Biometrical Letters vol. 45(2), 2008, pp. 9-22

In the context of the use of Joint Regression Analysis (JRA) for cultivars comparison, as in Pinto (2006), the environmental index is a variable which measures the productivity per block in field experiments. Initially, randomized blocks were used, and their mean yields (classical environmental indexes) were taken as measuring their environmental indexes. Later on, following Patterson and Williams (1976), alfa - designs, which have incomplete blocks, superseded randomized blocks. Then the environmental indexes for each block, couldn't be measured by the respective mean yields, which would lead to biased estimates. This problem was solved by the introduction of L₂ environmental indexes see Mexia et al. (1999). In that paper the authors, through the application of a Zig-Zag algorithm, showed how to adjust simultaneously the regression coefficients and the environmental indexes. This study presents an analysis of a with-covariates model for the Environmental Indexes (see Searle, 1987) using the adjusted L₂ environmental indexes as observations of the dependent variable. Concerning the explicative variables, we consider a general mean, the main effects of two qualitative factors, year and location and as covariates the "classical environmental indexes". The presence of multi-collinearity led to the use of principal components, see Judge et al. (1988), to perform the adjustment. Our results are applied to the data of a Wheat Plant Breeding Program in Portugal (1986-2000), kindly supplied by the Portuguese Plant Breeding Station. The sole significant result was for the covariate, which validates the linearity assumed in JRA.

linear models, covariates, dummy variables, multicollinearity, principal components, L₂ environmental indexes, JRA.