Four rules, three shapes, and no ordering that survives

The variance of the unadjusted treatment estimate under each rule, as a fraction of the variance a coin gives, over 450 trials of 200 units each. Against a covariate that enters linearly the rule that reads the number nearly halves it. Against a threshold at 1 it removes about a fifth. Against a quadratic every rule here is at or worse than a coin — they are all optimising a criterion that is one over the variance of an estimate in a model this outcome does not obey, and a constraint that helps nothing still costs something. Nothing in a trial says which column it is in.

The share of a coin's imbalance in an indicator 1{x > c} that survives a rule which balances the covariate itself. The smooth curve is 1 − ρ² with ρ = φ(c)/√(p(1−p)), a closed form with no trial in it; the points are counted over 500 trials of 200 units at each threshold. At the median the two agree that about a third survives — the removed share is exactly 2/π — and by two standard deviations 86.9% survives. The closed form is exact in the limit and optimistic by a few points at this many units, because the rule balances the sample's mean rather than the population's.

The variance of the treatment estimate relative to a coin's, for four things a rule might balance against three shapes the outcome might have, over 350 trials of 200 units. The diagonal is the easy part — a rule that reads the function the outcome uses removes about half the variance. What the table is for is the off-diagonal: reading the covariate alone is worth nothing against a quadratic (0.755), and reading all three is worth nearly as much against every shape as the matching rule is against its own (0.532, 0.493, 0.514).

At 100 units, over 2,000 trials, the variance of the imbalance a rule leaves in each of four shapes of the covariate, as a share of a coin's, as the rule reads one to six functions: the covariate, then its square, a threshold at one, the cube, a threshold at −1 and a median split. linear in x: 3.0%, 5.5%, 7.9%, 13.6%, 17.2%, 16.4%. a threshold at 1: 60.8%, 35.6%, 9.0%, 12.1%, 13.0%, 12.7%. quadratic in x: 105.9%, 7.8%, 10.6%, 19.9%, 21.4%, 24.6%. a median split: 38.1%, 37.8%, 38.9%, 30.3%, 27.0%, 8.8%. A shape the rule is not reading keeps most of a coin's imbalance; a shape it is reading keeps a share that grows with every function added.

The largest share of a coin's imbalance left in any of four shapes, measured on fresh trials, for each of the 63 subsets of six functions at 100 units. Subsets of one size spread widely: at four functions from 12.1% to 72.9%. The designer's order of addition gives 1: 94.4%, 2: 41.2%, 3: 39.0%, 4: 31.5%, 5: 27.0%, 6: 22.7%, against the best chosen subset of each size at 1: 83.6%, 2: 41.2%, 3: 25.4%, 4: 12.1%, 5: 16.2%, 6: 22.7%.