In statistics, Cook's distance or Cook's D is a commonly used estimate of the influence of a data point when performing a least-squares regression analysis. In a practical ordinary least squares analysis, Cook's distance can be used in several ways: to indicate influential data points that are particularly worth checking for validity; or to indicate regions of the design space where it would be good to be able to obtain more data points. It is named after the American statistician R. Dennis Cook, who introduced the concept in 1977.
Data points with large residuals (outliers) and/or high leverage may distort the outcome and accuracy of a regression. Cook's distance measures the effect of deleting a given observation. Points with a large Cook's distance are considered to merit closer examination in the analysis.
For the algebraic expression, first define
where is the error term, is the coefficient matrix, is the number of covariates or predictors for each observation, and is the design matrix including a constant. The least squares estimator then is , and consequently the fitted (predicted) values for the mean of are
where is the projection matrix (or hat matrix). The -th diagonal element of , given by , is known as the leverage of the -th observation. Similarly, the -th element of the residual vector is denoted by .
Cook's distance of observation is defined as the sum of all the changes in the regression model when observation is removed from it
Detecting highly influential observations
There are different opinions regarding what cut-off values to use for spotting highly influential points. A simple operational guideline of has been suggested. Others have indicated that , where is the number of observations, might be used.
Specifically can be interpreted as the distance one's estimates move within the confidence ellipsoid that represents a region of plausible values for the parameters.[clarification needed] This is shown by an alternative but equivalent representation of Cook's distance in terms of changes to the estimates of the regression parameters between the cases, where the particular observation is either included or excluded from the regression analysis.
- Mendenhall, William; Sincich, Terry (1996). A Second Course in Statistics: Regression Analysis (5th ed.). Upper Saddle River, NJ: Prentice-Hall. p. 422. ISBN 0-13-396821-9.
A measure of overall influence an outlying observation has on the estimated coefficients was proposed by R. D. Cook (1979). Cook's distance, Di, is calculated...
- Cook, R. Dennis (February 1977). "Detection of Influential Observations in Linear Regression". Technometrics. American Statistical Association. 19 (1): 15–18. doi:10.2307/1268249. JSTOR 1268249. MR 0436478.
- Cook, R. Dennis (March 1979). "Influential Observations in Linear Regression". Journal of the American Statistical Association. American Statistical Association. 74 (365): 169–174. doi:10.2307/2286747. JSTOR 2286747. MR 0529533.
- Hayashi, Fumio (2000). Econometrics. Princeton University Press. pp. 21–23.
- "Cook's Distance".
- "Statistics 512: Applied Linear Models" (PDF). Purdue University.
- Cook, R. Dennis; Weisberg, Sanford (1982). Residuals and Influence in Regression. New York, NY: Chapman & Hall. ISBN 0-412-24280-X.
- Bollen, Kenneth A.; Jackman, Robert W. (1990). Fox, John; Long, J. Scott, eds. Regression Diagnostics: An Expository Treatment of Outliers and Influential Cases. Modern Methods of Data Analysis. Newbury Park, CA: Sage. pp. 257–91. ISBN 0-8039-3366-5.
- Atkinson, Anthony; Riani, Marco (2000). "Deletion Diagnostics". Robust Diagnostics and Regression Analysis. New York: Springer. pp. 22–25. ISBN 0-387-95017-6.
- Heiberger, Richard M.; Holland, Burt (2013). "Case Statistics". Statistical Analysis and Data Display. Springer Science & Business Media. pp. 312–27. ISBN 9781475742848.
- Krasker, William S.; Kuh, Edwin; Welsch, Roy E. (1983). "Estimation for dirty data and flawed models". Handbook of Econometrics. 1. Elsevier. pp. 651–698. doi:10.1016/S1573-4412(83)01015-6.
- Aguinis, Herman; Gottfredson, Ryan K.; Joo, Harry (2013). "Best-Practice Recommendations for Defining Identifying and Handling Outliers" (PDF). Organizational Research Methods. Sage. 16 (2): 270–301. doi:10.1177/1094428112470848. Retrieved 4 December 2015.