Supplementary Materials s20

1Supplementary Materials

QTL9 Fig. S1. Effect of the parameter t of the scaled inverse chi-square prior of variance parameter of QTL effect on the power (black line ) and mean squared error (blue line) in the detection of QTL7 (closely linked with QTL8) and QTL9 (small-effect QTL). Based on the results of the 7th and 9th QTL detection, -3.5 is the best choice for the t . 2 3 4 Fig. S2. Effect of the bias correction coefficient (0.2, 0.4, 0.6, 0.8, 1.0 and 1.2) of the proposed method in this study on the detection of the 7th to 9th, 17th and 19th 5 QTL in the Monte Carlo simulation study. A) Power for QTL detection; B) mean squared error; C) absolute value of QTL effect; D) false positive rate (%). 6 7 8 Fig S3. Comparison of the eBayes method with the extended eBayes method using simulated datasets in a Monte Carlo simulation study. 9 Table S1

10 Main-effect QTL detected by the BSPML method under various bias correction 11 coefficients in the barley data analysis

kj = 0.2 kj = 1.0 kj = 1.2 QTL Ratio, % † Position, cM ǂ Effect LOD Ratio, % Position, cM Effect LOD Ratio, % Position, cM Effect LOD

1 38 0.2(0.61) 0.52(0.22) 19.07 96 0.3(0.83) 0.56(0.11) 37.73 100 0.8(1.44) 0.58(0.10) 36.90

2 21 95.7(2.18) 0.35(0.11) 9.07 96 96.0(1.91) 0.52(0.17) 12.58 98 90.6(1.95) 0.52(0.17) 9.26

3 34 328.3(0.00) 0.19(0.11) 13.99 34 328.3(0.00) 0.19(0.06) 10.98 33 323.2(5.05) 0.19(0.05) 9.58

4 11 387.9(0.00) -0.19(0.13) 12.29 40 387.9(0.00) -0.27(0.08) 18.26 40 389.5(1.85) -0.26(0.07) 14.33

5 26 424.3(0.00) -0.18(0.07) 17.32 55 424.3(0.00) -0.25(0.07) 15.57 55 424.3(0.00) -0.25(0.06) 12.53

6 15 678.2(0.00) 0.11(0.07) 10.13 38 678.2(0.00) 0.23(0.08) 12.34 44 678.2(0.00) 0.24(0.08) 11.95

7 12 839.3(0.00) 0.15(0.09) 6.19 78 839.3(0.00) 0.25(0.08) 16.96 80 839.3(0.00) 0.27(0.08) 16.65

8 16 1041.4(1.13) 0.73(0.35) 21.55 100 1041.4(1.06) 1.09(0.15) 15.89 100 1041.1(2.18) 1.08(0.13) 15.82

12†: The ratio of the number of detected QTL to the total number of imputed samples (100). ǂ: cent-Morgan. The numbers in parentheses are the standard deviations of the position or the effect

13 of QTL. The selected value of kj is highlighted in the ratio column. The notation also applies to Table S2. 14 Table S2 15 Epistatic QTL detected by the BSPML method under various bias correction coefficients in the barley data analysis 16

kj = 0.2 kj = 1.0 kj = 1.2

Interaction Position, cM Position, cM Position, cM Ratio, % Effect LOD Ratio, % Effect LOD Ratio, % Effect LOD QTL1 QTL2 QTL1 QTL2 QTL1 QTL2

1 17 0.0(0.0) 294.2(0.0) 0.28(0.12) 11.43 74 0.0(0.0) 294.2(0.0) 0.25(0.10) 16.74 77 0.0(0.0) 294.2(0.0) 0.23(0.08) 14.33

2 16 13.9(0.0) 828.8(0.0) -0.17(0.11) 9.56 72 13.9(0.0) 828.8(0.0) -0.20(0.08) 25.52 78 13.9(0.0) 828.8(0.0) -0.21(0.08) 10.43

3 9 37.2(0.0) 151.1(0.0) 0.18(0.10) 9.64 45 37.2(0.0) 151.1(0.0) 0.27(0.08) 12.37 46 37.2(0.0) 151.1(0.0) 0.28(0.06) 22.49

4 18 186.2(0.0) 880.8(2.81) 0.21(0.3) 7.28 55 186.2(0.0) 883.8(0.0) 0.25(0.15) 14.80 52 186.2(0.0) 883.8(0.0) 0.21(0.10) 11.08

5 19 579.3(0.0) 887.5(0.0) -0.07(0.10) 6.99 50 579.3(0.0) 887.5(0.0) -0.18(0.06) 7.57 44 579.3(0.0) 887.5(0.0) -0.16(0.05) 9.34

6 10 640.6(0.0) 1044.8(1.41) -0.09(0.12) 8.51 53 640.6(0.0) 1044.6(1.42) -0.16(0.09) 12.20 54 640.6(0.0) 1044.5(1.09) -0.19(0.06) 12.71

7 4 728.5(0.0) 1273.5(1.15) -0.22(0.09) 17.19 14 728.5(0.0) 1274.1(0.0) -0.17(0.03) 9.82 15 728.5(0.0) 1274.1(0.0) -0.17(0.06) 8.35

17 See legends of Table S1 for the definition of Ratio (%) and others. 18