Effect of dichotomizing a continuous variable on the model structure in multiple linear regression models

This manuscript studies analytically the consequences of changing the scale of measurement of a continuous independent variable in a multiple linear regression setting. Assuming the continuous outcome variable, a continuous exposure variable, and a continuous control variable follow a trivariate Gaussian distribution, we examine the effect upon the structure of the model of dichotomizing the continuous control variable. It is shown that. after dichotomization, the conditional expected value of the response is a quotient of two non-linear functions and hence is no longer linear in the exposure variable. Thus, when an underlying continuous independent variable is dichotomized in multiple linear regression, and one fits a linear model using the dichotomous variable, this model's linear structure is misspecified. The estimates obtained from this model are incorrect and potentially misleading.