ABSTRACT
A linear model consists of an outcome variable, generally denoted by Y, and several (a number denoted by p) predictor variables, generally denoted by X 1, X 2, …, X p. The outcome variable is also commonly referred to as the criterion or dependent variable, whereas the predictor variables are also commonly referred to as the explanatory or independent variables. The model takes on the general form: https://www.w3.org/1998/Math/MathML"> Y i = α + β 1 X 1 i + β 2 X 2 i + … β p X p i + ε i , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203108550/b6d82059-1f99-44d7-8387-6807b00f202e/content/math_3_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where α denotes the intercept and β 1, β 2, …, β p denote the regression coefficients, or slopes, associated with the predictor variables. The i subscript indicates an individual observation, and the total number of observations is typically denoted by n (so i = 1, 2, …, n). The term εi represents the residual or error of prediction, which is the difference between the actual value of Yi and the value of Yi predicted by the model (that is, α + β 1 X 1i + β 2 X 2i + … βpXpi for an individual observation. Finally, if the functional relationship between the outcome variable and the linear combination of predictors (i.e., the left and right sides of Equation 2.1) is not linear, Yi can be replaced by an optimal function of the outcome variable, g(Yi ), to establish a generalized linear model. Common models that fall under this general representation include linear multiple regression, analysis of variance, analysis of covariance, logistic regression, and others. Most of this chapter will focus on relative importance in the context of linear multiple regression. A discussion of other models is provided in the Extensions section of the chapter.
