Residual plots

Definition

Residual plots are graphical representations that display the residuals on the vertical axis and the fitted values (or another independent variable) on the horizontal axis. These plots help assess the quality of a regression model by showing how well the model's predictions align with actual data, and they can reveal patterns that may indicate problems such as non-linearity, heteroscedasticity, or outliers in the data.

5 Must Know Facts For Your Next Test

1. A well-behaved residual plot should display no obvious patterns; if there are clear patterns, it suggests that the model may not be capturing all underlying relationships in the data.
2. Residual plots are crucial for diagnosing issues like non-linearity or outliers, which can lead to misleading conclusions if not properly addressed.
3. When the residuals fan out or show a systematic curve, it indicates heteroscedasticity, suggesting that the variance of the errors is not constant.
4. In simple linear regression, examining residual plots helps ensure that the assumptions of linearity and equal variance are met.
5. By analyzing residual plots, one can determine if a more complex model is necessary, such as using polynomial regression or transforming variables.

Review Questions

• How can you use residual plots to evaluate the fit of a regression model?
  • Residual plots are used to assess how well a regression model fits the data by displaying residuals against fitted values. If the residuals show random scatter with no discernible pattern, it indicates a good fit. Conversely, if there is a visible pattern in the residuals, it suggests that the model may not adequately represent the underlying data structure and that adjustments may be needed.
• Discuss how identifying non-linearity in residual plots can influence your choice of regression models.
  • Identifying non-linearity in residual plots suggests that a simple linear regression model may not adequately capture the relationship between variables. This insight can prompt the use of more complex models, such as polynomial regression or transformations of variables, to better account for these relationships. Ignoring signs of non-linearity can lead to biased estimates and misleading conclusions.
• Evaluate the implications of heteroscedasticity revealed in residual plots for inferential statistics in regression analysis.
  • Heteroscedasticity indicated by residual plots suggests that the variance of errors is not constant across levels of an independent variable. This condition violates one of the key assumptions of ordinary least squares regression and can lead to inefficient estimates and unreliable hypothesis tests. To address this issue, one might need to transform data, use weighted least squares regression, or employ robust standard errors to ensure valid inference and improved model performance.

Related terms

residuals: The differences between observed values and the predicted values from a regression model.

homoscedasticity:

A condition in which the variance of residuals is constant across all levels of an independent variable.

fitted values: The predicted values obtained from a regression model, calculated based on the estimated coefficients and independent variables.