AMGE.AMMI computes the Sum Across Environments of Genotype-Environment Interaction (GEI) Modelled by AMMI (AMGE) (Sneller et al. 1997) considering all significant interaction principal components (IPCs) in the AMMI model. Using AMGE, the Simultaneous Selection Index for Yield and Stability (SSI) is also calculated according to the argument ssi.method.

AMGE.AMMI(model, n, alpha = 0.05, ssi.method = c("farshadfar", "rao"), a = 1)

Arguments

model

The AMMI model (An object of class AMMI generated by AMMI).

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" (See SSI).

a

The ratio of the weights given to the stability components for computation of SSI when method = "rao" (See SSI).

Value

A data frame with the following columns:

AMGE

The AMGE values.

SSI

The computed values of simultaneous selection index for yield and stability.

rAMGE

The ranks of AMGE 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 Sum Across Environments of GEI Modelled by AMMI (\(AMGE\)) (Sneller et al. 1997) is computed as follows:

\[AMGE = \sum_{j=1}^{E} \sum_{n=1}^{N'} \lambda_{n} \gamma_{in} \delta_{jn}\]

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; \(\gamma_{in}\) is the eigenvector value for \(i\)th genotype; and \(\delta{jn}\) is the eigenvector value for the \(j\)th environment.

References

Sneller CH, Kilgore-Norquest L, Dombek D (1997). “Repeatability of yield stability statistics in soybean.” Crop Science, 37(2), 383--390.

See also

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)
AMGE.AMMI(model)
#>                  AMGE SSI rAMGE rY    means
#> 102.18   1.598721e-14  48    25 23 26.31947
#> 104.22  -8.881784e-15  20     7 13 31.28887
#> 121.31   1.643130e-14  41    26 15 30.10174
#> 141.28  -4.440892e-15  11    10  1 39.75624
#> 157.26   3.241851e-14  33    28  5 36.95181
#> 163.9    3.108624e-15  45    18 27 21.41747
#> 221.19   8.881784e-15  48    22 26 22.98480
#> 233.11  -1.476597e-14  22     5 17 28.66655
#> 235.6   -2.975398e-14   5     1  4 38.63477
#> 241.2    7.105427e-15  42    20 22 26.34039
#> 255.7   -1.598721e-14  18     4 14 30.58975
#> 314.12  -1.776357e-15  31    13 18 28.17335
#> 317.6    1.776357e-15  26    17  9 35.32583
#> 319.20   8.437695e-15  24    21  3 38.75767
#> 320.16   1.154632e-14  45    24 21 26.34808
#> 342.15  -9.325873e-15  30     6 24 26.01336
#> 346.2   -3.552714e-15  36    11 25 23.84175
#> 351.26   1.110223e-15  24    16  8 36.11581
#> 364.21  -4.940492e-15  19     9 10 34.05974
#> 402.7   -4.163336e-16  33    14 19 27.47748
#> 405.2    8.881784e-16  31    15 16 28.98663
#> 406.12  -1.731948e-14  15     3 12 32.68323
#> 427.7   -2.553513e-15  19    12  7 36.19020
#> 450.3    1.021405e-14  29    23  6 36.19602
#> 506.2    6.439294e-15  30    19 11 33.26623
#> Canchan -7.993606e-15  28     8 20 27.00126
#> Desiree  1.754152e-14  55    27 28 16.15569
#> Unica   -2.042810e-14   4     2  2 39.10400

# With n = 4 and default ssi.method (farshadfar)
AMGE.AMMI(model, n = 4)
#>                  AMGE  SSI rAMGE rY    means
#> 102.18   1.643130e-14 48.0  25.0 23 26.31947
#> 104.22  -9.325873e-15 20.0   7.0 13 31.28887
#> 121.31   1.731948e-14 41.0  26.0 15 30.10174
#> 141.28  -4.218847e-15 11.5  10.5  1 39.75624
#> 157.26   3.019807e-14 33.0  28.0  5 36.95181
#> 163.9    2.664535e-15 45.0  18.0 27 21.41747
#> 221.19   8.271162e-15 48.0  22.0 26 22.98480
#> 233.11  -1.409983e-14 22.0   5.0 17 28.66655
#> 235.6   -2.797762e-14  5.0   1.0  4 38.63477
#> 241.2    6.883383e-15 42.0  20.0 22 26.34039
#> 255.7   -1.709743e-14 18.0   4.0 14 30.58975
#> 314.12  -2.664535e-15 31.0  13.0 18 28.17335
#> 317.6    2.220446e-15 26.0  17.0  9 35.32583
#> 319.20   7.549517e-15 24.0  21.0  3 38.75767
#> 320.16   1.243450e-14 45.0  24.0 21 26.34808
#> 342.15  -1.132427e-14 30.0   6.0 24 26.01336
#> 346.2   -4.440892e-15 34.0   9.0 25 23.84175
#> 351.26   1.110223e-15 23.0  15.0  8 36.11581
#> 364.21  -3.774758e-15 22.0  12.0 10 34.05974
#> 402.7   -9.159340e-16 33.0  14.0 19 27.47748
#> 405.2    1.165734e-15 32.0  16.0 16 28.98663
#> 406.12  -1.820766e-14 15.0   3.0 12 32.68323
#> 427.7   -4.218847e-15 17.5  10.5  7 36.19020
#> 450.3    9.992007e-15 29.0  23.0  6 36.19602
#> 506.2    6.522560e-15 30.0  19.0 11 33.26623
#> Canchan -6.994405e-15 28.0   8.0 20 27.00126
#> Desiree  1.743050e-14 55.0  27.0 28 16.15569
#> Unica   -2.220446e-14  4.0   2.0  2 39.10400

# With default n (N') and ssi.method = "rao"
AMGE.AMMI(model, ssi.method = "rao")
#>                  AMGE        SSI rAMGE rY    means
#> 102.18   1.598721e-14  -1.209920    25 23 26.31947
#> 104.22  -8.881784e-15   4.742740     7 13 31.28887
#> 121.31   1.643130e-14  -1.030703    26 15 30.10174
#> 141.28  -4.440892e-15   8.741371    10  1 39.75624
#> 157.26   3.241851e-14   0.184960    28  5 36.95181
#> 163.9    3.108624e-15  -9.937521    18 27 21.41747
#> 221.19   8.881784e-15  -2.973115    22 26 22.98480
#> 233.11  -1.476597e-14   3.173817     5 17 28.66655
#> 235.6   -2.975398e-14   2.370918     1  4 38.63477
#> 241.2    7.105427e-15  -3.794340    20 22 26.34039
#> 255.7   -1.598721e-14   3.065479     4 14 30.58975
#> 314.12  -1.776357e-15  19.531348    13 18 28.17335
#> 317.6    1.776357e-15 -17.460918    17  9 35.32583
#> 319.20   8.437695e-15  -2.654754    21  3 38.75767
#> 320.16   1.154632e-14  -2.004403    24 21 26.34808
#> 342.15  -9.325873e-15   4.393465     6 24 26.01336
#> 346.2   -3.552714e-15  10.083744    11 25 23.84175
#> 351.26   1.110223e-15 -28.602804    16  8 36.11581
#> 364.21  -4.940492e-15   7.802759     9 10 34.05974
#> 402.7   -4.163336e-16  80.310270    14 19 27.47748
#> 405.2    8.881784e-16 -36.280350    15 16 28.98663
#> 406.12  -1.731948e-14   2.974655     3 12 32.68323
#> 427.7   -2.553513e-15  14.127995    12  7 36.19020
#> 450.3    1.021405e-14  -2.056805    23  6 36.19602
#> 506.2    6.439294e-15  -4.049883    19 11 33.26623
#> Canchan -7.993606e-15   5.016556     8 20 27.00126
#> Desiree  1.754152e-14  -1.358068    27 28 16.15569
#> Unica   -2.042810e-14   2.893508     2  2 39.10400

# Changing the ratio of weights for Rao's SSI
AMGE.AMMI(model, ssi.method = "rao", a = 0.43)
#>                  AMGE          SSI rAMGE rY    means
#> 102.18   1.598721e-14  -0.03111319    25 23 26.31947
#> 104.22  -8.881784e-15   2.62088777     7 13 31.28887
#> 121.31   1.643130e-14   0.11624442    26 15 30.10174
#> 141.28  -4.440892e-15   4.49766702    10  1 39.75624
#> 157.26   3.241851e-14   0.76628938    28  5 36.95181
#> 163.9    3.108624e-15  -3.87508635    18 27 21.41747
#> 221.19   8.881784e-15  -0.85126241    22 26 22.98480
#> 233.11  -1.476597e-14   1.89751451     5 17 28.66655
#> 235.6   -2.975398e-14   1.73752955     1  4 38.63477
#> 241.2    7.105427e-15  -1.14202521    20 22 26.34039
#> 255.7   -1.598721e-14   1.88667228     4 14 30.58975
#> 314.12  -1.776357e-15   8.92208663    13 18 28.17335
#> 317.6    1.776357e-15  -6.85165762    17  9 35.32583
#> 319.20   8.437695e-15  -0.42122552    21  3 38.75767
#> 320.16   1.154632e-14  -0.37220928    24 21 26.34808
#> 342.15  -9.325873e-15   2.37265314     6 24 26.01336
#> 346.2   -3.552714e-15   4.77911338    11 25 23.84175
#> 351.26   1.110223e-15 -11.62798636    16  8 36.11581
#> 364.21  -4.940492e-15   3.98819325     9 10 34.05974
#> 402.7   -4.163336e-16  35.04409044    14 19 27.47748
#> 405.2    8.881784e-16 -15.06182868    15 16 28.98663
#> 406.12  -1.731948e-14   1.88652568     3 12 32.68323
#> 427.7   -2.553513e-15   6.74763968    12  7 36.19020
#> 450.3    1.021405e-14  -0.21171610    23  6 36.19602
#> 506.2    6.439294e-15  -1.12319038    19 11 33.26623
#> Canchan -7.993606e-15   2.65894277     8 20 27.00126
#> Desiree  1.754152e-14  -0.28371280    27 28 16.15569
#> Unica   -2.042810e-14   1.97096400     2  2 39.10400