18 Multiple Linear Regression

This chapter provides a comprehensive introduction to multiple regression analysis (MR), a highly flexible system for examining the relationship of a collection of independent variables (predictors) to a single dependent variable (criterion). The independent variables may be quantitative (e.g., personality traits, family income) or categorical (e.g., ethnic group, treatment conditions in an experiment). The present chapter explores ordinary least squares (OLS) regression, which requires a continuous dependent variable. The chapter emphasizes: (a) testing theoretical predictions through multiple regression, and (b) identifying problems with implementation of regression analysis. First the structure of MR is described, including the overall regression equation, estimation of partial regression coefficients for individual predictors, effect size measures for both overall model fit and for the contribution of individual predictors and sets of predictors to prediction accuracy. Treatment of categorical predictors through effects, dummy, and contrast coding is explained. Polynomial regression for capturing curvilinear relationships is explored. The specification, testing, and posthoc probing of interactions between continuous variables and between a continuous and a categorical variable are explicated. Second, detection of violations of MR assumptions is addressed, as this helps to identify problems with regression models. Regression diagnostics (case statistics) that are used to identify problematic cases which bias results are explained. Graphic displays for regression analysis that characterize the overall nature of the regression model, curvilinear, and interactive relationships among variables, and checking of assumptions are provided. An empirical example illustrates the interplay between theory and empirical findings in the specification, testing, and revision of regression models. Keywords: multiple regression analysis; ordinary least squares (OLS) regression; interactions in regression; model checking in multiple regression; graphics for multiple regression