R/AVAMGE.AMMI.R
AVAMGE.AMMI.RdAVAMGE.AMMI computes the Sum Across Environments of Absolute Value of
GEI Modelled by AMMI (AVAMGE)
(Zali et al. 2012)
considering all significant
interaction principal components (IPCs) in the AMMI model. Using AVAMGE, the
Simultaneous Selection Index for Yield and Stability (SSI) is also calculated
according to the argument ssi.method.
AVAMGE.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 AVAMGE values.
The computed values of simultaneous selection index for yield and stability.
The ranks of AVAMGE 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 Absolute Value of GEI Modelled by AMMI (\(AV_{(AMGE)}\)) (Zali et al. 2012) is computed as follows:
\[AV_{(AMGE)} = \sum_{j=1}^{E} \sum_{n=1}^{N'} \left |\lambda_{n} \gamma_{in} \delta_{jn} \right |\]
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.
Zali H, Farshadfar E, Sabaghpour SH, Karimizadeh R (2012). “Evaluation of genotype × environment interaction in chickpea using measures of stability from AMMI model.” Annals of Biological Research, 3(7), 3126--3136.
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)
AVAMGE.AMMI(model)
#> AVAMGE SSI rAVAMGE rY means
#> 102.18 30.229771 40 17 23 26.31947
#> 104.22 21.584579 21 8 13 31.28887
#> 121.31 27.893984 28 13 15 30.10174
#> 141.28 40.486706 24 23 1 39.75624
#> 157.26 44.055803 29 24 5 36.95181
#> 163.9 39.056228 48 21 27 21.41747
#> 221.19 17.905975 33 7 26 22.98480
#> 233.11 16.242635 21 4 17 28.66655
#> 235.6 39.840739 26 22 4 38.63477
#> 241.2 17.101113 28 6 22 26.34039
#> 255.7 29.306918 29 15 14 30.58975
#> 314.12 28.760304 32 14 18 28.17335
#> 317.6 22.700856 18 9 9 35.32583
#> 319.20 55.232023 30 27 3 38.75767
#> 320.16 30.717681 40 19 21 26.34808
#> 342.15 25.538281 34 10 24 26.01336
#> 346.2 46.236590 50 25 25 23.84175
#> 351.26 30.105573 24 16 8 36.11581
#> 364.21 6.742386 12 2 10 34.05974
#> 402.7 2.202291 20 1 19 27.47748
#> 405.2 35.890684 36 20 16 28.98663
#> 406.12 27.272847 24 12 12 32.68323
#> 427.7 16.756971 12 5 7 36.19020
#> 450.3 25.628188 17 11 6 36.19602
#> 506.2 15.760611 14 3 11 33.26623
#> Canchan 30.515224 38 18 20 27.00126
#> Desiree 69.096357 56 28 28 16.15569
#> Unica 47.204593 28 26 2 39.10400
# With n = 4 and default ssi.method (farshadfar)
AVAMGE.AMMI(model, n = 4)
#> AVAMGE SSI rAVAMGE rY means
#> 102.18 30.431550 39 16 23 26.31947
#> 104.22 21.176775 21 8 13 31.28887
#> 121.31 34.844853 34 19 15 30.10174
#> 141.28 40.382139 24 23 1 39.75624
#> 157.26 49.421992 31 26 5 36.95181
#> 163.9 38.846149 48 21 27 21.41747
#> 221.19 17.858564 33 7 26 22.98480
#> 233.11 17.449539 23 6 17 28.66655
#> 235.6 39.657410 26 22 4 38.63477
#> 241.2 17.225331 27 5 22 26.34039
#> 255.7 29.585043 28 14 14 30.58975
#> 314.12 28.801567 31 13 18 28.17335
#> 317.6 23.101824 18 9 9 35.32583
#> 319.20 55.695327 30 27 3 38.75767
#> 320.16 31.566364 39 18 21 26.34808
#> 342.15 26.310253 35 11 24 26.01336
#> 346.2 46.863568 50 25 25 23.84175
#> 351.26 29.920025 23 15 8 36.11581
#> 364.21 9.635146 12 2 10 34.05974
#> 402.7 3.665565 20 1 19 27.47748
#> 405.2 35.538076 36 20 16 28.98663
#> 406.12 26.916422 24 12 12 32.68323
#> 427.7 16.266701 11 4 7 36.19020
#> 450.3 25.622916 16 10 6 36.19602
#> 506.2 15.709209 14 3 11 33.26623
#> Canchan 30.908627 37 17 20 27.00126
#> Desiree 69.115600 56 28 28 16.15569
#> Unica 46.610186 26 24 2 39.10400
# With default n (N') and ssi.method = "rao"
AVAMGE.AMMI(model, ssi.method = "rao")
#> AVAMGE SSI rAVAMGE rY means
#> 102.18 30.229771 1.4579240 17 23 26.31947
#> 104.22 21.584579 1.8601746 8 13 31.28887
#> 121.31 27.893984 1.6314700 13 15 30.10174
#> 141.28 40.486706 1.7440938 23 1 39.75624
#> 157.26 44.055803 1.6163747 24 5 36.95181
#> 163.9 39.056228 1.1625489 21 27 21.41747
#> 221.19 17.905975 1.7619814 7 26 22.98480
#> 233.11 16.242635 2.0509293 4 17 28.66655
#> 235.6 39.840739 1.7147885 22 4 38.63477
#> 241.2 17.101113 1.9190480 6 22 26.34039
#> 255.7 29.306918 1.6160450 15 14 30.58975
#> 314.12 28.760304 1.5490150 14 18 28.17335
#> 317.6 22.700856 1.9504975 9 9 35.32583
#> 319.20 55.232023 1.5919808 27 3 38.75767
#> 320.16 30.717681 1.4493304 19 21 26.34808
#> 342.15 25.538281 1.5581219 10 24 26.01336
#> 346.2 46.236590 1.1695027 25 25 23.84175
#> 351.26 30.105573 1.7798138 16 8 36.11581
#> 364.21 6.742386 3.7995961 2 10 34.05974
#> 402.7 2.202291 9.1285592 1 19 27.47748
#> 405.2 35.890684 1.4502899 20 16 28.98663
#> 406.12 27.272847 1.7304443 12 12 32.68323
#> 427.7 16.756971 2.2619806 5 7 36.19020
#> 450.3 25.628188 1.8876432 11 6 36.19602
#> 506.2 15.760611 2.2350438 3 11 33.26623
#> Canchan 30.515224 1.4745437 18 20 27.00126
#> Desiree 69.096357 0.7891628 28 28 16.15569
#> Unica 47.204593 1.6590963 26 2 39.10400
# Changing the ratio of weights for Rao's SSI
AVAMGE.AMMI(model, ssi.method = "rao", a = 0.43)
#> AVAMGE SSI rAVAMGE rY means
#> 102.18 30.229771 1.1160597 17 23 26.31947
#> 104.22 21.584579 1.3813847 8 13 31.28887
#> 121.31 27.893984 1.2609787 13 15 30.10174
#> 141.28 40.486706 1.4888376 23 1 39.75624
#> 157.26 44.055803 1.3817977 24 5 36.95181
#> 163.9 39.056228 0.8979438 21 27 21.41747
#> 221.19 17.905975 1.1848289 7 26 22.98480
#> 233.11 16.242635 1.4146730 4 17 28.66655
#> 235.6 39.840739 1.4553938 22 4 38.63477
#> 241.2 17.101113 1.3147318 6 22 26.34039
#> 255.7 29.306918 1.2634156 15 14 30.58975
#> 314.12 28.760304 1.1896837 14 18 28.17335
#> 317.6 22.700856 1.4952513 9 9 35.32583
#> 319.20 55.232023 1.4048705 27 3 38.75767
#> 320.16 30.717681 1.1128962 19 21 26.34808
#> 342.15 25.538281 1.1534557 10 24 26.01336
#> 346.2 46.236590 0.9459897 25 25 23.84175
#> 351.26 30.105573 1.4365392 16 8 36.11581
#> 364.21 6.742386 2.2668332 2 10 34.05974
#> 402.7 2.202291 4.4359547 1 19 27.47748
#> 405.2 35.890684 1.1623466 20 16 28.98663
#> 406.12 27.272847 1.3515151 12 12 32.68323
#> 427.7 16.756971 1.6452535 5 7 36.19020
#> 450.3 25.628188 1.4843966 11 6 36.19602
#> 506.2 15.760611 1.5793281 3 11 33.26623
#> Canchan 30.515224 1.1358773 18 20 27.00126
#> Desiree 69.096357 0.6395966 28 28 16.15569
#> Unica 47.204593 1.4401668 26 2 39.10400