Comparing double minimization and zigzag algorithms in Joint Regression Analysis: the complete case

Abstract

Joint Regression Analysis is a widely used technique for cultivar comparison. For each cultivar a linear regression is adjusted on a non-observable regressor: the environmental index. This index measures, for each block, the corresponding productivity. When all cultivars are present in all the blocks in the field trials, the series of experiments is complete. To carry out the minimization of the sum of sums of squares of residuals in order to estimate the coefficients of the regressions and the environmental indexes an iterative algorithm, the zigzag algorithm, was introduced, see Mexia et al. [J.T. Mexia, L ₂ environmental indexes, Biom. Lett. 36 (1999), pp. 137–143.]. This algorithm performs well, see e.g. Mexia et al. [Weighted linear joint regression analysis, Biom. Lett. 38 (2001), pp. 33–40.] and Mexia and Pereira [J.T. Mexia, D.G. Pereira, Joint regression analysis for winter rye cultivars using L ₂ indexes, J. Colloquium Biometryczne 31 (2001), pp. 207–212.] but it has not been shown that it converges to the absolute minimum of the goal function. We present an alternative algorithm and show that, in the complete case, it converges to the absolute minimum. Through an example it is shown that the results obtained using both algorithms agree. We analyse the reason behind the agreement between both algorithms.

Keywords:

• Joint Regression Analysis
• linear regressions
• L ₂ environmental indexes
• double minimization
• zigzag algorithm

AMS Subject Classification :

• 68W40
• 62J05

Acknowledgements

We thank the referees for their suggestions that greatly improved the paper. The Portuguese National Plant Breeding Station (ENMP, Elvas) is thanked for allowing the use of their data in this study. The first author of this work is member of the CIMA-UE, research center financed by the Science and Technology Foundation – Portugal.