The Impact of Natural Constraints in Linear Regression of Log Transformed Response Variables

In linear regression, log transforming the response variable is the usual workaround regarding departures from the assumption of normality. However, the response variable is often subject to natural constraints, which can result in a truncated distribution of the residual errors on the log scale. In forestry, allometric relationships and tree growth are two typical examples a natural constraint; the response variable cannot be negative. Traditional least squares estimators do not account for constrained response variables. For this study, a modified maximum likelihood (MML) estimator that takes natural constraints into account was developed. This estimator was tested through a simulation study and showcased with black spruce tree diameter increment data. Results show that the ordinary least squares estimator underestimated large conditional expectations of the response variable on the original scale. In contrast, the MML estimator showed no evidence of bias for large sample sizes. Departures from distributional assumptions cannot be overlooked when the model is used for predictive purposes. Both Monte Carlo error propagation and prediction intervals rely on these assumptions. In this context, the MML estimator developed for this study can be used to properly propagate the errors and produce reliable prediction intervals.