LAUSR

The simple linear model is described by the mathematical straight line Y 1⁄4 b0 þ b1X, where b0 is called the intercept and b1 is called the slope. The focus in such models is on the slope b1 since, if equal to zero, there is no relationship between X and Y. Geometrically, any given straight line is determined by two points (x1,y1) and (x2,y2) that lie on the two-dimensional X–Y plane, so that the simple linear regression slope coefficient can be written as b1 1⁄4 y2 y1 ð Þ x2 x1 ð Þ , providing us the interpretation of the slope as the change in Y relative to a change in X. The geometric extension of a straight line in the twodimensional X–Y plane to the (k + 1)-dimension determined by k independent variables X1, X2, ... , Xk and Y is a hyperplane. Since high dimensions are difficult to visualize, we only present the geometry for the three-dimensional case, that is, when we have k = 2 independent variables X1 and X2 and we are interested in their relationship with a single dependent continuous variable Y. See Figure 1(a) for a hypothetical data example of n = 32 injured drivers with X1 = age, X2 = speed and Y = injury severity score (ISS). In the case of a linear relationship, the mathematical model, also called the linear predictor, is given by