MASV.AMMI computes the Modified AMMI Stability Value (MASV)
(Zali et al. 2012; Ajay et al. 2019)
(Please see Note) from a modified formula of AMMI Stability Value
(ASV) (Purchase 1997)
. This formula
calculates AMMI stability value considering all significant interaction
principal components (IPCs) in the AMMI model. Using MASV, the Simultaneous
Selection Index for Yield and Stability (SSI) is also calculated according to
the argument ssi.method.
MASV.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 MASV values.
The computed values of simultaneous selection index for yield and stability.
The ranks of MASV 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 Modified AMMI Stability Value (\(MASV\)) (Ajay et al. 2019) is computed as follows:
\[MASV = \sqrt{\sum_{n=1}^{N'-1}\left (\frac{SSIPC_{n}}{SSIPC_{n+1}} \times PC_{n} \right )^2 + \left (PC_{N'} \right )^2}\]
Where, \(SSIPC_{1}\), \(SSIPC_{2}\), \(\cdots\), \(SSIPC_{n}\) are the sum of squares of the 1st, 2nd, ..., and \(n\)th IPC; and \(PC_{1}\), \(PC_{2}\), \(\cdots\), \(PC_{n}\) are the scores of 1st, 2nd, ..., and \(n\)th IPC.
In Zali et al. (2012) , the formula for both AMMI stability value (ASV) was found to be erroneous, when compared with the original publications (Purchase 1997; Purchase et al. 1999; Purchase et al. 2000) .
ASV (Zali et al. 2012) \[ASV = \sqrt{\left ( \frac{SSIPC_{1}}{SSIPC_{2}} \right ) \times (PC_{1})^2 + \left (PC_{2} \right )^2}\]
ASV (Purchase 1997; Purchase et al. 1999; Purchase et al. 2000) \[ASV = \sqrt{\left (\frac{SSIPC_{1}}{SSIPC_{2}} \times PC_{1} \right )^2 + \left (PC_{2} \right )^2}\]
The authors believe that the proposed Modified AMMI stability value (MASV)
in Zali et al. (2012)
is also
erroneous and have implemented the corrected one in MASV.AMMI
(Ajay et al. 2019)
.
MASV (Zali et al. 2012) \[MASV = \sqrt{\sum_{n=1}^{N'-1}\left ( \frac{SSIPC_{n}}{SSIPC_{n+1}} \right ) \times (PC_{n})^2 + \left (PC_{N'} \right )^2}\]
Ajay BC, Aravind J, Fiyaz RA, Kumar N, Lal C, Gangadhar K, Kona P, Dagla MC, Bera SK (2019).
“Rectification of modified AMMI stability value (MASV).”
Indian Journal of Genetics and Plant Breeding (The), 79, 726--731.
Purchase JL (1997).
Parametric analysis to describe genotype × environment interaction and yield stability in winter wheat.
Ph.D. Thesis, University of the Orange Free State.
Purchase JL, Hatting H, van Deventer CS (1999).
“The use of the AMMI model and AMMI stability value to describe genotype x environment interaction and yield stability in winter wheat (Triticum aestivum L.).”
In Proceedings of the Tenth Regional Wheat Workshop for Eastern, Central and Southern Africa, 14-18 September 1998.
University of Stellenbosch, South Africa.
Purchase JL, Hatting H, van Deventer CS (2000).
“Genotype × environment interaction of winter wheat (Triticum aestivum L.) in South Africa: II. Stability analysis of yield performance.”
South African Journal of Plant and Soil, 17(3), 101--107.
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)
MASV.AMMI(model)
#> MASV SSI rMASV rY means
#> 102.18 4.7855876 42 19 23 26.31947
#> 104.22 3.8328358 25 12 13 31.28887
#> 121.31 4.0446758 29 14 15 30.10174
#> 141.28 5.1867706 21 20 1 39.75624
#> 157.26 7.6459224 29 24 5 36.95181
#> 163.9 4.4977055 43 16 27 21.41747
#> 221.19 2.1905344 31 5 26 22.98480
#> 233.11 3.1794345 26 9 17 28.66655
#> 235.6 8.4913020 29 25 4 38.63477
#> 241.2 2.0338659 26 4 22 26.34039
#> 255.7 4.7013868 32 18 14 30.58975
#> 314.12 3.1376678 26 8 18 28.17335
#> 317.6 2.3345492 15 6 9 35.32583
#> 319.20 8.6398087 30 27 3 38.75767
#> 320.16 3.8822326 34 13 21 26.34808
#> 342.15 3.6438425 34 10 24 26.01336
#> 346.2 5.3987165 47 22 25 23.84175
#> 351.26 5.4005468 31 23 8 36.11581
#> 364.21 1.4047546 12 2 10 34.05974
#> 402.7 0.3537818 20 1 19 27.47748
#> 405.2 4.1095727 31 15 16 28.98663
#> 406.12 5.3218165 33 21 12 32.68323
#> 427.7 2.4124676 14 7 7 36.19020
#> 450.3 4.6608954 23 17 6 36.19602
#> 506.2 1.9330143 14 3 11 33.26623
#> Canchan 3.6665608 31 11 20 27.00126
#> Desiree 9.0626072 56 28 28 16.15569
#> Unica 8.5447632 28 26 2 39.10400
# With n = 4 and default ssi.method (farshadfar)
MASV.AMMI(model, n = 4)
#> MASV SSI rMASV rY means
#> 102.18 4.8247593 39 16 23 26.31947
#> 104.22 4.0510711 23 10 13 31.28887
#> 121.31 5.2473236 34 19 15 30.10174
#> 141.28 5.9101338 23 22 1 39.75624
#> 157.26 8.7719153 30 25 5 36.95181
#> 163.9 4.5459209 41 14 27 21.41747
#> 221.19 2.7137861 29 3 26 22.98480
#> 233.11 3.7724279 26 9 17 28.66655
#> 235.6 8.6953084 28 24 4 38.63477
#> 241.2 2.8067193 26 4 22 26.34039
#> 255.7 5.0424601 32 18 14 30.58975
#> 314.12 3.4445298 25 7 18 28.17335
#> 317.6 2.8792321 14 5 9 35.32583
#> 319.20 8.8774217 30 27 3 38.75767
#> 320.16 4.1787768 33 12 21 26.34808
#> 342.15 4.1725070 35 11 24 26.01336
#> 346.2 5.8554350 46 21 25 23.84175
#> 351.26 6.4286626 31 23 8 36.11581
#> 364.21 1.6075453 12 2 10 34.05974
#> 402.7 0.5067415 20 1 19 27.47748
#> 405.2 4.2896919 29 13 16 28.98663
#> 406.12 5.3564283 32 20 12 32.68323
#> 427.7 2.9737174 13 6 7 36.19020
#> 450.3 4.7112537 21 15 6 36.19602
#> 506.2 3.6306466 19 8 11 33.26623
#> Canchan 4.8979104 37 17 20 27.00126
#> Desiree 9.1023670 56 28 28 16.15569
#> Unica 8.7835476 28 26 2 39.10400
# With default n (N') and ssi.method = "rao"
MASV.AMMI(model, ssi.method = "rao")
#> MASV SSI rMASV rY means
#> 102.18 4.7855876 1.4296717 19 23 26.31947
#> 104.22 3.8328358 1.7337655 12 13 31.28887
#> 121.31 4.0446758 1.6576851 14 15 30.10174
#> 141.28 5.1867706 1.8235808 20 1 39.75624
#> 157.26 7.6459224 1.5625443 24 5 36.95181
#> 163.9 4.4977055 1.3064192 16 27 21.41747
#> 221.19 2.1905344 1.9979910 5 26 22.98480
#> 233.11 3.1794345 1.7949089 9 17 28.66655
#> 235.6 8.4913020 1.5818054 25 4 38.63477
#> 241.2 2.0338659 2.2035784 4 22 26.34039
#> 255.7 4.7013868 1.5791422 18 14 30.58975
#> 314.12 3.1376678 1.7902786 8 18 28.17335
#> 317.6 2.3345492 2.3233562 6 9 35.32583
#> 319.20 8.6398087 1.5802761 27 3 38.75767
#> 320.16 3.8822326 1.5635888 13 21 26.34808
#> 342.15 3.6438425 1.5987650 10 24 26.01336
#> 346.2 5.3987165 1.2839782 22 25 23.84175
#> 351.26 5.4005468 1.6840095 23 8 36.11581
#> 364.21 1.4047546 3.0575043 2 10 34.05974
#> 402.7 0.3537818 8.6266993 1 19 27.47748
#> 405.2 4.1095727 1.6106479 15 16 28.98663
#> 406.12 5.3218165 1.5795802 21 12 32.68323
#> 427.7 2.4124676 2.3137009 7 7 36.19020
#> 450.3 4.6608954 1.7669921 17 6 36.19602
#> 506.2 1.9330143 2.4995588 3 11 33.26623
#> Canchan 3.6665608 1.6263253 11 20 27.00126
#> Desiree 9.0626072 0.8285565 28 28 16.15569
#> Unica 8.5447632 1.5950896 26 2 39.10400
# Changing the ratio of weights for Rao's SSI
MASV.AMMI(model, ssi.method = "rao", a = 0.43)
#> MASV SSI rMASV rY means
#> 102.18 4.7855876 1.1039112 19 23 26.31947
#> 104.22 3.8328358 1.3270288 12 13 31.28887
#> 121.31 4.0446758 1.2722512 14 15 30.10174
#> 141.28 5.1867706 1.5230171 20 1 39.75624
#> 157.26 7.6459224 1.3586506 24 5 36.95181
#> 163.9 4.4977055 0.9598080 16 27 21.41747
#> 221.19 2.1905344 1.2863130 5 26 22.98480
#> 233.11 3.1794345 1.3045842 9 17 28.66655
#> 235.6 8.4913020 1.3982110 25 4 38.63477
#> 241.2 2.0338659 1.4370799 4 22 26.34039
#> 255.7 4.7013868 1.2475474 18 14 30.58975
#> 314.12 3.1376678 1.2934270 8 18 28.17335
#> 317.6 2.3345492 1.6555805 6 9 35.32583
#> 319.20 8.6398087 1.3998375 27 3 38.75767
#> 320.16 3.8822326 1.1620273 13 21 26.34808
#> 342.15 3.6438425 1.1709323 10 24 26.01336
#> 346.2 5.3987165 0.9952142 22 25 23.84175
#> 351.26 5.4005468 1.3953434 23 8 36.11581
#> 364.21 1.4047546 1.9477337 2 10 34.05974
#> 402.7 0.3537818 4.2201550 1 19 27.47748
#> 405.2 4.1095727 1.2313006 15 16 28.98663
#> 406.12 5.3218165 1.2866435 21 12 32.68323
#> 427.7 2.4124676 1.6674932 7 7 36.19020
#> 450.3 4.6608954 1.4325166 17 6 36.19602
#> 506.2 1.9330143 1.6930696 3 11 33.26623
#> Canchan 3.6665608 1.2011435 11 20 27.00126
#> Desiree 9.0626072 0.6565359 28 28 16.15569
#> Unica 8.5447632 1.4126439 26 2 39.10400