Figures (2)  Tables (7)

    • Figure 1. 

      Selection and estimation accuracy of $ m_k(\cdot) $ for discrete $ {\boldsymbol{G}} $.

    • Figure 2. 

      Plot of environmental mixture index effect on birth weight.

    • $ \tau $ $ m(\cdot) $ p = 50 p = 100
      Oracle/% IMSE (Model) IMSE (Oracle) Oracle/% IMSE (Model) IMSE (Oracle)
      0.25 $ m_0(.) $ 100.0% 2.78E+00 2.96E−02 100.0% 2.67E+00 2.97E−02
      $ m_1(.) $ 100.0% 8.16E−02 6.16E−03 100.0% 7.07E−02 6.16E−03
      $ m_2(.) $ 100.0% 9.59E−02 1.30E−02 100.0% 7.80E−02 1.31E−02
      $ m_3(.) $ 100.0% 2.58E−02 8.77E−04 100.0% 2.19E−02 9.71E−04
      $ m_4(.) $ 100.0% 2.56E−02 1.02E−03 99.8% 2.12E−02 9.80E−04
      Zero 88.8% 1.03E−02 0 90.6% 7.27E−03 0
      0.5 $ m_0(.) $ 100.0% 7.49E−02 2.88E−02 100.0% 7.70E−02 2.91E−02
      $ m_1(.) $ 100.0% 4.98E−02 5.07E−03 100.0% 4.89E−02 5.31E−03
      $ m_2(.) $ 100.0% 5.72E−02 1.22E−02 100.0% 5.77E−02 1.23E−02
      $ m_3(.) $ 99.7% 1.25E−03 7.90E−04 99.9% 1.30E−03 7.90E−04
      $ m_4(.) $ 99.9% 1.50E−03 7.91E−04 99.8% 1.48E−03 8.29E−04
      Zero 98.7% 1.00E−04 0 99.1% 6.26E−05 0
      0.75 $ m_0(.) $ 100.0% 2.60E+00 3.01E−02 100.0% 2.60E+00 3.13E−02
      $ m_1(.) $ 100.0% 6.25E−02 6.08E−03 100.0% 6.17E−02 6.18E−03
      $ m_2(.) $ 100.0% 7.12E−02 1.35E−02 100.0% 7.14E−02 1.37E−02
      $ m_3(.) $ 99.9% 2.42E−02 8.58E−04 100.0% 2.18E−02 9.03E−04
      $ m_4(.) $ 99.9% 2.62E−02 9.93E−04 100.0% 2.20E−02 9.63E−04
      Zero 88.5% 1.05E−02 0 90.7% 7.19E−03 0

      Table 1. 

      Selection and estimation accuracy of $ m_k(\cdot) $ for continuous $ {\boldsymbol{G}} $.

    • $ \tau $$ \beta $p = 50p = 100
      Oracle/%MSE (Model)MSE (Oracle)Oracle/%MSE (Model)MSE (Oracle)
      0.25$ \beta_1 $100.0%1.20E−024.76E−05100.0%6.71E−034.36E−05
      $ \beta_2 $99.8%1.47E−024.75E−05100.0%7.69E−034.37E−05
      $ \beta_3 $99.9%7.15E−05099.9%5.81E−050
      $ \beta_4 $99.9%2.09E−04099.9%2.93E−040
      $ \beta_5 $99.9%9.60E−05099.9%9.53E−050
      0.5$ \beta_1 $100.0%5.29E−053.78E−05100.0%5.33E−053.54E−05
      $ \beta_2 $100.0%5.29E−053.78E−05100.0%5.34E−053.54E−05
      $ \beta_3 $98.6%1.52E−06098.3%1.23E−060
      $ \beta_4 $98.3%3.33E−06098.8%3.24E−060
      $ \beta_5 $98.7%1.85E−06098.9%1.45E−060
      0.75$ \beta_1 $100.0%4.40E−034.81E−05100.0%3.60E−034.89E−05
      $ \beta_2 $100.0%4.51E−034.80E−05100.0%3.49E−034.88E−05
      $ \beta_3 $95.3%4.14E−05096.3%1.87E−050
      $ \beta_4 $95.9%5.31E−05096.2%4.23E−050
      $ \beta_5 $95.8%3.85E−05095.8%2.32E−050

      Table 2. 

      Selection and estimation accuracy of $ {\boldsymbol{\beta}} $.

    • $ m(\cdot) $ function MAF of $ {\boldsymbol{G}}_k $
      $ m_0(u) = 2sin(2\pi u) $ −
      $ m_1(u)= 2cos(\pi u) + 2 $ 0.5
      $ m_2(u) = sin(2\pi u) + cos(\pi u) + 1 $ 0.5
      $ m_3(u)= 2cos(\pi u) + 2 $ 0.3
      $ m_4(u) = sin(2\pi u) + cos(\pi u) + 1 $ 0.3
      $ m_5(u)= 2cos(\pi u) + 2 $ 0.1
      $ m_6(u) = sin(2\pi u) + cos(\pi u) + 1 $ 0.1
      $ m_7(u)= 2 $ 0.5
      $ m_8(u)= 2 $ 0.3
      $ m_9(u)= 2 $ 0.1
      $ m_k(u)= 0, k>9 $ Unif (0.05, 0.5)

      Table 3. 

      Function $ m_k(\cdot) $ and the corresponding MAF for $ {\boldsymbol{G}}_k $.

    • $ \tau $ Type $ p = 50 $ $ p = 100 $
      Oracle/% IMSE (Model) IMSE (Oracle) Oracle/% IMSE (Model) IMSE (Oracle)
      0.25 $ m_0(\cdot) $ 100.0% 7.54E−01 2.47E−02 100.0% 7.98E−01 2.44E−02
      V 97.5% 1.44E−01 2.30E−02 91.4% 2.23E−01 2.32E−02
      C 94.2% 1.66E−02 3.19E−03 95.5% 1.66E−02 3.22E−03
      Z 93.9% 4.71E−03 0 96.0% 3.05E−03 0
      0.5 $ m_0(\cdot) $ 100.0% 4.84E−02 2.35E−02 100.0% 4.97E−02 2.30E−02
      V 99.9% 9.14E−02 2.00E−02 99.3% 9.84E−02 1.98E−02
      C 95.4% 7.52E−03 2.68E−03 95.5% 9.30E−03 2.76E−03
      Z 93.6% 2.87E−03 0 93.8% 2.87E−03 0
      0.75 $ m_0(\cdot) $ 100.0% 1.02E+00 2.53E−02 100.0% 1.09E+00 2.48E−02
      V 90.8% 2.29E−01 2.33E−02 84.1% 3.35E−01 2.30E−02
      C 96.7% 1.35E−02 3.19E−03 98.6% 1.26E−02 3.14E−03
      Z 96.8% 2.09E−03 0 98.5% 9.24E−04 0

      Table 4. 

      Selection and estimation accuracy for $ m_k(\cdot) $ with discrete $ {\boldsymbol{G}} $.

    • $ \tau $ $ \beta $ $ p = 50 $ $ p = 100 $
      Oracle/% MSE (Model) MSE (Oracle) Oracle/% MSE (Model) MSE (Oracle)
      0.25 $ \beta_1 $ 100.0% 3.98E−04 4.42E−05 100.0% 4.63E−04 4.75E−05
      $ \beta_2 $ 100.0% 4.04E−04 4.43E−05 100.0% 4.69E−04 4.74E−05
      $ \beta_3 $ 100.0% 0 0 100.0% 0 0
      $ \beta_4 $ 100.0% 0 0 100.0% 0 0
      $ \beta_5 $ 100.0% 0 0 100.0% 0 0
      0.5 $ \beta_1 $ 100.0% 5.20E−05 3.96E−05 100.0% 5.30E−05 3.97E−05
      $ \beta_2 $ 100.0% 5.21E−05 3.97E−05 100.0% 5.30E−05 3.97E−05
      $ \beta_3 $ 99.1% 2.27E−07 0 99.1% 1.00E−06 0
      $ \beta_4 $ 98.9% 3.43E−07 0 98.4% 8.12E−07 0
      $ \beta_5 $ 98.9% 5.45E−07 0 98.7% 5.87E−07 0
      0.75 $ \beta_1 $ 100.0% 2.75E−04 5.13E−05 100.0% 3.70E−04 4.69E−05
      $ \beta_2 $ 100.0% 2.84E−04 5.15E−05 100.0% 3.59E−04 4.69E−05
      $ \beta_3 $ 93.6% 2.52E−05 0 93.0% 2.26E−05 0
      $ \beta_4 $ 94.3% 2.23E−05 0 93.9% 1.80E−05 0
      $ \beta_5 $ 93.5% 3.46E−05 0 93.3% 3.08E−05 0

      Table 5. 

      Selection and estimation accuracy for $ {\boldsymbol{\beta}} $ with discrete $ {\boldsymbol{G}} $.

    • SNP ID $ \tau = 0.25 $ $ \tau = 0.50 $ $ \tau = 0.75 $
      rs13267049 0.0260 0 0
      rs2736860 0 0 −0.0290
      rs2142306 0 0.0720 0
      rs6986303 0.1129 0 0
      rs6990329 0.1411 0 0
      rs9643299 0 0 −0.0489
      rs7460764 0 0 −0.1077
      rs7831227 −0.0294 −0.0801 −0.1007

      Table 6. 

      Effect of SNPs in gene ST3GAL1.

    • $ \tau $ $ \beta_1 $ $ \beta_2 $ $ \beta_3 $
      0.25 0.272 0.707 0.653
      0.5 0.895 0.000 −0.445
      0.75 0.637 −0.288 −0.715

      Table 7. 

      Estimated loading parameters corresponding to gene ST3GAL1.