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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