Normal Distribution
nothing is called 95% until it has been counted
Concept

Influence — where it appears

How much a fitted result changes when one observation is removed, which combines where the observation sits with how far it falls from the fit. It is measured by refitting without the row, and two rows standing together can each look uninfluential while jointly deciding the answer.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Twenty points and one more, at leverage 0.74. Without the distant point the slope is 0.495; with it the slope is -0.389. Its leverage is 0.737 and its Cook's distance is 24.1, against a conventional threshold of 1.

The line that one point drew

A single observation among twenty-one reverses the sign of a fitted relationship. Its leverage is known from its x value before the outcome is looked at, so this is a property of the design rather than a surprise in the data.

regression · Leverage What a design does to the residuals of a correct model. Every residual has standard deviation sigma times the square root of one minus its leverage. On this design the leverages run from 0.045 to 0.663, so the residual spreads differ by a factor of 1.68 — and the model is exactly right. The high-leverage point's residual averages 0.46 of the fitted spread where a typical point's averages 0.79.

Residuals are not the errors

A residual's standard deviation is σ√(1 − hᵢᵢ), so a design whose leverages run from 0.045 to 0.663 produces residuals whose spreads differ by a factor of 1.68 with the model exactly right. On the samples where the high-leverage point really did have the largest error, a raw residual plot shows it as the largest on 0.0% of them.

lineup · Qq Two far rows, and the line with one of them deleted. Twenty clean points and two rows near x = 9. The slope is −0.511 with every row, −0.376 with one far row deleted, and 0.495 with both deleted. Deleting one of them barely moves the line, because the other is still there.

Two points that hide each other

One far observation among twenty-one has a Cook's distance of 24.1. Put a second beside it and the two read 0.966 and 0.772, neither crossing 1, while together they reverse the slope and deleting both moves the fit by 53.3.

regression · Leverage

Named alongside it

The objects these essays reach for when they reach for this one.

LeverageCook's distanceLeast squaresModel diagnosticsOutliersStudentised residualBonferroniClosed formExperimental designGroup deletionHat matrixLeave-one-out

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