Question

Question: Suppose you fit the interaction model $\mathbf{y}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}\mathbf{\epsilon }$ to n = 32 data points and obtain the following results:${\mathbf{SS}}_{\mathbf{yy}}\mathbf{=}\mathbf{479}\mathbf{,} \mathbf{SSE}\mathbf{=}\mathbf{21}\mathbf{,} {\hat{\mathbf{\beta }}}_{\mathbf{3}}\mathbf{=}\mathbf{10}\mathbf{,}$ and $\mathbf{s}{\hat{\mathbf{\beta }}}_{\mathbf{3}}\mathbf{=}\mathbf{4}$

a. Find ${\mathbf{R}}^{\mathbf{2}}$and interpret its value.

b. Is the model adequate for predicting y? Test at $\mathbf{\alpha }\mathbf{=}\mathbf{.}\mathbf{05}$

c. Use a graph to explain the contribution of the ${\mathbf{x}}_{\mathbf{1}}$ , ${\mathbf{x}}_{\mathbf{2}}$ term to the model.

d. Is there evidence that ${\mathbf{x}}_{\mathbf{1}}$and ${\mathbf{x}}_{\mathbf{2}}$ interact? Test at $\mathbf{\alpha }\mathbf{=}\mathbf{.}\mathbf{05}$ .

Short answer

a. The value of ${\mathbf{R}}^{\mathbf{2}}$close to 95% indicates that almost 95% of the variation in the variables can be explained by the model.

b. At 95% confidence interval, it can be concluded that ${\mathbf{\beta }}_{\mathbf{1}}\mathbf{=}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{=}{\mathbf{\beta }}_{\mathbf{3}}\mathbf{=}\mathbf{0}$

c. Graph

d. At 95% confidence interval,${\mathbf{\beta }}_{\mathbf{3}}\ne \mathbf{\theta }$ . Hence it can be concluded with enough evidence that ${\mathbf{x}}_{\mathbf{1}}$and ${\mathbf{x}}_{\mathbf{2}}$interact in the model.

Step-by-step solution

Determining  R2

$$\begin{gathered} {\mathbf{R}}^{\mathbf{2}}\mathbf{=}\mathbf{1}\mathbf{-}\frac{\mathbf{SSE}}{{\mathbf{SS}}_{\mathbf{yy}}} \\ \mathbf{=}\mathbf{1}\mathbf{-}\frac{\mathbf{21}}{\mathbf{479}} \\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{95615} \end{gathered}$$

The value of${\mathbf{R}}^{\mathbf{2}}$close to 95% indicates that almost 95% of the variation in the variables can be explained by the model.

Goodness of the model fit 

${\mathbf{H}}_{\mathbf{0}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{1}}\mathbf{=}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{=}{\mathbf{\beta }}_{\mathbf{3}}\mathbf{=}\mathbf{0}$

${\mathbf{H}}_{\mathbf{a}}:$At least one of the parameters ${\mathbf{\beta }}_{\mathbf{1}},{\mathbf{\beta }}_{\mathbf{2}},{\mathbf{\beta }}_{\mathbf{3}}$ is non zero

Here, F test statistic $\mathbf{=}\frac{\mathbf{SSe}}{\mathbf{n}\mathbf{-}\mathbf{(}\mathbf{k}\mathbf{+}\mathbf{1}\mathbf{)}}\mathbf{=}\frac{\mathbf{21}}{\mathbf{32}\mathbf{-}\mathbf{4}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{75}$

Value of ${\mathbf{F}}_{\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{,}\mathbf{28}\mathbf{,}\mathbf{28}}$ is 1.915

${\mathbf{H}}_{\mathbf{0}}$is rejected if F statistic > ${\mathbf{F}}_{\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{,}\mathbf{28}\mathbf{,}\mathbf{28}}$. For $\mathbf{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$, since F < ${\mathbf{F}}_{\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{,}\mathbf{28}\mathbf{,}\mathbf{28}}$

We do not have sufficient evidence to reject ${\mathbf{H}}_{\mathbf{0}}$ at a 95% confidence interval.

$\mathbf{Therefore}\mathbf{,}$ ${\mathbf{\beta }}_{\mathbf{1}}\mathbf{=}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{=}{\mathbf{\beta }}_{\mathbf{3}}\mathbf{=}\mathbf{0}$

Graph

When there’s an interaction term in the model, the model is estimated assuming one variable constant, and the relationship between the dependent and the other independent variable is plotted given the value of an independent variable.

In this case, two lines are plotted, where one line is drawn assuming ${\mathbf{x}}_{\mathbf{2}}$ as constant ${\mathbf{K}}_{\mathbf{1}}$ and the other line is drawn assuming ${\mathbf{x}}_{\mathbf{1}}$ constant as ${\mathbf{t}}_{\mathbf{1}}$ Since these two variables are interacting, it can be seen in the graph that the regression lines will interact at some point on the Cartesian graph.

Significance of  β3

${\mathbf{H}}_{\mathbf{0}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{3}}\mathbf{=}\mathbf{0}$

${\mathbf{H}}_{\mathbf{0}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{3}}\ne \mathbf{0}$

Here, t-test statistic $\mathbf{=}\frac{{\hat{\mathbf{\beta }}}_{\mathbf{3}}}{\mathbf{s}{\hat{\mathbf{\beta }}}_{\mathbf{3}}}\mathbf{=}\frac{\mathbf{10}}{\mathbf{4}}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{5}$

Value of ${\mathbf{t}}_{\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{,}\mathbf{31}}$ is 1.6955

${\mathbf{x}}_{\mathbf{1}}{\mathbf{H}}_{\mathbf{0}}$is rejected if statistic >${\mathbf{t}}_{\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{,}\mathbf{24}\mathbf{,}\mathbf{24}}$. For, Since t > ${\mathbf{t}}_{\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{,}\mathbf{31}}$

Sufficient evidence to reject ${\mathbf{H}}_{\mathbf{0}}$ at 95% confidence interval.

Therefore,${\mathbf{\beta }}_{\mathbf{3}}\ne \mathbf{4}$ . Hene it can be concluded with enough evidence that ${\mathbf{x}}_{\mathbf{1}}$ and ${\mathbf{x}}_{\mathbf{2}}$ interact in the model.