Concept

Outlier — where it appears

An observation that disagrees with the pattern the others follow by more than their spread explains. Whether it is a bad measurement or a sign the model is wrong cannot be told from the observation alone, and a design without replicates cannot tell at all.

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

A discrepancy of 4 σ at the top of the range: one bad run, or a line that bends there, in two designs of twelve runs. The same twelve errors in every panel. With one run at the top (upper panels) a bad observation and a bent line produce identical data, point for point. With three runs at the top (lower panels) the bad observation leaves one run far from its two neighbours and the bent line moves all three together. The dashed line is the true line; the grey line is the least-squares fit.

Three runs at the end of the line

With one run at the top of a regression's range, a bad observation there and a line that bends there produce data with exactly the same distribution, so no residual, influence measure or test can say which happened. Move six of twelve runs to the ends, three at each, and a discrepancy of four σ is named correctly as a bad run 84.1% of the time and as a bend 91.6% — while the slope's standard error falls from 0.0836 to 0.0709. The design, not the diagnostic, decides whether the question has an answer.

lineup · Qq

Three summaries of a fitted line's geometry on Anscombe's four datasets, with each rule's threshold. Largest leverage: 0.318, 0.318, 0.318, 1.000 against a threshold of 0.364. Largest Cook's distance: 0.489, 0.808, 1.393, undefined, the point fits itself against 1 and against 0.364. Largest studentised deleted residual: 2.1, 2.2, 1192.2, undefined against a Bonferroni threshold of 3.90.

Two numbers for the fit's geometry

No summary of the dependence between x and y can flag Anscombe's third and fourth datasets. A summary of the fit's own geometry can, and it takes two numbers of different kinds: the largest leverage, which reads the design before any y is seen and flags the fourth with certainty, and the largest deleted residual against a Bonferroni threshold, which flags the third at 1,192 and raises a false alarm on 4.78% of honest datasets. The rule of thumb most often quoted, Cook's distance over 4/n, flags the honest line too — and 66.45% of honest datasets like it.

spread · Summary

Named alongside it

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

Leverage Model diagnostics Anscombe quartet Bonferroni Cook's distance Experimental design False positive Influence Lack of fit Pure-error Residual plot Studentised residual

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