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, \[CF = e^{\sigma^2_v / 2} = e^{\text{MSE}/2}, \] where MSE is the mean square error of a model. For the log–log model, the corrected prediction is thus

\[\begin{align} \hat{Y} &= e^a \log(X+1)^b CF \\ \log \hat{Y} &= a + b \log (\log (X+1)) + \log{CF}. \end{align}\]

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:

\[\begin{align} \log \hat{Y} &= (a + \log{CF}) + b \log (\log (X+1)) \\ &= a' + b \log (\log (X+1)). \end{align}\]

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

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

\[\begin{align} \hat{Y} &= e^{a + bX} CF \\ &= e^{a + bX + \log{CF}} \\ \log \hat{Y} &= a + bX + \log{CF} \\ &= (a + \log{CF}) + bX \\ &= a' + bX, \end{align}\] thus yielding the same reported \(a' = a + \log{CF}\).