Correction factor

The following is a discussion note during the making of our paper:

Song, X. P., Lai, H. R., Wijedasa, L. S., Tan, P. Y., Edwards, P. J., & Richards, D. R. (2020). Height–diameter allometry for the management of city trees in the tropics. Environmental Research Letters, 15(11), 114017. doi.org/10.1088/1748-9326/abbbad

To correct for back-transformation bias in log-transformed allometric models, we multiply the predictions, $\hat{Y}$ by the correction factor, $$CF=e^{{\sigma }_v^2/2}=e^{\text{MSE}/2},$$ where MSE is the mean square error of a model. For the log–log model, the corrected prediction is thus

$$\begin{array}{rl}\hat{Y}& =e^a\log (X+1{)}^{b}CF\\ \log \hat{Y}& =a+b\log (\log (X+1))+\log CF.\end{array}$$

Previously I wrongly stated that the correction factor cannot be built into the log–log or exp equations for reporting, but this can actually be done. Rearranging the above we get:

$$\begin{array}{rl}\log \hat{Y}& =(a+\log CF)+b\log (\log (X+1))\\ & =a^{\prime }+b\log (\log (X+1)).\end{array}$$

We can then just report the newly defined $a^{\prime }=a+\log CF$, which already has the correction factor built in.

The same operation can be performed on the exponential model (and more easily so):

$$\begin{array}{rl}\hat{Y}& =e^{a+bX}CF\\ & =e^{a+bX+\log CF}\\ \log \hat{Y}& =a+bX+\log CF\\ & =(a+\log CF)+bX\\ & =a^{\prime }+bX,\end{array}$$ thus yielding the same reported $a^{\prime }=a+\log CF$.