DZ.AMMI computes the Zhang's D Parameter values or AMMI statistic
coefficient or AMMI distance or AMMI stability index
(\(\textrm{D}_{\textrm{z}}\))
(Zhang et al. 1998)
considering all significant
interaction principal components (IPCs) in the AMMI model. It is the distance
of IPC point from origin in space. Using
\(\textrm{D}_{\textrm{z}}\), the Simultaneous Selection Index for Yield
and Stability (SSI) is also calculated according to the argument
ssi.method.
DZ.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 DZ values.
The computed values of simultaneous selection index for yield and stability.
The ranks of DZ 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 Zhang's D Parameter value (\(D_{z}\)) (Zhang et al. 1998) is computed as follows:
\[D_{z} = \sqrt{\sum_{n=1}^{N'}\gamma_{in}^{2}}\]
Where, \(N'\) is the number of significant IPCs (number of IPC that were retained in the AMMI model via F tests); and \(\gamma_{in}\) is the eigenvector value for \(i\)th genotype.
Zhang Z, Lu C, Xiang Z (1998). “Analysis of variety stability based on AMMI model.” Acta Agronomica Sinica, 24(3), 304--309.
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)
DZ.AMMI(model)
#> DZ SSI rDZ rY means
#> 102.18 0.26393535 37 14 23 26.31947
#> 104.22 0.22971564 21 8 13 31.28887
#> 121.31 0.32031744 34 19 15 30.10174
#> 141.28 0.39838535 23 22 1 39.75624
#> 157.26 0.53822924 33 28 5 36.95181
#> 163.9 0.26659011 42 15 27 21.41747
#> 221.19 0.19563325 29 3 26 22.98480
#> 233.11 0.25167755 27 10 17 28.66655
#> 235.6 0.46581370 28 24 4 38.63477
#> 241.2 0.21481887 28 6 22 26.34039
#> 255.7 0.30862904 31 17 14 30.58975
#> 314.12 0.22603261 25 7 18 28.17335
#> 317.6 0.20224771 14 5 9 35.32583
#> 319.20 0.50675112 29 26 3 38.75767
#> 320.16 0.23280596 30 9 21 26.34808
#> 342.15 0.25989774 36 12 24 26.01336
#> 346.2 0.37125512 45 20 25 23.84175
#> 351.26 0.43805896 31 23 8 36.11581
#> 364.21 0.07409309 12 2 10 34.05974
#> 402.7 0.02004533 20 1 19 27.47748
#> 405.2 0.26238837 29 13 16 28.98663
#> 406.12 0.28179394 28 16 12 32.68323
#> 427.7 0.20176581 11 4 7 36.19020
#> 450.3 0.25465368 17 11 6 36.19602
#> 506.2 0.30899851 29 18 11 33.26623
#> Canchan 0.37201039 41 21 20 27.00126
#> Desiree 0.52005815 55 27 28 16.15569
#> Unica 0.48083049 27 25 2 39.10400
# With n = 4 and default ssi.method (farshadfar)
DZ.AMMI(model, n = 4)
#> DZ SSI rDZ rY means
#> 102.18 0.28722309 33 10 23 26.31947
#> 104.22 0.25160706 21 8 13 31.28887
#> 121.31 0.60785568 42 27 15 30.10174
#> 141.28 0.40268829 21 20 1 39.75624
#> 157.26 0.70597721 33 28 5 36.95181
#> 163.9 0.29151868 39 12 27 21.41747
#> 221.19 0.19743603 29 3 26 22.98480
#> 233.11 0.25722999 26 9 17 28.66655
#> 235.6 0.52269682 29 25 4 38.63477
#> 241.2 0.22585722 26 4 22 26.34039
#> 255.7 0.31747123 30 16 14 30.58975
#> 314.12 0.22646067 23 5 18 28.17335
#> 317.6 0.24329787 16 7 9 35.32583
#> 319.20 0.56961794 29 26 3 38.75767
#> 320.16 0.38533472 40 19 21 26.34808
#> 342.15 0.36788692 41 17 24 26.01336
#> 346.2 0.42725798 46 21 25 23.84175
#> 351.26 0.43813521 30 22 8 36.11581
#> 364.21 0.19569373 12 2 10 34.05974
#> 402.7 0.08624291 20 1 19 27.47748
#> 405.2 0.28808268 27 11 16 28.98663
#> 406.12 0.29573097 26 14 12 32.68323
#> 427.7 0.23651352 13 6 7 36.19020
#> 450.3 0.29177451 19 13 6 36.19602
#> 506.2 0.30918827 26 15 11 33.26623
#> Canchan 0.37244277 38 18 20 27.00126
#> Desiree 0.52017037 52 24 28 16.15569
#> Unica 0.50357109 25 23 2 39.10400
# With default n (N') and ssi.method = "rao"
DZ.AMMI(model, ssi.method = "rao")
#> DZ SSI rDZ rY means
#> 102.18 0.26393535 1.5536988 14 23 26.31947
#> 104.22 0.22971564 1.8193399 8 13 31.28887
#> 121.31 0.32031744 1.5545939 19 15 30.10174
#> 141.28 0.39838535 1.7570779 22 1 39.75624
#> 157.26 0.53822924 1.5459114 28 5 36.95181
#> 163.9 0.26659011 1.3869397 15 27 21.41747
#> 221.19 0.19563325 1.6878048 3 26 22.98480
#> 233.11 0.25167755 1.6641025 10 17 28.66655
#> 235.6 0.46581370 1.6538090 24 4 38.63477
#> 241.2 0.21481887 1.7134093 6 22 26.34039
#> 255.7 0.30862904 1.5922105 17 14 30.58975
#> 314.12 0.22603261 1.7307783 7 18 28.17335
#> 317.6 0.20224771 2.0595024 5 9 35.32583
#> 319.20 0.50675112 1.6259792 26 3 38.75767
#> 320.16 0.23280596 1.6476346 9 21 26.34808
#> 342.15 0.25989774 1.5545233 12 24 26.01336
#> 346.2 0.37125512 1.2718506 20 25 23.84175
#> 351.26 0.43805896 1.5966462 23 8 36.11581
#> 364.21 0.07409309 3.5881882 2 10 34.05974
#> 402.7 0.02004533 10.0539968 1 19 27.47748
#> 405.2 0.26238837 1.6447637 13 16 28.98663
#> 406.12 0.28179394 1.7171135 16 12 32.68323
#> 427.7 0.20176581 2.0898536 4 7 36.19020
#> 450.3 0.25465368 1.9010808 11 6 36.19602
#> 506.2 0.30899851 1.6787677 18 11 33.26623
#> Canchan 0.37201039 1.3738642 21 20 27.00126
#> Desiree 0.52005815 0.8797586 27 28 16.15569
#> Unica 0.48083049 1.6568004 25 2 39.10400
# Changing the ratio of weights for Rao's SSI
DZ.AMMI(model, ssi.method = "rao", a = 0.43)
#> DZ SSI rDZ rY means
#> 102.18 0.26393535 1.1572429 14 23 26.31947
#> 104.22 0.22971564 1.3638258 8 13 31.28887
#> 121.31 0.32031744 1.2279220 19 15 30.10174
#> 141.28 0.39838535 1.4944208 22 1 39.75624
#> 157.26 0.53822924 1.3514985 28 5 36.95181
#> 163.9 0.26659011 0.9944318 15 27 21.41747
#> 221.19 0.19563325 1.1529329 3 26 22.98480
#> 233.11 0.25167755 1.2483375 10 17 28.66655
#> 235.6 0.46581370 1.4291726 24 4 38.63477
#> 241.2 0.21481887 1.2263072 6 22 26.34039
#> 255.7 0.30862904 1.2531668 17 14 30.58975
#> 314.12 0.22603261 1.2678419 7 18 28.17335
#> 317.6 0.20224771 1.5421234 5 9 35.32583
#> 319.20 0.50675112 1.4194898 26 3 38.75767
#> 320.16 0.23280596 1.1981670 9 21 26.34808
#> 342.15 0.25989774 1.1519083 12 24 26.01336
#> 346.2 0.37125512 0.9899993 20 25 23.84175
#> 351.26 0.43805896 1.3577771 23 8 36.11581
#> 364.21 0.07409309 2.1759278 2 10 34.05974
#> 402.7 0.02004533 4.8338929 1 19 27.47748
#> 405.2 0.26238837 1.2459704 13 16 28.98663
#> 406.12 0.28179394 1.3457828 16 12 32.68323
#> 427.7 0.20176581 1.5712389 4 7 36.19020
#> 450.3 0.25465368 1.4901748 11 6 36.19602
#> 506.2 0.30899851 1.3401295 18 11 33.26623
#> Canchan 0.37201039 1.0925852 21 20 27.00126
#> Desiree 0.52005815 0.6785528 27 28 16.15569
#> Unica 0.48083049 1.4391795 25 2 39.10400