91Ó°ÊÓ

The equation of an interaction model for E(y) as a function of x₁ and x₂ would have an additional variable (x₁x₂) to represent the interaction amongst the variable.

Therefore, the equation would be $\mathit{E}\left(y\right)\mathbf{=}{\mathit{β}}_{\mathbf{0}}\mathbf{+}{\mathit{β}}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{β}}_{\mathbf{2}}{\mathit{x}}_{\mathbf{2}}\mathbf{+}{\mathit{β}}_{\mathbf{3}}{\mathit{x}}_{\mathbf{1}}{\mathit{x}}_{\mathbf{2}}\mathbf{+}\mathit{Î\mu }$ .

The interaction model equation is $E\left(y\right)={\mathrm{β}}_0+{\mathrm{β}}_1{x}_1+{\mathrm{β}}_2{x}_2+{\mathrm{β}}_3{x}_1{x}_2+\mathrm{Î\mu }$.

Here, holding x₂ value constant at 2.5, the model becomes

$E\left(y\right)={\mathrm{β}}_0+{\mathrm{β}}_1{x}_1+{\mathrm{β}}_2\left(2.5\right)+{\mathrm{β}}_3{x}_1\left(2.5\right)+\mathrm{Î\mu }$

$E\left(y\right)=\left({\mathrm{β}}_0+2.5{\mathrm{β}}_2\right)+\left({\mathrm{β}}_1+2.5{\mathrm{β}}_3\right)x_1+\mathrm{Î\mu }$

The equation becomes$\mathit{E}\left(y\right)\mathbf{=}\left({\mathrm{β}}_0+2.5{\mathrm{β}}_2\right)\mathbf{+}\left({\mathrm{β}}_1+2.5{\mathrm{β}}_3\right){\mathit{x}}_{\mathbf{1}}\mathbf{+}\mathit{Î\mu }$ Hence, revenue (y) changes by localid="1651173294032" $\left({\mathrm{β}}_1+2.5{\mathrm{β}}_3\right)$ for every 1-unit increase in the tweet rate (x₁) holding PN-ratio (x₂) constant.

The interaction model equation is $E\left(y\right)={\mathrm{β}}_0+{\mathrm{β}}_1{x}_1+{\mathrm{β}}_2{x}_2+{\mathrm{β}}_3{x}_1{x}_2+\mathrm{Î\mu }$.

Here, holding x₂ value constant at 5, the model becomes

$E\left(y\right)={\mathrm{β}}_0+{\mathrm{β}}_1{x}_1+{\mathrm{β}}_2\left(5\right)+{\mathrm{β}}_3{x}_1\left(5\right)+\mathrm{Î\mu }$

$E\left(y\right)=\left({\mathrm{β}}_0+5{\mathrm{β}}_2\right)+\left({\mathrm{β}}_1+5{\mathrm{β}}_3\right)x_1+\mathrm{Î\mu }$

The equation becomes,localid="1651173183555" $\mathit{E}\left(y\right)\mathbf{=}\left({\mathrm{β}}_0+5{\mathrm{β}}_2\right)\mathbf{+}\left({\mathrm{β}}_1+5{\mathrm{β}}_3\right){\mathit{x}}_{\mathbf{1}}\mathbf{+}\mathit{Î\mu }$. Hence, revenue (y) changes bylocalid="1651173324259" ${\mathit{β}}_{\mathbf{2}}\mathbf{+}\mathbf{100}{\mathit{β}}_{\mathbf{3}}$ for every 1-unit increase in the tweet rate (x₁) holding PN-ratio (x₂) constant.

The interaction model equation is $E\left(y\right)={\mathrm{β}}_0+{\mathrm{β}}_1{x}_1+{\mathrm{β}}_2{x}_2+{\mathrm{β}}_3{x}_1{x}_2+\mathrm{Î\mu }$.

Here, holding x₁ value constant at 100, the model becomes

$E\left(y\right)={\mathrm{β}}_0+{\mathrm{β}}_1\left(100\right)+{\mathrm{β}}_2{x}_2+{\mathrm{β}}_3\left(100\right)x_2+\mathrm{Î\mu }$

$E\left(y\right)=\left({\mathrm{β}}_0+100{\mathrm{β}}_1\right)+\left({\mathrm{β}}_2+100{\mathrm{β}}_3\right)x_2+\mathrm{Î\mu }$

The equation becomes, role="math" localid="1651173154939" $\mathit{E}\left(y\right)\mathbf{=}\left({\mathrm{β}}_0+100{\mathrm{β}}_1\right)\mathbf{+}\left({\mathrm{β}}_2+100{\mathrm{β}}_3\right){\mathit{x}}_{\mathbf{2}}\mathbf{+}\mathit{Î\mu }$. Hence, revenue (y) changes by ${\mathit{β}}_{\mathbf{2}}\mathbf{+}\mathbf{100}{\mathit{β}}_{\mathbf{3}}$ for every 1-unit increase in PN-ratio x₂ holding the tweet rate x₁ constant.

To test whether x₁ and x₂ variables interact in the model, the presence of the model parameter ${\mathrm{β}}_3$ is tested.

Hence the null hypothesis would be the absence of the model parameter ${\mathrm{β}}_3$and the alternate hypothesis would be the presence of ${\mathrm{β}}_3$

Mathematically, ${\mathit{H}}_{\mathbf{0}}\mathbf{:}{\mathit{β}}_{\mathbf{3}}\mathbf{=}\mathbf{0}\mathbf{;}{\mathit{H}}_{\mathbf{a}}\mathbf{:}{\mathit{β}}_{\mathbf{3}}\mathbf{≠}\mathbf{0}$