| X1 X2 |
| X1 | Left-skewed covariate |
| X2 | Right- skewed covariate |
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| D1 binary |
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| (skewness, kurtosis) | Mean | SD | (S, K) Coefficient | SE | Power | Mean | SD | (S, K) Coefficient | SE | Power | Coefficient SE | SE | Power |
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| 70 | 8 | β1 = 0.05 |
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| 160 | 25 | β2 = 0.02 |
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| β2 = 0.83 |
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| Correlation of (X1, X2) = 0.0 |
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| (0.0, 0.0) (0.0, 0.0) | 70.0 | 8.0 | (0.0, 0.0) 0.052 | 0.019 | 0.774 | 160.0 | 24.9 | (0.0, 0.0) 0.020 | 0.006 | 0.931 | 0.860 | 0.304 | 0.801 |
| (−0.5, 0.5) (0.0, 0.0) | 70.0 | 8.0 | (−0.5, 0.5) 0.052 | 0.020 | 0.754 | 160.0 | 24.9 | (0.0, 0.0) 0.020 | 0.006 | 0.933 | 0.864 | 0.304 | 0.815 |
| (−1.0, 1.0) (0.0, 0.0) | 70.0 | 8.0 | (−1.0, 1.0) 0.053 | 0.021 | 0.727 | 160.0 | 24.9 | (0.0, 0.0) 0.020 | 0.006 | 0.933 | 0.864 | 0.304 | 0.818 |
| (0.0, 0.0) (0.4, 0.3) | 70.0 | 8.0 | (0.0, 0.0) 0.052 | 0.019 | 0.764 | 160.0 | 24.9 | (0.4, 0.3) 0.020 | 0.006 | 0.948 | 0.861 | 0.305 | 0.805 |
| (0.0, 0.0) (0.8, 0.6) | 70.0 | 8.0 | (0.0, 0.0) 0.052 | 0.019 | 0.766 | 160.0 | 24.9 | (0.8, 0.6) 0.020 | 0.006 | 0.957 | 0.861 | 0.306 | 0.805 |
| (−0.5, 0.5) (0.4, 0.3) | 70.0 | 8.0 | (−0.5, 0.5) 0.052 | 0.020 | 0.753 | 160.0 | 24.9 | (0.4, 0.3) 0.020 | 0.006 | 0.948 | 0.863 | 0.305 | 0.812 |
| (−1.0, 1.0) (0.8, 0.6) | 70.0 | 8.0 | (−1.0, 1.0) 0.053 | 0.021 | 0.723 | 160.0 | 24.9 | (0.8, 0.6) 0.020 | 0.006 | 0.955 | 0.861 | 0.305 | 0.816 |
| Correlation of (X1, X2) = 0.3 |
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| (0.0, 0.0) (0.0, 0.0) | 70.0 | 8.0 | (0.0, 0.0) 0.051 | 0.020 | 0.742 | 160.0 | 25.0 | (0.0, 0.0) 0.021 | 0.007 | 0.910 | 0.871 | 0.305 | 0.820 |
| (−0.5, 0.5) (0.0, 0.0) | 70.0 | 8.0 | (−0.5, 0.5) 0.052 | 0.021 | 0.704 | 160.0 | 25.0 | (0.0, 0.0) 0.021 | 0.007 | 0.910 | 0.871 | 0.304 | 0.830 |
| (−1.0, 1.0) (0.0, 0.0) | 70.0 | 8.0 | (−1.0, 1.0) 0.052 | 0.022 | 0.665 | 160.0 | 25.0 | (0.0, 0.0) 0.021 | 0.007 | 0.906 | 0.875 | 0.306 | 0.826 |
| (0.0, 0.0) (0.4, 0.3) | 70.0 | 8.0 | (0.0, 0.0) 0.052 | 0.020 | 0.747 | 160.0 | 25.0 | (0.4, 0.3) 0.020 | 0.006 | 0.922 | 0.873 | 0.303 | 0.827 |
| (0.0, 0.0) (0.8, 0.6) | 70.0 | 8.0 | (0.0, 0.0) 0.052 | 0.020 | 0.755 | 160.0 | 25.0 | (0.8, 0.6) 0.020 | 0.006 | 0.935 | 0.874 | 0.305 | 0.823 |
| (−0.5, 0.5) (0.4, 0.3) | 70.0 | 8.0 | (−0.5, 0.5) 0.052 | 0.021 | 0.704 | 160.0 | 25.0 | (0.4, 0.3) 0.020 | 0.006 | 0.926 | 0.874 | 0.305 | 0.829 |
| (−1.0, 1.0) (0.8, 0.6) | 70.0 | 8.0 | (−1.0, 1.0) 0.052 | 0.023 | 0.662 | 160.0 | 25.0 | (0.8, 0.6) 0.020 | 0.006 | 0.938 | 0.874 | 0.305 | 0.829 |
| Correlation of (X1, X2) = 0.6 |
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| (0.0, 0.0) (0.0, 0.0) | 70.0 | 8.0 | (0.0, 0.0) 0.052 | 0.024 | 0.587 | 160.0 | 25.0 | (0.0, 0.0) 0.021 | 0.008 | 0.779 | 0.873 | 0.305 | 0.817 |
| (−0.5, 0.5) (0.0, 0.0) | 70.0 | 8.0 | (−0.5, 0.5) 0.052 | 0.025 | 0.542 | 160.0 | 25.0 | (0.0, 0.0) 0.020 | 0.008 | 0.780 | 0.875 | 0.304 | 0.823 |
| (−0.1, 1.0) (0.0, 0.0) | 70.0 | 8.0 | (−1.0, 1.0) 0.052 | 0.027 | 0.508 | 160.0 | 25.0 | (0.0, 0.0) 0.021 | 0.008 | 0.805 | 0.875 | 0.303 | 0.819 |
| (0.0, 0.0) (0.4, 0.3) | 70.0 | 8.0 | (0.0, 0.0) 0.052 | 0.024 | 0.595 | 160.0 | 25.0 | (0.4, 0.3) 0.020 | 0.007 | 0.812 | 0.873 | 0.307 | 0.822 |
| (0.0, 0.0) (0.8, 0.6) | 70.0 | 8.0 | (0.0, 0.0) 0.052 | 0.024 | 0.608 | 160.0 | 25.0 | (0.8, 0.6) 0.020 | 0.007 | 0.844 | 0.873 | 0.308 | 0.815 |
| (−0.5, 0.5) (0.4, 0.3) | 70.0 | 8.0 | (−0.5, 0.5) 0.052 | 0.025 | 0.557 | 160.0 | 25.0 | (0.4, 0.3) 0.020 | 0.007 | 0.815 | 0.872 | 0.305 | 0.818 |
| (−1.0, 1.0) (0.8, 0.6) | 70.0 | 8.0 | (−1.0, 1.0) 0.052 | 0.026 | 0.530 | 160.0 | 25.0 | (0.8, 0.6) 0.020 | 0.007 | 0.878 | 0.869 | 0.305 | 0.812 |