Regression On …

The phrase “regression on” refers to the act of projecting the independent variable vector (sometimes referred to as the “regressand”) onto the Euclidean subspace spanned by the DEPENDENT VARIABLES (sometimes collectively referred to as “REGRESSORS”). Although the phrase refers to the underlying geometry of least squares, it is often used to indicate the set of independent variables used to model the observed variation in the dependent variable.

For example, if a researcher is interested in the simple linear regression of Y = b0 + b1X + e1, we say that “Y is regressed on X”—indicating that the simple regression recovers the set of coefficients {b0,b1}such that the projection of Y onto the subspace spanned by X (in this case, a line) is given by b0 + b1X. Because different values of b0 and b1 (say, b'0 and b'1) project Y onto the subspace of X differently (i.e., b'0 + b'1X), there are (infinitely) many possible projections/regressions. To determine which projection/regression is of most interest, researchers must assess the “fit” according to a given criterion. A commonly used criterion is that of least squares, which denotes the “best” projection as the choice of {b0,b1} for which |Y −(b0 + b1X)|2 is minimized.

In the MULTIPLE REGRESSION ANALYSIS context, “regression on” refers to the projection of the dependent variable vector into the subspace spanned by the set of independent variables—with the REGRESSION[Page 943]PLANE defining the plane that minimizes the squared differences between the observed values of the dependent variable and its projection into the regression plane.