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