AMGE.AMMI computes the Sum Across Environments of Genotype-Environment
Interaction (GEI) Modelled by AMMI (AMGE)
(Sneller et al. 1997)
considering all
significant interaction principal components (IPCs) in the AMMI model. Using
AMGE, the Simultaneous Selection Index for Yield and Stability (SSI) is also
calculated according to the argument ssi.method.
Usage
AMGE.AMMI(model, n, alpha = 0.05, ssi.method = c("farshadfar", "rao"), a = 1)Arguments
- model
The AMMI model (An object of class
AMMIgenerated byAMMI).- n
The number of principal components to be considered for computation. The default value is the number of significant IPCs.
- alpha
Type I error probability (Significance level) to be considered to identify the number of significant IPCs.
- ssi.method
The method for the computation of simultaneous selection index. Either
"farshadfar"or"rao"(SeeSSI).- a
The ratio of the weights given to the stability components for computation of SSI when
method = "rao"(SeeSSI).
Value
A data frame with the following columns:
- AMGE
The AMGE values.
- SSI
The computed values of simultaneous selection index for yield and stability.
- rAMGE
The ranks of AMGE values.
- rY
The ranks of the mean yield of genotypes.
- means
The mean yield of the genotypes.
The names of the genotypes are indicated as the row names of the data frame.
Details
The Sum Across Environments of GEI Modelled by AMMI (\(AMGE\)) (Sneller et al. 1997) is computed as follows:
\[AMGE = \sum_{j=1}^{E} \sum_{n=1}^{N'} \lambda_{n} \gamma_{in} \delta_{jn}\]
Where, \(N'\) is the number of significant IPCs (number of IPC that were retained in the AMMI model via F tests); \(\lambda_{n}\) is the singular value for \(n\)th IPC and correspondingly \(\lambda_{n}^{2}\) is its eigen value; \(\gamma_{in}\) is the eigenvector value for \(i\)th genotype; and \(\delta{jn}\) is the eigenvector value for the \(j\)th environment.
References
Sneller CH, Kilgore-Norquest L, Dombek D (1997). “Repeatability of yield stability statistics in soybean.” Crop Science, 37(2), 383–390.
Examples
library(agricolae)
data(plrv)
# AMMI model
model <- with(plrv, AMMI(Locality, Genotype, Rep, Yield, console = FALSE))
# ANOVA
model$ANOVA
#> Analysis of Variance Table
#>
#> Response: Y
#> Df Sum Sq Mean Sq F value Pr(>F)
#> ENV 5 122284 24456.9 257.0382 9.08e-12 ***
#> REP(ENV) 12 1142 95.1 2.5694 0.002889 **
#> GEN 27 17533 649.4 17.5359 < 2.2e-16 ***
#> ENV:GEN 135 23762 176.0 4.7531 < 2.2e-16 ***
#> Residuals 324 11998 37.0
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
# IPC F test
model$analysis
#> percent acum Df Sum.Sq Mean.Sq F.value Pr.F
#> PC1 56.3 56.3 31 13368.5954 431.24501 11.65 0.0000
#> PC2 27.1 83.3 29 6427.5799 221.64069 5.99 0.0000
#> PC3 9.4 92.7 27 2241.9398 83.03481 2.24 0.0005
#> PC4 4.3 97.1 25 1027.5785 41.10314 1.11 0.3286
#> PC5 2.9 100.0 23 696.1012 30.26527 0.82 0.7059
# Mean yield and IPC scores
model$biplot
#> type Yield PC1 PC2 PC3 PC4
#> 102.18 GEN 26.31947 -1.50828851 1.258765244 -0.19220309 0.48738861
#> 104.22 GEN 31.28887 0.32517729 -1.297024517 -0.63695749 -0.44159957
#> 121.31 GEN 30.10174 0.95604605 1.143461054 -1.28777348 2.22246913
#> 141.28 GEN 39.75624 2.11153737 0.817810467 1.45527701 0.25257620
#> 157.26 GEN 36.95181 1.05139017 2.461179974 -1.97208942 -1.96538800
#> 163.9 GEN 21.41747 -2.12407441 -0.284381234 -0.21791137 -0.50743629
#> 221.19 GEN 22.98480 -0.84981828 0.347983673 -0.82400783 -0.11451944
#> 233.11 GEN 28.66655 0.07554203 -1.046497338 1.04040485 0.22868362
#> 235.6 GEN 38.63477 1.20102029 -2.816581184 0.80975361 1.02013062
#> 241.2 GEN 26.34039 -0.79948495 0.220768053 -0.98538801 0.30004421
#> 255.7 GEN 30.58975 -1.49543817 -1.186549449 0.92552519 -0.32009239
#> 314.12 GEN 28.17335 1.39335380 -0.332786322 -0.73226877 0.05987348
#> 317.6 GEN 35.32583 1.05170769 0.002555823 -0.81561907 0.58180433
#> 319.20 GEN 38.75767 3.08338144 1.995946966 0.87971668 -1.11908943
#> 320.16 GEN 26.34808 -1.55737097 0.732314249 -0.41432567 1.32097009
#> 342.15 GEN 26.01336 -1.35880873 -0.741980068 0.87480105 -1.12013125
#> 346.2 GEN 23.84175 -2.48453928 -0.397045286 1.07091711 -0.90974484
#> 351.26 GEN 36.11581 1.22670345 1.537183139 1.79835728 -0.03516368
#> 364.21 GEN 34.05974 0.27328985 -0.447941156 0.03139543 0.77920500
#> 402.7 GEN 27.47748 -0.12907269 -0.080086669 0.01934016 -0.36085862
#> 405.2 GEN 28.98663 -1.90936369 0.309047963 0.57682642 0.51163370
#> 406.12 GEN 32.68323 0.90781100 -1.733433781 -0.24223050 -0.38596144
#> 427.7 GEN 36.19020 0.42791957 -0.723190970 -0.85381724 -0.53089914
#> 450.3 GEN 36.19602 1.38026196 1.279525147 0.16025163 0.61270137
#> 506.2 GEN 33.26623 -0.33054261 -0.302588536 -1.58471588 -0.04659416
#> Canchan GEN 27.00126 1.47802905 0.380553178 1.67423900 0.07718375
#> Desiree GEN 16.15569 -3.64968796 1.720025405 0.43761089 0.04648011
#> Unica GEN 39.10400 1.25331924 -2.817033826 -0.99510845 -0.64366599
#> Ayac ENV 23.70254 -2.29611851 0.966037760 1.95959116 2.75548057
#> Hyo-02 ENV 45.73082 3.85283195 -5.093371615 1.16967118 -0.08985538
#> LM-02 ENV 34.64462 -1.14575146 -0.881093222 -4.56547274 0.55159099
#> LM-03 ENV 53.83493 5.34625518 4.265275487 -0.14143931 -0.11714533
#> SR-02 ENV 14.95128 -2.58678337 0.660309540 0.89096920 -3.25055305
#> SR-03 ENV 11.15328 -3.17043379 0.082842050 0.68668051 0.15048221
#> PC5
#> 102.18 -0.04364115
#> 104.22 0.95312506
#> 121.31 -1.30661916
#> 141.28 -0.25996142
#> 157.26 -0.59719268
#> 163.9 0.18563390
#> 221.19 -0.57504816
#> 233.11 0.65754266
#> 235.6 -0.40273415
#> 241.2 0.07555258
#> 255.7 -0.46344763
#> 314.12 0.54406154
#> 317.6 0.39627052
#> 319.20 0.29657050
#> 320.16 2.29506737
#> 342.15 -0.10776433
#> 346.2 -0.12738693
#> 351.26 0.30191335
#> 364.21 -0.95811256
#> 402.7 -0.28473777
#> 405.2 -0.34397623
#> 406.12 -0.49796296
#> 427.7 1.00677993
#> 450.3 -0.34325251
#> 506.2 0.87807441
#> Canchan 0.49381313
#> Desiree -0.86767477
#> Unica -0.90489253
#> Ayac 1.67177210
#> Hyo-02 0.01540152
#> LM-02 0.52350416
#> LM-03 -0.40285728
#> SR-02 1.37283488
#> SR-03 -3.18065538
# G*E matrix (deviations from mean)
array(model$genXenv, dim(model$genXenv), dimnames(model$genXenv))
#> ENV
#> GEN Ayac Hyo-02 LM-02 LM-03 SR-02
#> 102.18 5.5726162 -12.4918224 1.7425251 -2.7070438 2.91734869
#> 104.22 -2.8712076 7.1684102 3.9336218 -4.0358373 0.47881580
#> 121.31 0.3255230 -3.8666836 4.3182811 10.4366135 -11.88343843
#> 141.28 -0.9451837 5.6454825 -9.7806639 14.6463104 -4.80337115
#> 157.26 -10.3149711 -10.6241677 4.2336365 16.8683612 2.71710210
#> 163.9 3.0874931 -6.9416721 3.4963790 -12.5533271 7.01688164
#> 221.19 -0.6041752 -6.0090018 4.0648518 -2.6974743 1.27671246
#> 233.11 2.5837535 6.8277609 -3.4440645 -4.4985717 0.19989490
#> 235.6 -1.7541523 19.8225025 -2.2394463 -5.6643239 -8.11400542
#> 241.2 1.0710975 -5.3831118 5.4253097 -3.2588271 0.46433086
#> 255.7 2.4443155 1.3860497 -1.8857757 -12.9626594 4.31373929
#> 314.12 -3.8812099 6.2098482 2.3577759 5.9071782 -3.92419060
#> 317.6 -1.7450319 3.0388540 3.0448064 5.5211634 -4.79271565
#> 319.20 -6.0155949 2.8477540 -9.7697504 24.8850017 -1.82949467
#> 320.16 10.9481796 -10.2982108 4.9608280 -6.2233088 2.99984918
#> 342.15 0.8508002 -0.3338618 -2.4575390 -10.3783871 7.29753151
#> 346.2 4.7000495 -6.2178087 -2.2612391 -14.9700672 9.90123888
#> 351.26 2.6002030 -0.9918665 -10.8315931 12.7429121 -0.02713985
#> 364.21 -0.4533734 3.2864208 -0.1335527 -0.1592533 -4.82292664
#> 402.7 -1.2134573 -0.0387229 -0.2179557 -0.8774011 1.08032472
#> 405.2 6.6477681 -8.3071271 -0.6159895 -8.8927189 3.52179705
#> 406.12 -6.1296667 12.0703469 1.1195092 -2.2601009 -3.13776595
#> 427.7 -3.1340922 4.3967072 4.2792028 -1.0194744 0.76266844
#> 450.3 -0.5047010 -1.0720791 -3.2821761 12.8806007 -5.04562407
#> 506.2 -1.2991912 -1.5682154 8.3142802 -3.1819279 0.60021498
#> Canchan 1.2929442 5.7152780 -9.3713622 9.0803035 -1.65332869
#> Desiree 9.5767845 -22.3280421 0.2396387 -11.8935722 9.62433886
#> Unica -10.8355195 18.0569790 4.7604622 -4.7341684 -5.13878822
#> ENV
#> GEN SR-03
#> 102.18 4.9663762
#> 104.22 -4.6738028
#> 121.31 0.6697043
#> 141.28 -4.7625741
#> 157.26 -2.8799609
#> 163.9 5.8942454
#> 221.19 3.9690870
#> 233.11 -1.6687730
#> 235.6 -2.0505746
#> 241.2 1.6812008
#> 255.7 6.7043306
#> 314.12 -6.6694018
#> 317.6 -5.0670763
#> 319.20 -10.1179157
#> 320.16 -2.3873373
#> 342.15 5.0214562
#> 346.2 8.8478267
#> 351.26 -3.4925156
#> 364.21 2.2826853
#> 402.7 1.2672123
#> 405.2 7.6462704
#> 406.12 -1.6623226
#> 427.7 -5.2850119
#> 450.3 -2.9760204
#> 506.2 -2.8651608
#> Canchan -5.0638348
#> Desiree 14.7808522
#> Unica -2.1089651
# With default n (N') and default ssi.method (farshadfar)
AMGE.AMMI(model)
#> AMGE SSI rAMGE rY means
#> 102.18 -5.329071e-15 26.0 3.0 23 26.31947
#> 104.22 1.776357e-15 29.5 16.5 13 31.28887
#> 121.31 8.881784e-16 29.0 14.0 15 30.10174
#> 141.28 3.552714e-15 24.0 23.0 1 39.75624
#> 157.26 -2.664535e-15 10.5 5.5 5 36.95181
#> 163.9 -5.329071e-15 30.0 3.0 27 21.41747
#> 221.19 -2.220446e-15 33.0 7.0 26 22.98480
#> 233.11 2.553513e-15 36.0 19.0 17 28.66655
#> 235.6 3.552714e-15 27.0 23.0 4 38.63477
#> 241.2 -1.998401e-15 30.0 8.0 22 26.34039
#> 255.7 -1.776357e-15 24.0 10.0 14 30.58975
#> 314.12 3.552714e-15 41.0 23.0 18 28.17335
#> 317.6 2.664535e-15 29.0 20.0 9 35.32583
#> 319.20 7.105427e-15 30.0 27.0 3 38.75767
#> 320.16 -1.776357e-15 31.0 10.0 21 26.34808
#> 342.15 -1.776357e-15 34.0 10.0 24 26.01336
#> 346.2 -5.329071e-15 28.0 3.0 25 23.84175
#> 351.26 2.220446e-15 26.0 18.0 8 36.11581
#> 364.21 1.221245e-15 25.0 15.0 10 34.05974
#> 402.7 -3.330669e-16 32.0 13.0 19 27.47748
#> 405.2 -2.664535e-15 21.5 5.5 16 28.98663
#> 406.12 3.552714e-15 35.0 23.0 12 32.68323
#> 427.7 3.552714e-15 30.0 23.0 7 36.19020
#> 450.3 1.776357e-15 22.5 16.5 6 36.19602
#> 506.2 -7.632783e-16 23.0 12.0 11 33.26623
#> Canchan 3.996803e-15 46.0 26.0 20 27.00126
#> Desiree -1.243450e-14 29.0 1.0 28 16.15569
#> Unica 8.881784e-15 30.0 28.0 2 39.10400
# With n = 4 and default ssi.method (farshadfar)
AMGE.AMMI(model, n = 4)
#> AMGE SSI rAMGE rY means
#> 102.18 -1.065814e-14 27.0 4.0 23 26.31947
#> 104.22 5.773160e-15 36.0 23.0 13 31.28887
#> 121.31 -1.465494e-14 16.0 1.0 15 30.10174
#> 141.28 8.881784e-16 17.0 16.0 1 39.75624
#> 157.26 1.509903e-14 33.0 28.0 5 36.95181
#> 163.9 -1.776357e-15 37.0 10.0 27 21.41747
#> 221.19 -4.440892e-16 38.0 12.0 26 22.98480
#> 233.11 -2.775558e-16 30.0 13.0 17 28.66655
#> 235.6 0.000000e+00 18.5 14.5 4 38.63477
#> 241.2 -4.662937e-15 29.0 7.0 22 26.34039
#> 255.7 0.000000e+00 28.5 14.5 14 30.58975
#> 314.12 5.329071e-15 39.5 21.5 18 28.17335
#> 317.6 -2.220446e-15 18.0 9.0 9 35.32583
#> 319.20 1.243450e-14 30.0 27.0 3 38.75767
#> 320.16 -1.332268e-14 23.0 2.0 21 26.34808
#> 342.15 7.993606e-15 49.0 25.0 24 26.01336
#> 346.2 1.776357e-15 42.0 17.0 25 23.84175
#> 351.26 2.220446e-15 26.0 18.0 8 36.11581
#> 364.21 -4.773959e-15 16.0 6.0 10 34.05974
#> 402.7 2.553513e-15 38.0 19.0 19 27.47748
#> 405.2 -9.769963e-15 21.0 5.0 16 28.98663
#> 406.12 6.661338e-15 36.0 24.0 12 32.68323
#> 427.7 5.329071e-15 28.5 21.5 7 36.19020
#> 450.3 -2.664535e-15 14.0 8.0 6 36.19602
#> 506.2 -1.026956e-15 22.0 11.0 11 33.26623
#> Canchan 3.996803e-15 40.0 20.0 20 27.00126
#> Desiree -1.243450e-14 31.0 3.0 28 16.15569
#> Unica 1.065814e-14 28.0 26.0 2 39.10400
# With default n (N') and ssi.method = "rao"
AMGE.AMMI(model, ssi.method = "rao")
#> AMGE SSI rAMGE rY means
#> 102.18 -5.329071e-15 3.3050451 3.0 23 26.31947
#> 104.22 1.776357e-15 -6.3204566 16.5 13 31.28887
#> 121.31 8.881784e-16 -13.6998126 14.0 15 30.10174
#> 141.28 3.552714e-15 -2.3740484 23.0 1 39.75624
#> 157.26 -2.664535e-15 6.0986020 5.5 5 36.95181
#> 163.9 -5.329071e-15 3.1452123 3.0 27 21.41747
#> 221.19 -2.220446e-15 6.6219522 7.0 26 22.98480
#> 233.11 2.553513e-15 -4.1718482 19.0 17 28.66655
#> 235.6 3.552714e-15 -2.4106145 23.0 4 38.63477
#> 241.2 -1.998401e-15 7.3838654 8.0 22 26.34039
#> 255.7 -1.776357e-15 8.3380459 10.0 14 30.58975
#> 314.12 3.552714e-15 -2.7517153 23.0 18 28.17335
#> 317.6 2.664535e-15 -3.7419460 20.0 9 35.32583
#> 319.20 7.105427e-15 -0.5714451 27.0 3 38.75767
#> 320.16 -1.776357e-15 8.1997438 10.0 21 26.34808
#> 342.15 -1.776357e-15 8.1888301 10.0 24 26.01336
#> 346.2 -5.329071e-15 3.2242576 3.0 25 23.84175
#> 351.26 2.220446e-15 -4.6949415 18.0 8 36.11581
#> 364.21 1.221245e-15 -9.5667691 15.0 10 34.05974
#> 402.7 -3.330669e-16 40.0460472 13.0 19 27.47748
#> 405.2 -2.664535e-15 5.8388923 5.5 16 28.98663
#> 406.12 3.552714e-15 -2.6046683 23.0 12 32.68323
#> 427.7 3.552714e-15 -2.4903213 23.0 7 36.19020
#> 450.3 1.776357e-15 -6.1604559 16.5 6 36.19602
#> 506.2 -7.632783e-16 18.1683574 12.0 11 33.26623
#> Canchan 3.996803e-15 -2.3821184 26.0 20 27.00126
#> Desiree -1.243450e-14 1.5754301 1.0 28 16.15569
#> Unica 8.881784e-15 -0.1931203 28.0 2 39.10400
# Changing the ratio of weights for Rao's SSI
AMGE.AMMI(model, ssi.method = "rao", a = 0.43)
#> AMGE SSI rAMGE rY means
#> 102.18 -5.329071e-15 1.9103218 3.0 23 26.31947
#> 104.22 1.776357e-15 -2.1362868 16.5 13 31.28887
#> 121.31 8.881784e-16 -5.3314728 14.0 15 30.10174
#> 141.28 3.552714e-15 -0.2819635 23.0 1 39.75624
#> 157.26 -2.664535e-15 3.3091554 5.5 5 36.95181
#> 163.9 -5.329071e-15 1.7504890 3.0 27 21.41747
#> 221.19 -2.220446e-15 3.2746163 7.0 26 22.98480
#> 233.11 2.553513e-15 -1.2611213 19.0 17 28.66655
#> 235.6 3.552714e-15 -0.3185295 23.0 4 38.63477
#> 241.2 -1.998401e-15 3.6646033 8.0 22 26.34039
#> 255.7 -1.776357e-15 4.1538760 10.0 14 30.58975
#> 314.12 3.552714e-15 -0.6596304 23.0 18 28.17335
#> 317.6 2.664535e-15 -0.9524995 20.0 9 35.32583
#> 319.20 7.105427e-15 0.4745974 27.0 3 38.75767
#> 320.16 -1.776357e-15 4.0155740 10.0 21 26.34808
#> 342.15 -1.776357e-15 4.0046603 10.0 24 26.01336
#> 346.2 -5.329071e-15 1.8295343 3.0 25 23.84175
#> 351.26 2.220446e-15 -1.3476056 18.0 8 36.11581
#> 364.21 1.221245e-15 -3.4807038 15.0 10 34.05974
#> 402.7 -3.330669e-16 17.7304745 13.0 19 27.47748
#> 405.2 -2.664535e-15 3.0494457 5.5 16 28.98663
#> 406.12 3.552714e-15 -0.5125833 23.0 12 32.68323
#> 427.7 3.552714e-15 -0.3982364 23.0 7 36.19020
#> 450.3 1.776357e-15 -1.9762860 16.5 6 36.19602
#> 506.2 -7.632783e-16 8.4306530 12.0 11 33.26623
#> Canchan 3.996803e-15 -0.5224874 26.0 20 27.00126
#> Desiree -1.243450e-14 0.9776915 1.0 28 16.15569
#> Unica 8.881784e-15 0.6437136 28.0 2 39.10400