Question

Factors that impact an auditor’s judgment. A study was conducted to determine the effects of linguistic delivery style and client credibility on auditors’ judgments (Advances in Accounting and Behavioural Research, 2004). Two hundred auditors from Big 5 accounting firms were each asked to perform an analytical review of a fictitious client’s financial statement. The researchers gave the auditors different information on the client’s credibility and linguistic delivery style of the client’s explanation. Each auditor then provided an assessment of the likelihood that the client-provided explanation accounted for the fluctuation in the financial statement. The three variables of interest—credibility (x₁), linguistic delivery style (x₂) , and likelihood (y) —were all measured on a numerical scale. Regression analysis was used to fit the interaction model,$y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_1{x}_2+\epsilon$ . The results are summarized in the table at the bottom of page.

a) Interpret the phrase client credibility and linguistic delivery style interact in the words of the problem.

b) Give the null and alternative hypotheses for testing the overall adequacy of the model.

c) Conduct the test, part b, using the information in the table.

d) Give the null and alternative hypotheses for testing whether client credibility and linguistic delivery style interact.

e) Conduct the test, part d, using the information in the table.

f) The researchers estimated the slope of the likelihood–linguistic delivery style line at a low level of client credibility 1x1 = 222. Obtain this estimate and interpret it in the words of the problem.

g) The researchers also estimated the slope of the likelihood–linguistic delivery style line at a high level of client credibility 1x1 = 462. Obtain this estimate and interpret it in the words of the problem.

Short answer

a) In the model, variables x₁ and x₂ are said to have some interaction amongst them indicating that there is a relationship between the two variables which means that the client’s credibility might be related to the linguistic delivery style the client had chosen. This dependency is expressed using the term ‘x₁ x₂ ’.

b) $H_0:{\beta }_1={\beta }_2={\beta }_3=0$ while H_a At least one of the parameters ${\beta }_1,{\beta }_2,{\beta }_3$is non zero

c) At 95% significance level, it can be concluded that ${\beta }_1\ne {\beta }_2\ne {\beta }_3\ne 0$

d) The null hypothesis and alternate hypothesis are $H_0:{\beta }_3=0$ while $H_a:{\beta }_3\ne 0$

e) At 95% significance level localid="1651179551636" ${\beta }_3\ne 0$ . Hence it can be concluded with enough evidence that x₁ and x₂do not interact in the model.

f) The slope of the line relating y to x₂ when x₁ = 22 is 1.47. The positive value denotes a positive relationship amongst the two variables and a low value means that the relation is not so strong.

g) The slope of the line relating y to x₂ when x₁ = 46 is 2.334. The positive value denotes a positive relationship amongst the two variables and a low value means that the relation is not so strong.

Step-by-step solution

Interaction amongst independent variables

The model is trying to explain the likelihood that the client-provided explanation accounted for the fluctuation in the financial statement where the independent variables are client credibility (x₁) and linguistic delivery style (x₂) . In the model, variables x₁ and x₂ are said to have some interaction amongst them indicating that there is a relationship between the two variables which means that the client’s credibility might be related to the linguistic delivery style the client had chosen. This dependency is expressed using the term ‘x₁ x₂ ’.

Overall goodness of the fit of the model

To check the overall goodness of the fit, the null hypothesis is whether the model parameters are explaining the model where the beta values are zero and the alternate hypothesis is whether the beta values are non-zero.

Mathematically,

$H_0:{\beta }_1={\beta }_2={\beta }_3=0$

H_a At least one of the parameters ${\beta }_1,{\beta }_2,{\beta }_3$is non zero

Goodness of the model fit

$H_0:{\beta }_1={\beta }_2={\beta }_3=0$

At least one of the parameters${\beta }_1,{\beta }_2,{\beta }_3$is non zero

Here, F test statistic =$\frac{SSE}{n-\left(k+1\right)}=55.35$

$H_0$is rejected if P - value < 0.01. For , since P - value < 0.0005

Sufficient evidence to reject $H_0$ at 95% confidence interval.

Therefore,${\beta }_1\ne {\beta }_2\ne {\beta }_3\ne 0$

Significance of  β3

To test whether client credibility and linguistic delivery style interact, the value of ${\beta }_3$ is tested

Mathematically,

localid="1651179732490" $$\begin{gathered} H_0:{\beta }_3=0 \\ {H}_a:{\beta }_3\ne 0 \end{gathered}$$

Significance ofβ3

$$\begin{gathered} H_0:{\beta }_3=0 \\ {H}_a:{\beta }_3\ne 0 \end{gathered}$$

Here, t-test statistic $\frac{\overbrace{{\beta }_3}}{s{\beta }_3}=\frac{0.036}{0.009}$

Value of $t_{0.05,199}$ is 1.645

is rejected if t static$>t_{0.05,199}$ . For$\alpha =0.05$, since t > $t_{0.05,199}$

Sufficient evidence to reject H₀ at 95% confidence interval.

Therefore,${\beta }_3\ne 0$. Hence it can be concluded with enough evidence thatx₁and x₂do not interact in the model.

Interpretation of slope

Given, $E\left(y\right)=15.865+0.037x_1-0.0678x_2+0.036x_1{x}_2$for x₁=22

localid="1651181050324" $$\begin{gathered} E\left(y\right)=15.865+0.037\left(22\right)-0.0678x_2+0.036\left(22\right)x_2 \\ E\left(y\right)=16.679+1.47x_2 \end{gathered}$$

The slope of the line relating y tox₂whenx₁= 22 is 1.47. The positive value denotes a positive relationship amongst the two variables and a low value means that the relation is not so strong.

Interpretation of slope

Given, $E\left(y\right)=15.865+0.037x_1-0.0678x_2+0.036x_1{x}_2$ for x₁=46

$$\begin{gathered} E\left(y\right)=15.865+0.037\left(46\right)-0.0678x_2+0.036\left(46\right)x_2 \\ E\left(y\right)=17.567+2.334x_2 \end{gathered}$$

The slope of the line relating y tox₂whenx₁= 46 is 2.334. The positive value denotes a positive relationship amongst the two variables and a low value means that the relation is not so strong.