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)
| model | The AMMI model (An object of class |
|---|---|
| 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 |
| a | The ratio of the weights given to the stability components for
computation of SSI when |
A data frame with the following columns:
The AMGE values.
The computed values of simultaneous selection index for yield and stability.
The ranks of AMGE values.
The ranks of the mean yield of genotypes.
The mean yield of the genotypes.
The Sum Across Environments of GEI Modelled by AMMI (AMGE) is computed as follows:
AMGE = ∑Ej=1∑N'n=1 λnγinδjn
Where, N' is the number of significant IPCs (number of IPC that were retained in the AMMI model via F tests); λn is the singular value for nth IPC and correspondingly λ2n is its eigen value; γin is the eigenvector value for ith genotype; and δjn is the eigenvector value for the jth environment.
Sneller CH, Kilgore-Norquest L, Dombek D (1997). “Repeatability of yield stability statistics in soybean.” Crop Science, 37(2), 383--390. doi: 10.2135/cropsci1997.0011183X003700020013x .
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 -8.659740e-15 28.0 5.0 23 26.31947 #> 104.22 1.110223e-15 28.0 15.0 13 31.28887 #> 121.31 4.440892e-16 29.0 14.0 15 30.10174 #> 141.28 1.021405e-14 27.5 26.5 1 39.75624 #> 157.26 2.220446e-15 22.5 17.5 5 36.95181 #> 163.9 -1.243450e-14 28.0 1.0 27 21.41747 #> 221.19 -4.440892e-15 35.0 9.0 26 22.98480 #> 233.11 2.275957e-15 36.0 19.0 17 28.66655 #> 235.6 5.773160e-15 26.5 22.5 4 38.63477 #> 241.2 -5.329071e-15 30.0 8.0 22 26.34039 #> 255.7 -3.774758e-15 24.0 10.0 14 30.58975 #> 314.12 5.773160e-15 40.5 22.5 18 28.17335 #> 317.6 2.220446e-15 26.5 17.5 9 35.32583 #> 319.20 1.731948e-14 31.0 28.0 3 38.75767 #> 320.16 -6.217249e-15 27.0 6.0 21 26.34808 #> 342.15 -2.442491e-15 35.0 11.0 24 26.01336 #> 346.2 -1.110223e-14 28.0 3.0 25 23.84175 #> 351.26 1.021405e-14 34.5 26.5 8 36.11581 #> 364.21 1.415534e-15 26.0 16.0 10 34.05974 #> 402.7 -3.885781e-16 31.0 12.0 19 27.47748 #> 405.2 -1.088019e-14 20.0 4.0 16 28.98663 #> 406.12 3.108624e-15 32.0 20.0 12 32.68323 #> 427.7 1.110223e-16 20.0 13.0 7 36.19020 #> 450.3 6.439294e-15 30.0 24.0 6 36.19602 #> 506.2 -5.773160e-15 18.0 7.0 11 33.26623 #> Canchan 9.325873e-15 45.0 25.0 20 27.00126 #> Desiree -1.132427e-14 30.0 2.0 28 16.15569 #> Unica 5.329071e-15 23.0 21.0 2 39.10400# With n = 4 and default ssi.method (farshadfar) AMGE.AMMI(model, n = 4)#> AMGE SSI rAMGE rY means #> 102.18 -9.992007e-15 28 5 23 26.31947 #> 104.22 2.886580e-15 31 18 13 31.28887 #> 121.31 -3.996803e-15 25 10 15 30.10174 #> 141.28 9.992007e-15 27 26 1 39.75624 #> 157.26 8.881784e-15 29 24 5 36.95181 #> 163.9 -1.065814e-14 29 2 27 21.41747 #> 221.19 -4.718448e-15 35 9 26 22.98480 #> 233.11 1.387779e-15 32 15 17 28.66655 #> 235.6 3.108624e-15 23 19 4 38.63477 #> 241.2 -6.550316e-15 29 7 22 26.34039 #> 255.7 -3.774758e-15 25 11 14 30.58975 #> 314.12 6.217249e-15 41 23 18 28.17335 #> 317.6 0.000000e+00 22 13 9 35.32583 #> 319.20 2.087219e-14 31 28 3 38.75767 #> 320.16 -1.021405e-14 25 4 21 26.34808 #> 342.15 2.053913e-15 41 17 24 26.01336 #> 346.2 -7.993606e-15 31 6 25 23.84175 #> 351.26 9.159340e-15 33 25 8 36.11581 #> 364.21 -8.881784e-16 22 12 10 34.05974 #> 402.7 2.983724e-16 33 14 19 27.47748 #> 405.2 -1.326717e-14 17 1 16 28.98663 #> 406.12 3.552714e-15 32 20 12 32.68323 #> 427.7 1.887379e-15 23 16 7 36.19020 #> 450.3 5.107026e-15 27 21 6 36.19602 #> 506.2 -5.592748e-15 19 8 11 33.26623 #> Canchan 1.010303e-14 47 27 20 27.00126 #> Desiree -1.043610e-14 31 3 28 16.15569 #> Unica 5.773160e-15 24 22 2 39.10400# With default n (N') and ssi.method = "rao" AMGE.AMMI(model, ssi.method = "rao")#> AMGE SSI rAMGE rY means #> 102.18 -8.659740e-15 0.5673198 5.0 23 26.31947 #> 104.22 1.110223e-15 3.2887624 15.0 13 31.28887 #> 121.31 4.440892e-16 6.6529106 14.0 15 30.10174 #> 141.28 1.021405e-14 1.5428597 26.5 1 39.75624 #> 157.26 2.220446e-15 2.3391212 17.5 5 36.95181 #> 163.9 -1.243450e-14 0.4957785 1.0 27 21.41747 #> 221.19 -4.440892e-15 0.1822906 9.0 26 22.98480 #> 233.11 2.275957e-15 2.0413097 19.0 17 28.66655 #> 235.6 5.773160e-15 1.6959735 22.5 4 38.63477 #> 241.2 -5.329071e-15 0.3862254 8.0 22 26.34039 #> 255.7 -3.774758e-15 0.3301705 10.0 14 30.58975 #> 314.12 5.773160e-15 1.3548726 22.5 18 28.17335 #> 317.6 2.220446e-15 2.2861050 17.5 9 35.32583 #> 319.20 1.731948e-14 1.4091383 28.0 3 38.75767 #> 320.16 -6.217249e-15 0.4539931 6.0 21 26.34808 #> 342.15 -2.442491e-15 -0.1829870 11.0 24 26.01336 #> 346.2 -1.110223e-14 0.5505176 3.0 25 23.84175 #> 351.26 1.021405e-14 1.4241614 26.5 8 36.11581 #> 364.21 1.415534e-15 2.8898091 16.0 10 34.05974 #> 402.7 -3.885781e-16 -5.5857093 12.0 19 27.47748 #> 405.2 -1.088019e-14 0.7136396 4.0 16 28.98663 #> 406.12 3.108624e-15 1.8758598 20.0 12 32.68323 #> 427.7 1.110223e-16 23.8657048 13.0 7 36.19020 #> 450.3 6.439294e-15 1.5713258 24.0 6 36.19602 #> 506.2 -5.773160e-15 0.6484020 7.0 11 33.26623 #> Canchan 9.325873e-15 1.1504601 25.0 20 27.00126 #> Desiree -1.132427e-14 0.3043571 2.0 28 16.15569 #> Unica 5.329071e-15 1.7476282 21.0 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 -8.659740e-15 0.7330999 5.0 23 26.31947 #> 104.22 1.110223e-15 1.9956774 15.0 13 31.28887 #> 121.31 4.440892e-16 3.4201982 14.0 15 30.10174 #> 141.28 1.021405e-14 1.4023070 26.5 1 39.75624 #> 157.26 2.220446e-15 1.6925787 17.5 5 36.95181 #> 163.9 -1.243450e-14 0.6112325 1.0 27 21.41747 #> 221.19 -4.440892e-15 0.5055618 9.0 26 22.98480 #> 233.11 2.275957e-15 1.4105366 19.0 17 28.66655 #> 235.6 5.773160e-15 1.4473033 22.5 4 38.63477 #> 241.2 -5.329071e-15 0.6556181 8.0 22 26.34039 #> 255.7 -3.774758e-15 0.7104896 10.0 14 30.58975 #> 314.12 5.773160e-15 1.1062024 22.5 18 28.17335 #> 317.6 2.220446e-15 1.6395625 17.5 9 35.32583 #> 319.20 1.731948e-14 1.3262482 28.0 3 38.75767 #> 320.16 -6.217249e-15 0.6849012 6.0 21 26.34808 #> 342.15 -2.442491e-15 0.4047789 11.0 24 26.01336 #> 346.2 -1.110223e-14 0.6798261 3.0 25 23.84175 #> 351.26 1.021405e-14 1.2836086 26.5 8 36.11581 #> 364.21 1.415534e-15 1.8756248 16.0 10 34.05974 #> 402.7 -3.885781e-16 -1.8911807 12.0 19 27.47748 #> 405.2 -1.088019e-14 0.8455870 4.0 16 28.98663 #> 406.12 3.108624e-15 1.4140438 20.0 12 32.68323 #> 427.7 1.110223e-16 10.9348548 13.0 7 36.19020 #> 450.3 6.439294e-15 1.3483801 24.0 6 36.19602 #> 506.2 -5.773160e-15 0.8970722 7.0 11 33.26623 #> Canchan 9.325873e-15 0.9965214 25.0 20 27.00126 #> Desiree -1.132427e-14 0.4311301 2.0 28 16.15569 #> Unica 5.329071e-15 1.4782355 21.0 2 39.10400