Regression coefficients--Units of Measurement

Regression coefficients--Units of Measurement
Gerard E. Dallal, Ph.D.

When a predictor in a regression equation undergoes a linear transformation, its regression coefficient undergoes a corresponding transformation so that the predictions will be the same for both equations.

Consider a situation in which total cholesterol (CHOL, measured in mg/dL) in adult males is predicted from weight in pounds (WTLB). Suppose the least squares equation is

CHOL = 36 + 0.80 WTLB .

Since the response is in mg/dL, it follows that each term on the right-hand side of the equation is in mg/dL. Therefore,

the units attached to 36 are mg/dL, while
the units attached to 0.80 are (mg/dL)/lb.

Suppose that weight is transformed from pounds to kilograms, WTKG. A kilogram is roughly 2.2 pounds. Since the units of the product of weight in kilograms and its regression coefficient must still be mg/dL, the units of the regression coefficient must be (mg/dL)/kg.

It follows that

(mg/dL)/lb = (mg/dL)/(lb kg/(2.2 lb)) = 2.2 (mg/dL)/kg and 0.8 (mg/dL)/lb = 0.8

2.2 (mg/dL)/lb = 1.76 (mg/dL)/kg .

Therefore, when weight is recorded in kilograms, the equation becomes

CHOL = 36 + 1.76 WTKG .