Question

Question: Job performance under time pressure. Time pressure is common at firms that must meet hard and fast deadlines. How do employees working in teams perform when they perceive time pressure? And, can this performance improve with a strong team leader? These were the research questions of interest in a study published in the Academy of Management Journal (October, 2015). Data were collected on n = 139 project teams working for a software company in India. Among the many variables recorded were team performance (y, measured on a 7-point scale), perceived time pressure (, measured on a 7-point scale), and whether or not the team had a strong and effective team leader (x₂ = 1 if yes, 0 if no). The researchers hypothesized a curvilinear relationship between team performance (y) and perceived time pressure (), with different-shaped curves depending on whether or not the team had an effective leader (x₂). A model for E(y) that supports this theory is the complete second-order model:$\mathit{E}\mathit{\left(}\mathit{y}\mathit{\right)}\mathit{}\mathit{=}{\mathit{\beta }}_{\mathit{0}}\mathit{+}{\mathit{\beta }}_{\mathit{1}}{\mathit{x}}_{\mathit{1}}\mathit{+}{\mathit{\beta }}_{\mathit{2}}{{\mathit{x}}_{\mathit{1}}}^{\mathit{2}}\mathit{+}{\mathit{\beta }}_{\mathit{3}}{\mathit{x}}_{\mathit{2}}\mathit{+}{\mathit{\beta }}_{\mathit{4}}{\mathit{x}}_{\mathit{1}}{\mathit{x}}_{\mathit{2}}\mathit{+}{\mathit{\beta }}_{\mathit{5}}{{\mathit{x}}_{\mathit{1}}}^{\mathit{2}}{\mathit{x}}_{\mathit{2}}$

a. Write the equation for E(y) as a function of x₁ when the team leader is not effective (x₂= 0).

b. Write the equation for E(y) as a function ofwhen the team leader is effective (x₂= 1).

c. The researchers reported the following b-estimates:.

$\widehat{{\mathbf{\beta }}_{\mathbf{0}}}=4.5,\widehat{{\mathbf{\beta }}_{\mathbf{1}}}=0.13,\widehat{{\mathbf{\beta }}_{\mathbf{3}}}=0.15,\widehat{{\mathbf{\beta }}_{\mathbf{4}}}=0.15and\widehat{{\mathbf{\beta }}_{\mathbf{5}}}=0.29$Use these estimates to sketch the two equations, parts a and b. What is the nature of the curvilinear relationship when the team leaders is not effective? Effective?

Short answer

Answer

1. When x₂= 0, the equation of E(y) can be written as$\mathit{E}\mathit{\left(}\mathit{y}\mathit{\right)}\mathit{}\mathit{=}{\mathit{\beta }}_{\mathit{0}}\mathit{+}{\mathit{\beta }}_{\mathit{1}}{\mathit{x}}_{\mathit{1}}\mathit{+}{\mathit{\beta }}_{\mathit{2}}{{\mathit{x}}_{\mathit{1}}}^{\mathit{2}}$
2. Whenx₂= 1, the equation of E(y) can be written as$\mathit{E}\mathit{\left(}\mathit{y}\mathit{\right)}\mathit{}\mathit{=}\mathit{(}{\mathit{\beta }}_{\mathit{0}}\mathit{+}{\mathit{\beta }}_{\mathit{3}}\mathit{)}\mathit{+}\mathit{(}{\mathit{\beta }}_{\mathit{1}}\mathit{+}{\mathit{\beta }}_{\mathit{4}}\mathit{)}{\mathit{x}}_{\mathit{1}}\mathit{+}\mathit{(}{\mathit{\beta }}_{\mathit{2}}\mathit{+}{\mathit{\beta }}_{\mathit{5}}\mathbf{)}{{\mathit{x}}_{\mathit{1}}}^{\mathit{2}}$
3. The curvilinear relationship changes when team leaders are effective and not effective. When the team leaders are effective, there is a downward sloping curve observed meaning that there is a negative relationship between team performance and effectiveness of team leader. While when the team leaders are not effective, there is an upward sloping curve observed indicating positive relationship between team performance and team leader not being effective.

Step-by-step solution

Equation for E(y)

When x₂= 0, the equation of E(y) can be written as

$$\begin{gathered} \mathit{E}\mathit{\left(}\mathit{y}\mathit{\right)}\mathit{}\mathit{=}{\mathit{\beta }}_{\mathit{0}}\mathit{+}{\mathit{\beta }}_{\mathit{1}}{\mathit{x}}_{\mathit{1}}\mathit{+}{\mathit{\beta }}_{\mathit{2}}{{\mathit{x}}_{\mathit{1}}}^{\mathit{2}}\mathit{+}{\mathit{\beta }}_{\mathit{3}}{\mathit{x}}_{\mathit{2}}\mathit{+}{\mathit{\beta }}_{\mathit{4}}{\mathit{x}}_{\mathit{1}}{\mathit{x}}_{\mathit{2}}\mathit{+}{\mathit{\beta }}_{\mathit{5}}{{\mathit{x}}_{\mathit{1}}}^{\mathit{2}}{\mathit{x}}_{\mathit{2}} \\ \mathit{E}\mathit{\left(}\mathit{y}\mathit{\right)}\mathit{}\mathit{=}{\mathit{\beta }}_{\mathbf{0}}\mathit{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathit{+}{\mathit{\beta }}_{\mathbf{2}}{{\mathbf{x}}_{\mathbf{1}}}^{\mathbf{2}}\mathit{+}{\mathit{\beta }}_{\mathbf{3}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathit{+}{\mathit{\beta }}_{\mathbf{4}}{\mathit{x}}_{\mathbf{1}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathit{+}{\mathit{\beta }}_{\mathbf{5}}{{\mathbf{x}}_{\mathbf{1}}}^{\mathbf{2}}\mathbf{\left(}\mathbf{0}\mathbf{\right)} \\ \mathit{E}\mathit{\left(}\mathit{y}\mathit{\right)}\mathit{}\mathit{=}{\mathit{\beta }}_{\mathbf{0}}\mathit{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathit{+}{\mathit{\beta }}_{\mathbf{2}}{{\mathbf{x}}_{\mathbf{1}}}^{\mathbf{2}} \end{gathered}$$

Calculation for E(y)

Whenx₂= 1, the equation of E(y) can be written as

$$\begin{gathered} \mathit{E}\mathit{\left(}\mathit{y}\mathit{\right)}\mathit{}\mathit{=}{\mathit{\beta }}_{\mathit{0}}\mathit{+}{\mathit{\beta }}_{\mathit{1}}{\mathit{x}}_{\mathit{1}}\mathit{+}{\mathit{\beta }}_{\mathit{2}}{{\mathit{x}}_{\mathit{1}}}^{\mathit{2}}\mathit{+}{\mathit{\beta }}_{\mathit{3}}{\mathit{x}}_{\mathit{2}}\mathit{+}{\mathit{\beta }}_{\mathit{4}}{\mathit{x}}_{\mathit{1}}{\mathit{x}}_{\mathit{2}}\mathit{+}{\mathit{\beta }}_{\mathit{5}}{{\mathit{x}}_{\mathit{1}}}^{\mathit{2}}{\mathit{x}}_{\mathit{2}} \\ \mathit{E}\mathit{\left(}\mathit{y}\mathit{\right)}\mathit{}\mathit{=}{\mathit{\beta }}_{\mathbf{0}}\mathit{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathit{+}{\mathit{\beta }}_{\mathbf{2}}{{\mathbf{x}}_{\mathbf{1}}}^{\mathbf{2}}\mathit{+}{\mathit{\beta }}_{\mathbf{3}}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathit{+}{\mathit{\beta }}_{\mathbf{4}}{\mathit{x}}_{\mathbf{1}}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathit{+}{\mathit{\beta }}_{\mathbf{5}}{{\mathbf{x}}_{\mathbf{1}}}^{\mathbf{2}}\mathbf{\left(}\mathbf{1}\mathbf{\right)} \\ \mathit{E}\mathit{\left(}\mathit{y}\mathit{\right)}\mathit{}\mathit{=}\mathit{(}{\mathit{\beta }}_{\mathbf{0}}\mathit{+}{\mathit{\beta }}_{\mathbf{3}}\mathbf{)}\mathbf{+}\mathit{(}{\mathit{\beta }}_{\mathbf{1}}\mathit{+}{\mathit{\beta }}_{\mathbf{4}}\mathbf{)}{\mathit{x}}_{\mathbf{1}}\mathit{+}\mathit{(}{\mathit{\beta }}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{5}}\mathbf{)}{{\mathbf{x}}_{\mathbf{1}}}^{\mathbf{2}} \end{gathered}$$

Graph

When x₂= 0, the equation of E(y) can be written as$\mathit{E}\mathit{\left(}\mathit{y}\mathit{\right)}\mathit{}\mathit{=}\mathit{4}\mathit{.}\mathit{5}\mathit{-}\mathit{0}\mathit{.}\mathit{13}{\mathit{x}}_{\mathit{1}}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{17}{{\mathit{x}}_{\mathit{1}}}^{\mathit{2}}$.

Whenx₂ = 1, the equation of E(y) can be written as$\mathit{E}\mathit{\left(}\mathit{y}\mathit{\right)}\mathit{}\mathit{=}\mathit{4}\mathit{.}\mathit{65}\mathit{+}\mathit{0}\mathit{.}\mathit{02}{\mathit{x}}_{\mathit{1}}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{12}{{\mathit{x}}_{\mathit{1}}}^{\mathit{2}}$

The curvilinear relationship changes when team leaders are effective and not effective. When the team leaders are effective, there is a downward sloping curve observed meaning that there is a negative relationship between team performance and effectiveness of team leader.

While when the team leaders are not effective, there is an upward sloping curve observed indicating positive relationship between team performance and team leader not being effective.