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 .
where is the fitted response value obtained when excluding , and is the mean squared error of the regression model.
Detecting highly influential observations
There are different opinions regarding what cut-off values to use for spotting highly influential points. Since Cook's distance is in the metric of an F distribution with and (as defined for the design matrix above) degrees of freedom, the median point (i.e., ) can be used as a cut-off. Since this value is close to 1 for large , a simple operational guideline of has been suggested. Note that the Cook's distance measure does not always correctly identify influential observations.
Relationship to other influence measures (and interpretation)
The benefit in the last formulation is that it clearly shows the relationship between and to (while p and n are the same for all observations). If is large then it (for non-extreme values of ) will increase . If is close to 0 than will be small, while if is close to 1 then will become very large (as long as , i.e.: that the observation is not exactly on the regression line that was fitted without observation ).
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.
High dimensional Influence Measure (HIM), is an alternative to Cook's distance for when (i.e.: more predictors than observations) . While the Cook’s distance quantifies the individual observation’s influence on the least squares regression coefficient estimate, the HIM measures the influence of an observation on the marginal correlations.
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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...
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- High-dimensional influence measure
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