DA.AMMI computes the Annicchiarico's D Parameter values
(\(\textrm{D}_{\textrm{a}}\))
(Annicchiarico 1997)
considering all
significant interaction principal components (IPCs) in the AMMI model. It is
the unsquared Euclidean distance from the origin of significant IPC axes in
the AMMI model. Using \(\textrm{D}_{\textrm{a}}\), the Simultaneous
Selection Index for Yield and Stability (SSI) is also calculated according to
the argument ssi.method.
DA.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 DA values.
The computed values of simultaneous selection index for yield and stability.
The ranks of DA 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 Annicchiarico's D Parameter value (\(D_{a}\)) (Annicchiarico 1997) is computed as follows:
\[D_{a} = \sqrt{\sum_{n=1}^{N'}(\lambda_{n}\gamma_{in})^2}\]
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.
Annicchiarico P (1997). “Joint regression vs AMMI analysis of genotype-environment interactions for cereals in Italy.” Euphytica, 94(1), 53--62.
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)
DA.AMMI(model)
#> DA SSI rDA rY means
#> 102.18 15.040431 39 16 23 26.31947
#> 104.22 9.798867 22 9 13 31.28887
#> 121.31 12.917859 26 11 15 30.10174
#> 141.28 19.659222 23 22 1 39.75624
#> 157.26 21.459064 29 24 5 36.95181
#> 163.9 17.499098 48 21 27 21.41747
#> 221.19 8.507426 31 5 26 22.98480
#> 233.11 8.981297 24 7 17 28.66655
#> 235.6 21.941275 29 25 4 38.63477
#> 241.2 8.453875 26 4 22 26.34039
#> 255.7 15.423064 32 18 14 30.58975
#> 314.12 12.222308 28 10 18 28.17335
#> 317.6 9.592839 17 8 9 35.32583
#> 319.20 28.986374 30 27 3 38.75767
#> 320.16 13.835583 34 13 21 26.34808
#> 342.15 13.025230 36 12 24 26.01336
#> 346.2 21.230207 48 23 25 23.84175
#> 351.26 17.269543 28 20 8 36.11581
#> 364.21 3.781576 12 2 10 34.05974
#> 402.7 1.191312 20 1 19 27.47748
#> 405.2 16.027557 35 19 16 28.98663
#> 406.12 13.989359 26 14 12 32.68323
#> 427.7 7.507408 10 3 7 36.19020
#> 450.3 14.270920 21 15 6 36.19602
#> 506.2 8.954538 17 6 11 33.26623
#> Canchan 15.138085 37 17 20 27.00126
#> Desiree 32.114860 56 28 28 16.15569
#> Unica 22.343936 28 26 2 39.10400
# With n = 4 and default ssi.method (farshadfar)
DA.AMMI(model, n = 4)
#> DA SSI rDA rY means
#> 102.18 15.185880 39 16 23 26.31947
#> 104.22 9.981329 22 9 13 31.28887
#> 121.31 16.071287 33 18 15 30.10174
#> 141.28 19.689228 23 22 1 39.75624
#> 157.26 23.064716 31 26 5 36.95181
#> 163.9 17.634737 48 21 27 21.41747
#> 221.19 8.521680 30 4 26 22.98480
#> 233.11 9.035019 24 7 17 28.66655
#> 235.6 22.375871 28 24 4 38.63477
#> 241.2 8.551852 27 5 22 26.34039
#> 255.7 15.484417 31 17 14 30.58975
#> 314.12 12.225021 28 10 18 28.17335
#> 317.6 9.913993 17 8 9 35.32583
#> 319.20 29.383463 30 27 3 38.75767
#> 320.16 14.957211 35 14 21 26.34808
#> 342.15 13.888046 35 11 24 26.01336
#> 346.2 21.587939 48 23 25 23.84175
#> 351.26 17.270205 28 20 8 36.11581
#> 364.21 5.053446 12 2 10 34.05974
#> 402.7 1.956846 20 1 19 27.47748
#> 405.2 16.177987 35 19 16 28.98663
#> 406.12 14.087553 24 12 12 32.68323
#> 427.7 7.847138 10 3 7 36.19020
#> 450.3 14.512302 19 13 6 36.19602
#> 506.2 8.956781 17 6 11 33.26623
#> Canchan 15.141726 35 15 20 27.00126
#> Desiree 32.115482 56 28 28 16.15569
#> Unica 22.514867 27 25 2 39.10400
# With default n (N') and ssi.method = "rao"
DA.AMMI(model, ssi.method = "rao")
#> DA SSI rDA rY means
#> 102.18 15.040431 1.4730947 16 23 26.31947
#> 104.22 9.798867 1.9640618 9 13 31.28887
#> 121.31 12.917859 1.6974593 11 15 30.10174
#> 141.28 19.659222 1.7667347 22 1 39.75624
#> 157.26 21.459064 1.6358359 24 5 36.95181
#> 163.9 17.499098 1.2268624 21 27 21.41747
#> 221.19 8.507426 1.8365835 5 26 22.98480
#> 233.11 8.981297 1.9644804 7 17 28.66655
#> 235.6 21.941275 1.6812376 25 4 38.63477
#> 241.2 8.453875 1.9528811 4 22 26.34039
#> 255.7 15.423064 1.5970737 18 14 30.58975
#> 314.12 12.222308 1.6753281 10 18 28.17335
#> 317.6 9.592839 2.1159612 8 9 35.32583
#> 319.20 28.986374 1.5827930 27 3 38.75767
#> 320.16 13.835583 1.5275780 13 21 26.34808
#> 342.15 13.025230 1.5582533 12 24 26.01336
#> 346.2 21.230207 1.2130205 23 25 23.84175
#> 351.26 17.269543 1.7131362 20 8 36.11581
#> 364.21 3.781576 3.5563052 2 10 34.05974
#> 402.7 1.191312 8.6595018 1 19 27.47748
#> 405.2 16.027557 1.5221857 19 16 28.98663
#> 406.12 13.989359 1.7267910 14 12 32.68323
#> 427.7 7.507408 2.4119665 3 7 36.19020
#> 450.3 14.270920 1.8282838 15 6 36.19602
#> 506.2 8.954538 2.1175331 6 11 33.26623
#> Canchan 15.138085 1.4913580 17 20 27.00126
#> Desiree 32.114860 0.8147588 28 28 16.15569
#> Unica 22.343936 1.6889406 26 2 39.10400
# Changing the ratio of weights for Rao's SSI
DA.AMMI(model, ssi.method = "rao", a = 0.43)
#> DA SSI rDA rY means
#> 102.18 15.040431 1.1225831 16 23 26.31947
#> 104.22 9.798867 1.4260562 9 13 31.28887
#> 121.31 12.917859 1.2893541 11 15 30.10174
#> 141.28 19.659222 1.4985733 22 1 39.75624
#> 157.26 21.459064 1.3901660 24 5 36.95181
#> 163.9 17.499098 0.9255986 21 27 21.41747
#> 221.19 8.507426 1.2169078 5 26 22.98480
#> 233.11 8.981297 1.3775000 7 17 28.66655
#> 235.6 21.941275 1.4409668 25 4 38.63477
#> 241.2 8.453875 1.3292801 4 22 26.34039
#> 255.7 15.423064 1.2552580 18 14 30.58975
#> 314.12 12.222308 1.2439983 10 18 28.17335
#> 317.6 9.592839 1.5664007 8 9 35.32583
#> 319.20 28.986374 1.4009197 27 3 38.75767
#> 320.16 13.835583 1.1465427 13 21 26.34808
#> 342.15 13.025230 1.1535122 12 24 26.01336
#> 346.2 21.230207 0.9647024 23 25 23.84175
#> 351.26 17.269543 1.4078678 20 8 36.11581
#> 364.21 3.781576 2.1622181 2 10 34.05974
#> 402.7 1.191312 4.2342600 1 19 27.47748
#> 405.2 16.027557 1.1932619 19 16 28.98663
#> 406.12 13.989359 1.3499442 14 12 32.68323
#> 427.7 7.507408 1.7097474 3 7 36.19020
#> 450.3 14.270920 1.4588721 15 6 36.19602
#> 506.2 8.954538 1.5287986 6 11 33.26623
#> Canchan 15.138085 1.1431075 17 20 27.00126
#> Desiree 32.114860 0.6506029 28 28 16.15569
#> Unica 22.343936 1.4529998 26 2 39.10400