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

Decided before the data

Every other field here repairs an analysis after the fact. These figures are about the arrangement that makes the repair unnecessary: blocking removes exactly the variance the blocks carry, randomisation buys a known reference distribution rather than balance, and varying every factor at once estimates each of them from every run.
The same 40 units, arranged two ways. Both designs estimate the same effect of 0.5 and both are unbiased — 0.488 and 0.497. The blocked design's estimate has standard deviation 0.318 against 0.692, a variance ratio of 0.21 where the model predicts 0.20.

The variance removed before the data

Arranging forty units in pairs rather than assigning them at random cuts the variance of the estimated effect to a fifth — and the fifth is knowable in advance, because it is exactly the share of the variance the pairs do not carry.

 Every way of splitting 16 units into two halves. All 12,870 assignments, enumerated. The spread of the standardised imbalance is exactly 2/√n = 0.500, whatever the covariate's own distribution, and 33.3% of assignments differ by more than 0.5 standard deviations. Randomisation does not deliver balance; it delivers a known distribution of imbalance.

Randomisation is not balance

A third of all ways to split sixteen units leave the two halves more than half a standard deviation apart on a covariate. What randomisation delivers is not balance but a known reference distribution — and it makes a test exact with no assumption about the data's shape at all.

 Two factors that interact — where each design looks. The true response at the four corners is -10, 6, 4, 0. One factor at a time visits three of them, sees that raising either factor alone helps, and recommends raising both — a corner it never ran, and one that is worse than either single change. It picks the best corner 0.0% of the time against the factorial design's 99.4%, on the same number of runs.

One factor at a time

Changing one thing per experiment estimates each effect from two conditions; changing everything at once estimates each from every run. The ratio is (k+1)/2 and it is exact — and when two factors interact, the one-at-a-time design recommends a setting it never tried.

 Power at an effect of 0.5 standard deviations. The curve is the non-central t on 2n − 2 degrees of freedom with δ = d√(n/2); the dots are 4,000 experiments run at each size. Reaching 80% power needs 64 per arm.

How many subjects

Sixty-four per arm for 80% power at half a standard deviation — a power figure that could only be simulated, with nothing to disagree with, until the non-central t was written. Two routes now, agreeing to within the simulation's own error.

 What a 8-run fraction of 4 factors confounds. The defining relation is I = ABCD, so the resolution is 4. A is estimated as A + BCD; B is estimated as B + ACD; C is estimated as C + ABD; D is estimated as D + ABC. Each of those is an identity about the design rather than an approximation about the data.

The word a fraction costs

A half fraction estimates each main effect as an exact sum of that effect and everything it is confounded with — no error term, no sample-size argument. With every interaction at 0.8 the design reports a true effect of −1 as −0.20, and the design cannot test the assumption that makes the number mean anything.

 The Box–Behnken design in three factors: 15 runs, none at a corner. Twelve runs at the midpoints of the cube's edges and 3 at its centre. Every run holds one factor at zero, so no run puts all three factors at an extreme — which is what makes it runnable where a corner is not. The three panels are the design's coordinate projections, with repeated positions marked.

The design that refuses the corners

Box–Behnken runs three factors in fifteen runs and puts none of them at a corner, which is what makes it usable where a corner cannot be run. It predicts the corner 1.84 times worse than the seventeen-run design that goes there, and 1.31 times worse at the middle of a face, and all three numbers are matrix computations with no simulation in them.

 What one lost run costs a 16-run factorial fitting 11 coefficients. Every run is worth the same: dropping any one multiplies every coefficient's variance by 1.2000, which is 1 + 1/(N − p) with N = 16 and p = 11, and gives every pair of coefficients a correlation of 0.1667 where the complete design had exactly zero.

The run that did not happen

Lose one run from any orthogonal design and every coefficient's variance is multiplied by exactly 1 + 1/(N − p), and every pair of coefficients acquires a correlation of exactly 1/(N − p + 1) where there was none. The price is set by the design's spare capacity and by nothing else, and a saturated design cannot survive it at all.

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