\[
\newcommand{\mY}{\mathbf{Y}}
\newcommand{\mZ}{\mathbf{Z}}
\newcommand{\mX}{\mathbf{X}}
\newcommand{\mU}{\mathbf{U}}
\newcommand{\mV}{\mathbf{V}}
\newcommand{\mD}{\mathbf{D}}
\newcommand{\vg}{\boldsymbol{\gamma}}
\newcommand{\vd}{\boldsymbol{\delta}}
\newcommand{\ybar}{\bar{y}}
\newcommand{\vgs}{\mathbf{g}}
\newcommand{\ves}{\mathbf{e}}
\newcommand{\va}{\mathbf{a}}
\newcommand{\vx}{\mathbf{x}}
\]
Abstract
The AMMI model, its biplots and the stability values built on its axes.
The model
am <-
ammi(
.data = ge_data
, .y = Yield
, .rep = Rep
, .gen = Gen
, .env = Env
)
am$pc_anova
PC SingVal Percent AccumPerc Df SS MeanSq Fvalue
<char> <num> <num> <num> <num> <num> <num> <num>
1: PC1 4496.142 20.081104 20.08110 68 40430588 594567.47 5.9956010
2: PC2 4174.531 17.311027 37.39213 66 34853411 528081.98 5.3251633
3: PC3 4046.566 16.266000 53.65813 64 32749392 511709.24 5.1600611
4: PC4 3489.555 12.096162 65.75429 62 24353985 392806.22 3.9610464
5: PC5 3380.195 11.349876 77.10417 60 22851440 380857.33 3.8405542
6: PC6 3113.224 9.627826 86.73199 58 19384326 334212.52 3.3701893
7: PC7 2297.953 5.245534 91.97753 56 10561173 188592.38 1.9017601
8: PC8 2030.472 4.095450 96.07298 54 8245635 152696.94 1.5397915
9: PC9 1521.907 2.300825 98.37380 52 4632399 89084.60 0.8983265
10: PC10 1279.478 1.626196 100.00000 50 3274127 65482.53 0.6603240
ProbF
<num>
1: 2.428622e-36
2: 1.511108e-30
3: 1.150112e-28
4: 3.679112e-19
5: 7.571781e-18
6: 4.018722e-14
7: 1.448407e-04
8: 9.520167e-03
9: 6.770319e-01
10: 9.659157e-01
plot(x = am, which = "scree")
AMMI analysis
AMMI8: 8 interaction axes are significant; the first two axes explain 20.1%
and 17.3% of the interaction.
Findings:
- The interaction is significant (p < 0.001).
- Genotypes with a mean above average and the smallest first-axis scores
(stable): NIBGE GANDUM3, PR119, V14154, V14170 and TWS12464.
- Environments with the strongest interaction (largest first-axis scores):
Gujranwala, FSD and KSK; the weakest: FSDR, Sahiwal and Khanewal.
- Genotypes and environments with first-axis scores of the same sign interact
positively (the genotype does better there than its mean predicts).
Recommendations:
- Use AMMI8 for predictions and the stability indices (stab_asv(),
stab_masv() with .m = 8).
- Look at the AMMI1 biplot (plot(x, which = "ammi1")) to see mean and
interaction together.
Cautions:
- The first two axes explain less than half of the interaction; biplots show
it only partly.
Biplots
ab <-
ammi_biplot(
.data = ge_data
, .y = Yield
, .rep = Rep
, .gen = Gen
, .env = Env
)
plot(x = ab, which = "ammi1")
plot(x = ab, which = "ammi2")
Stability values
asv <-
stab_asv(
.data = ge_data
, .y = Yield
, .rep = Rep
, .gen = Gen
, .env = Env
)
head(x = asv$ASD, n = 6)
Gen Mean ASD rMean rASD YSI PC1 PC2
<char> <num> <num> <int> <int> <int> <num> <num>
1: PR115 4444.273 1.1235033 3 2 5 -0.9031507 0.4057644
2: V14154 4364.727 2.6556975 5 5 10 -0.1647526 -2.6488118
3: MSH3 4247.045 1.9014683 12 3 15 1.2330175 -1.2529001
4: V14168 4342.045 5.6958779 7 12 19 4.7800620 -1.3024826
5: PR119 4156.727 0.6095297 24 1 25 0.1556221 -0.5821833
6: LOCAL CHECK 4280.136 7.7084442 9 19 28 1.6384767 -7.4704480
masv <-
stab_masv(
.data = ge_data
, .y = Yield
, .rep = Rep
, .gen = Gen
, .env = Env
, .m = 4
)
head(x = masv$MASD, n = 6)
Gen Mean MASD rMean rMASD MYSI
<char> <num> <num> <int> <int> <int>
1: V14154 4364.727 3.893937 5 1 6
2: PR115 4444.273 12.339474 3 11 14
3: 013BT034 4247.500 7.787213 11 3 14
4: LOCAL CHECK 4280.136 12.868956 9 14 23
5: UOS1 4239.091 11.462978 16 9 25
6: V14225 4194.182 8.494185 21 5 26
dst <-
stab_dist(
.data = ge_data
, .y = Yield
, .rep = Rep
, .gen = Gen
, .env = Env
, .m = 4
)
head(x = dst$stabDist, n = 6)
Gen Mean Dist rMean rDist YSIDist
<char> <num> <num> <int> <int> <int>
1: V14154 4364.727 3.696609 5 1 6
2: PR115 4444.273 11.318617 3 12 15
3: 013BT034 4247.500 7.394034 11 4 15
4: LOCAL CHECK 4280.136 12.514985 9 16 25
5: UOS1 4239.091 11.101746 16 10 26
6: V14225 4194.182 7.893190 21 5 26
plot(x = asv, which = "ranks")
stab_masv() is the MASV of Zali et al. (2012). The earlier versions computed it on two axes; the function now takes any number of axes .m, from 2 to the number of non-zero interaction axes. The ratio of the sums of squares of each axis to the next weighs the first axes heavily. stab_dist() is the AMMI stability distance developed by the package author: the Euclidean distance from the origin over the first .m axes of the AMMI scores, in which every axis enters only through its scores. Use for .m the number of significant axes found by ammi().
Modified AMMI stability value
Best on the yield stability index (yield and stability ranks together):
V14154, PR115, 013BT034, LOCAL CHECK and UOS1.
Findings:
- Smallest modified AMMI stability value (most stable in the AMMI sense):
V14154, NIBGE GANDUM4, 013BT034, PR119 and V14225.
- 17 genotype(s) combine a mean above average with a modified AMMI stability
value below the median.
- The rank correlation between mean and modified AMMI stability value is
0.03.
Recommendations:
- Consider V14154, PR115 and 013BT034; check their adaptation in the AMMI or
GGE biplot.
Cautions:
- The index rests on the AMMI axes it uses; when more axes are significant
(see ammi()), stab_masv() with more axes is more complete.
Summary
Many significant axes mean the interaction has several independent patterns; the first two show only part of it, and indices over more axes (stab_masv()) describe the stability more fully.
References
Gauch, H. G. (1992). Statistical Analysis of Regional Yield Trials: AMMI Analysis of Factorial Designs. Elsevier, Amsterdam.
Gollob, H. F. (1968). A statistical model which combines features of factor analytic and analysis of variance techniques. Psychometrika, 33(1), 73–115.
Purchase, J. L., Hatting, H. and van Deventer, C. S. (2000). Genotype x environment interaction of winter wheat in South Africa: II. Stability analysis of yield performance. South African Journal of Plant and Soil, 17(3), 101–107.
Zali, H., Farshadfar, E., Sabaghpour, S. H. and Karimizadeh, R. (2012). Evaluation of genotype x environment interaction in chickpea using measures of stability from AMMI model. Annals of Biological Research, 3(7), 3126–3136.
