5.6 What to do about influential observations

Influence diagnostics locate the points that drive the fit; they do not say what to do with them. The first step is always to check the data: a point that is both influential and wrong should be corrected or removed. Beyond that:

(i)

A sound influential point may show that the model is inadequate. In Example 5.5.1, the warehouse store may genuinely behave differently, and the right model for the chain may need a regressor for the type of store, or a curve that flattens at large areas. One store at \(x = 20\) cannot tell these apart.

(ii)

If the point is sound but different in kind, the honest course may be to restrict the model to the range where it is supported, here branches of up to \(700\) m\(^2\), and say so.

(iii)

Never delete a point simply because it is influential. Report the fit with and without it. If the conclusions agree, the point does not matter; if they differ, as here, the data cannot settle the question and more observations are needed in the gap.

A design lesson follows too. A single point far from the others is a weakness in the data, and when the \(x\) values can be chosen they should be spread so that no one observation carries the estimate. Methods of robust regression, which limit the weight any one point can have, exist for data in which some influential points are unavoidable.

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