Final answer:
When residual plots show strong nonlinear patterns, a linear regression model becomes less reliable, and nonlinear regression methods with transformations of the response and predictor variables should be considered. These methods can improve the model's fit to the data and make predictions more accurate.
Step-by-step explanation:
If residual plots exhibit strong nonlinear patterns, the inferences made by a linear regression model can be quite misleading. In such instances, we should employ nonlinear regression methods based on simple transformations of the response variables and the predictor variables.
Regression analysis is a powerful statistical method that allows us to examine the relationship between two or more variables of interest. While the most commonly applied form of regression analysis is linear regression, it has some assumptions that must be met to provide accurate predictions. These include a linear relationship between the variables, residuals that are independent and normally distributed, and equal variance of residuals.
The least-squares regression line is often used to find the line of best fit in linear regression because it minimizes the SSE, or the sum of squared errors. This line facilitates predicting the expected value of y (the response variable) based on x (the predictor variable), but its reliability depends significantly on the appropriateness of the linear model.
A residual plot helps to verify whether a linear model is appropriate by displaying the residuals (differences between observed and predicted values) on the vertical axis against the predictor variable on the horizontal axis. When residual plots show a random dispersion of points around the horizontal axis, this suggests that the model’s predictions are consistent with a linear relationship. However, if there is a discernible pattern or curve in the residual plot, this is a sign that a linear model may not be suitable, and a nonlinear model should be considered instead.
To address situations where a linear model does not fit well, we can apply transformations to the response and predictor variables. These transformations can be logarithmic, exponential, square root, or others, depending on the nature of the nonlinearity. By transforming the data, we aim to linearize the relationship thus making the linear regression model more applicable and the resultant predictions more accurate.