Question

Service workers and customer relations. A study in Industrial Marketing Management (February 2016) investigated the impact of service workers’ (e.g., waiters and waitresses) personal resources on the quality of the firm’s relationship with customers. The study focused on four types of personal resources: flexibility in dealing with customers(x₁), service worker reputation(x₂), empathy for the customer(x₃), and service worker’s task alignment(x₄). A multiple regression model was employed used to relate these four independent variables to relationship quality (y). Data were collected for n = 220 customers who had recent dealings with a service worker. (All variables were measured on a quantitative scale, based on responses to a questionnaire.)

a) Write a first-order model for E(y) as a function of the four independent variables. Refer to part

Which β coefficient measures the effect of flexibility(x₁)on relationship quality (y), independently of the other

b) independent variables in the model?

c) Repeat part b for reputation(x₂), empathy(x₃), and task alignment(x₄).

d) The researchers theorize that task alignment(x₄)“moderates” the effect of each of the other x’s on relationship quality (y) — that is, the impact of eachx, x₁,x₂, orx₃on y depends on(x₄). Write an interaction model for E(y) that matches the researchers’ theory.

e) Refer to part d. What null hypothesis would you test to determine if the effect of flexibility(x₁)on relationship quality (y) depends on task alignment(x₄)?

f) Repeat part e for the effect of reputation(x₂)and the effect of empathy(x₃).

g) None of the t-tests for interaction were found to be “statistically significant”. Given these results, the researchers concluded that their theory was not supported. Do you agree?

Short answer

a) The equation is y=$y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3+{\beta }_4{x}_4+\epsilon$.

b) The $\beta$coefficient that measures changes in y for a given 1-unit increase in flexibility is measured by.

c) According to the equation the $\beta$ coefficients associated with reputation (x₂), empathy (x₃) , and task alignment (x₄) are${\beta }_1,{\beta }_2$, and ${\beta }_3$ respectively.

d) $E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3+{\beta }_4{x}_4+{\beta }_5{x}_1{x}_4+{\beta }_6{x}_2{x}_4+{\beta }_7{x}_3{x}_4+\epsilon$

e) The null hypothesis would be $H_0:{\beta }_5=0$against the alternate hypothesis $H_a:{\beta }_5\ne 0$;

f) To test whether the effect of reputation (x₂) and task alignment (x₄) interact, the value of ${\beta }_6$is tested. Mathematically, the null hypothesis would belocalid="1651190898603" $H_0:{\beta }_6=0$; against the alternate hypothesislocalid="1651190913791" $H_a:{\beta }_6\ne 0$

g) To test whether the effect of empathy (x₃) and task alignment (x₄) interact, the value of ${\beta }_7$is tested. Mathematically, the null hypothesis would be localid="1651190931488" $H_0:{\beta }_7=0$; against the alternate hypothesislocalid="1651190948413" $H_a:{\beta }_7\ne 0$

Step-by-step solution

First-order model equation

The first-order model equation here is$E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3+{\beta }_4{x}_4+\epsilon$

Where,$x_1$= flexibility in dealing with customers

$x_2$= service worker reputation

$x_3$= empathy for the customer

x₄= service worker’s task alignment

Interpretation of  β coefficient

The $\beta$coefficient that measures changes in y for a given 1-unit increase in flexibility is measured by${\beta }_1$ .

Clarification of β coefficient

The $\beta$coefficients associated with different variables represent the changes in y due to a 1-unit change in the respective variable.

Therefore, according to the equation the $\beta$ coefficients associated with reputation (x₂) , empathy (X₃), and task alignment (x4) are ${\beta }_2$,${\beta }_3$ , and ${\beta }_4$ respectively.

Interaction model

The interaction model where task alignment (x₄) impacts the relationship of each (x₁, X₂, and X₃) can be written as,

$E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3+{\beta }_4{x}_4+{\beta }_5{x}_1{x}_4+{\beta }_6{x}_2{x}_4+{\beta }_7{x}_3{x}_4+\epsilon$

Here the added variables x₁x_4,a, x₂x₄and x₃x₄represent the interaction between x₁and x₄, x₂and x₄, x₃ and x₄respectively.

Significance of β5

To test whether the effect of flexibility x₁ and x₄ task alignmentinteract, the value is tested

Mathematically,

The null hypothesis would be $H_0:{\beta }_5=0$against the alternate hypothesis;

Importance of β6

To test whether the effect of reputation (x₂) and task alignment (x₄) interact, the value is tested

Mathematically,

The null hypothesis would be $H_0:{\beta }_6=0$; against the alternate hypothesis $H_a:{\beta }_6\ne 0$

To test whether the effect of empathy (x₃) and task alignment (x₄) interact, the value is tested

Mathematically,

The null hypothesis would be $H_0:{\beta }_7=0$; against the alternate hypothesis $H_a:{\beta }_7\ne 0$

 Conclusion about the interaction model

When it is concluded from the t-test that none of the tests are statistically significant for interaction then it can be said that the researcher’s theory that there are some interactions in the model is not true.