er_anova() gives Eberhart and Russell's (1966) analysis of variance of a multi-environment trial, which
splits the environments and the genotype-by-environment interaction into a linear part (the regression of each
genotype on the environment index) and the deviations from regression, with a test of the deviations of every
genotype, together with the stability parameters of each genotype.
Usage
er_anova(.data, .y, .rep, .gen, .env)
# Default S3 method
er_anova(.data, .y, .rep, .gen, .env)Arguments
- .data
a data frame (or
data.table) with one row per plot.- .y
response: a column of
.datagiven by name (.y = Yieldor.y = "Yield") or a numeric vector of lengthnrow(.data).- .rep
replicate (block within environment): a column of
.dataor a vector, as for.y.- .gen
genotype: a column of
.dataor a vector, as for.y.- .env
environment: a column of
.dataor a vector, as for.y.
Value
An object of class c("stab_er_anova", "stab_result", "list") with the components of earlier
versions:
add_anovathe combined analysis of variance of
add_anova.er_anovaEberhart and Russell's analysis of variance (class
anova): linesTotal,Gen,Env + (Gen x Env),Env (linear),Gen x Env(linear),Pooled deviation, one line per genotype,Pooled error.
and a new one:
Parametersdata.table, one row per genotype:Gen,Mean,Slope(\(\hat{\beta}_i\)),SE,p.b1(test of \(\beta_i = 1\)),S2d(\(S^2_{d_i}\)),p.S2d(test of \(S^2_{d_i} = 0\)).
Methods: print(), summary(), plot() (mean against slope, which = "slope"; residuals,
which = "residuals" or "qq"), stab_interpret().
Details
With the table of means \(\mathbf{Y} = (\bar{y}_{ij})\) (\(G\) genotypes by \(E\) environments) and the environment index \(I_j = \bar{y}_{\cdot j} - \bar{y}\), every genotype is regressed on the index: $$\bar{y}_{ij} = \mu_i + \beta_i I_j + \delta_{ij}, \qquad \hat{\beta}_i = \frac{\sum_j \bar{y}_{ij}I_j}{\sum_j I_j^2}, \qquad S^2_{d_i} = \frac{\sum_j \hat{\delta}_{ij}^2}{E - 2} - \frac{s^2}{B},$$ where \(s^2\) is the pooled error mean square of the plots and \(B\) the number of replicates. On the scale of the means the analysis of variance is
| Total | \(GE - 1\) | \(\sum_{ij}(\bar{y}_{ij} - \bar{y})^2\) |
| Gen | \(G - 1\) | \(E\sum_i(\bar{y}_{i\cdot} - \bar{y})^2\) |
| Env + (Gen x Env) | \(G(E - 1)\) | \(\sum_{ij}(\bar{y}_{ij} - \bar{y}_{i\cdot})^2\) |
| Env (linear) | \(1\) | \(G\sum_j I_j^2\) |
| Gen x Env (linear) | \(G - 1\) | \(\sum_i(\sum_j\bar{y}_{ij}I_j)^2/\sum_j I_j^2 - G\sum_j I_j^2\) |
| Pooled deviation | \(G(E - 2)\) | \(\sum_{ij}\hat{\delta}_{ij}^2\) |
| Pooled error | \(f_e\) | \(f_e s^2/B\) |
Genotypes and the linear interaction are tested against the pooled deviation, the deviation of each genotype (\(E - 2\) degrees of freedom) against the pooled error. A stable genotype in the sense of Eberhart and Russell has a high mean, \(\beta_i = 1\) and \(S^2_{d_i} = 0\).
Changes from the earlier version. The lines Total and Gen used the replicate sum of squares; the deviation sums of squares were divided by the number of replicates (they are already on the scale of the means); the pooled error had \(EG(B - 1)\) instead of \(E(G - 1)(B - 1)\) degrees of freedom. All three are corrected, and the regression is computed in closed form (lme4 is no longer needed).
References
Eberhart, S. A. and Russell, W. A. (1966). Stability parameters for comparing varieties. Crop Science, 6(1), 36–40. doi:10.2135/cropsci1966.0011183X000600010011x
Singh, R. K. and Chaudhary, B. D. (2004). Biometrical Methods in Quantitative Genetic Analysis. Kalyani Publishers, New Delhi.
See also
stab_reg, add_anova, stab_interpret
Other stability analyses:
add_anova(),
ammi(),
ammi_biplot(),
ge_effects(),
ge_means(),
gge_biplot(),
indiv_anova(),
stab_asv(),
stab_dist(),
stab_eskridge(),
stab_fox(),
stab_huehn(),
stab_kang(),
stab_masv(),
stab_measures(),
stab_par(),
stab_reg(),
stab_reliability(),
stab_superiority()
Author
Muhammad Yaseen (myaseen208@gmail.com), Kent M. Eskridge (keskridge1@unl.edu)
Examples
data(ge_data)
fit <-
er_anova(
.data = ge_data
, .y = Yield
, .rep = Rep
, .gen = Gen
, .env = Env
)
fit
#> Eberhart and Russell's analysis: 60 genotypes, 11 environments, 2 replicates
#>
#> Df Sum Sq Mean Sq F value Pr(>F)
#> Total 659 268469701
#> Gen 59 35831215 607309 3.777 <2e-16 ***
#> Env + (Gen x Env) 600 232638486
#> Env (linear) 1 131970248
#> Gen x Env(linear) 59 13847665 234706 1.460 0.0178 *
#> Pooled deviation 540 86820573 160779
#> Pooled error 649 32179784 49584
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> (60 genotype deviation lines: x$er_anova; parameters: x$Parameters)
#>
#> Gen Mean Slope SE p.b1 S2d p.S2d
#> <char> <num> <num> <num> <num> <num> <num>
#> 1: DN126 4481 1.7108 0.2412 0.01629 78341 6.340e-03
#> 2: V14170 4464 1.2798 0.3003 0.37579 148757 5.436e-05
#> 3: PR115 4444 0.8154 0.2401 0.46177 77256 6.794e-03
#> 4: 12B.2511 4414 1.6375 0.2844 0.05170 128304 2.261e-04
#> 5: V14154 4365 0.6902 0.1384 0.05201 -7447 5.703e-01
#> 6: FAISALABAD08 4361 1.5737 0.3009 0.08897 149584 5.128e-05
#> 7: V14168 4342 0.9673 0.2755 0.90807 117372 4.787e-04
#> 8: NR491 4320 1.5681 0.2642 0.05997 103888 1.191e-03
#> 9: LOCAL CHECK 4280 1.2221 0.2561 0.40827 94621 2.204e-03
#> 10: 122559 4259 0.9834 0.3015 0.95730 150380 4.849e-05
head(fit$Parameters)
#> Gen Mean Slope SE p.b1 S2d p.S2d
#> <char> <num> <num> <num> <num> <num> <num>
#> 1: 013BT034 4247.500 0.9539767 0.2181489 0.83760871 55088.39 2.670478e-02
#> 2: 122557 4062.364 1.5159430 0.1702936 0.01425298 14201.74 2.406599e-01
#> 3: 122559 4258.591 0.9834026 0.3015181 0.95730445 150380.28 4.849012e-05
#> 4: 12B.2511 4414.409 1.6375025 0.2843871 0.05170192 128303.52 2.261371e-04
#> 5: 12FJ26 4057.227 0.7025845 0.2633342 0.28792236 102940.80 1.268556e-03
#> 6: 14B.1030 3995.409 0.9951608 0.2598803 0.98554993 98965.95 1.653205e-03
plot(x = fit, which = c("slope", "residuals", "qq")[1])
