Question

Question: Personality traits and job performance. Refer to the Journal of Applied Psychology (Jan. 2011) study of the relationship between task performance and conscientiousness, Exercise 12.54 (p. 747). Recall that the researchers used a quadratic model to relate y = task performance score (measured on a 30-point scale) to x₁ = conscientiousness score (measured on a scale of -3 to +3). In addition, the researchers included job complexity in the model, where x₂ = {1 if highly complex job, 0 if not}. The complete model took the form

$$\begin{gathered} \mathit{E}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{{\mathbf{x}}_{\mathbf{1}}}^{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{3}}{\mathit{x}}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{4}}{\mathit{x}}_{\mathbf{1}}{\mathit{x}}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{5}}{{\mathbf{x}}_1}^{\mathbf{2}}{\mathit{x}}_{\mathbf{2}\mathbf{}}\mathbf{}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\mathbf{}{\mathit{x}}_{\mathbf{2}}\mathbf{=}\mathbf{1} \\ \mathbf{}\mathit{E}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{{\mathbf{x}}_{\mathbf{1}}}^{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{3}}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{+}{\mathit{\beta }}_{\mathbf{4}}{\mathit{x}}_{\mathbf{1}}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{+}{\mathit{\beta }}_{\mathbf{5}}{\mathbf{\left(}\mathbf{1}\mathbf{\right)}}^{\mathbf{2}}\mathbf{\left(}\mathbf{1}\mathbf{\right)} \\ \mathbf{}\mathbf{}\mathit{E}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{=}\mathbf{(}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{3}}\mathbf{)}\mathbf{}\mathbf{+}\mathbf{(}{\mathit{\beta }}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{4}}\mathbf{)}{\mathit{x}}_{\mathbf{1}}\mathbf{+}\mathbf{(}{\mathit{\beta }}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{5}}\mathbf{)}{\mathbf{\left(}{\mathbf{x}}_{\mathbf{1}}\mathbf{\right)}}^{\mathbf{2}} \end{gathered}$$

a. For jobs that are not highly complex, write the equation of the model for E1y2 as a function of x₁. (Substitute x2 = 0 into the equation.)

b. Refer to part a. What do each of the b’s represent in the model?

c. For highly complex jobs, write the equation of the model for E(y) as a function of x₁. (Substitute x₂ = 1 into the equation.)

d. Refer to part c. What do each of the b’s represent in the model?

e. Does the model support the researchers’ theory that the curvilinear relationship between task performance score (y) and conscientiousness score (x₁) depends on job complexity (x₂)? Explain.

Short answer

Answer

a. For jobs that are not highly complex, write the equation of the model for E(y) as a function of x₁ can be written as$\mathbf{}\mathbf{}\mathit{E}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{\mathbf{\left(}{\mathbf{x}}_{\mathbf{1}}\mathbf{\right)}}^{\mathbf{2}}$

b. In part a, the equation here is $\mathit{E}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{{\mathbf{x}}_{\mathbf{1}}}^{\mathbf{2}}$where β₁ = y-intercept of the model, β₂ = represents the 1-unit change in y due to changes in x_1,and β₃ = represents the changes in y due to change in curvature of the arc

c. For highly complex jobs, the equation of the model for E(y) as a function of x₁ can be written as$\mathit{E}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{=}\mathbf{(}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{3}}\mathbf{)}\mathbf{}\mathbf{+}\mathbf{(}{\mathit{\beta }}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{4}}\mathbf{)}{\mathit{x}}_{\mathbf{1}}\mathbf{+}\mathbf{(}{\mathit{\beta }}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{5}}\mathbf{)}{\mathbf{\left(}{\mathbf{x}}_{\mathbf{1}}\mathbf{\right)}}^{\mathbf{2}}$

d. In part c, the equation here is here,$\mathbf{}\mathbf{}\mathit{E}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{=}\mathbf{(}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{3}}\mathbf{)}\mathbf{}\mathbf{+}\mathbf{(}{\mathit{\beta }}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{4}}\mathbf{)}{\mathit{x}}_{\mathbf{1}}\mathbf{+}\mathbf{(}{\mathit{\beta }}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{5}}\mathbf{)}{\mathbf{\left(}{\mathbf{x}}_{\mathbf{1}}\mathbf{\right)}}^{\mathbf{2}}$ $\mathbf{(}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{3}}\mathbf{)}\mathbf{}$represents the y-intercept for variable x₂ when x₂ = 1 (for highly complex jobs), $\mathbf{(}{\mathit{\beta }}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{4}}\mathbf{)}$represents the 1-unit change in y due to changes in x₁when x₂ = 1, and $\mathbf{(}{\mathit{\beta }}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{5}}\mathbf{)}$ represents the changes in y due to change in curvature of the arc when x₂ = 1.

e. In the interaction model of part c, the te represents the curvilinear relationship between task performance score (y) and conscientiousness score (x₁) and x₂ term represents job complexity.

Step-by-step solution

Equation for model

For jobs that are not highly complex, write the equation of the model for E(y) as a function of x₁ can be written as

$$\begin{gathered} \mathit{E}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{{\mathbf{x}}_{\mathbf{1}}}^{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{3}}{\mathit{x}}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{4}}{\mathit{x}}_{\mathbf{1}}{\mathit{x}}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{5}}{{\mathbf{x}}_1}^{\mathbf{2}}{\mathit{x}}_{\mathbf{2}\mathbf{}}\mathbf{}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\mathbf{}{\mathit{x}}_{\mathbf{2}}\mathbf{=}\mathbf{0} \\ \mathbf{}\mathit{E}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{{\mathbf{x}}_{\mathbf{1}}}^{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{3}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{+}{\mathit{\beta }}_{\mathbf{4}}{\mathit{x}}_{\mathbf{1}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{+}{\mathit{\beta }}_{\mathbf{5}}{\mathbf{\left(}{\mathit{x}}_{\mathbf{1}}\mathbf{\right)}}^{\mathbf{2}}\mathbf{\left(}\mathbf{0}\mathbf{\right)} \\ \mathbf{}\mathbf{}\mathit{E}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{{\mathbf{x}}_{\mathbf{1}}}^{\mathbf{2}} \end{gathered}$$

Interpretation of β

In part a, the equation here is

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

Whereβ₁ = y-intercept of the model

β₂ = represents the 1-unit change in y due to changes in x₁

β₃ = represents the changes in y due to change in curvature of the arc

Equation for model

For highly complex jobs, the equation of the model for E(y) as a function of x₁ can be written as

$$\begin{gathered} \mathit{E}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{{\mathbf{x}}_{\mathbf{1}}}^{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{3}}{\mathit{x}}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{4}}{\mathit{x}}_{\mathbf{1}}{\mathit{x}}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{5}}{{\mathbf{x}}_1}^{\mathbf{2}}{\mathit{x}}_{\mathbf{2}\mathbf{}}\mathbf{}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\mathbf{}{\mathit{x}}_{\mathbf{2}}\mathbf{=}\mathbf{1} \\ \mathbf{}\mathit{E}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{{\mathbf{x}}_{\mathbf{1}}}^{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{3}}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{+}{\mathit{\beta }}_{\mathbf{4}}{\mathit{x}}_{\mathbf{1}}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{+}{\mathit{\beta }}_{\mathbf{5}}{\mathbf{\left(}\mathbf{1}\mathbf{\right)}}^{\mathbf{2}}\mathbf{\left(}\mathbf{1}\mathbf{\right)} \\ \mathbf{}\mathbf{}\mathit{E}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{=}\mathbf{(}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{3}}\mathbf{)}\mathbf{}\mathbf{+}\mathbf{(}{\mathit{\beta }}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{4}}\mathbf{)}{\mathit{x}}_{\mathbf{1}}\mathbf{+}\mathbf{(}{\mathit{\beta }}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{5}}\mathbf{)}{\mathbf{\left(}{\mathbf{x}}_{\mathbf{1}}\mathbf{\right)}}^{\mathbf{2}} \end{gathered}$$

Clarification of β

In part c, the equation here is$\mathit{E}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{=}\mathbf{(}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{3}}\mathbf{)}\mathbf{}\mathbf{+}\mathbf{(}{\mathit{\beta }}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{4}}\mathbf{)}{\mathit{x}}_{\mathbf{1}}\mathbf{+}\mathbf{(}{\mathit{\beta }}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{5}}\mathbf{)}{\mathbf{\left(}{\mathbf{x}}_{\mathbf{1}}\mathbf{\right)}}^{\mathbf{2}}$

Here, $\mathbf{(}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{3}}\mathbf{)}\mathbf{}$represents the y-intercept for variable x₂ when x₂ = 1 (for highly complex jobs)

$\mathbf{(}{\mathit{\beta }}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{4}}\mathbf{)}$representsthe 1-unit change in y due to changes in x₁when x₂ = 1

$\mathbf{(}{\mathit{\beta }}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{5}}\mathbf{)}$represents the changes in y due to change in curvature of the arcwhen x₂ = 1

Model interpretation

In the interaction model of part c, the term${\mathit{\beta }}_{\mathbf{5}}{{\mathbf{x}}_1}^{\mathbf{2}}{\mathit{x}}_{\mathbf{2}\mathbf{}}$ represents the curvilinear relationship between task performance score (y) and conscientiousness score (x₁) and x₂ term represents job complexity.