SIPC.AMMI computes the Sums of the Absolute Value of the IPC Scores (ASI) (Sneller et al. 1997) considering all significant interaction principal components (IPCs) in the AMMI model. Using SIPC, the Simultaneous Selection Index for Yield and Stability (SSI) is also calculated according to the argument ssi.method.

SIPC.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:

SIPC

The SIPC values.

SSI

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

rSIPC

The ranks of SIPC 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 Sums of the Absolute Value of the IPC Scores (\(SIPC\)) (Sneller et al. 1997) is computed as follows:

\[SIPC = \sum_{n=1}^{N'} \left | \lambda_{n}^{0.5}\gamma_{in} \right |\]

OR

\[SIPC = \sum_{n=1}^{N'}\left | PC_{n} \right |\]

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 \(PC_{1}\), \(PC_{2}\), \(\cdots\), \(PC_{n}\) are the scores of 1st, 2nd, ..., and \(n\)th IPC.

The closer the SIPC scores are to zero, the more stable the genotypes are across test environments.

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)
SIPC.AMMI(model)
#>              SIPC SSI rSIPC rY    means
#> 102.18  2.9592568  39    16 23 26.31947
#> 104.22  2.2591593  22     9 13 31.28887
#> 121.31  3.3872806  33    18 15 30.10174
#> 141.28  4.3846248  23    22  1 39.75624
#> 157.26  5.4846596  31    26  5 36.95181
#> 163.9   2.6263670  38    11 27 21.41747
#> 221.19  2.0218098  32     6 26 22.98480
#> 233.11  2.1624442  24     7 17 28.66655
#> 235.6   4.8273551  28    24  4 38.63477
#> 241.2   2.0056410  27     5 22 26.34039
#> 255.7   3.6075128  34    20 14 30.58975
#> 314.12  2.4584089  28    10 18 28.17335
#> 317.6   1.8698826  12     3  9 35.32583
#> 319.20  5.9590451  31    28  3 38.75767
#> 320.16  2.7040109  33    12 21 26.34808
#> 342.15  2.9755899  41    17 24 26.01336
#> 346.2   3.9525017  46    21 25 23.84175
#> 351.26  4.5622439  31    23  8 36.11581
#> 364.21  0.7526264  12     2 10 34.05974
#> 402.7   0.2284995  20     1 19 27.47748
#> 405.2   2.7952381  29    13 16 28.98663
#> 406.12  2.8834753  27    15 12 32.68323
#> 427.7   2.0049278  11     4  7 36.19020
#> 450.3   2.8200387  20    14  6 36.19602
#> 506.2   2.2178470  19     8 11 33.26623
#> Canchan 3.5328212  39    19 20 27.00126
#> Desiree 5.8073242  55    27 28 16.15569
#> Unica   5.0654615  27    25  2 39.10400

# With n = 4 and default ssi.method (farshadfar)
SIPC.AMMI(model, n = 4)
#>              SIPC SSI rSIPC rY    means
#> 102.18  3.4466455  38    15 23 26.31947
#> 104.22  2.7007589  23    10 13 31.28887
#> 121.31  5.6097497  38    23 15 30.10174
#> 141.28  4.6372010  22    21  1 39.75624
#> 157.26  7.4500476  33    28  5 36.95181
#> 163.9   3.1338033  38    11 27 21.41747
#> 221.19  2.1363292  29     3 26 22.98480
#> 233.11  2.3911278  23     6 17 28.66655
#> 235.6   5.8474857  29    25  4 38.63477
#> 241.2   2.3056852  27     5 22 26.34039
#> 255.7   3.9276052  31    17 14 30.58975
#> 314.12  2.5182824  26     8 18 28.17335
#> 317.6   2.4516869  16     7  9 35.32583
#> 319.20  7.0781345  30    27  3 38.75767
#> 320.16  4.0249810  39    18 21 26.34808
#> 342.15  4.0957211  43    19 24 26.01336
#> 346.2   4.8622465  47    22 25 23.84175
#> 351.26  4.5974075  28    20  8 36.11581
#> 364.21  1.5318314  12     2 10 34.05974
#> 402.7   0.5893581  20     1 19 27.47748
#> 405.2   3.3068718  29    13 16 28.98663
#> 406.12  3.2694367  24    12 12 32.68323
#> 427.7   2.5358269  16     9  7 36.19020
#> 450.3   3.4327401  20    14  6 36.19602
#> 506.2   2.2644412  15     4 11 33.26623
#> Canchan 3.6100050  36    16 20 27.00126
#> Desiree 5.8538044  54    26 28 16.15569
#> Unica   5.7091275  26    24  2 39.10400

# With default n (N') and ssi.method = "rao"
SIPC.AMMI(model, ssi.method = "rao")
#>              SIPC       SSI rSIPC rY    means
#> 102.18  2.9592568 1.5124653    16 23 26.31947
#> 104.22  2.2591593 1.8772594     9 13 31.28887
#> 121.31  3.3872806 1.5531093    18 15 30.10174
#> 141.28  4.3846248 1.7378762    22  1 39.75624
#> 157.26  5.4846596 1.5578664    26  5 36.95181
#> 163.9   2.6263670 1.4355650    11 27 21.41747
#> 221.19  2.0218098 1.7071153     6 26 22.98480
#> 233.11  2.1624442 1.8300896     7 17 28.66655
#> 235.6   4.8273551 1.6608098    24  4 38.63477
#> 241.2   2.0056410 1.8242469     5 22 26.34039
#> 255.7   3.6075128 1.5341245    20 14 30.58975
#> 314.12  2.4584089 1.7062126    10 18 28.17335
#> 317.6   1.8698826 2.1873134     3  9 35.32583
#> 319.20  5.9590451 1.5886436    28  3 38.75767
#> 320.16  2.7040109 1.5751613    12 21 26.34808
#> 342.15  2.9755899 1.4988930    17 24 26.01336
#> 346.2   3.9525017 1.2672546    21 25 23.84175
#> 351.26  4.5622439 1.6019853    23  8 36.11581
#> 364.21  0.7526264 3.6831976     2 10 34.05974
#> 402.7   0.2284995 9.3696848     1 19 27.47748
#> 405.2   2.7952381 1.6378227    13 16 28.98663
#> 406.12  2.8834753 1.7371554    15 12 32.68323
#> 427.7   2.0049278 2.1457493     4  7 36.19020
#> 450.3   2.8200387 1.8667975    14  6 36.19602
#> 506.2   2.2178470 1.9576974     8 11 33.26623
#> Canchan 3.5328212 1.4284673    19 20 27.00126
#> Desiree 5.8073242 0.8601813    27 28 16.15569
#> Unica   5.0654615 1.6572552    25  2 39.10400

# Changing the ratio of weights for Rao's SSI
SIPC.AMMI(model, ssi.method = "rao", a = 0.43)
#>              SIPC       SSI rSIPC rY    means
#> 102.18  2.9592568 1.1395125    16 23 26.31947
#> 104.22  2.2591593 1.3887312     9 13 31.28887
#> 121.31  3.3872806 1.2272836    18 15 30.10174
#> 141.28  4.3846248 1.4861641    22  1 39.75624
#> 157.26  5.4846596 1.3566391    26  5 36.95181
#> 163.9   2.6263670 1.0153407    11 27 21.41747
#> 221.19  2.0218098 1.1612364     6 26 22.98480
#> 233.11  2.1624442 1.3197119     7 17 28.66655
#> 235.6   4.8273551 1.4321829    24  4 38.63477
#> 241.2   2.0056410 1.2739673     5 22 26.34039
#> 255.7   3.6075128 1.2281898    20 14 30.58975
#> 314.12  2.4584089 1.2572786    10 18 28.17335
#> 317.6   1.8698826 1.5970821     3  9 35.32583
#> 319.20  5.9590451 1.4034355    28  3 38.75767
#> 320.16  2.7040109 1.1670035    12 21 26.34808
#> 342.15  2.9755899 1.1279873    17 24 26.01336
#> 346.2   3.9525017 0.9880230    21 25 23.84175
#> 351.26  4.5622439 1.3600729    23  8 36.11581
#> 364.21  0.7526264 2.2167818     2 10 34.05974
#> 402.7   0.2284995 4.5396387     1 19 27.47748
#> 405.2   2.7952381 1.2429858    13 16 28.98663
#> 406.12  2.8834753 1.3544008    15 12 32.68323
#> 427.7   2.0049278 1.5952740     4  7 36.19020
#> 450.3   2.8200387 1.4754330    14  6 36.19602
#> 506.2   2.2178470 1.4600692     8 11 33.26623
#> Canchan 3.5328212 1.1160645    19 20 27.00126
#> Desiree 5.8073242 0.6701345    27 28 16.15569
#> Unica   5.0654615 1.4393751    25  2 39.10400