Our primary analysis estimates a linear regression of several IDPs on alcohol intake in log(1 + daily units), including various control variables and interactions. Given the slight concavity of the LOWESS regression lines in the descriptive analysis of the global IDPs, we included both linear and quadratic terms for alcohol intake and age in the regression:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{\rm{IDP}}}}}}_{i}={\beta }_{0}+{\beta }_{1}{X}_{i}+{\beta }_{2}{X}_{i}^{2}+{\beta }_{3}{X}_{i}{{{{\rm{x}}}}}\,{{{{{\rm{SEX}}}}}}_{i}+{\beta }_{4}{X}_{i}{{{{\rm{x}}}}}\,{{{{{\rm{AGE}}}}}}_{i}+{\gamma} \,{Z}_{i}+{e}_{i},$$\end{document}IDPi=β0+β1Xi+β2Xi2+β3XixSEXi+β4XixAGEi+γZi+ei,where IDPi is the IDP normalized for head size, Xi is the standardized alcohol intake in log(1 + daily units), AGEi is standardized age, Zi is a vector of control variables, and ei is an error term.
