# Nonparametric measures of phenotypic stability. Part 1: Theory

## Summary

For an estimation of phenotypic stability of genotypes grown in different environments three stability parameters have been proposed which are based upon the ranks of the genotypes in each environment: In a two-way table with K rows (genotypes) and N columns (environments) the original data xij (=phenotypic value of the i th genotype in the j th environment (i=1,2,...,K;j=1,2,...,N)) are transformed into ranks for each of the N environments separately. We denote: rij=rank of genotype i in environment j. Then, a genotype i may be considered to be stable over environments if its ranks are similar over environments (maximum stability = equal ranks over environments). Each statistic for the similarity of the ranks in each row = genotype may be used as a stability parameter. Three different measures are proposed and discussed.

One of these nonparametric measures is defined as a ratio between ‘variability of the rij's’ and ‘mean of the rij's’ and, therefore, it represents a confounding and simultaneous consideration of stability and yield.

Differences among genotypes have an effect on the stability measures and may lead to differences in stability among genotypes when in fact there is no genotype-environment interaction. To avoid this ambiguity one may correct the xij values for the genotypic effects and the nonparametric measures may be computed using the ranks based on the corrected values xij *=xij−(\-xi.−\-x..)where \-xi.=marginal mean of genotype i and \-x\2=overall mean.

Finally, approximate tests of significance based on the normal distribution are discussed for the two nonparametric measures ‘mean absolute rank difference’ and ‘variance of the ranks’ for 1) testing the stability of a certain genotype and 2) comparing the stabilities of different genotypes.

This is a preview of subscription content, access via your institution.

## References

1. Becker, H.C., 1981. Biometrical and empirical relations between different concepts of phenotypic stability. In: Quantitative Genetics and Breeding Methods, Proc. 4th Meeting of the EUCARPIA section ‘Biometrics in Plant Breeding’, Poitiers, France 1981: 307–314.

2. Becker, H.C. & J.Léon, 1988. Stability Analysis in Plant Breeding. Plant Breeding 101: 1–23.

3. Böhm, H. & W.Schuster, 1985. Untersuchungen zur Leistungsstabilität des Kornertrages von Mais (Zea mays L.). Z. Acker- und Pflanzenbau 154: 222–231.

4. Cox, D.R., 1984. Interaction. Int. Statist. Rev. 52: 1–31.

5. Fatunla, T. & K.J.Frey, 1976. Repeatability of regression stability indexes for grain yield of oats (Avena sativa L.). Euphytica 25: 21–28.

6. Freeman, G.H., 1973. Statistical methods for the analysis of genotype-environment interactions. Heredity 31: 339–354.

7. Freeman, G.H., 1985. The analysis and interpretation of interactions. Journal of Applied Statistics 12: 3–10.

8. Hattemer, H.H., 1967. Bemerkungen zur Auswertung der Milieuinteraktionen in Sortenversuchsserien. DLG-Arbeitsgemeinschaft für Biometrie in der Pflanzenzüchtung. Rundschreiben 1/1967: 46–47.

9. Heine, H. & W.E.Weber, 1982. Die Aussagekraft statistischer Maßzahlen für die phänotypische Stabilität in amtlichen Sortenprüfungen bei Winterweizen und Körnermais. Z. Pflanzenzüchtg. 89: 89–99.

10. Hill, J., 1975. Genotype-environment interactions—a challenge for plant breeding. Journal of Agric. Science 85: 477–493.

11. Hühn, M. & J.Léon, 1985. Genotype-environment interactions and phenotypic stability of Brassica napus L. Z. Pflanzenzüchtg. 95: 135–146.

12. Huehn, M., 1990. Nonparametric measures of phenotypic stability. Part 2: Applications. Euphytica 47: 195–201.

13. Krumphuber, Ch., 1982. Versuche zur Bildung eines einfachen multiplen Index in der Hybridzüchtung. In: Bericht über die Arbeitstagung 1982 der Arbeitsgemeinschaft der Saatzuchtleiter in Gumpenstein, Verlag der Bundesanstalt für alpenländische Landwirtschaft Gumpenstein: 167–173.

14. Léon, J., 1986. Methods of simultaneous estimation of yield and yield stability. In: Biometrics in Plant Breeding. Proc. 6th Meeting EUCARPIA-section ‘Biometrics in Plant Breeding’, Birmingham, UK: 299–308.

15. Léon, J. & H.C.Becker, 1988. Repeatability of some statistical measures of phenotypic stability—Correlations between single year results and multi years results. Plant Breeding 100: 137–142.

16. Lin, C.S., M.R.Binns & L.P.Lefkovitch, 1986. Stability analysis: Where do we stand? Crop Sci. 26: 894–900.

17. Nachit, M.M. & H. Ketata, 1986. Breeding strategy for improving durum wheat in mediterranean rainfed areas. Proc. ‘Int. wheat conf.’, Rabat (Marocco).

18. Nassar, R. & M.Hühn, 1987. Studies on estimation of phenotypic stability: Tests of significance for nonparametric measures of phenotypic stability. Biometrics 43: 45–53.

19. Schuster, W. & K.H.Zschoche, 1981. Wie ertragsstabil sind unsere Rapssorten? DLG-Mitteilungen 12: 670–671.

20. Snoad, B. & A.E.Arthur, 1976. The use of regression techniques for predicting the response of peas to environment. Theor. Appl. Genet. 47: 9–19.

21. Thomson, N.J. & R.B.Cunningham, 1979. Genotype x environment interactions and evaluation of cotton cultivars. Austr. J. Agric. Res. 30: 105–112.

22. Utz, H.F., 1972. Die Zerlegung der Genotyp x Umwelt-Interaktionen. EDV in Medizin und Biologie 3: 52–59.

23. Westcott, B., 1986. Some methods of analysing genotype-environment interaction. Heredity 56: 243–253.

24. Wricke, G. & W.E.Weber, 1980. Erweiterte Analyse von Wechselwirkungen in Versuchsserien. In: W.Köpcke & K.Überla (Eds), Biometrie-heute und morgen. Springer Verlag 094 Berlin-Heidelberg-New York, pp. 87–95.

Authors

## Rights and permissions

Reprints and Permissions

Huehn, M. Nonparametric measures of phenotypic stability. Part 1: Theory. Euphytica 47, 189–194 (1990). https://doi.org/10.1007/BF00024241