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Description of data set simulated for the pig scenario example. Relative breeding values for ADG (BVuADG), number of piglets born alive (BVuTBA), and ratio of N excretion (BVuRNE) are shown for each population. Average values for overall preference simulated (OCPui) and corresponding average ranking values (RPi) are also shown for the scenarios simulated (i = 1) independence among traits; (i = 2) coalition between BVADG and BVRNE

 
Pig population BVuADG BVuTBA BVuRNE OCPu1 RP1 OCPu2 RP2
1 0.000 0.664 0.000 0.03 0.00 0.33 0.19
2 0.948 0.293 0.251 0.32 0.40 0.81 0.55
3 0.871 0.307 0.670 0.68 0.60 0.96 0.80
4 0.363 0.445 0.214 0.24 0.20 0.78 0.34
5 0.496 1.000 0.737 0.73 0.71 0.79 0.38
6 0.649 0.856 0.236 0.29 0.30 0.81 0.53
7 0.647 0.337 1.000 0.96 0.90 0.16 0.00
8 0.553 0.000 0.427 0.43 0.50 0.99 0.88
9 0.788 0.283 0.031 0.11 0.10 0.29 0.11
10 1.000 0.774 0.717 0.75 0.79 0.93 0.72



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Data set considered in the dairy cattle example

 
PTA1
Bull P F PL SCS DPR DCE DSB No_offs2
1 42 49 4.7 2.79 1.2 4.1 3.7 615
2 54 95 5 2.98 0.2 5 4.8 474
3 59 63 9.2 2.9 1.8 5.9 5 428
4 65 81 4.2 3.07 −1.1 6.2 6.4 396
5 44 59 5.1 2.97 0.9 4.5 6.9 388
6 44 64 6.4 2.94 1.4 3.3 3.2 322
7 55 65 7.7 2.88 2.6 5.9 7.4 289
8 53 89 8.1 2.83 1.6 5.6 5.1 281
9 57 102 4.6 2.88 −1 5.3 7.1 278
10 57 48 5.4 2.89 2.8 6.1 5.5 271
11 50 53 6.8 2.87 1.6 6.2 5.2 260
12 61 72 6.6 2.73 0.6 4.7 4.7 245
13 36 57 3.3 2.82 0.1 5.8 6.2 242
14 79 80 6.8 3 1.7 4.6 4.4 212
15 42 59 7.9 2.76 2.6 5.5 5 182
16 24 29 5.6 2.88 4.4 4.3 6.7 159
17 20 34 7.2 2.49 6 5.4 4.7 157
18 62 72 5.6 2.89 1.6 3.4 3.9 149
19 39 79 4.9 2.85 0.2 5.1 5.9 142
20 66 100 5.7 3.07 −0.4 7.2 6.3 141
21 50 66 7.5 2.7 2.7 4.6 4.1 140
22 55 100 6.8 3 0.8 4.9 4.6 134
23 27 70 4.7 2.82 0.6 6 6.5 134
24 2 7 0 2.74 −1 4 5.9 134
25 43 79 3.8 2.92 1.3 6.7 6.5 125
26 39 68 4 2.88 0.4 6.1 7.2 120
27 50 57 6.3 2.74 0.6 5.3 5.8 119
28 41 51 8.2 2.77 2.5 7.4 5.7 118
29 75 94 5.1 2.92 0.4 4.8 3.4 w114
30 51 72 4.8 2.91 1.3 6.9 5.8 108
31 48 64 3.9 2.74 1.1 5.2 4.5 107
32 51 76 4.1 2.89 −0.2 6.9 7.4 105
33 41 91 6.2 2.73 1.1 4.7 6.9 90
34 59 75 5 2.93 1.3 3.3 4.5 85
35 48 67 4.9 2.8 0.8 4 5.9 84
36 −19 −10 −0.6 3.14 −1.7 9.7 10.4 82
37 32 64 5.2 2.76 5.4 5.4 8 81
38 68 74 6.6 2.93 2.5 4.5 6.2 78
39 37 33 6.6 2.53 3.4 5.6 6.3 75
40 37 68 4.6 2.52 −0.6 6 6.4 67
41 32 44 5.8 2.86 2.3 5.9 6.8 67
42 34 42 0.2 2.87 −3.4 7.2 9.2 63
1Predicted transmitting ability (PTA) for milk protein (P), milk fat (F), productive life (PL), somatic cell score (SCS), daughter pregnancy rate (DPR), daughter calving ease (DCE), and daughter stillbirth (DSB).
2Number of offspring for 42 highly employed bulls over the period considered.



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Shapley values and interaction indexes estimated in the pig example when considering overall preference (OCPui) or ranking values (RPi) simulated assuming (i = 1) independence among traits or (i = 2) coalition between breeding values for ADG (BVADG) and breeding values for the ratio of N excretion (BVRNE)

 
k = 2
k = 1
Shapley value
Interaction index
Shapley value
Simulation Trait1 TBA RNE
OCPu1 ADG 0.082 −0.007 −0.004 0.091
TBA 0.030 −0.018 0.025
RNE 0.888 0.883
OCPu2 ADG 0.754 −0.568 0.489 0.570
TBA 0.229 −0.455 0.430
RNE 0.017 0.000
RP1 ADG 0.186 −0.062 −0.001 0.155
TBA 0.018 0.097 0.000
RNE 0.797 0.845
RP2 ADG 0.677 −0.161 0.817 0.414
TBA 0.068 −0.306 0.151
RNE 0.256 0.436
1TBA, number of piglets born alive; RNE, ratio of N excreted over N intake.



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Shapley values and interaction indexes estimated in the dairy cattle example

 
Interaction indexes
Trait1 Shapley values F PL -SCS DPR -DCE -DSB
P 0 0 0 0 0 0 0
F 0.0834 0 0 0.1667 0 0
PL 0 0 0 0 0
-SCS 0.4166 0.8333 0 0
DPR 0.5000 0 0
-DCE 0 0
-DSB 0
1P, milk protein; F, milk fat; PL, productive life; SCS, somatic cell score; DPR, daughter pregnancy rate; DCE, daughter calving ease; DSB, daughter stillbirth.