Influential Data Points

Graphical Methods

In the case of simple linear regression (p = 2), a contingency plot of the response versus predictor values might disclose influential observations, which will fall well outside the general two-dimensional trend of the data. Observations with high leverage as a result of the joint effects of multiple explanatory variables, however, are difficult to reveal by graphical means. Although simple graphing is effective in identifying extreme outliers and nonsensical values, and is valuable as an initial screen, the eyeball might not correctly discern less obvious influential points, especially when the data are sparse (i.e., small n).

Leverage

Observations whose influence is derived from explanatory values are known as leverage points. The leverage of the ith observation is defined as hi = xi(X′X)-1x′i, where xi is the ith row of the n × p design matrix X for p predictors and sample size n. Larger values of hi, where 0≤hi≤1, are indicative of greater leverage. For reasonably large data sets (n – p > 50), a value of hi greater than 2p/n is a standard criterion for classification as a leverage point, where ∑ni=1hi = p and thus the mean of hi = p / n.

Standardized Residuals

An objective test for outliers is available in the form of standardized residuals. The Studentized deleted residuals,

Estimates of Influence

Several additional measures assess influence on the basis of effect on the model fit and estimated regression parameters. The standardized change in fit, DFFITSi =

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