Statistical Applications for Environmental Analysis and Risk Assessment

Chapter 15Multiple Linear Regression

15.1 Introduction and Overview

Multiple linear regression (MLR) extends the concepts of simple linear regression (SLR) described in Chapter 12 to the case where the dependent or response (i.e., Y) variable is correlated with more than one predictor or explanatory (i.e., X) variable. The X variable is described as “predictor” or “explanatory,” but not “independent” because, although undesirable, some of the X variables in an MLR model may be correlated with one another (in addition to being correlated with the Y variable), in which case describing such X variables as “independent” would be misleading. Intercorrelations among the X variables in an MLR model (referred to as multicollinearity) is undesirable because they can result in irrational regression coefficients, inflated standard errors and incorrect tests of significance for the coefficients, and even failure to achieve a solution of the MLR model in extreme cases. Multicollinearity is further described in Section 15.5.2, including possible remedies.

As with SLR, the multiple regression is described as linear if the relationship between the Y variable and each X variable is linear or can be linearized through data transformation or other means. In Section 12.2, it was shown how multiple regression can still be characterized as linear even if Y is not linearly related with some of the X variables, as long as the regression coefficients are linearly expressed. The polynomial model described ...