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Table 1 Expressions of effect size and the 95% confidence interval estimation for each model and set of change criteria

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
\( \widehat{{ \log}_a(Y)}=\widehat{\alpha}+\widehat{\beta}\cdotp { \log}_b(X) \)
Absolute change in Y for an absolute change of c units in X \( \begin{array}{c}\hfill c\cdotp \widehat{\beta}\hfill \\ {}\hfill \mathrm{c}\cdotp \left[\widehat{\beta}\pm 1.96\cdotp \mathrm{se}\left(\widehat{\beta}\right)\right]\hfill \end{array} \)
(5), (6)
\( \begin{array}{c}\hfill { \log}_b\left(1+\frac{c}{E\left[ X\right]}\right)\cdotp \widehat{\beta}\hfill \\ {}\hfill { \log}_b\left(1+\frac{c}{E\left[ X\right]}\right)\cdotp \left[\widehat{\beta}\pm 1.96\cdotp \mathrm{se}\left(\widehat{\beta}\right)\right]\hfill \end{array} \)
(2), (13), (14)
\( \begin{array}{c}\hfill \left({a}^{c\cdotp \widehat{\beta}}-1\right)\cdotp E\left[ Y\right]\hfill \\ {}\hfill \left({a}^{c\cdotp \left[\widehat{\beta}\pm 1.96\cdotp \mathrm{se}\left(\widehat{\beta}\right)\right]}-1\right)\cdotp E\left[ Y\right]\hfill \end{array} \)
(4), (22), (23)
\( \begin{array}{c}\hfill \left({a}^{\log_b\left(1+\frac{c}{E\left[ X\right]}\right)\cdotp \widehat{\beta}}-1\right)\cdotp E\left[ Y\right]\hfill \\ {}\hfill \left({a}^{\log_b\left(1+\frac{c}{E\left[ X\right]}\right)\cdotp \left[\widehat{\beta}\pm 1.96\cdotp \mathrm{se}\left(\widehat{\beta}\right)\right]}-1\right)\cdotp E\left[ Y\right]\hfill \end{array} \)
(2), (33), (34)
Absolute change in Y for a relative change of k times in X \( \begin{array}{c}\hfill \left( k-1\right)\cdotp \mathrm{E}\left[ X\right]\cdotp \widehat{\beta}\hfill \\ {}\hfill \left( k-1\right)\cdotp \mathrm{E}\left[ X\right]\cdotp \left[\widehat{\beta}\pm 1.96\cdotp \mathrm{se}\left(\widehat{\beta}\right)\right]\hfill \end{array} \)
(1), (5), (6)
\( \begin{array}{c}\hfill { \log}_b(k)\cdotp \widehat{\beta}\hfill \\ {}\hfill { \log}_b(k)\cdotp \left[\widehat{\beta}\pm 1.96\cdotp \mathrm{se}\left(\widehat{\beta}\right)\right]\hfill \end{array} \)
(13), (14)
\( \begin{array}{c}\hfill \left({a}^{\left( k-1\right)\cdotp \mathrm{E}\left[ X\right]\cdotp \widehat{\beta}}-1\right)\cdotp E\left[ Y\right]\hfill \\ {}\hfill \left({a}^{\left( k-1\right)\cdotp \mathrm{E}\left[ X\right]\cdotp \left[\widehat{\beta}\pm 1.96\cdotp \mathrm{se}\left(\widehat{\beta}\right)\right]}-1\right)\cdotp E\left[ Y\right]\hfill \end{array} \)
(1), (4), (26)
\( \begin{array}{c}\hfill \left({a}^{\log_b(k)\cdotp \widehat{\beta}}-1\right)\cdotp E\left[ Y\right]\hfill \\ {}\hfill \left({a}^{\log_b(k)\cdotp \left[\widehat{\beta}\pm 1.96\cdotp \mathrm{se}\left(\widehat{\beta}\right)\right]}-1\right)\cdotp E\left[ Y\right]\hfill \end{array} \)
(4), (29), (30)
Relative change in Y for an absolute change of c units in X \( \begin{array}{c}\hfill 1+\frac{c\cdotp \widehat{\beta}}{E\left[ Y\right]}\hfill \\ {}\hfill \frac{c}{E\left[ Y\right]}\left\{\frac{E\left[ Y\right]}{c}+\widehat{\beta}\pm 1.96\cdotp se\left(\widehat{\beta}\right)\right\}\hfill \end{array} \)
(3), (5), (6)
\( \begin{array}{c}\hfill 1+\frac{{ \log}_b\left(1+\frac{c}{E\left[ X\right]}\right)\cdotp \widehat{\beta}}{E\left[ Y\right]}\hfill \\ {}\hfill \frac{{ \log}_b\left(1+\frac{c}{E\left[ X\right]}\right)}{E\left[ Y\right]}\left\{\frac{E\left[ Y\right]}{{ \log}_b\left(1+\frac{c}{E\left[ X\right]}\right)}+\widehat{\beta}\pm 1.96\cdotp se\left(\widehat{\beta}\right)\right\}\hfill \end{array} \)
(2), (17), (19)
\( \begin{array}{c}\hfill {a}^{c\cdotp \widehat{\beta}}\hfill \\ {}\hfill {a}^{c\cdotp \left[\widehat{\beta}\pm 1.96\cdotp \mathrm{se}\left(\widehat{\beta}\right)\right]}\hfill \end{array} \)
(22), (23)
\( \begin{array}{c}\hfill {a}^{\log_b\left(1+\frac{c}{E\left[ X\right]}\right)\cdotp \widehat{\beta}}\hfill \\ {}\hfill {a}^{\log_b\left(1+\frac{c}{E\left[ X\right]}\right)\cdotp \left[\widehat{\beta}\pm 1.96\cdotp \mathrm{se}\left(\widehat{\beta}\right)\right]}\hfill \end{array} \)
(2), (29), (30)
Relative change in Y for a relative change of k times in X \( \begin{array}{c}\hfill 1+\frac{\left( k-1\right)\cdotp \mathrm{E}\left[ X\right]\cdotp \widehat{\beta}}{E\left[ Y\right]}\hfill \\ {}\hfill \frac{\left( k-1\right)\cdotp \mathrm{E}\left[ X\right]}{E\left[ Y\right]}\left\{\frac{E\left[ Y\right]}{\left( k-1\right)\cdotp \mathrm{E}\left[ X\right]}+\widehat{\beta}\pm 1.96\cdotp se\left(\widehat{\beta}\right)\right\}\hfill \end{array} \)
(1), (3), (10)
\( \begin{array}{c}\hfill 1+\frac{{ \log}_b(k)\cdotp \widehat{\beta}}{E\left[ Y\right]}\hfill \\ {}\hfill \frac{{ \log}_b(k)}{E\left[ Y\right]}\left\{\frac{E\left[ Y\right]}{{ \log}_b(k)}+\widehat{\beta}\pm 1.96\cdotp se\left(\widehat{\beta}\right)\right\}\hfill \end{array} \)
(3), (13), (14)
\( \begin{array}{c}\hfill {a}^{\left( k-1\right)\cdotp \mathrm{E}\left[ X\right]\cdotp \widehat{\beta}}\hfill \\ {}\hfill {a}^{\left( k-1\right)\cdotp \mathrm{E}\left[ X\right]\cdotp \left[\widehat{\beta}\pm 1.96\cdotp \mathrm{se}\left(\widehat{\beta}\right)\right]}\hfill \end{array} \)
(1), (22), (23)
\( \begin{array}{c}\hfill {a}^{\log_b(k)\cdotp \widehat{\beta}}\hfill \\ {}\hfill {a}^{\log_b(k)\cdotp \left[\widehat{\beta}\pm 1.96\cdotp \mathrm{se}\left(\widehat{\beta}\right)\right]}\hfill \end{array} \)
(29), (30)
  1. Note: Numbers in the bottom of cells indicate equations involved in derivation. Formulae (1, 2, 3 and 4) are found in the main text and (5) to (34) are found in Additional file 1