Multiple Linear Regression

●Volume10 30 50 70● ●●● ●● ● ● ● ●●● ●●●●● ● ●●●●●●●● ● ●8 10 12 14 16 18 20GirthFigure 3: Scatterplot of Volume versus Girth for the trees data4 Polynomial Regression4.1 Quadratic Regression ModelAbove we assumed that µ was a linear function of the explanatory variables, that is we assumedµ(x 1 , x 2 ) = β 0 + β 1 x 1 + β 2 x 2 . In every case the scatterplots indicated that our assumption wasreasonable. Sometimes, however, plots of the data suggest that the linear model is incomplete andshould be modified.For example, let us examine a scatterplot of Volume versus Girth a little more closely. SeeFigure 3. There might be a slight curvature to the data; the volume curves ever so slightly upwardas the girth increases. After looking at the plot we might try to capture the curvature with a meanresponse such asµ(x 1 ) = β 0 + β 1 x 1 + β 2 x 2 1 , (41)which is associated with the modelY = β 0 + β 1 x 1 + β 2 x 2 1 + ɛ. (42)The regression assumptions are the same.14