An examination of the lognormal and box and cox family of transformations in fitting environmental data

This paper investigates the fit of the lognormal model and Box and Cox (BC) family of transformation models to several data sets that arise in environmental contexts where the data may be heavily left‐censored with multiple detection limits. A model fitting protocol is described that selects an optimal BC model, identifies outlying observations, and verifies the final model using a normal probability plot correlation statistic. This protocol is used to fit these models to data collected on seventeen groundwater chemistry variables in Ingham County, Michigan in 1986‐1987 as part of the Michigan Groundwater Survey. Six of these variables are adequately modelled by an optimally‐chosen BC model and five of these six are adequately modelled by the lognormal model. Ten of the study variables are marginally modelled by a BC model, whereas only six of the study variables are marginally fit by the lognormal model. The remaining variables fail the fitting criteria due to the significance of the probability plot correlation statistic or because of the presence of outlying observations in the final model. On the basis of these results, the data analyst is cautioned against the automatic use of the lognormal model for fitting environmental data in every circumstance. What is recommended is checking the adequacy of the lognormal model prior to use. If the lognormal model is not an adequate model, then perhaps some other member of the Box and Cox family will provide an adequate model.