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

Saddle point — where it appears

A stationary point of a fitted surface that is a maximum in one direction and a minimum in another, so the fitted optimum is not an optimum at all. Reporting a stationary point without saying which kind it is is how a response-surface analysis recommends a setting nobody should run.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

Four runs, and the term they cannot reach. Every run sits at a corner, so x₁² and x₂² are 1 at every run and both columns are copies of the intercept. The normal matrix is singular: the design has no information about curvature at all, and no analysis can recover it.

The design that cannot see a curve

A two-level factorial has every run at a corner, where every squared term equals one — so the column that would estimate curvature is a copy of the intercept, and the design has no information about it at all. A few runs at the centre buy one number back, and only one.

surface · Factorial Where the maximum is, from 15 runs. One dataset, one fitted quadratic, and two answers to "where is the best setting". The delta method reports 0.80 ± 0.46, a finite interval it will report whatever the data does. Fieller's set is 0.49 to 1.76, because the curvature here has t = -4.04. The true optimum is at 0.75.

The optimum is a ratio, and its interval is sometimes the whole line

The best setting is −b₁/2b₂: a ratio of two estimates whose denominator is a curvature the design can often barely see. The delta method reports a finite interval every time and covers 68.8% where the curvature is weak; Fieller's set covers 95% and says so by being unbounded.

surface · Optimum What the fit calls the shape, against what it is. One eigenvalue held at −3 and the other swept from −2 to 2, so the truth is a maximum on the left and a saddle on the right and the change happens at exactly zero. At an eigenvalue of −0.25 — a genuine maximum — the fit reports a saddle on 26.4% of studies; at +0.25 — a genuine saddle — it reports a maximum on 25.1%. The standard error of a squared coefficient under this design is 0.3791, and the region of confusion is about that wide either side of zero.

The sign the curvature has

A fitted surface reports a maximum, a minimum or a saddle, and the report is a comparison of two estimated eigenvalues against zero. At a true second eigenvalue of −0.25 the fit calls a genuine maximum a saddle on 26.4% of studies, and at +0.25 it calls a genuine saddle a maximum on 25.1%.

surface · Optimum The ridge, when the fitted optimum is outside the region. One fitted surface. Its stationary point is at a radius of 2.289 and the fit calls the shape a maximum. The ridge is the best setting at each radius, found by the Lagrange condition (B̂ − μI)x = −ĝ/2; the fitted response rises along it from 59.93 at the centre to 62.10 at the edge. The true optimum is at (0.4, 0.3).

When the best setting is outside the region

On a flat surface at twice the noise the fitted optimum lands outside the experimental region on 24.9% of studies and more than three coded units out on 11.8%. The answer is a ridge — the best setting at each radius, with a closed form — and the two obvious rules for using it turn out to be within four per cent of each other.

surface · Optimum The law is the eigenvalues, and nothing else. The mean and the skewness of n(ĝ − g) at the stationary point, measured over 40,000 draws, against the closed forms: half the sum of the Hessian's eigenvalues for the mean, and for the skewness twice root two times the sum of their cubes over the sum of their squares to the power three halves. The worst disagreement anywhere is 0.028. In one variable the second-order law is a single chi-square with one degree of freedom and its sign is the sign of the second derivative; here it is a weighted sum with the Hessian's eigenvalues as weights, so a bowl and a valley differ in both moments and a saddle has both equal to zero.

A flat point with more than one direction

At a stationary point of a function of several means the second-order law is ½ Z′HZ, so the bias is half the Hessian's trace — 2.008 for a bowl, 5.028 for a valley, and −0.006 for a saddle, where the eigenvalues cancel. The saddle's coverage is the worst of the three.

normal · Clt Testing for flatness at 5% before choosing the interval for a squared mean. Coverage of four nominal 95% intervals for μ², exact, against the true mean's distance δ from the flat point in standard errors. The tangent interval Z² ± 2z|Z| has its worst, 85.98%, at δ = 1.45. The second-order interval read off χ²₁ is right only at δ = 0 and falls to 34.22% at δ = 2. The two-stage rule, which uses the second-order interval when a 5% test does not reject flatness, has its worst, 49.72%, at δ = 1.96. The squared exact interval never falls below 95%.

Testing for a flat point first

Test whether the gradient is zero, and if the test does not reject, read the interval off the second-order law that holds at a stationary point. For a squared mean that two-stage rule covers about 50% at its worst whatever the pretest's level — 49.72% with a 5% test, where the tangent interval it was meant to rescue never falls below 85.98% — because its worst sits exactly where the true distance equals the test's critical value, half the samples are declared flat there, and the second-order interval cannot reach the truth from any of them.

normal · Clt

Named alongside it

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

Stationary pointCurvatureEigenvalueConfidence intervalCoverageDelta methodExperimental designMonte CarloCanonical analysisCentral composite designChi-squareClosed form

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