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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.

$D_i=\frac{\sum _{j=1}^n({\hat{Y}}_j-{\hat{Y}}_{j(i)}{)}^2}{p\mathrm{M}\mathrm{S}\mathrm{E}}.$

The following are the algebraically equivalent expressions (in case of simple linear regression):

$D_i=\frac{e_i^2}{p\mathrm{M}\mathrm{S}\mathrm{E}}\left[\frac{h_{ii}}{(1-h_{ii}{)}^2}\right],$

$D_i=\frac{(\hat{\beta }-{\hat{\beta }}^{(-i)}{)}^T(X^{T}X)(\hat{\beta }-{\hat{\beta }}^{(-i)})}{(1+p)s^2}.$

In the above equations:

${\hat{Y}}_j$ 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;

$h_{ii}$ is the i-th diagonal element of the hat matrix $\mathbf{X}{\left({\mathbf{X}}^T\mathbf{X}\right)}^{-1}{\mathbf{X}}^T$;

$e_i$ is the crude residual (i.e., the difference between the observed value and the value fitted by the proposed model);

$\mathrm{M}\mathrm{S}\mathrm{E}$ is the mean square error of the regression model;

$p$ is the number of fitted parameters in the model

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 ${\hat{\beta }}_{[-i]}$.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. 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.Template:Clarify 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

References

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