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

Leverage — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · regression
  2. 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.

    part 2 · regression
  3. 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.

    part 3 · regression
  4. 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.

    part 4 · regression
  5. Two halves of one dataset with 8 far rows of twenty: the trimmed fit's winner and the best half with the other sign. Twenty rows, 8 of them near x = 9 on a line with a reversed slope. The exact trimmed fit keeps the eleven rows whose own line has the smallest residual sum: 5 far rows and 6 clean ones, slope -0.629, residual sum 0.587. The best eleven rows whose line slopes the other way are all clean, slope 0.398, residual sum 0.657 — 1.12 times the winner's.

    The other half a trimmed fit rejected

    The exact trimmed fit visits every half of the data, so it can report more than its winner: the best half whose line slopes the other way. Whenever the winner is reversed by a far cluster, that other half is the clean line. Printed beside the winner when its residual sum is within 1.70 times — a rule that raises five per cent of clean datasets — it flags 65% of the reversed fits at seven far rows of twenty, 55% at eight and 30% at nine, and no rule built on fit can do better at nine, where the far cluster has become the tighter half by every measure the data hold.

    part 5 · regression

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