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.
Usage
DZ.AMMI(model, n, alpha = 0.05, ssi.method = c("farshadfar", "rao"), a = 1)Arguments
- model
The AMMI model (An object of class
AMMIgenerated byAMMI).- n
The number of principal components to be considered for computation. The default value is the number of significant IPCs.
- alpha
Type I error probability (Significance level) to be considered to identify the number of significant IPCs.
- ssi.method
The method for the computation of simultaneous selection index. Either
"farshadfar"or"rao"(SeeSSI).- a
The ratio of the weights given to the stability components for computation of SSI when
method = "rao"(SeeSSI).
Value
A data frame with the following columns:
- DZ
The DZ values.
- SSI
The computed values of simultaneous selection index for yield and stability.
- rDZ
The ranks of DZ values.
- rY
The ranks of the mean yield of genotypes.
- means
The mean yield of the genotypes.
The names of the genotypes are indicated as the row names of the data frame.
Details
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.
References
Zhang Z, Lu C, Xiang Z (1998). “Analysis of variety stability based on AMMI model.” Acta Agronomica Sinica, 24(3), 304–309.
Examples
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