Table 4 Simulation results when Y has 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 | \( \widehat{\beta} \) = 0.625 se(\( \widehat{\beta} \)) = 0.254 | \( \widehat{\beta} \) = 5.834 se(\( \widehat{\beta} \)) = 2.441 | \( \widehat{\beta} \) = 0.011 se(\( \widehat{\beta} \)) = 0.005 | \( \widehat{\beta} \) = 0.103 se(\( \widehat{\beta} \)) = 0.044 | |
Absolute change in Y for an absolute change of c units in X | Effect size | 0.625 | 0.557 | 0.551 | 0.493 |
95% CI | (0.128–1.122) | (0.101–1.013) | (0.100–1.006) | (0.080–0.909) | |
Absolute change in Y for a relative change of k times in X | Effect size | 0.625 | 0.557 | 0.551 | 0.493 |
95% CI | (0.128–1.122) | (0.101–1.013) | (0.100–1.006) | (0.080–0.909) | |
Relative change in Y for an absolute change of c units in X | Effect size | 1.0125 | 1.0111 | 1.0110 | 1.0099 |
95% CI | (1.0026–1.0224) | (1.0020–1.0203) | (1.0020–1.0201) | (1.0016–1.0182) | |
Relative change in Y for a relative change of k times in X | Effect size | 1.0125 | 1.0111 | 1.0110 | 1.0099 |
95% CI | (1.0026–1.0224) | (1.0020–1.0203) | (1.0020–1.0201) | (1.0016–1.0182) | |