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Table 3 Simulation results when X and Y have an asymmetric distribution

From: Standardizing effect size from linear regression models with log-transformed variables for meta-analysis

  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