Which assumption of linear regression is violated if the residuals exhibit a pattern when plotted against the predicted values?

The assumption of linear regression that is violated when the residuals exhibit a pattern when plotted against the predicted values is known as the homoscedasticity assumption.

Explanation

1. Homoscedasticity means that the variance of the errors (residuals) should be constant across all levels of the independent variable(s). If the residuals display a systematic pattern (for instance, forming a funnel shape), this indicates that the variability of the residuals changes with the level of the predicted values.

2. Implications of Violation: Such a violation suggests that the model may be incorrectly specified or that there is a non-linear relationship that has not been accounted for, which could lead to invalid statistical inference.

3. Recommendations: To address this, one can use various methods, such as transforming the dependent variable, using weighted least squares, or adding polynomial terms to capture the non-linear patterns.

Thus, when you observe patterns in the residuals, it is a clear indication that the assumption of homoscedasticity has been violated, warranting a reevaluation of the model's specifications or the data transformation techniques employed.