. 2017 Mar 17;17:44. doi: 10.1186/s12874-017-0322-8

Table 1.

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

	Model A $\hat{Y} = \hat{α} + \hat{β} \cdot X$	Model B $\hat{Y} = \hat{α} + \hat{β} \cdot {log}_{b} (X)$	Model C $\hat{{log}_{a} (Y)} = \hat{α} + \hat{β} \cdot X$	Model D $\hat{{log}_{a} (Y)} = \hat{α} + \hat{β} \cdot {log}_{b} (X)$
Absolute change in Y for an absolute change of c units in X	$\begin{array}{c} c \cdot \hat{β} \\ c \cdot [\hat{β} \pm 1.96 \cdot s e (\hat{β})] \end{array}$ (5), (6)	$\begin{array}{c} {log}_{b} (1 + \frac{c}{E [X]}) \cdot \hat{β} \\ {log}_{b} (1 + \frac{c}{E [X]}) \cdot [\hat{β} \pm 1.96 \cdot s e (\hat{β})] \end{array}$ (2), (13), (14)	$\begin{array}{c} (a^{c \cdot \hat{β}} - 1) \cdot E [Y] \\ (a^{c \cdot [\hat{β} \pm 1.96 \cdot s e (\hat{β})]} - 1) \cdot E [Y] \end{array}$ (4), (22), (23)	$\begin{array}{c} (a^{{log}_{b} (1 + \frac{c}{E [X]}) \cdot \hat{β}} - 1) \cdot E [Y] \\ (a^{{log}_{b} (1 + \frac{c}{E [X]}) \cdot [\hat{β} \pm 1.96 \cdot s e (\hat{β})]} - 1) \cdot E [Y] \end{array}$ (2), (33), (34)
Absolute change in Y for a relative change of k times in X	$\begin{array}{c} (k - 1) \cdot E [X] \cdot \hat{β} \\ (k - 1) \cdot E [X] \cdot [\hat{β} \pm 1.96 \cdot s e (\hat{β})] \end{array}$ (1), (5), (6)	$\begin{array}{c} {log}_{b} (k) \cdot \hat{β} \\ {log}_{b} (k) \cdot [\hat{β} \pm 1.96 \cdot s e (\hat{β})] \end{array}$ (13), (14)	$\begin{array}{c} (a^{(k - 1) \cdot E [X] \cdot \hat{β}} - 1) \cdot E [Y] \\ (a^{(k - 1) \cdot E [X] \cdot [\hat{β} \pm 1.96 \cdot s e (\hat{β})]} - 1) \cdot E [Y] \end{array}$ (1), (4), (26)	$\begin{array}{c} (a^{{log}_{b} (k) \cdot \hat{β}} - 1) \cdot E [Y] \\ (a^{{log}_{b} (k) \cdot [\hat{β} \pm 1.96 \cdot s e (\hat{β})]} - 1) \cdot E [Y] \end{array}$ (4), (29), (30)
Relative change in Y for an absolute change of c units in X	$\begin{array}{c} 1 + \frac{c \cdot \hat{β}}{E [Y]} \\ \frac{c}{E [Y]} \{\frac{E [Y]}{c} + \hat{β} \pm 1.96 \cdot s e (\hat{β})\} \end{array}$ (3), (5), (6)	$\begin{array}{c} 1 + \frac{{log}_{b} (1 + \frac{c}{E [X]}) \cdot \hat{β}}{E [Y]} \\ \frac{{log}_{b} (1 + \frac{c}{E [X]})}{E [Y]} \{\frac{E [Y]}{{log}_{b} (1 + \frac{c}{E [X]})} + \hat{β} \pm 1.96 \cdot s e (\hat{β})\} \end{array}$ (2), (17), (19)	$\begin{array}{c} a^{c \cdot \hat{β}} \\ a^{c \cdot [\hat{β} \pm 1.96 \cdot s e (\hat{β})]} \end{array}$ (22), (23)	$\begin{array}{c} a^{{log}_{b} (1 + \frac{c}{E [X]}) \cdot \hat{β}} \\ a^{{log}_{b} (1 + \frac{c}{E [X]}) \cdot [\hat{β} \pm 1.96 \cdot s e (\hat{β})]} \end{array}$ (2), (29), (30)
Relative change in Y for a relative change of k times in X	$\begin{array}{c} 1 + \frac{(k - 1) \cdot E [X] \cdot \hat{β}}{E [Y]} \\ \frac{(k - 1) \cdot E [X]}{E [Y]} \{\frac{E [Y]}{(k - 1) \cdot E [X]} + \hat{β} \pm 1.96 \cdot s e (\hat{β})\} \end{array}$ (1), (3), (10)	$\begin{array}{c} 1 + \frac{{log}_{b} (k) \cdot \hat{β}}{E [Y]} \\ \frac{{log}_{b} (k)}{E [Y]} \{\frac{E [Y]}{{log}_{b} (k)} + \hat{β} \pm 1.96 \cdot s e (\hat{β})\} \end{array}$ (3), (13), (14)	$\begin{array}{c} a^{(k - 1) \cdot E [X] \cdot \hat{β}} \\ a^{(k - 1) \cdot E [X] \cdot [\hat{β} \pm 1.96 \cdot s e (\hat{β})]} \end{array}$ (1), (22), (23)	$\begin{array}{c} a^{{log}_{b} (k) \cdot \hat{β}} \\ a^{{log}_{b} (k) \cdot [\hat{β} \pm 1.96 \cdot s e (\hat{β})]} \end{array}$ (29), (30)

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