Using Shukla’s mixed model and the related joint regression model in analyses of stability and adaptation of genotypes
Part I. Theoretical considerations
Wiesław Mądry
wieslaw_madry@sggw.edu.plKatedra Statystyki Matematycznej i Doświadczalnictwa, Szkoła Główna Gospodarstwa Wiejskiego w Warszawie (Poland)
Abstract
The most important theoretical problems of the mixed Shukla’s model and a joint regression model called Eberhart-Russell-Shukla model (E-R-S model) — (Piepho, 1999) are presented in this paper. Rather simple and efficient estimators and tests for parameters of the models are shown for a case of complete genotype*environment classification. Genotypic means and other parameters of the models, called stability measures, are considered. In the Shukla’s model a stability variance (σ2i) is stability measure and in the joint regression model E-R-S, stability measures are regression coefficient (βi or bi), residual variance (σ2d(i) and σ2δ(i)) as well as determination coefficient (R2i). The given estimators of these stability measures in the models have been obtained using MINQUE method in Shukla’s model (1972) or ordinary approximated minimum least squares method (OALS) in the E-R-S model (Eberhart and Russell, 1966; Shukla, 1972; Mądry, 2002). These tools are useful only for data in balanced (complete) two-way genotype × environment classifications. Tests F, both usual and approximated ones, are recommended for testing hypothesis on stability measures in the models. The statistical tools in Shukla’s model have optimal properties. The tools for all considered stability parameters in the E-R-S model could be almost optimal if the number of genotypes would be large and environmental variance σ2e would seriously dominate variances of all other random effects in the model (Piepho, 1998; Mądry, 2002). These conditions are usually fulfilled in practice. The considerations in the paper and in literature show that the models, both Shukla’s and E-R-S ones could be useful in a study on stability and adaptation of genotypes in variety trials.
Keywords:
joint regression model of Eberhart-Russell-Shukla (E-R-S model), MINQUE method, ordinary approximate minimum least squares method (OALS method), series of variety trials, Shukla’s model, stability and adaptation analyses of genotypesReferences
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Authors
Wiesław Mądrywieslaw_madry@sggw.edu.pl
Katedra Statystyki Matematycznej i Doświadczalnictwa, Szkoła Główna Gospodarstwa Wiejskiego w Warszawie Poland
Statistics
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