# 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)
1. Note: c = 1 and k = 1.1 