EV.AMMI computes the Sums of the Averages of the Squared Eigenvector
Values (EV) (Zobel 1994)
considering all
significant interaction principal components (IPCs) in the AMMI model. Using
EV, the Simultaneous Selection Index for Yield and Stability (SSI) is also
calculated according to the argument ssi.method.
EV.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 EV values.
The computed values of simultaneous selection index for yield and stability.
The ranks of EV 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 Averages of the Squared Eigenvector Values (\(EV\)) (Zobel 1994) is computed as follows:
\[EV = \sum_{n=1}^{N'}\frac{\gamma_{in}^2}{N'}\]
Where, \(N'\) is the number of significant IPCs (number of IPC that were retained in the AMMI model via F tests); and \(\gamma_{in}\) is the eigenvector value for \(i\)th genotype.
Zobel RW (1994). “Stress resistance and root systems.” In Proceedings of the Workshop on Adaptation of Plants to Soil Stress. 1-4 August, 1993. INTSORMIL Publication 94-2, 80--99. Institute of Agriculture and Natural Resources, University of Nebraska-Lincoln.
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)
EV.AMMI(model)
#> EV SSI rEV rY means
#> 102.18 0.0232206231 37 14 23 26.31947
#> 104.22 0.0175897578 21 8 13 31.28887
#> 121.31 0.0342010876 34 19 15 30.10174
#> 141.28 0.0529036285 23 22 1 39.75624
#> 157.26 0.0965635719 33 28 5 36.95181
#> 163.9 0.0236900961 42 15 27 21.41747
#> 221.19 0.0127574566 29 3 26 22.98480
#> 233.11 0.0211138628 27 10 17 28.66655
#> 235.6 0.0723274691 28 24 4 38.63477
#> 241.2 0.0153823821 28 6 22 26.34039
#> 255.7 0.0317506280 31 17 14 30.58975
#> 314.12 0.0170302467 25 7 18 28.17335
#> 317.6 0.0136347120 14 5 9 35.32583
#> 319.20 0.0855988994 29 26 3 38.75767
#> 320.16 0.0180662044 30 9 21 26.34808
#> 342.15 0.0225156118 36 12 24 26.01336
#> 346.2 0.0459434537 45 20 25 23.84175
#> 351.26 0.0639652186 31 23 8 36.11581
#> 364.21 0.0018299284 12 2 10 34.05974
#> 402.7 0.0001339385 20 1 19 27.47748
#> 405.2 0.0229492190 29 13 16 28.98663
#> 406.12 0.0264692745 28 16 12 32.68323
#> 427.7 0.0135698145 11 4 7 36.19020
#> 450.3 0.0216161656 17 11 6 36.19602
#> 506.2 0.0318266934 29 18 11 33.26623
#> Canchan 0.0461305761 41 21 20 27.00126
#> Desiree 0.0901534938 55 27 28 16.15569
#> Unica 0.0770659860 27 25 2 39.10400
# With n = 4 and default ssi.method (farshadfar)
EV.AMMI(model, n = 4)
#> EV SSI rEV rY means
#> 102.18 0.020624276 33 10 23 26.31947
#> 104.22 0.015826528 21 8 13 31.28887
#> 121.31 0.092372131 42 27 15 30.10174
#> 141.28 0.040539465 21 20 1 39.75624
#> 157.26 0.124600955 33 28 5 36.95181
#> 163.9 0.021245785 39 12 27 21.41747
#> 221.19 0.009745247 29 3 26 22.98480
#> 233.11 0.016541818 26 9 17 28.66655
#> 235.6 0.068302992 29 25 4 38.63477
#> 241.2 0.012752871 26 4 22 26.34039
#> 255.7 0.025196996 30 16 14 30.58975
#> 314.12 0.012821109 23 5 18 28.17335
#> 317.6 0.014798464 16 7 9 35.32583
#> 319.20 0.081116150 29 26 3 38.75767
#> 320.16 0.037120712 40 19 21 26.34808
#> 342.15 0.033835196 41 17 24 26.01336
#> 346.2 0.045637346 46 21 25 23.84175
#> 351.26 0.047990616 30 22 8 36.11581
#> 364.21 0.009574009 12 2 10 34.05974
#> 402.7 0.001859460 20 1 19 27.47748
#> 405.2 0.020747907 27 11 16 28.98663
#> 406.12 0.021864201 26 14 12 32.68323
#> 427.7 0.013984661 13 6 7 36.19020
#> 450.3 0.021283092 19 13 6 36.19602
#> 506.2 0.023899346 26 15 11 33.26623
#> Canchan 0.034678404 38 18 20 27.00126
#> Desiree 0.067644303 52 24 28 16.15569
#> Unica 0.063395960 25 23 2 39.10400
# With default n (N') and ssi.method = "rao"
EV.AMMI(model, ssi.method = "rao")
#> EV SSI rEV rY means
#> 102.18 0.0232206231 0.9920136 14 23 26.31947
#> 104.22 0.0175897578 1.1968926 8 13 31.28887
#> 121.31 0.0342010876 1.0723629 19 15 30.10174
#> 141.28 0.0529036285 1.3550266 22 1 39.75624
#> 157.26 0.0965635719 1.2370234 28 5 36.95181
#> 163.9 0.0236900961 0.8295284 15 27 21.41747
#> 221.19 0.0127574566 0.9930645 3 26 22.98480
#> 233.11 0.0211138628 1.0818975 10 17 28.66655
#> 235.6 0.0723274691 1.3026828 24 4 38.63477
#> 241.2 0.0153823821 1.0609011 6 22 26.34039
#> 255.7 0.0317506280 1.0952885 17 14 30.58975
#> 314.12 0.0170302467 1.1011148 7 18 28.17335
#> 317.6 0.0136347120 1.3797760 5 9 35.32583
#> 319.20 0.0855988994 1.3000274 26 3 38.75767
#> 320.16 0.0180662044 1.0311353 9 21 26.34808
#> 342.15 0.0225156118 0.9862240 12 24 26.01336
#> 346.2 0.0459434537 0.8450255 20 25 23.84175
#> 351.26 0.0639652186 1.2261684 23 8 36.11581
#> 364.21 0.0018299284 2.8090292 2 10 34.05974
#> 402.7 0.0001339385 24.1014741 1 19 27.47748
#> 405.2 0.0229492190 1.0805609 13 16 28.98663
#> 406.12 0.0264692745 1.1830798 16 12 32.68323
#> 427.7 0.0135698145 1.4090495 4 7 36.19020
#> 450.3 0.0216161656 1.3239797 11 6 36.19602
#> 506.2 0.0318266934 1.1823230 18 11 33.26623
#> Canchan 0.0461305761 0.9477687 21 20 27.00126
#> Desiree 0.0901534938 0.5612418 27 28 16.15569
#> Unica 0.0770659860 1.3153400 25 2 39.10400
# Changing the ratio of weights for Rao's SSI
EV.AMMI(model, ssi.method = "rao", a = 0.43)
#> EV SSI rEV rY means
#> 102.18 0.0232206231 0.9157183 14 23 26.31947
#> 104.22 0.0175897578 1.0961734 8 13 31.28887
#> 121.31 0.0342010876 1.0205626 19 15 30.10174
#> 141.28 0.0529036285 1.3215387 22 1 39.75624
#> 157.26 0.0965635719 1.2186766 28 5 36.95181
#> 163.9 0.0236900961 0.7547449 15 27 21.41747
#> 221.19 0.0127574566 0.8541946 3 26 22.98480
#> 233.11 0.0211138628 0.9979893 10 17 28.66655
#> 235.6 0.0723274691 1.2781883 24 4 38.63477
#> 241.2 0.0153823821 0.9457286 6 22 26.34039
#> 255.7 0.0317506280 1.0394903 17 14 30.58975
#> 314.12 0.0170302467 0.9970866 7 18 28.17335
#> 317.6 0.0136347120 1.2498410 5 9 35.32583
#> 319.20 0.0855988994 1.2793305 26 3 38.75767
#> 320.16 0.0180662044 0.9330723 9 21 26.34808
#> 342.15 0.0225156118 0.9075396 12 24 26.01336
#> 346.2 0.0459434537 0.8064645 20 25 23.84175
#> 351.26 0.0639652186 1.1984717 23 8 36.11581
#> 364.21 0.0018299284 1.8408895 2 10 34.05974
#> 402.7 0.0001339385 10.8743081 1 19 27.47748
#> 405.2 0.0229492190 1.0033632 13 16 28.98663
#> 406.12 0.0264692745 1.1161483 16 12 32.68323
#> 427.7 0.0135698145 1.2784931 4 7 36.19020
#> 450.3 0.0216161656 1.2420213 11 6 36.19602
#> 506.2 0.0318266934 1.1266582 18 11 33.26623
#> Canchan 0.0461305761 0.9093641 21 20 27.00126
#> Desiree 0.0901534938 0.5415905 27 28 16.15569
#> Unica 0.0770659860 1.2923516 25 2 39.10400