jpg" align=absmiddle>FSn , where P Sn-1 is the phenotypic value of the plant Sn-1; <img border=0 width=13 height=15 src="../../../../img/revistas/asagr/v31n2/a03img01.jpg" align=absmiddle>FSn is the phenotypic value of the Sn family, descendant of the Sn-1 plant; b1 is the weight of the individual phenotypic value, and b2 it is the weight of the phenotypic value of Sn family. The weights were obtained by derivation, assuming that they minimize the variance of the difference between the index and the value of the plant genetic additive Sn-1. To compare the methods, families and inbred plants were used, obtained from the population of Viçosa. A selection was made for popping expansion, using the three methods in S1, S2, S3 and S4 generations, and the efficiency of the methods of comparing averages in popping expansion of descendants of families selected by each method in S2, S3, S4 and S5 was assessed. For comparison of averages, the t-test at 5% of probability was used. It is concluded that the combined selection proposal was superior to the mass selection and was, in general, higher than the selection among and within. The selection among and within was, in general, higher than the mass selection.

The prediction of gains from selection allows the comparison of breeding methods and selection strategies, although these estimates may be biased. The objective of this study was to investigate the extent of such bias in predicting genetic gain. For this, we simulated 10 cycles of a hypothetical breeding program that involved seven traits, three population classes, three experimental conditions and two breeding methods (mass and half-sib selection). Each combination of trait, population, heritability, method and cycle was repeated 10 times. The predicted gains were biased, even when the genetic parameters were estimated without error. Gain from selection in both genders is twice the gain from selection in a single gender only in the absence of dominance. The use of genotypic variance or broad sense heritability in the predictions represented an additional source of bias. Predictions based on additive variance and narrow sense heritability were equivalent, as were predictions based on genotypic variance and broad sense heritability. The predictions based on mass and family selection were suitable for comparing selection strategies, whereas those based on selection within progenies showed the largest bias and lower association with the realized gain.

O objetivo deste trabalho foi avaliar a eficiência da seleção recorrente recíproca em produzir híbridos de milho-pipoca (Zea mays) de progênies endógamas superiores, com famílias de irmãos-completos. O programa de seleção recorrente recíproca envolveu as populações de milho-pipoca 'Viçosa' e 'Beija-Flor'. Os testes dos híbridos S0xS0 e S1xS1 foram conduzidos em delineamento látice, nos anos agrícolas de 2002/2003 e 2004/2005. Os dois ensaios incluíram testemunhas comerciais comuns, que permitiram a comparação do desempenho dos híbridos. Analisaram-se a produtividade e a capacidade de expansão. Foram preditos ganhos direto e indireto com a seleção em capacidadede expansão, avaliada em pipoqueira de ar quente e na pipocadora "Metric Weight Volume Tester". Os ganhos observados foram calculados para avaliar a eficiência da seleção recorrente recíproca. As análises de covariância mostraram variabilidade genotípica nas populações. Em relação às testemunhas comerciais, os híbridos S0xS0 foram inferiores em qualidade e equivalentes em produtividade, e os híbridos S1xS1 foram equivalentes nas duas características. O ganho observado na capacidade de expansão foi substancial, cerca de 4 mL g-1, superior ao ganho predito (2,8 mL g-1). A redução observada na produtividade não foi relevante. Verificou-se redução na variabilidade genotípica quanto à capacidade de expansão. O método de seleção recorrente recíproca foi eficiente em produzir híbridos S1xS1 superiores aos S0xS0.

The objective of this work was to evaluate the efficiency of reciprocal recurrent selection in producing hybrids of popcorn (Zea mays) from superior endogamous progenies, with full-sibling families. The reciprocal recurrent selection program involved the 'Viçosa' and 'Beija-Flor' popcorn populations. Tests of the hybrids S0xS0 and S1xS1 were carried out in lattice design, in the agricultural years 2002/2003 and 2004/2005. The two assays included common commercial controls, allowing for the comparison of the hybrids performance. Productivity and expansion capacity were evaluated. Direct and indirect gains with selection were predicted on expansion capacity assessed in hot air and Metric Weight Volume Tester popcorn makers. The gains observed were calculated to evaluate the efficiency of the reciprocal recurrent selection. The covariance analyses showed genotypic variability in the populations. When compared to the commercial controls, the S0xS0 hybrids were inferior in quality and equivalent in productivity; and the S1xS1 hybrids were equivalent in both traits. The observed gain in expansion capacity was substantial (around 4 mL g-1) and higher than the predicted gain (2.8 mL g-1). The reduction observed in productivity was not relevant. Reduction in the genotypic variability was verified for expansion capacity. The reciprocal recurrent selection method was efficient in producing S1xS1 hybrids superior to S0xS0 hybrids.

Os objetivos deste trabalho foram analisar ganhos genéticos teóricos de milho como resultado da seleção entre famílias de irmãos completos e de meios-irmãos, obtidas pelos Delineamentos I, de Irmãos Completos e de Meios-Irmãos, e a variabilidade genotípica e perda de genes com seleção a longo prazo. Os delineamentos foram avaliados por simulação, com base nos ganhos médios estimados após dez ciclos de seleção. O processo de simulação considerou sete sistemas gênicos com dez genes (com distintos graus de dominância), três classes de populações (com diferentes freqüências gênicas), sob três condições de ambiente (valores de herdabilidade), e quatro estratégias de seleção. Cada combinação foi repetida dez vezes, totalizando 25.200 simulações. Seleção entre famílias de irmãos completos é geralmente mais eficiente do que seleção com base em progênies de meios-irmãos. Empregar famílias de irmãos completos obtidas pelo Delineamento I é geralmente mais eficiente que usar progênies derivadas do Delineamento de Irmãos Completos. O Delineamento I, com 50 machos e 200 fêmeas (tamanho efetivo de 160), não determinou população melhorada com variabilidade genotípica mínima. Nas populações melhoradas com menor tamanho efetivo (160 e 400), a perda de genes favoráveis é restrita àqueles recessivos em baixa freqüência.

The objectives of this work were to analyze theoretical genetic gains of maize due to recurrent selection among full-sib and half-sib families, obtained by Design I, Full-Sib Design and Half-Sib Design, and genotypic variability and gene loss with long term selection. The designs were evaluated by simulation, based on average estimated gains after ten selection cycles. The simulation process was based on seven gene systems with ten genes (with distinct degrees of dominance), three population classes (with different gene frequencies), under three environmental conditions (heritability values), and four selection strategies. Each combination was repeated ten times, amounting to 25, 200 simulations. Full-sib selection is generally more efficient than half-sib selection, mainly with favorable dominant genes. The use of full-sib families derived by Design I is generally more efficient than using progenies obtained by Full-Sib Design. Using Design I with 50 males and 200 females (effective size of 160) did not result in improved populations with minimum genotypic variability. In the populations with lower effective size (160 and 400) the loss of favorable genes was restricted to recessive genes with reduced frequencies.

Este estudo discute do ponto de vista teórico as análises de heterose e capacidade de combinação a partir de dialelo parcial. Foram realizados estudos de simulação com o objetivo de investigar as informações proporcionadas pelos parâmetros genéticos. O modelo genético ajustado considerou um gene, grupos de seis populações, três graus de dominância (|d/a| igual a 0,5, 1 e 2, representando dominâncias parcial e completa, e sobredominância) e freqüências alélicas aleatórias. Para cada grau de dominância foram realizadas 50 simulações. O modelo para análise de heterose não precisa ser restrito. O modelo irrestrito proporciona as mesmas informações do restrito. O modelo para análise de capacidade de combinação é restrito. As restrições do modelo partial-G2 de GERALDI e MIRANDA-FILHO não satisfazem aos valores paramétricos dos efeitos de capacidade específica de combinação. Como na análise de dialelo completo, do ponto de vista teórico não há diferenças entre as análises de heterose e de capacidade de combinação. A análise das diferenças de freqüências gênicas entre as populações dos dois grupos é mais eficiente quando são consideradas as heteroses e não as heteroses específicas - ou os efeitos de heterose específica ou os efeitos de capacidade específica de combinação - principalmente quanto a genes cujas freqüências são contrastantes nos grupos.

This study discusses from the theoretical standpoint the analyses of heterosis and combining ability from the partial diallel. Aiming at to investigate the information provided by the genetic parameters, simulation studies were realized. The genetic model adjusted considered one gene, groups of six populations, three degrees of dominance (|d/a| equal to 0.5, 1 and 2, representing partial, complete and overdominance) and random allelic frequencies. For each degree of dominance 50 replications were performed. The model for heterosis analysis does not need to be restricted. The unrestricted model gives the same information as the restricted one. The model for combining ability is restricted. The restrictions of the partial-G2 model of GERALDI and MIRANDA-FILHO do not satisfy the parametric values of the specific combining ability effects. As in the complete diallel analysis, from the theoretical point of view there is no differences between the analysis of heterosis and combining ability. The analysis of differences in gene frequencies among the populations of the two groups is more efficient when it considers the heterosis and not the specific heterosis - or the specific heterosis effects or the effects of specific combining ability - mainly for genes whose frequencies are contrasting in the groups.

Although epistasis is common in gene systems that determine quantitative traits, it is usually not possible to estimate the epistatic components of genotypic variance because experiments in breeding programs include only one type of progeny. As the study of this phenomenon is complex, there is a lack of theoretical knowledge on the contribution of the epistatic variances when predicting gains from selection and on the bias in estimating genetic parameters when fitting the additive-dominant model. The objective of this paper is to discuss these aspects. Regarding a non-inbred population, the genetic value due to dominance and the epistatic components of the genotypic value are not indicators of the number of favorable genes present in an individual. Thus, the efficiency of a selection process should be based on the narrow-sense heritability, a function only of additive variance. If there is no epistasis, generally it is satisfactory to assess the selection efficiency and to predict gain based on the broad-sense heritability. Regardless of the selection unit or type of epistasis, the bias in the estimate of the additive variance when assuming the additive-dominant model is considerable. This implies overestimation of the heritabilities at half sib family mean, plant within family and plant levels, and underestimation if the selection units are full sib progenies. The predicted gains will have a bias proportional to that of the heritability.