Cook's Distance

Definition

Cook's distance is a measure used in regression analysis to identify influential observations, or outliers, that have a disproportionate impact on the regression model. It quantifies the change in the regression coefficients that would result from the deletion of a particular data point.

5 Must Know Facts For Your Next Test

1. Cook's distance is a diagnostic tool used to identify influential observations in a regression model that may have a significant impact on the model's parameter estimates.
2. A high Cook's distance value for a particular observation indicates that the removal of that observation would substantially change the regression coefficients, suggesting it is an influential data point.
3. Cook's distance is calculated as the normalized change in the regression coefficients that would result from deleting a specific observation from the data set.
4. Observations with a Cook's distance greater than 1 are generally considered to be highly influential and should be investigated further.
5. Cook's distance is sensitive to both the leverage of an observation and the size of its residual, making it a useful tool for identifying both outliers and high-leverage points.

Review Questions

• Explain the purpose of using Cook's distance in regression analysis.
  • The purpose of using Cook's distance in regression analysis is to identify observations that have a disproportionate influence on the regression model. Cook's distance quantifies the change in the regression coefficients that would result from the deletion of a particular data point. By identifying influential observations, researchers can assess the robustness of their regression model and make informed decisions about whether to retain or exclude certain data points from the analysis.
• Describe the relationship between Cook's distance, leverage, and residuals.
  • Cook's distance is a function of both the leverage of an observation and the size of its residual. Leverage measures how much an individual observation influences the regression line based on its position in the predictor space. Residuals represent the difference between the observed and predicted values for each observation. Cook's distance combines these two factors to identify observations that are both highly leveraged and have large residuals, indicating that they have a significant impact on the regression model. Observations with high Cook's distance values are considered influential and may need to be investigated further or excluded from the analysis.
• Discuss how the interpretation of Cook's distance can inform decisions about model selection and data cleaning in regression analysis.
  • The interpretation of Cook's distance can provide valuable insights for model selection and data cleaning in regression analysis. Observations with high Cook's distance values, typically greater than 1, indicate that they are highly influential and may be driving the regression model. Researchers can use this information to assess the robustness of their model and decide whether to retain or exclude these influential observations. If the removal of an influential observation significantly changes the regression coefficients, it may suggest that the model is overly sensitive to certain data points and that alternative model specifications or data transformations should be considered. Additionally, identifying and addressing influential observations through data cleaning can improve the overall fit and reliability of the regression model.

Related terms

Regression Analysis: A statistical technique used to model the relationship between a dependent variable and one or more independent variables.

Outlier: An observation that lies an abnormal distance from other values in a data set, and may indicate a data entry error or suggest that the observation comes from a different population than the rest of the data.

Leverage: A measure of how much an individual observation influences the regression line, based on its position in the predictor space.