Numerical transformation of geochemical data: 2. Stabilizing measurement error to facilitate data interpretation

Abstract

Traditional uses of data transformations in geoscience have typically been motivated by three objectives: (1) creating normally distributed data; (2) creating data that are additive; and (3) making errors constant across the data range. Part 1 of this paper discussed the transformation applicability for the first two reasons in geochemical applications, and introduced a fourth motivation whereby transformation maximizes the variance (or geochemical contrast) in the transformed data; it did not discuss transformation to stabilize measurement error. Although transformation to stabilizing errors in geochemical data is not common, this is a useful attribute in geochemical data analysis. The transformation is dependent on the model describing the magnitude of measurement (sampling and analysis) error as a function of concentration. Models describing measurement error as a function of concentration can be used to derive a transformation that will stabilize the measurement error in the transformed variable. Poisson, binomial and hypergeometric models are typically used to describe sampling errors, whereas straight line (constant, proportional and affine) models are used to describe analytical errors. The associated variance stabilizing transformations, derived from these models, have constant propagated errors. As a result, these transformations create a ‘level playing field’ for subsequent data analysis, enabling the discovery of additional information in the data. Homoscedastic measurement error allows the geochemist to justify use of a specific transformation based not on the subsequent data analysis results (circular reasoning; ‘the end justifies the means’), but on optimal properties created by the transformation. In this way, objective results can be achieved scientifically, providing another motivation to collect geochemical quality control data.