The Impact of Correlated and/or Interacting Predictor Omission on Estimated Regression Coefficients in Linear Regression

We examine cases of predictor omission defined by the relationship between the set of omitted predictor(s) and a set of remaining predictor(s), both of which are included in the full model. We consider a wider range of omitted predictors than previously studied by systematically accounting for both interaction and correlation between the included and the omitted predictors. Our study highlights the impact of predictor omission on the resulting estimated regression coefficients and their squared standard errors. Theoretical and simulated results are presented to illustrate the impact of predictor omission among cases of interaction and correlation. In our simulated results, bias diverged as correlation increased from zero to one. On its own, interaction amplified bias, but the impact of interaction was worse when combined with correlation. Overall, our discussions surround the known problem of predictor omission with a rigorous framework to quantify bias in the included predictor's estimated regression coefficient and squared standard error.