Compositional and Noncompositional Covariates in Multiple Linear Regression

Empirical studies in the social sciences often include covariates that are compositional in nature—vectors of shares that sum to a constant—alongside noncompositional covariates. A common approach to handling compositional data in linear regression is to omit one component to avoid perfect multicollinearity. I demonstrate why these coefficients (along with standard errors and t-statistics) can be highly sensitive to the choice of omitted category. Transforming the compositional data using additive logarithmic ratios (ALR) yields permutation-invariant regressions and permits counterfactual changes in the implied composition that remain within the simplex. Furthermore, I demonstrate that applying a simple scale factor to the ALR coefficients generates the coefficients and standard errors associated with isometric logarithmic ratios (ILR) for the variables of interest. Finally, using log-ratios does not exacerbate inherent problems of multicollinearity associated with compositional data. Economic growth regressions incorporating compositional and noncompositional covariates are used to illustrate.