Cook's distance

In statistics, Cook's distance or Cook's D is a commonly used estimate of the influence of a data point when performing least squares regression analysis. In a practical ordinary least squares analysis, Cook's distance can be used in several ways: to indicate data points that are particularly worth checking for validity; 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.

Definition

Cook's distance measures the effect of deleting a given observation. Data points with large residuals (outliers) and/or high leverage may distort the outcome and accuracy of a regression. Points with a large Cook's distance are considered to merit closer examination in the analysis. For the algebraic expression, first define

$\mathbf {H} \equiv \mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }$

as the $(n\times n)$ hat matrix (or projection matrix) of the $n$ observations of each explanatory variables. Then let ${\hat {\beta }}^{(-i)}$ be the OLS estimate of $\beta$ that results from omitting the $i$-th observation ($i=1,2,\dots ,n$). Then we have

${\hat {\beta }}-{\hat {\beta }}^{(-i)}=\left({\frac {1}{1-h_{i}}}\right)(\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {x_{i}} \cdot e_{i}$

where $e_{i}\,$ is the residual (i.e., the difference between the observed value and the value fitted by the proposed model), and $h_{ii}\,$, defined as

$h_{ii}\equiv \mathbf {x} _{i}^{\top }(\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {x} _{i}$

is the leverage, i.e., the $i$-th diagonal element of $\mathbf {H} \,$. With this, we can define Cook's distance as

$D_{i}={\frac {e_{i}^{2}}{k\ \mathrm {MSE} }}\left[{\frac {h_{ii}}{(1-h_{ii})^{2}}}\right],$

where $k$ is the number of fitted parameters, and $\mathrm {MSE} \,$ is the mean square error of the regression model. Algebraically equivalent is the following expression

$D_{i}={\frac {({\hat {\beta }}-{\hat {\beta }}^{(-i)})^{\top }\mathbf {X} ^{\top }\mathbf {X} ({\hat {\beta }}-{\hat {\beta }}^{(-i)})}{(1+k)s^{2}}},$

where $s^{2}$ is the OLS estimate of the variance of the error term, defined as

$s^{2}\equiv {\frac {\mathbf {e} ^{\top }\mathbf {e} }{n-k}}$

And a third equivalent expression is

$D_{i}={\frac {\sum _{j=1}^{n}({\hat {Y}}_{j}\ -{\hat {Y}}_{j(i)})^{2}}{k\ \mathrm {MSE} }},$

where:

${\hat {Y}}_{j}\,$ is the prediction from the full regression model for observation j;

${\hat {Y}}_{j(i)}\,$ is the prediction for observation j from a refitted regression model in which observation i has been omitted;

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 $D_{i}>1$ has been suggested. Others have indicated that $D_{i}>4/n$, where $n$ is the number of observations, might be used.

A conservative approach relies on the fact that Cook's distance has the form W/p, where W is formally identical to the Wald statistic that one uses for testing that $H_{0}:\beta _{i}=\beta _{0}$ using some ${\hat {\beta }}_{[-i]}$. Recalling that W/p has an $F_{p,n-p}$ distribution (with p and n-p degrees of freedom), we see that Cook's distance is equivalent to the F statistic for testing this hypothesis, and we can thus use $F_{p,n-p,1-\alpha }$ as a threshold.

Interpretation

Specifically $D_{i}$ can be interpreted as the distance one's estimates move within the confidence ellipsoid that represents a region of plausible values for the parameters. 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.

See also

• Outlier
• Leverage (statistics)
• Partial leverage
• DFFITS
• Studentized residual

Further reading

• Atkinson, Anthony; Riani, Marco (2000). "Deletion Diagnostics". Robust Diagnostics and Regression Analysis. New York: Springer. pp. 22–25. ISBN 0-387-95017-6. {{cite book}}: External link in |chapterurl= (help); Unknown parameter |chapterurl= ignored (|chapter-url= suggested) (help)
• Heiberger, Richard M.; Holland, Burt (2013). "Case Statistics". Statistical Analysis and Data Display. Springer Science & Business Media. pp. 312–27. ISBN 9781475742848. {{cite book}}: External link in |chapterurl= (help); Unknown parameter |chapterurl= ignored (|chapter-url= suggested) (help)
• Krasker, William S.; Kuh, Edwin; Welsch, Roy E. (1983). "Estimation for dirty data and flawed models". Handbook of Econometrics. Vol. 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. 16 (2). Sage: 270–301. doi:10.1177/1094428112470848. Retrieved 4 December 2015.