FA.AMMI computes the Stability Measure Based on Fitted AMMI Model (FA)
(Raju 2002)
considering all significant
interaction principal components (IPCs) in the AMMI model. Using FA, the
Simultaneous Selection Index for Yield and Stability (SSI) is also calculated
according to the argument ssi.method.
FA.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 FA values.
The computed values of simultaneous selection index for yield and stability.
The ranks of FA 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 Stability Measure Based on Fitted AMMI Model (\(FA\)) (Raju 2002) is computed as follows:
\[FA = \sum_{n=1}^{N'}\lambda_{n}^{2}\gamma_{in}^{2}\]
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; and \(\gamma_{in}\) is the eigenvector value for \(i\)th genotype.
When \(N'\) is replaced by 1 (only first IPC axis is considered for computation), then the parameter \(FP\) can be estimated (Zali et al. 2012) .
\[FP = \lambda_{1}^{2}\gamma_{i1}^{2}\]
When \(N'\) is replaced by 2 (only first two IPC axes are considered for computation), then the parameter \(B\) can be estimated (Zali et al. 2012) .
\[B = \sum_{n=1}^{2}\lambda_{n}^{2}\gamma_{in}^{2}\]
When \(N'\) is replaced by \(N\) (All the IPC axes are considered for computation), then the parameter estimated is equivalent to Wricke's ecovalence (\(W_{(AMMI)}\)) (Wricke 1962; Zali et al. 2012) .
\[W_{(AMMI)} = \sum_{n=1}^{N}\lambda_{n}^{2}\gamma_{in}^{2}\]
Raju BMK (2002).
“A study on AMMI model and its biplots.”
Journal of the Indian Society of Agricultural Statistics, 55(3), 297--322.
Wricke G (1962).
“On a method of understanding the biological diversity in field research.”
Zeitschrift fur Pflanzenzuchtung, 47, 92--146.
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)
FA.AMMI(model)
#> FA SSI rFA rY means
#> 102.18 226.214559 39 16 23 26.31947
#> 104.22 96.017789 22 9 13 31.28887
#> 121.31 166.871081 26 11 15 30.10174
#> 141.28 386.485026 23 22 1 39.75624
#> 157.26 460.491413 29 24 5 36.95181
#> 163.9 306.218437 48 21 27 21.41747
#> 221.19 72.376305 31 5 26 22.98480
#> 233.11 80.663694 24 7 17 28.66655
#> 235.6 481.419528 29 25 4 38.63477
#> 241.2 71.468008 26 4 22 26.34039
#> 255.7 237.870912 32 18 14 30.58975
#> 314.12 149.384801 28 10 18 28.17335
#> 317.6 92.022551 17 8 9 35.32583
#> 319.20 840.209886 30 27 3 38.75767
#> 320.16 191.423345 34 13 21 26.34808
#> 342.15 169.656627 36 12 24 26.01336
#> 346.2 450.721670 48 23 25 23.84175
#> 351.26 298.237108 28 20 8 36.11581
#> 364.21 14.300314 12 2 10 34.05974
#> 402.7 1.419225 20 1 19 27.47748
#> 405.2 256.882577 35 19 16 28.98663
#> 406.12 195.702153 26 14 12 32.68323
#> 427.7 56.361179 10 3 7 36.19020
#> 450.3 203.659148 21 15 6 36.19602
#> 506.2 80.183743 17 6 11 33.26623
#> Canchan 229.161607 37 17 20 27.00126
#> Desiree 1031.364210 56 28 28 16.15569
#> Unica 499.251489 28 26 2 39.10400
# With n = 4 and default ssi.method (farshadfar)
FA.AMMI(model, n = 4)
#> FA SSI rFA rY means
#> 102.18 230.610963 39 16 23 26.31947
#> 104.22 99.626933 22 9 13 31.28887
#> 121.31 258.286270 33 18 15 30.10174
#> 141.28 387.665704 23 22 1 39.75624
#> 157.26 531.981114 31 26 5 36.95181
#> 163.9 310.983953 48 21 27 21.41747
#> 221.19 72.619025 30 4 26 22.98480
#> 233.11 81.631564 24 7 17 28.66655
#> 235.6 500.679624 28 24 4 38.63477
#> 241.2 73.134171 27 5 22 26.34039
#> 255.7 239.767170 31 17 14 30.58975
#> 314.12 149.451148 28 10 18 28.17335
#> 317.6 98.287259 17 8 9 35.32583
#> 319.20 863.387913 30 27 3 38.75767
#> 320.16 223.718164 35 14 21 26.34808
#> 342.15 192.877830 35 11 24 26.01336
#> 346.2 466.039106 48 23 25 23.84175
#> 351.26 298.259992 28 20 8 36.11581
#> 364.21 25.537314 12 2 10 34.05974
#> 402.7 3.829248 20 1 19 27.47748
#> 405.2 261.727258 35 19 16 28.98663
#> 406.12 198.459140 24 12 12 32.68323
#> 427.7 61.577580 10 3 7 36.19020
#> 450.3 210.606905 19 13 6 36.19602
#> 506.2 80.223923 17 6 11 33.26623
#> Canchan 229.271862 35 15 20 27.00126
#> Desiree 1031.404193 56 28 28 16.15569
#> Unica 506.919240 27 25 2 39.10400
# With default n (N') and ssi.method = "rao"
FA.AMMI(model, ssi.method = "rao")
#> FA SSI rFA rY means
#> 102.18 226.214559 0.9902913 16 23 26.31947
#> 104.22 96.017789 1.3314840 9 13 31.28887
#> 121.31 166.871081 1.1606028 11 15 30.10174
#> 141.28 386.485026 1.3736129 22 1 39.75624
#> 157.26 460.491413 1.2697440 24 5 36.95181
#> 163.9 306.218437 0.7959379 21 27 21.41747
#> 221.19 72.376305 1.1624072 5 26 22.98480
#> 233.11 80.663694 1.3052353 7 17 28.66655
#> 235.6 481.419528 1.3217963 25 4 38.63477
#> 241.2 71.468008 1.2770668 4 22 26.34039
#> 255.7 237.870912 1.1230515 18 14 30.58975
#> 314.12 149.384801 1.1186933 10 18 28.17335
#> 317.6 92.022551 1.4766266 8 9 35.32583
#> 319.20 840.209886 1.2992910 27 3 38.75767
#> 320.16 191.423345 1.0152386 13 21 26.34808
#> 342.15 169.656627 1.0243579 12 24 26.01336
#> 346.2 450.721670 0.8436895 23 25 23.84175
#> 351.26 298.237108 1.2777984 20 8 36.11581
#> 364.21 14.300314 3.2006702 2 10 34.05974
#> 402.7 1.419225 21.9563817 1 19 27.47748
#> 405.2 256.882577 1.0614812 19 16 28.98663
#> 406.12 195.702153 1.2183859 14 12 32.68323
#> 427.7 56.361179 1.7103246 3 7 36.19020
#> 450.3 203.659148 1.3269556 15 6 36.19602
#> 506.2 80.183743 1.4574286 6 11 33.26623
#> Canchan 229.161607 1.0108222 17 20 27.00126
#> Desiree 1031.364210 0.5557465 28 28 16.15569
#> Unica 499.251489 1.3348781 26 2 39.10400
# Changing the ratio of weights for Rao's SSI
FA.AMMI(model, ssi.method = "rao", a = 0.43)
#> FA SSI rFA rY means
#> 102.18 226.214559 0.9149776 16 23 26.31947
#> 104.22 96.017789 1.1540477 9 13 31.28887
#> 121.31 166.871081 1.0585058 11 15 30.10174
#> 141.28 386.485026 1.3295309 22 1 39.75624
#> 157.26 460.491413 1.2327465 24 5 36.95181
#> 163.9 306.218437 0.7403010 21 27 21.41747
#> 221.19 72.376305 0.9270120 5 26 22.98480
#> 233.11 80.663694 1.0940246 7 17 28.66655
#> 235.6 481.419528 1.2864071 25 4 38.63477
#> 241.2 71.468008 1.0386799 4 22 26.34039
#> 255.7 237.870912 1.0514284 18 14 30.58975
#> 314.12 149.384801 1.0046453 10 18 28.17335
#> 317.6 92.022551 1.2914868 8 9 35.32583
#> 319.20 840.209886 1.2790139 27 3 38.75767
#> 320.16 191.423345 0.9262367 13 21 26.34808
#> 342.15 169.656627 0.9239372 12 24 26.01336
#> 346.2 450.721670 0.8058900 23 25 23.84175
#> 351.26 298.237108 1.2206726 20 8 36.11581
#> 364.21 14.300314 2.0092951 2 10 34.05974
#> 402.7 1.419225 9.9519184 1 19 27.47748
#> 405.2 256.882577 0.9951589 19 16 28.98663
#> 406.12 195.702153 1.1313300 14 12 32.68323
#> 427.7 56.361179 1.4080414 3 7 36.19020
#> 450.3 203.659148 1.2433009 15 6 36.19602
#> 506.2 80.183743 1.2449536 6 11 33.26623
#> Canchan 229.161607 0.9364771 17 20 27.00126
#> Desiree 1031.364210 0.5392276 28 28 16.15569
#> Unica 499.251489 1.3007530 26 2 39.10400