Huehn's nonparametric stability statistics

stab_huehn() computes Huehn's (1979) nonparametric stability statistics $S^{(1)}_i$ (mean absolute rank difference over pairs of environments) and $S^{(2)}_i$ (variance of the ranks) from the ranks of the genotypes in each environment after removing the genotype main effect, with the tests of Nassar and Huehn (1987). A stable genotype keeps its rank: its statistics are small.

Usage

stab_huehn(.data, .y, .rep, .gen, .env)

# Default S3 method
stab_huehn(.data, .y, .rep, .gen, .env)

Arguments

.data: a data frame (or data.table) with one row per plot.
.y: response: a column of .data given by name (.y = Yield or .y = "Yield") or a numeric vector of length nrow(.data).
.rep: replicate: not used (kept for compatibility); may be omitted.
.gen: genotype: a column of .data or a vector, as for .y.
.env: environment: a column of .data or a vector, as for .y.

Value

An object of class c("stab_huehn", "stab_result", "list"):

Huehn: data.table sorted by S1: Gen, Mean, MeanRank, S1, Z1, p.Z1, S2, Z2, p.Z2, rS1, rS2.
Tests: data frame: the sums of $Z$ for $S^{(1)}$ and $S^{(2)}$, their degrees of freedom and p-values.

Methods: print(), summary(), plot(), stab_interpret().

Details

The ranks $r_{ij}$ (1 = smallest) are those of the corrected values $\bar{y}_{ij} - \bar{y}_{i\cdot} + \bar{y}$ within environment $j$, so that only the interaction moves a genotype's rank; with $\bar{r}_i$ the mean rank of genotype $i$, $$S^{(1)}_i = \frac{2\sum_{j<j'}|r_{ij} - r_{ij'}|}{E(E - 1)}, \qquad S^{(2)}_i = \frac{\sum_j(r_{ij} - \bar{r}_i)^2}{E - 1}.$$ Under the hypothesis of equal stability (Nassar and Huehn 1987) $$E(S^{(1)}) = \frac{G^2 - 1}{3G}, \quad \mathrm{Var}(S^{(1)}) = \frac{(G^2 - 1)[(G^2 - 4)(E - 1) + 30]}{45G^2E}, \quad E(S^{(2)}) = \frac{G^2 - 1}{12}, \quad \mathrm{Var}(S^{(2)}) = \frac{(G^2 - 1)[2(G^2 - 4)(E - 1) + 5(G^2 - 1)]}{360E(E - 1)},$$ and $Z_i = (S_i - E(S))^2/\mathrm{Var}(S)$ is approximately chi-square with one degree of freedom; the sum of the $Z_i$ over genotypes, chi-square with $G$ degrees of freedom, tests whether the genotypes differ in stability.

References

Huehn, M. (1979). Beitrage zur Erfassung der phanotypischen Stabilitat I. Vorschlag einiger auf Ranginformationen beruhenden Stabilitatsparameter. EDV in Medizin und Biologie, 10, 112–117.

Nassar, R. and Huehn, M. (1987). Studies on estimation of phenotypic stability: tests of significance for nonparametric measures of phenotypic stability. Biometrics, 43(1), 45–53.

Author

Muhammad Yaseen (myaseen208@gmail.com), Kent M. Eskridge (keskridge1@unl.edu)

Examples

data(ge_data)
fit <-
  stab_huehn(
      .data = ge_data
    , .y    = Yield
    , .rep  = Rep
    , .gen  = Gen
    , .env  = Env
    )
head(fit$Huehn)
#>              Gen     Mean MeanRank       S1         Z1      p.Z1       S2
#>           <char>    <num>    <num>    <num>      <num>     <num>    <num>
#> 1: NIBGE GANDUM4 3586.545 30.45455 12.50909 0.77084781 0.3799549 112.8727
#> 2:         WBG14 3765.455 31.09091 16.32727 0.18501481 0.6670985 201.0909
#> 3:         PR119 4156.727 28.90909 16.83636 0.13721149 0.7110691 220.4909
#> 4:        V14154 4364.727 29.63636 16.94545 0.12789571 0.7206239 214.4545
#> 5:      TWS12245 4100.455 28.54545 16.98182 0.12486322 0.7238186 216.0727
#> 6:         DN117 4045.409 31.27273 17.52727 0.08374189 0.7722891 225.2182
#>           Z2       p.Z2   rS1   rS2
#>        <num>      <num> <int> <int>
#> 1: 4.2812291 0.03853547     1     1
#> 2: 1.1951475 0.27429370     2     2
#> 3: 0.7719762 0.37960639     3     5
#> 4: 0.8937757 0.34445618     4     3
#> 5: 0.8602497 0.35366882     5     4
#> 6: 0.6828177 0.40861814     6     7
fit$Tests
#>   Statistic     Chisq Df   p.value  Expected   Variance
#> 1        S1  6.525828 60 1.0000000  19.99444   72.68687
#> 2        S2 46.577429 60 0.8977291 299.91667 8171.82033