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

Masking — where it appears

The concealment of influential observations from single-row diagnostics by other observations like them. Deleting one row leaves its partners holding the fit where they all wanted it, so each looks ordinary while together they reverse the result.

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

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 One row at x = 9 on a wrong line, fitted three ways. Least squares gives slope −0.389, Huber 0.171 with the extra row at weight 0.115, and least trimmed squares 0.420, fitted to the 12 rows it keeps. The twenty clean rows alone give 0.495. Open circles are the rows the trimmed fit leaves out.

A robust loss and a far x

One far row drags least squares to a slope of −0.389. Huber's loss, the standard robust line, reaches only 0.171, and carried further out the same row gets its full weight back. Least trimmed squares reads 0.420 at every distance, and at the normal model keeps 7.13% of least squares' efficiency to do it.

regression · Leverage One row at x = 9 on a wrong line, fitted three ways. Least squares gives slope −0.389, Huber 0.171 with the extra row at weight 0.115, and least trimmed squares 0.420, fitted to the 12 rows it keeps. The MM-estimator carried on from the trimmed fit gives 0.479, with the extra row at weight 0.000 and the scale fixed at 0.319. The twenty clean rows alone give 0.495. Open circles are the rows the trimmed fit leaves out.

The start an efficient robust line inherits

The MM-estimator carries a trimmed fit on through a redescending loss, and it does what it promises on one far row: slope 0.479 at every distance, the row at weight exactly zero, and 87.2% of least squares' efficiency at twenty rows. What it cannot do is choose. At eight far rows of twenty the exact trimmed fit picks the wrong half on 111 datasets; the efficient step repairs none of them, spoils none of the other 89, and ends nearer the wrong line than the start did.

regression · Leverage

Named alongside it

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

Closed formLeverageOutliersBreakdown pointEfficiencyIteratively reweighted least squaresLeast trimmed squaresLocal minimumM estimatorRobust regressionBonferroniCook's distance

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