Article

# Comparison of rank-based stability statistics for grain yield in rainfed durum wheat

Authors:
• Dryland Agricultural Research Institute
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## Abstract

Several rank-based stability statistics are in use, but little information is available on their repeatability and accuracy. The main objective was to examine the repeatability, similarity and accuracy of the estimates of mean yields and 10 rank-based stability statistics calculated from 25 rainfed durum wheat genotypes grown in 21 diversified rainfed environments. Spearman's rank-correlation coefficients and Kendall's coefficient of concordance were used to estimate the repeatability and similarity of stability statistics, while the coefficient of variation from resampling correlations obtained by bootstrap method was used to quantify their accuracy. Correlations between the estimates from each of the rank-based stability statistics obtained from single-year results were very low and non-significant, suggesting that single-year results could not serve as a basis to quantify phenotypic stability. The accuracy for the repeatability of stability statistics was found to be poor, whereas the accuracy for similarities of stability statistics was high.

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... , Coefficient of concordance, Biplot analysis Significant genotype-by-environment (G×E) interaction had been reported in large number of multi environmental studies (Pour et al. 2019). Cross over interaction masks the correlation between genotypic and phenotypic values of genotypes and hinders the selection of the promising genotypes (Mohammadi et al. 2016). Large number of statistical approaches, based on univariate and multivariate models, have been observed in literature to estimate 324 Print ISSN : 1974ISSN : -1712 Online ISSN : 2230-732X stable performance of genotypes (Vaezi et al. 2018). ...
... The loadings of measures based on first two significant principal components were reflected in table 12 (Mohammadi et al. 2016). Both significant PAC's accounting for 85.9% of the variance of the original variables (Fig. 1) ...
... Multi-environment trials (MET) have been established as inevitable to recommend promising genotypes for different locations of the country (Vaezi et al., 2018). Genotype x Environment (GxE) interaction effects decrease the association between genotypic and phenotypic values and also masks the selection of the desirable genotypes (Mohammadi et al., 2016). Stability analysis methods are categorized into two parametric and non-parametric groups (Huhn & Leon, 1995;Farshadfar et al., 2014;Golkar et al., 2020). ...
... Biplot analysis of nonparametric measures had been carried out to explore any type of association among studied measures. Loadings of the first two principal components axes (PCA) of ranks of nonparametric measures were shown in figure 1 (Mohammadi et al., 2016). Two separate groups of Yield with GAI and MR with Med were displayed in the graphical analysis. ...
... Both genotypes could be regarded as good candidates for other environments of the country. Nonparametric tests for interactions provided more specific information about the nature of GE interactions (Rasoli et al., 2015;Mohammadi et al., 2016). The main purpose of barley improvement programs is to sustain higher yield with stability. ...
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Chapter
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Die Entwicklung verteilungsfreier Rangtests zur Überwindung der Normalverteilungsannahme in den so häufig gebrauchten varianzanalytischen Modellen hat sich in zweierlei Weise vollzogen. Einmal werden sie als bedingte Tests, zum anderen als asymptotische Verfahren vorgestellt. Dabei zeigt es sich, daß die Methodik des bedingten Rangtests zwar in einer Vielzahl von speziellen Versuchsplänen anwendbar ist, wie das Buch von Puri und Sen (1971) angibt, daß aber diese Verfahrensweise nicht auf das Testen linearer Hypothesen in allgemeinen linearen Modellen zu erweitern ist. Asymptotische Untersuchungen haben hier hingegen in letzter Zeit zu befriedigenden Ergebnissen geführt. Dabei wurde von folgender koordinatengebundener Formulierung ausgegangen: In einem linearen Modell $$y = x_1b_1 + x_2b_2 + e$$ (1.1) betrachte man das Testproblem $$H:b_2 = 0 \; \text{gegen} \; K:b_2 \neq 0$$ (1.2)
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A suggestion is given of how to prove main and interaction effects in two-way layouts independent of each other even if the data are just ordinally scaled. Starting from HILDEBRAND'S (1980a, b) non-parametric approach which presupposes interval-scaled data, transformations of ranks are settled before analysis per analogy to the H-test takes place. That is, the same formula of an asymptotically X2-distributed test-statistic results but mean ranks are used instead of mean scores in order to partialize, for instance, main effects while testing interaction effect. Finally an allusion is given of how to handle ties as well as unequal sample sizes.
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Cultivar evaluation and mega-environment identification are among the most important objectives of multi-environment trials (MET). Although the measured yield is a combined result of effects of genotype (G), environment (E), and genotype x environment interaction (GE), only G and GE are relevant to cultivar evaluation and mega-environment identification. This paper presents a GGE (i.e., G + GE) biplot, which is constructed by the first two symmetrically scaled principal components (PC1 and PC2) derived from singular value decomposition of environment-centered MET data. The GGE biplot graphically displays G plus GE of a MET in a way that facilitates visual cultivar evaluation and mega-environment identification. When applied to yield data of the 1989 through 1998 Ontario winter wheat (Triticum aestivum L.) performance trials, the GGE biplots clearly identified yearly winning genotypes and their winning niches. Collective analysis of the yearly biplots suggests two winter wheat mega-environments in Ontario: a minor mega-environment (eastern Ontario) and a major one (southern and western Ontario), the latter being traditionally divided into three subareas. There were frequent crossover GE interactions within the major mega-environment but the location groupings were variable across years. It therefore could not be further divided into meaningful subareas. It was revealed that in most years PC1 represents a proportional cultivar response across locations, which leads to noncrossover GE interactions, while PC2 represents a disproportional cultivar response across locations, which is responsible for any crossover GE interactions. Consequently, genotypes with large PC1 scores tend to give higher average yield, and locations with large PC1 scores and near-zero PC2 scores facilitates identification of such genotypes.
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There are many statistics to assess cultivar yield stability: some deal with genotype x environment interaction (GEI) patterns (b1, and interaction principal component scores), some with GEI noise (s(di)/2), some with total GEI sum of squares (δ(i)/2), and others with rank changes [S(1)]. Our objective was to assess the repeatability in soybean [Glycine max (L.) Merr.] of different yield stability statistics. We evaluated nine statistics derived from additive main effects and multiplicative interaction (AMMI) analyses, regression coefficients from analyses using either yield-based or plant height-based environmental indices, as well as s(di)/2, δ(i)/2, S(1). Yield data were from Arkansas trials of maturity group V cultivars from 1987 to 1994. Stability analyses were warranted because GEl was significant, partitioning of GEI sum of squares by AMMI or regression techniques was generally significant, and all stability statistics found significant differences among the cultivars. The AMMI-derived stability statistics produced information on cultivar stability that was similar to that produced by s(di)/2 and δ(i)/2. The repeatability of cultivar rankings for s(di)/2, δ(i)/2, S(1), and most AMMI statistics, estimated in different single- or two-year periods were low, indicating that these statistics would not be useful to breeders or growers selecting for stability. The repeatability of two regression coefficient statistics and one AMMI-derived statistic were moderate, particularly when estimated over environments from 2 yr. Each of the three statistics relates to a different concept of stability and may be useful to growers and breeders attempting to select cultivars with predictable yield across environments.
Article
Cultivar performance trials are usually analyzed by analysis of variance techniques. In the field of practical applications, the required simplifying assumptions may often not be fulfilled. Nonparametric metbods based on ranks avoid this difficulty and provide a viable alternative to parametric procedures. The objective of this study was to compare four nonparametric procedures (Bredenkamp, Hildebrand, Kubinger, and van der Laan-de Kroon) and the parametric analysis of variance with each other for extensive data sets from German official registration trials (1985–1989) with winter oilseed rape (Brassica napus L.), fababean (Vicia faba L.), oat (Avena sativa L.), fodder beet (Beta vulgaris L.), and sugar beet (B. vulgaris). The van der Laan-de Kroon method is based on the crossover concept of interaction (different rank orders), while the other methods are based on the concept of deviations from additivity of main effects. The methods were compared using the numerical values of the test statistics. For the sources cultivars, locations, and interactions one obtains these relationships: ANOVA ≥ Hildebrand ≅ Kubinger > van der Laan-de Kroon > Bredenkamp. If the assumptions for parametric methods cannot be accepted, the van der Laan-de Kroon method is recommended for the crossover interaction concept, while the Hildebrand or Kubinger method (both of which are approximately equivalent) should be applied for the usual interaction concept. The Bredenkamp method cannot be recommended. Please view the pdf by using the Full Text (PDF) link under 'View' to the left. Copyright © . .
Article
To study the importance of the effects of genotype - Environment interactions on the yield of pigeonpea (Cajanus cajan L. Millsp.), 10 early-maturing genotypes were grown in a randomized complete block design with three replications in a total of seven environments spread over five regions of Kenya between 1987 and 1988. Results indicated the presence of a substantial genotype - Environment interaction effect on grain yield. The observed significant genotype - Environment interaction effect is discussed in relation to its importance in pigeonpea grain yield evaluation studies. It is noted that the best genotype in one environment is not always so in other environments. Results from regression analysis indicated that this method of analysis is appropriate for describing the response of pigeonpea genotypes grown in a number of locations. Analysis of variance showed significant additive and multiplicative genotype - Environment interaction effects. Only the first interaction principal component axis (IPCA) was found to be important in describing the multiplicative interaction effects. The additive main effects and multiplicative effects (AMMI) model allowed the partitioning of interaction variance into agronomically important sources (genotype groups), and the specific genotype × environment patterns that are the basis of these sources of variance were examined.
Article
High and stable yield is a very desirable attribute of soybean (Glycine max (L.) Merr.) cultivars. Stable yield of a cultivar means that its rank relative to other cultivars remains unchanged in a given set of environments. To characterize 12 soybean cultivars chosen from performance trials, data were obtained from 10 environments (five locations in 2 years). Six stability parameters from four statistical models were derived for each cultivar. Regression coefficients were significantly and positively correlated only with coefficients of variation; they are useful in characterizing whether cultivars responded well in favourable or poor environments. Nassar and Huhn's nonparametric measures, Si(1) and Si(2), were significantly and positively correlated with Eberhart and Russell's sdi2 and Wricke's ecovalence (Wi). The stability measures are useful in characterizing cultivars by showing their relative performance in various environments. Results revealed that high-yielding cultivars also can be stable cultivars. Correlations between stability parameters obtained from individual years over the same set of locations and cultivars were very low and nonsignificant, suggesting that single-year data are not reliable as basis for selection. To provide an additional guide for selection, Kang's rank-sum approach was applied, in which both yield (in rank) and measured nonparametric stability (in rank) were considered. In general, selection for yield only would sacrifice stability to some degree, and selection for stability only would sacrifice a certain amount of yield. The rank-sum approach reconciles the two and appeared to provide a useful means to characterize soybean cultivars.
Article
The stability and genotypic mean of four traits, grain yield, grain protein content, alveograph W and bread volume, were evaluated in three multi-location trials, each covering two years. The stability of each genotype was evaluated by environmental variance (s2 E), interaction variance (s2 W) and variance of the ranks of the phenotypic values corrected for the genotypic effect (s2 R). The bootstrap method was used to study correlations between the genotypic mean and the three stability statistics and to calculate their accuracy. The repeatability of the stability statistics was measured by correlations between the values obtained in each of the two years. In addition, theoretical smaller trials were generated by random sampling and the stability values calculated were correlated with those of the original trial. Environmental variance appears to be usable both for yield and for quality traits, but there is a risk of counter-selecting a high genotypic mean of W. Whatever the trait and statistic envisaged, stability is poorly repeatable and its evaluation requires several years and a large number of locations per year to minimise sampling and environmental effects.
Article
An experiment was undertaken to determine the stability of seed yield in 21 common bean genotypes representing three growth habits. Seven genotypes in each growth habit (determinate bush, indeterminate bush and indeterminate prostrate) were evaluated in replicated trials at three locations for three years under rain fed conditions in Ethiopia. A combined analysis of variance, stability statistics and rank correlations among stability statistics and yield-stability statistic were determined. The genotypes differed significantly for seed yield and genotype × environment (year by location) interaction (GE). The different stability statistics namely Type1, Type 2 and Type 3 measured the different aspects of stability. This was substantiated by rank correlation coefficient. There were strong rank correlations among Si 2d, Wi 2,σi 2 and Si 2, where as there was weak correlation between biand Ri 2, Si 2d, Wi 2, σi 2 and Si 2. R2 was significantly and negatively correlated with Wi 2, σi 2 and Si 2. σi 2 is significantly correlated with Wi 2.Yield is significantly correlated with bi and Ri 2.None of the statistics per se was useful for selecting high yielding and stable genotypes except the YS(yield-stability statistic). Most of the high yielding genotypes were relatively stable. Of the 21 genotypes, only 11genotypes were selected for their high yielding and stable performance. Genotypes with growth habit III and I (in determinate prostrate and determinate bush) were generally more stable than in determinate bush.
Article
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.
Article
Twenty parametric and non-parametric measures derived from grain yield of 15 advanced durum genotypes evaluated across 12 variable environments during the 2004–2006 growing seasons were used to assess performance stability and adaptability of the genotypes and to study interrelationship among these measures. The combined ANOVA and the non-parametric tests of Genotype×environment interaction indicated the presence of significant crossover and non-crossover interactions, and of significant differences among genotypes. Principal component analysis based on the rank correlation matrix indicated that most non-parametric measures were significantly inter-correlated with parametric measures and therefore can be used as alternatives. The results also revealed that stability measures can be classified into three groups based on static and dynamic concepts of stability. The group related to the dynamic concept and strongly correlated with mean grain yield of stability included the parameters of TOP (proportion of environments in which a genotype ranked in the top third), superiority index (P i) and geometric adaptability index. The second group reflecting the concept of static stability included, Wricke’s ecovalence, the variance in regression deviation (S 2 di), AMMI stability value, the Huehn’s parameters [S i(1), S i(2)], Tennarasua’s parameter [NPi(1)], Kang’s parameter (RS) and yield reliability index (I i) which were not correlated with mean grain yield. The third group influenced simultaneously by grain yield and stability included the measures S i(3), S i(6), NPi(2), NPi(3), environmental variance (S 2 xi), coefficient of variability and coefficient of regression (b i). Based on the concept of dynamic stability, genotypes G6, G4, and G3 were found to be the most adapted to favorable environments, whereas genotypes G8, G9, and G12 were more stable and are related to the concept of static stability.
Article
Average commercial maize yield in the US has increased from about 1 Mg/ha in the 1930s to about 7 Mg/ha in the 1990s. Although the increase has been the result of both genetic and agronomic-management improvements, we contend that most of this improvement is the result of the genotype×management interaction. The genetic improvement in maize yield is associated neither with yield potential per se, nor with heterosis per se, but it is associated with increased stress tolerance, which is consistent with the improvement in the genotype×management interaction. The potential for future yield improvement through increased stress tolerance of maize in the US is large, as yield potential is approximately three times greater than current commercial maize yields. The mechanism by which maize breeders have improved stress tolerance is not known, but we speculate that increased stress tolerance may have resulted from the selection for yield stability. Stability analyses were performed on a number of high-yielding maize hybrids, including three hybrids that have been involved in some of the highest maize yields recorded in producers’ fields, to examine the relationship between yield and yield stability. Results of the stability analyses showed that high-yielding maize hybrids can differ in yield stability, but results do not support the contention that yield stability and high grain yield are mutually exclusive.
Article
Twenty-two different methods (parametric, nonparametric and multivariate) used for analysing genotype×environment (G×E) interaction were compared by applying them to two sets of experimental data (15 faba bean cultivars×12 environments and 11 pea cultivars×16 environments). A principal components analysis was performed on the rank correlation matrix arising from the application of each method. The 22 methods can be categorized, in both sets of experimental data, in three groups: (1) those which are mostly associated with yield level and show little or no correlation with stability parameters; (2) those in which both yield level and stability of performance are considered simultaneously to reduce the effect of G×E interaction; and (3) those methods which only measure stability. This analysis also separated those methods based on an agronomic concept of stability from those which are based on a biological one, as well as distinguishing between dynamic' and static' stability-based methods.