Joint Regression Analysis applied to genotype stability evaluation over years

Most genotype differences connected with yield stability are due to genotype × environment interaction. The presence and dimension of this interaction are the factors that determine the performance of genotypes in distinct environments. The environmental factors, like annual rainfall, temperature, diseases or soil fertility, can only explain part of this interaction. Many statistical tools have been developed with the aim to explain the information contained in the GE interaction data matrix. In our work we use the Joint Regression Analysis (JRA), the Zig-Zag Algorithm to estimate the regression coefficients and the multiple comparison tests of Scheffé, Tukey and Bonferroni. We point out not just the limitations of the JRA when used year by year, but also genotype selection advantage from general JRA over years. Data of the Portuguese Plant Breeding Board were used to carry the year and over years analyses of yielding stability of 22 different genotypes of oat ( Avena sativa L . ) at six different locations in the years 2002, 2003 and 2004.


INTRODUCTION
To perform successfully, a certain genotype should be dominant or at least not dominated in consecutive years of evaluation. Then, the stability of parameters is a very important issue in plant breeding programs, and methods for testing and interpreting the genotype × environment interaction have been a matter of concern not only in agronomic literature, but also from the point of view of statistical tools development. Joint regression analysis (JRA) is a powerful tool to analyse this interaction, as was shown for example by Aastveit, and Mejza (1992).
The yield data, obtained for 22 different genotypes of oat (Avena sativa L.) at six locations in the years 2002, 2003 and 2004 kindly provided by the Portuguese Plant Breeding Board, were used: -to study and evaluate genotype stability in three consecutive years, year by year; -to compare the results over years and to draw conclusions on genotype selection.
We used the JRA year by year and over years. The Zig-Zag Algorithm was be used to estimate the regression coefficients. Multiple comparison tests of Scheffé, Tuckey and Bonferroni were also done.

REGRESSION ADJUSTMENT AND UPPER CONTOUR
In the JRA, the environmental index is a synthetic variable used to measure the productive capacity of each block by year combination.
Since in our problem every genotype is present in each block, this is the complete case. For this case the corresponding environmental indexes are estimated through their average yields, as it can be see in Gusmão (1985 a, b). The next step is to adjust linear regressions, one per genotype, of yields on environmental indexes.
Considering J genotypes occurring in each of the b blocks, where b is the number of locations multiplied by the number of replications, yi,j as the yield for the j th genotype on the i th block, j =1,…,J; i=1,…,b, then to adjust the regression coefficients we minimize the objective function The pairs (αj, βj, j=1,…,J, represent the regression coefficients for the J genotypes, x b = (x1,….,xb) is the vector of environmental indexes, α J = (α1,….,αJ) and β J = (β1,…,βJ). In the general case pi,j, I = 1,…,b; j =1,…,J , represent the genotype weight in the block. We will assume pi,j, = 1 since we are in the complete case -all genotypes are present in all blocks. The minimization procedure is carried out using the Zig-Zag Algorithm, (see Pereira and Mexia, 2002).
After the adjustment procedure we need to carry on joint representation of regression lines and make the interpretation of upper contour.
The upper contour of the adjusted regression lines on genotypes selection and comparison was introduced by Mexia et. al. (1997), and it is shown to be a convex polygonal.
The upper contour is constituted by segments of the adjusted regression lines where each of them corresponds to a range of variation of the environmental indexes in which the corresponding genotype has maximum adjusted yield.
The genotypes whose regressions partake in the upper contour are the dominant ones. The other genotypes must be compared with the dominant ones to check if they are significantly dominated throughout the entire range of environmental indexes.  Table 1. Table 2 presents the names of genotypes investigated.

Upper contour analysis
According to Mexia et al (1997), the representation of all the adjusted regression lines defines a convex polygonal as the upper contour. The genotypes whose adjusted regressions partake of the upper contour are dominant, and each dominant genotype has a dominance range. Non dominant genotypes should be compared with the dominant ones to test if they are significantly dominated, or not. In Figures 1, 2  To see which genotypes are dominated by AE9303 in the leftmost range and by 87SA29 in the rightmost range, we made out the following multiple comparisons: -at point 73 . 1823 we had to compare AE9303 with the genotypes with lesser adjusted slopes;        Table 7 contains the adjusted coefficients and the corresponding R². These results were obtained using the Zig-Zag algorithm, after 4 iterations. The genotypes are presented according to decreasing adjusted slopes order.  Figure 4 shows a graphic for the adjusted linear regressions over the years 2002, 2003 and 2004. Considering the entire dominance range of environmental indexes over years, [701.45 ; 5225.95], we find two dominant genotypes: AE9303 dominant in [701.45 ;2534.68] and87SA29 dominant in [2534.68 ;5225.95].  To see which genotypes are dominated by AE9303 in the leftmost range and by 87SA29 in the rightmost range, we made the following multiple comparisons: -at point 701.45 we had to compare AE9303 with the genotypes with lesser adjusted slopes; -at point 2534.68 we had to compare AE9303 with the genotypes with larger adjusted slopes; -at point 2534.68 we had to compare 87SA29 with the genotypes with lesser adjusted slopes; Table 8 contains the results on genotype comparison using Scheffé, Tukey and Bonferroni methods, at the 5% level of significance.

Synthesis on multiple comparisons
A global analysis on multiple comparisons year by year and over years is presented in Table  9. Table 9 Global analysis on the multiple comparisons Synteza na podstawie testów wielokrotnych genotypes, while the over years analysis resulted in 20 dominated genotypes, at the 5% level of significance.

CONCLUSION
According to the JRA year by year, in the year 2002 the genotypes S.VICENTE*AE8303 and 87SA29 could be recommended as the preferable ones in program evaluation; in 2003 the best genotypes were AE9403 and AE9303; in 2004 we should select 87SA29 and AE9303. In the over years analysis we can see that 87SA29 and AE9303 are the recommendable genotypes.
It can be concluded that it is not a good way to select genotypes based on the year by year analysis alone. It is seen that, for example in 2002, the genotype 87SA29 was dominant at the leftmost range, whereas in 2003 it was neither dominant nor dominated, not too relevant. However, in 2004, this genotype could be again dominant, and according to the over years on analysis it is one of the best two genotypes.
Genotype AE9303 was neither dominant nor dominated in 2002, and it was dominant in 2003 and 2004 and can be recommended in the over years analysis.
It is also to be pointed out that the over years analysis is likely to be very discriminate in what concerns significantly dominated genotypes, but very important in selection of dominant genotypes. The results of the over year analysis show that only two types of genotypes have been identified: dominant genotypes and dominated genotypes.
In a plant breeding program we should take into account the results of both analyses to draw proper conclusions. The year by year analysis is important, but when taken alone, it may cause the risk of discarding good genotypes. Comparatively, over years analysis is very important in selecting the most promising genotypes, but in the case of our studies it was not flexible enough to distinguish intermediate oat genotypes.