A phenotypic stability analysis of common bean (Phaseolus vulgaris L.) grain yield (kg.ha-1) was carried out using data of the Regional Trial of the State of Parana during the year/harvest 1996/97. A non-linear model using the parameters proposed by Toler, Yij = ai + [Zjb1i + (1 - Zj) b2i] mj + eij, was adopted. The author proposed tests of the hypotheses H(b1i = b2i) and H(b1i = b2i = bi = 1) according to the principles of regression analysis, that allow the genotypes to be placed in five groups (A, B, C, D, and E) based on their response pattern. The parameters of the model are simultaneously estimated using iterative (non-linear) least squares through the modified Gauss-Newton method. The model was useful in classifying the genetic materials according to their yield and response patterns, contributing to a greater understanding of their yield behavior. The mean of the A and E groups was influenced by a negative association between and . The majority of genotypes with a double favorable response pattern (A group, convex pattern) had lower yields. Genotypes with higher yields were placed mostly in the E group (concave pattern, double unfavorable). Those from groups with a single-segment pattern (B, C, and D) showed variable yields. The difficulty in finding genotypes with both high yields and convex response pattern was evident.

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Modelo de regressão não-linear aplicado ao estudo da estabilidade fenotípica de genótipos de feijão no Estado do Paraná

O objetivo deste trabalho foi comparar dois metodos de regressao empregados para avaliar a estabilidade fenotipica em plantas de cana-de-acucar, o linear, de Cruz et al., e o nao-linear, de Toler & Burrows. Utilizaram-se as informacoes do ensaio de caracterizacao de 1994 do programa de melhoramento genetico da Copersucar, compreendendo 13 locais e 20 genotipos de cana-de-acucar no Estado de Sao Paulo. Os resultados permitiram concluir que ambos os modelos se ajustaram satisfatoriamente aos dados. As estimativas da qualidade ambiental pelos dois modelos mostraram diferencas entre seus valores, porem foram concordantes para inferir sobre os melhores e piores ambientes. Ao classificar os genotipos com base na produtividade e no padrao de resposta, no entanto, apenas dez foram coincidentes nos dois metodos. As vantagens estatisticas do metodo nao-linear torna-o recomendavel para pesquisa sobre estabilidade.

https://www.scielo.br/pdf/pab/v37n1/7544.pdf

Comparação de métodos de regressão para avaliar a estabilidade fenotípica em cana-de-açúcar

Analysis of groups of experiments or multi-environment trials (MET) has been traditionally based on simple models assuming error variance homogeneity between trials, independent error within trials, genotype x environment (g x e) effects as a set of independent random effects. The combined analysis of MET data through realistic models is a complex statistical problem which requires extensions to the standard linear mixed model. The relaxation of the assumption concerning the independence of g x e effects can be achieved with the use of multiplicative models. Such models have been popularised as additive main effects and multiplicative interaction effects (AMMI) and a number of applications have been found. However, AMMI analysis presents at least five great limitations: it considers the genotype and g x e effects as fixed; it is suitable only for balanced data sets; it does not consider spatial variation within trials; it does not consider the heterogeneity of variance between trials; it does not consider the different number of replications across sites. These features are not realistic in analysing field data. In a mixed model setting, Piepho (1998) presented a factor analytic multiplicative mixed (FAMM) model with random genotype and g x e effects which is conceptually and functionally better than AMMI. In the same context, Smith et al. (2001) presented a general class of FAMM models that encompass the approach of Piepho (1998) and include separate spatial errors for each environment (FAMMS). Such general class of models provides a full realistic approach for analysing MET data. The present paper deals with the application of FAMM and FAMMS models in two large unbalanced data sets (on eucalypt and tea plant) aiming at emphasising their advantages over AMMI models in terms of the assumptions of error variance homogeneity between trials and independent error within trials. Also, the ability of FAMM models in providing parsimonious models is also stressed. Parsimonious FAMM models were found for the two data sets. There were great advantages of heterogeneous variance FAMM models over homogeneous variance FAMM models. This reveals the superiority of FAMM models over AMMI models. It was noted that there was heterogeneity among the specific variances in individual environments; therefore, factor analytic models with common specific variances for all sites were not suitable. FAMM models provided estimates of the full correlation structure, facilitating practical decisions to be made. FAMM models with heterogeneous variance among traits and spatial errors within traits were advantageous over FAMM models with variance homogeneity and non-spatial error. This also shows the superiority of FAMM models over AMMI models, which do not allow for dependent or spatial errors. For analysing multi-environment data sets with longitudinal data, FAMMS models proved to be a very useful tool.

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Factor analytic multiplicative mixed models in the analysis of multiple experiments

Considering the climatic seasons in which lettuce is cultivated, probably a high interaction genotype x environment could be obtained. The yield, adaptability and phenotypic stability of six lettuce cultivars of the crisp group (Deyse, Elba, Sabrina, Summer Green, Vera and Veronica) were evaluated, on hydroponic 'NFT'. Nine experiments were carried out in three seasons (autumn= Outn, winter= Invn and spring= Prin), in a completely randomized design with four replications, evaluating four plants in each teatment. There occurred significant differences among environments (sowing date). Planting in autumn presented higher yield (Out1 = 3,1; Out2 = 5,3 and; Out3 = 4,6 kg m2) than winter (Inv1 = 3,6; Inv2 = 3,5 and; Inv = 3,7 kg m2) or spring (Pri1 = 4,0; Pri2 = 3,6 and; Pri3 = 3,2 kg m2). Although the significant differences in the interaction cultivars x environment for yield, except for Inv3 and Pri1, there were no significant differences among cultivars in each planting season. Significant differences were found among cultivars only in Out3, for the number of leaves in each plant. Cultivars Deyse and Veronica were the only ones well adapted in both environments for yield, the other cultivars being unstable. Only Summer Green presented behavior highly previsible (stable) for number of leaves per plant, nevertheless, showed adaptability only in unfavorable environments.

/pdf/adaptabilidade-e-estabilidade-fenotipica-de-cultivares-de-bbuz64gbc7.pdf

Adaptabilidade e estabilidade fenotípica de cultivares de alface do grupo crespa em cultivo hidropônico

/pdf/estabilidade-e-adaptabilidade-fenotipica-atraves-da-2jnv59gwwu.pdf

Estabilidade e adaptabilidade fenotípica através da reamostragem Bootstrap no modelo AMMI.

Summary A non-linear model for examining genotypic responses across an array of environments is contrasted with the 'joint regression' formulation, and a rigorous approach to hypothesis testing using the conditional error principle is demonstrated. The model is extended to cater for situations where single straight-line response patterns fail to characterize genotypic behaviors over an environmental array: a combination of two straight lines, with slope in below-average and in above-average environments, is offered as the 1 2 simplest representation of convex and concave patterns. A protocol for classifying genotypes according to the results of hypothesis tests, i.e. H( = ) and H( = = = 1), is 1 2 1 2 presented . A doubly desirable response pattern is convex ( 1> ). 1 2

P. M. Burrows

Papers

Genotypic performance over environmental arrays: A non-linear grouping protocol