Question

What diagnostic plot can you use to determine whether the assumption of equal variance has been violated? What should the plot look like when the variances are equal for all values of \(x ?\)

Short answer

Answer: The diagnostic plot used to check the assumption of equal variances is the residual plot, which plots the residuals against the predicted (fitted) values. When the variances are equal for all values of \(x\), the residual plot should display a random scatter of points around the 0 horizontal line, without any visible pattern, and with the spread of the residuals approximately constant across the entire range of the predicted values.

Step-by-step solution

Identify the diagnostic plot to check equal variance assumption

The most common diagnostic plot for checking the assumption of equal variances is the residual plot, where the residuals (the difference between the observed values and the predicted values) are plotted against the predicted values, which are also known as the fitted values.

Describe how the residual plot should look when the assumption of equal variances is met

When the variances are equal for all values of \(x\), the residual plot should display a random scatter of points around 0 horizontal line, without any visible pattern. The spread of the residuals should be approximately constant across the entire range of the predicted values.

Indications of violation of the equal variance assumption in the residual plot

If the residual plot shows a pattern, like a cone shape (indicating heteroscedasticity), or a curve, this would suggest that the assumption of equal variances has been violated. In such cases, alternative models or data transformation methods can be used to address the issue.