How Do You Interpret Your Regression Coefficients?

This note is in response to David C. Hoaglin’s provocative statement in The Stata Journal (2016) that “Regressions are commonly misinterpreted”. “Citing the preliminary edition of Tukey’s classic Exploratory Data Analysis (1970, chap. 23), Hoaglin argues that the correct interpretation of a regression coefficient is that it “tells us how Y responds to change in X2 after adjusting for simultaneous linear change in the other predictors in the data at hand”. He contrasts this with what he views as the common misinterpretation of the coefficient as “the average change in Y for a 1-unit increase in X2 when the other Xs are held constant”. He asserts that this interpretation is incorrect because “[i]t does not accurately reflect how multiple regression works”. We find that Hoaglin’s characterization of common practice is often inaccurate and that his narrow view of proper interpretation is too limiting to fully exploit the potential of regression models. His article rehashes debates that were settled long ago, confuses the estimator of an effect with what is estimated, ignores modern approaches, and rejects a basic goal of applied research.” (Long and Drukker, 2016:25). This note broadly agrees with the comments that followed his article in the same issue of The Stata Journal (2016) and seeks to present an argument in favour of the commonly held interpretation that Hoaglin unfortunately marks as misinterpretation.