Table 2 Simulation results when X and Y are normally distributed
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 | \( \widehat{\beta} \) = 0.995 se(\( \widehat{\beta} \)) = 0.054 | \( \widehat{\beta} \) = 9.587 se(\( \widehat{\beta} \)) = 0.520 | \( \widehat{\beta} \) = 0.020 se(\( \widehat{\beta} \)) = 0.001 | \( \widehat{\beta} \) = 0.193 se(\( \widehat{\beta} \)) = 0.011 | |
Absolute change in Y for an absolute change of c units in X | Effect size | 0.995 | 0.914 | 1.006 | 0.928 |
95% CI | (0.889–1.101) | (0.817–1.011) | (0.895–1.118) | (0.827–1.029) | |
Absolute change in Y for a relative change of k times in X | Effect size | 0.995 | 0.914 | 1.006 | 0.928 |
95% CI | (0.889–1.101) | (0.817–1.011) | (0.895–1.118) | (0.827–1.029) | |
Relative change in Y for an absolute change of c units in X | Effect size | 1.0199 | 1.0183 | 1.0201 | 1.0186 |
95% CI | (1.0178–1.0220) | (1.0163–1.0202) | (1.0179–1.0224) | (1.0165–1.0206) | |
Relative change in Y for a relative change of k times in X | Effect size | 1.0199 | 1.0183 | 1.0201 | 1.0186 |
95% CI | (1.0178–1.0220) | (1.0163–1.0202) | (1.0179–1.0224) | (1.0165–1.0206) | |