Table 3 Simulation results when X and Y have an asymmetric distribution
Model A \( \widehat{Y}=\widehat{\alpha}+\widehat{\beta}\cdotp X \) | Model B \( \widehat{Y}=\widehat{\alpha}+\widehat{\beta}\cdotp { \log}_b(X) \) | Model C \( \widehat{{ \log}_a(Y)}=\widehat{\alpha}+\widehat{\beta}\cdotp X \) | Model D \( \widehat{{ \log}_a(Y)}=\widehat{\alpha}+\widehat{\beta}\cdotp { \log}_b(X) \) | ||
|---|---|---|---|---|---|
Beta-hat coefficient and standard error from regression model | \( \widehat{\beta} \) = 0.997 se(\( \widehat{\beta} \)) = 0.009 | \( \widehat{\beta} \) = 6.071 se(\( \widehat{\beta} \)) = 0.213 | \( \widehat{\beta} \) = 0.018 se(\( \widehat{\beta} \)) = 0.0002 | \( \widehat{\beta} \) = 0.115 se(\( \widehat{\beta} \)) = 0.003 | |
Absolute change in Y for an absolute change of c units in X | Effect size | 0.997 | 0.579 | 0.894 | 0.551 |
95% CI | (0.980–1.014) | (0.539–0.618) | (0.874–0.915) | (0.518–0.584) | |
Absolute change in Y for a relative change of k times in X | Effect size | 0.997 | 0.579 | 0.894 | 0.551 |
95% CI | (0.980–1.014) | (0.539–0.618) | (0.874–0.915) | (0.518–0.584) | |
Relative change in Y for an absolute change of c units in X | Effect size | 1.0199 | 1.0116 | 1.0179 | 1.0110 |
95% CI | (1.0196–1.0203) | (1.0108–1.0124) | (1.0175–1.0183) | (1.0100–1.0117) | |
Relative change in Y for a relative change of k times in X | Effect size | 1.0199 | 1.0116 | 1.0179 | 1.0110 |
95% CI | (1.0196–1.0203) | (1.0108–1.0124) | (1.0175–1.0183) | (1.0100–1.0117) | |