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Table 5 Simulation results when X has 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 \( \widehat{\beta} \) = 0.288
se(\( \widehat{\beta} \)) = 0.015
\( \widehat{\beta} \) = 1.517
se(\( \widehat{\beta} \)) = 0.133
\( \widehat{\beta} \) = 0.005
se(\( \widehat{\beta} \)) = 0.0003
\( \widehat{\beta} \) = 0.028
se(\( \widehat{\beta} \)) = 0.003
Absolute change in Y for an absolute change of c units in X Effect size 0.288 0.145 0.263 0.133
95% CI (0.259–0.317) (0.120–0.169) (0.236–0.291) (0.109–0.156)
Absolute change in Y for a relative change of k times in X Effect size 0.288 0.145 0.263 0.133
95% CI (0.259–0.317) (0.120–0.169) (0.236–0.291) (0.109–0.156)
Relative change in Y for an absolute change of c units in X Effect size 1.0058 1.0029 1.0053 1.0027
95% CI (1.0052–1.0063) (1.0024–1.0034) (1.0047–1.0058) (1.0022–1.0031)
Relative change in Y for a relative change of k times in X Effect size 1.0058 1.0029 1.0053 1.0027
95% CI (1.0052–1.0063) (1.0024–1.0034) (1.0047–1.0058) (1.0022–1.0031)
  1. Note: c = 1 and k = 1.1