# 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)
1. Note: c = 1 and k = 1.1