Use of Ratios and Logarithms in Statistical Regression Models

In many regression models, we use logarithmic transformations of either the regression summary measure (a log link), the regression response variable (e.g., when analyzing geometric means), or one or more of the predictors. In this manuscript, I discuss the rationale for using logarithmic transformations, the interpretation of ratios, and the general properties of logarithms. 1 Use of Logarithmic Transformations Logarithmic transformations of data and/or parameters are used extensively in statistics. The fundamental reason for this stems from the following logic: 1. We are most often interested in using statistics to detect associations between two variables. 2. By an association, we mean that the distribution of one variable (we call this the “response variable”) is different in some way between groups that are homogeneous for the other variable (we call this the “predictor of interest” or POI). 3. If the two variables are associated, then that means that some aspect of the distribution (some “summary measure” like the mean, geometric mean, etc.) is unequal between two groups that differ in their value of the POI. In describing the association, we will want to describe (a) How the POI differs between the two (conceptual) groups being compared, and (b) How the summary measure of response compares across two groups. 4. There are two simple arithmetic ways to tell if two numbers (e.g., the level of POI in two separate groups, or the mean of the “response variable” in two separate groups) are unequal: (a) their difference is not 0, or (b) their ratio is not 1. 5. We choose between differences and ratios as methods for comparisons based on a variety of criteria that are sometimes competing: (a) People understand differences more than they understand ratios. i. Part of this is because natural language (e.g., English) is not very precise when describing ratios.