Diagnostic influential observations

Influential observations are ones that significantly affect the values of the parameter estimates, their standard errors, and the predicted values. One statistic used to detect influential observations has already been presented, the HAT matrix. An obvious way to detect these observations is to remove an observation one at a time and examine how the recalculated parameter estimates compare to their original values. This is the row deletion approach to influence diagnostics and on first glance it would appear that this process requires n-iter-ations—a numerically intensive procedure. Statisticians, however, have derived equations that directly reflect the influence of the ith observation without iteration. One useful diagnostic is DFFITS... 

Apart from the analysis of residuals, the recognition of outUers and of influential observations is important for the selection of a regression model. We will raise those questions for the generalized regression diagnostics in Section 6.2. 

Figure 6.10 Regression diagnostics Residual plot in dependence on for influential observations and out- the calculated y values, (c) Jack-liers. (a) Cook s distance for recogni- knifed residuals according to tion of influential observations, (b) Eq. (6.102).

Previously in this chapter, we discussed Cook s distance for simple linear regression, a regression diagnostic that is used to detect an extreme value and its influence by removing it from the analysis and then observing the results. In multiple linear regression, the same approach is used, except that a data set is removed. Cook s distance lets the researcher determine just how influential the ith value set is. 

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