Question

Question: Write a second-order model relating the mean of y, E(y), to

a. one quantitative independent variable

b. two quantitative independent variables

c. three quantitative independent variables [Hint: Include allpossible two- way cross-product terms and squared terms.]

Short answer

a. The second-order model equation in one quantitative variable is

$\mathbf{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{+}\mathbf{\epsilon }$

b. The second-order model equation in two quantitative variables is

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

c. The second-order model equation in three quantitative variables is

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

Step-by-step solution

Subsequent-sequence model equation

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

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

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

Second-order model equation

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

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

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

Secondary-series model equation

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

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

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