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 x_ij (=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: r_ij=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 r_ij's’ and ‘mean of the r_ij'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 x_ij values for the genotypic effects and the nonparametric measures may be computed using the ranks based on the corrected values x_ij ^*=x_ij−(\-x_i.−\-x..)where \-x_i.=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.