Question

Goal congruence in top management teams. Do chief executive officers (CEOs) and their top managers always agree on the goals of the company? Goal importance congruence between CEOs and vice presidents (VPs) was studied in the Academy of Management Journal (Feb. 2008). The researchers used regression to model a VP’s attitude toward the goal of improving efficiency (y) as a function of the two quantitative independent variables level of CEO $\left(x_1\right)$leadership and level of congruence between the CEO and the VP $\left(x_2\right)$. A complete second-order model in ${\mathbf{x}}_{\mathbf{1}}$and ${\mathbf{x}}_{\mathbf{2}}$was fit to data collected for n = 517 top management team members at U.S. credit unions.

a. Write the complete second-order model for E(y).

b. The coefficient of determination for the model, part a, was reported as${\mathbf{R}}^{\mathbf{2}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{14}$. Interpret this value.

c. The estimate of the$\mathbf{\beta }$-value for the$\mathbf{(}{\mathbf{x}}_{\mathbf{2}}{\mathbf{)}}^{\mathbf{2}}$term in the model was found to be negative. Interpret this result, practically.

d. A t-test on the$\mathbf{\beta }$-value for the interaction term in the model,${\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}$, resulted in a p-value of 0.02. Practically interpret this result, using$\mathbf{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$.

Short answer

a. The complete second-order model equation for ${\mathbf{x}}_{\mathbf{1}}$and ${\mathbf{x}}_{\mathbf{2}}$is.$\mathbf{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{4}}{\mathbf{x}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{5}}{\mathbf{x}}_{\mathbf{2}}^{\mathbf{2}}$

b. 14% is a very low value for ${\mathbf{R}}^{\mathbf{2}}$and thus the model is not an ideal fit for the data.

c. The value of ${\mathbf{\beta }}_{\mathbf{5}}$indicates the curvature of the parabola due to the changes in the value of ${\mathbf{x}}_{\mathbf{2}}$. Here a negative value means that the parabola will be a downward shaping curve.

d. At 95% confidence interval, ${\mathbf{\beta }}_{\mathbf{3}}\ne \mathbf{0}$.

Step-by-step solution

Second-order equation

The complete second-order model equation for ${\mathbf{x}}_{\mathbf{1}}$and ${\mathbf{x}}_{\mathbf{2}}$is$\mathbf{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{4}}{\mathbf{x}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{5}}{\mathbf{x}}_{\mathbf{2}}^{\mathbf{2}}$

Interpretation of R2

The value of ${\mathbf{R}}^{\mathbf{2}}$is said to be 0.14 which indicates that almost 14% of the variation in the variables is explained by the model. A higher value denotes that the model is a good fit for the data while a lower value denotes that the model is not an ideal fit for the data. 14% is a very low value for ${\mathbf{R}}^{\mathbf{2}}$and thus the model is not an ideal fit for the data.

Analysis of β5

The value of ${\mathbf{\beta }}_{\mathbf{5}}$indicates the curvature of the parabola due to the changes in the value of ${\mathbf{x}}_{\mathbf{2}}$. Here a negative value means that the parabola will be a downward shaping curve.

Simplification of β3

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

The p-value of ${\mathbf{\beta }}_{\mathbf{3}}$is 0.02 while $\mathbf{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$.

${\mathbf{H}}_{\mathbf{0}}$is rejected ifp-value < $\mathbf{\alpha }$. For $\mathbf{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$, since$\mathbf{0}\mathbf{.}\mathbf{02}\mathbf{<}\mathbf{0}\mathbf{.}\mathbf{05}$

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

Therefore,${\mathbf{\beta }}_{\mathbf{3}}\ne \mathbf{0}$.