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.
AMGE.AMMI(model, n, alpha = 0.05, ssi.method = c("farshadfar", "rao"), a = 1)The AMMI model (An object of class AMMI generated by
AMMI).
The number of principal components to be considered for computation. The default value is the number of significant IPCs.
Type I error probability (Significance level) to be considered to identify the number of significant IPCs.
The method for the computation of simultaneous selection
index. Either "farshadfar" or "rao" (See
SSI).
The ratio of the weights given to the stability components for
computation of SSI when method = "rao" (See
SSI).
A data frame with the following columns:
The AMGE values.
The computed values of simultaneous selection index for yield and stability.
The ranks of AMGE values.
The ranks of the mean yield of genotypes.
The mean yield of the genotypes.
The names of the genotypes are indicated as the row names of the data frame.
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.
Sneller CH, Kilgore-Norquest L, Dombek D (1997). “Repeatability of yield stability statistics in soybean.” Crop Science, 37(2), 383--390.
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 1.598721e-14 48 25 23 26.31947
#> 104.22 -8.881784e-15 20 7 13 31.28887
#> 121.31 1.643130e-14 41 26 15 30.10174
#> 141.28 -4.440892e-15 11 10 1 39.75624
#> 157.26 3.241851e-14 33 28 5 36.95181
#> 163.9 3.108624e-15 45 18 27 21.41747
#> 221.19 8.881784e-15 48 22 26 22.98480
#> 233.11 -1.476597e-14 22 5 17 28.66655
#> 235.6 -2.975398e-14 5 1 4 38.63477
#> 241.2 7.105427e-15 42 20 22 26.34039
#> 255.7 -1.598721e-14 18 4 14 30.58975
#> 314.12 -1.776357e-15 31 13 18 28.17335
#> 317.6 1.776357e-15 26 17 9 35.32583
#> 319.20 8.437695e-15 24 21 3 38.75767
#> 320.16 1.154632e-14 45 24 21 26.34808
#> 342.15 -9.325873e-15 30 6 24 26.01336
#> 346.2 -3.552714e-15 36 11 25 23.84175
#> 351.26 1.110223e-15 24 16 8 36.11581
#> 364.21 -4.940492e-15 19 9 10 34.05974
#> 402.7 -4.163336e-16 33 14 19 27.47748
#> 405.2 8.881784e-16 31 15 16 28.98663
#> 406.12 -1.731948e-14 15 3 12 32.68323
#> 427.7 -2.553513e-15 19 12 7 36.19020
#> 450.3 1.021405e-14 29 23 6 36.19602
#> 506.2 6.439294e-15 30 19 11 33.26623
#> Canchan -7.993606e-15 28 8 20 27.00126
#> Desiree 1.754152e-14 55 27 28 16.15569
#> Unica -2.042810e-14 4 2 2 39.10400
# With n = 4 and default ssi.method (farshadfar)
AMGE.AMMI(model, n = 4)
#> AMGE SSI rAMGE rY means
#> 102.18 1.643130e-14 48.0 25.0 23 26.31947
#> 104.22 -9.325873e-15 20.0 7.0 13 31.28887
#> 121.31 1.731948e-14 41.0 26.0 15 30.10174
#> 141.28 -4.218847e-15 11.5 10.5 1 39.75624
#> 157.26 3.019807e-14 33.0 28.0 5 36.95181
#> 163.9 2.664535e-15 45.0 18.0 27 21.41747
#> 221.19 8.271162e-15 48.0 22.0 26 22.98480
#> 233.11 -1.409983e-14 22.0 5.0 17 28.66655
#> 235.6 -2.797762e-14 5.0 1.0 4 38.63477
#> 241.2 6.883383e-15 42.0 20.0 22 26.34039
#> 255.7 -1.709743e-14 18.0 4.0 14 30.58975
#> 314.12 -2.664535e-15 31.0 13.0 18 28.17335
#> 317.6 2.220446e-15 26.0 17.0 9 35.32583
#> 319.20 7.549517e-15 24.0 21.0 3 38.75767
#> 320.16 1.243450e-14 45.0 24.0 21 26.34808
#> 342.15 -1.132427e-14 30.0 6.0 24 26.01336
#> 346.2 -4.440892e-15 34.0 9.0 25 23.84175
#> 351.26 1.110223e-15 23.0 15.0 8 36.11581
#> 364.21 -3.774758e-15 22.0 12.0 10 34.05974
#> 402.7 -9.159340e-16 33.0 14.0 19 27.47748
#> 405.2 1.165734e-15 32.0 16.0 16 28.98663
#> 406.12 -1.820766e-14 15.0 3.0 12 32.68323
#> 427.7 -4.218847e-15 17.5 10.5 7 36.19020
#> 450.3 9.992007e-15 29.0 23.0 6 36.19602
#> 506.2 6.522560e-15 30.0 19.0 11 33.26623
#> Canchan -6.994405e-15 28.0 8.0 20 27.00126
#> Desiree 1.743050e-14 55.0 27.0 28 16.15569
#> Unica -2.220446e-14 4.0 2.0 2 39.10400
# With default n (N') and ssi.method = "rao"
AMGE.AMMI(model, ssi.method = "rao")
#> AMGE SSI rAMGE rY means
#> 102.18 1.598721e-14 -1.209920 25 23 26.31947
#> 104.22 -8.881784e-15 4.742740 7 13 31.28887
#> 121.31 1.643130e-14 -1.030703 26 15 30.10174
#> 141.28 -4.440892e-15 8.741371 10 1 39.75624
#> 157.26 3.241851e-14 0.184960 28 5 36.95181
#> 163.9 3.108624e-15 -9.937521 18 27 21.41747
#> 221.19 8.881784e-15 -2.973115 22 26 22.98480
#> 233.11 -1.476597e-14 3.173817 5 17 28.66655
#> 235.6 -2.975398e-14 2.370918 1 4 38.63477
#> 241.2 7.105427e-15 -3.794340 20 22 26.34039
#> 255.7 -1.598721e-14 3.065479 4 14 30.58975
#> 314.12 -1.776357e-15 19.531348 13 18 28.17335
#> 317.6 1.776357e-15 -17.460918 17 9 35.32583
#> 319.20 8.437695e-15 -2.654754 21 3 38.75767
#> 320.16 1.154632e-14 -2.004403 24 21 26.34808
#> 342.15 -9.325873e-15 4.393465 6 24 26.01336
#> 346.2 -3.552714e-15 10.083744 11 25 23.84175
#> 351.26 1.110223e-15 -28.602804 16 8 36.11581
#> 364.21 -4.940492e-15 7.802759 9 10 34.05974
#> 402.7 -4.163336e-16 80.310270 14 19 27.47748
#> 405.2 8.881784e-16 -36.280350 15 16 28.98663
#> 406.12 -1.731948e-14 2.974655 3 12 32.68323
#> 427.7 -2.553513e-15 14.127995 12 7 36.19020
#> 450.3 1.021405e-14 -2.056805 23 6 36.19602
#> 506.2 6.439294e-15 -4.049883 19 11 33.26623
#> Canchan -7.993606e-15 5.016556 8 20 27.00126
#> Desiree 1.754152e-14 -1.358068 27 28 16.15569
#> Unica -2.042810e-14 2.893508 2 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 1.598721e-14 -0.03111319 25 23 26.31947
#> 104.22 -8.881784e-15 2.62088777 7 13 31.28887
#> 121.31 1.643130e-14 0.11624442 26 15 30.10174
#> 141.28 -4.440892e-15 4.49766702 10 1 39.75624
#> 157.26 3.241851e-14 0.76628938 28 5 36.95181
#> 163.9 3.108624e-15 -3.87508635 18 27 21.41747
#> 221.19 8.881784e-15 -0.85126241 22 26 22.98480
#> 233.11 -1.476597e-14 1.89751451 5 17 28.66655
#> 235.6 -2.975398e-14 1.73752955 1 4 38.63477
#> 241.2 7.105427e-15 -1.14202521 20 22 26.34039
#> 255.7 -1.598721e-14 1.88667228 4 14 30.58975
#> 314.12 -1.776357e-15 8.92208663 13 18 28.17335
#> 317.6 1.776357e-15 -6.85165762 17 9 35.32583
#> 319.20 8.437695e-15 -0.42122552 21 3 38.75767
#> 320.16 1.154632e-14 -0.37220928 24 21 26.34808
#> 342.15 -9.325873e-15 2.37265314 6 24 26.01336
#> 346.2 -3.552714e-15 4.77911338 11 25 23.84175
#> 351.26 1.110223e-15 -11.62798636 16 8 36.11581
#> 364.21 -4.940492e-15 3.98819325 9 10 34.05974
#> 402.7 -4.163336e-16 35.04409044 14 19 27.47748
#> 405.2 8.881784e-16 -15.06182868 15 16 28.98663
#> 406.12 -1.731948e-14 1.88652568 3 12 32.68323
#> 427.7 -2.553513e-15 6.74763968 12 7 36.19020
#> 450.3 1.021405e-14 -0.21171610 23 6 36.19602
#> 506.2 6.439294e-15 -1.12319038 19 11 33.26623
#> Canchan -7.993606e-15 2.65894277 8 20 27.00126
#> Desiree 1.754152e-14 -0.28371280 27 28 16.15569
#> Unica -2.042810e-14 1.97096400 2 2 39.10400