91Ó°ÊÓ

A Subsequent-sequence model relating mean of y, E(y) to one quantitative independent variable can be written as

$\mathbf{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathbf{β}}_{\mathbf{0}}\mathbf{+}{\mathbf{β}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{β}}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{β}}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{3}}\mathbf{+}{\mathbf{β}}_{\mathbf{4}}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{β}}_{\mathbf{5}}{\mathbf{x}}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{3}}\mathbf{+}{\mathbf{β}}_{\mathbf{6}}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{3}}\mathbf{+}{\mathbf{β}}_{\mathbf{7}}{\mathbf{x}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{+}{\mathbf{β}}_{\mathbf{8}}{\mathbf{x}}_{\mathbf{2}}^{\mathbf{2}}\mathbf{+}{\mathbf{β}}_{\mathbf{9}}{\mathbf{x}}_{\mathbf{3}}^{\mathbf{2}}$

Here, ${\mathbf{β}}_{\mathbf{0}}$denotes the y-intercept, ${\mathbf{β}}_{\mathbf{1}}$denotes the slope of the regression line, and ${\mathbf{β}}_{\mathbf{3}}$denotes the curvature of the parabola.

A second-order model relating the mean of y, E(y) to two quantitative independent variables can be written as

$\mathbf{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathbf{β}}_{\mathbf{0}}\mathbf{+}{\mathbf{β}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{β}}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{β}}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{+}{\mathbf{β}}_{\mathbf{4}}{\mathbf{x}}_{\mathbf{2}}^{\mathbf{2}}$

Here, ${\mathbf{β}}_{\mathbf{0}}$denotes the y-intercept, ${\mathbf{β}}_{\mathbf{1}}$denotes changes in y due to ${\mathbf{x}}_{\mathbf{1}}$holding${\mathbf{x}}_{\mathbf{2}}$ fixed, ${\mathbf{β}}_{\mathbf{2}}$denotes changes in y due to ${\mathbf{x}}_{\mathbf{2}}$holding ${\mathbf{x}}_{\mathbf{1}}$fixed, ${\mathbf{β}}_{\mathbf{3}}$denotes the curvature of the parabola relating y to ${\mathbf{x}}_{\mathbf{1}}$when ${\mathbf{x}}_{\mathbf{2}}$is held fixed, and${\mathbf{β}}_{\mathbf{4}}$ denotes the curvature of the parabola relating y to ${\mathbf{x}}_{\mathbf{2}}$when ${\mathbf{x}}_{\mathbf{1}}$is held fixed.

A secondaryseries model relating mean of y, E(y) to three quantitative independent variables can be written as

$\mathbf{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathbf{β}}_{\mathbf{0}}\mathbf{+}{\mathbf{β}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{β}}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{β}}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{3}}\mathbf{+}{\mathbf{β}}_{\mathbf{4}}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{β}}_{\mathbf{5}}{\mathbf{x}}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{3}}\mathbf{+}{\mathbf{β}}_{\mathbf{6}}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{3}}\mathbf{+}{\mathbf{β}}_{\mathbf{7}}{\mathbf{x}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{+}{\mathbf{β}}_{\mathbf{8}}{\mathbf{x}}_{\mathbf{2}}^{\mathbf{2}}\mathbf{+}{\mathbf{β}}_{\mathbf{9}}{\mathbf{x}}_{\mathbf{3}}^{\mathbf{2}}$

Here, ${\mathbf{β}}_{\mathbf{0}}$denotes the y-intercept ${\mathbf{β}}_{\mathbf{1}}\mathbf{,}\mathbf{}{\mathbf{β}}_{\mathbf{2}}$and ${\mathbf{β}}_{\mathbf{3}}$denotes changes in y due to changes in x holding other x constant, ${\mathbf{β}}_{\mathbf{4}}\mathbf{}\mathbf{,}\mathbf{}{\mathbf{β}}_{\mathbf{5}}$and ${\mathbf{β}}_{\mathbf{6}}$denotes the interaction variables and ${\mathbf{β}}_{\mathbf{7}}\mathbf{,}\mathbf{}{\mathbf{β}}_{\mathbf{8}}$and ${\mathbf{β}}_{\mathbf{9}}$denotes the curvature of the parabola relating y to one x whenanother axis held fixed.