One zero holds and one does not

Three rules, at a correlation of 0.5, against the skewness of the covariate. A rule balancing the mean of each covariate removes exactly nothing of their product when the marginal is symmetric — including the heavy-tailed symmetric one at skewness zero, which is what says the guarantee needs symmetry rather than normality — and removes up to 29.7% when it is not. A rule balancing a median split of each removes exactly nothing of the product of the splits under every marginal here, to 1e-30: both sides are functions of the sign of the latent normal, and a monotone transformation moves neither. A rule balancing a threshold at a value on the covariate's own scale removes between 4.9% and 22.5% — it never had a zero to lose, under any marginal at all.

Six covariates, each a monotone transformation of the same latent normal. The vertical line at zero is where every median split sits, on every one of them, because a monotone map preserves order: the median of the covariate is the image of the median of the latent normal. The marks on the curves are where a threshold at 1 on the covariate's scale falls — 1.000, 0.881, 0.875, 0.783, 0.713, 0.337 — and none of them is at zero. That is the whole of the difference. A function of the sign of the latent normal is odd, and a rule made of odd functions removes exactly nothing of an interaction between two of them; a threshold anywhere else is neither odd nor even and removes something.

Rows are what the rule is told to balance and columns are what the outcome depends on, at a correlation of 0.5 on a normal covariate of skewness -0.00. Every entry is the share of that shape a rule holding those functions removes, computed exactly. The worst case of each row is what a trial can act on, because an outcome model is not known before the trial: the mean of each 0.00%, a median split of each 0.00%, the mean and the median split 0.00%, the mean and the square 6.84%. On a symmetric covariate the darkest cells are exact zeros put there by parity; on a skewed one they are small numbers whose size depends on the marginal.

With strongly skewed covariates (skewness 4.75) at a correlation of 0.5, and a rule holding the mean of each, the share of a coin's imbalance left in each of six outcome shapes: by the closed form, which assumes the dictionary balanced exactly, and in trials of 100 units, over 2,000 trials. linear in one covariate: 0.0% exact, 11.9% in the trial; the square of one: 55.7% exact, 76.8% in the trial; a median split of one: 67.2% exact, 82.1% in the trial; the product of the covariates: 70.3% exact, 93.9% in the trial; the product of the two median splits: 98.7% exact, 102.1% in the trial; one covariate times the other's square: 93.7% exact, 101.5% in the trial.

The share of a coin's imbalance left at 400 units, by marginal and by what the rule reads. normal: linear 1.1% on values and 5.7% on ranks; split 37.4% on values and 25.4% on ranks. skew 2.26: linear 1.5% on values and 28.6% on ranks; split 61.9% on values and 25.0% on ranks. skew 4.75: linear 2.5% on values and 52.2% on ranks; split 95.5% on values and 25.1% on ranks. exponential: linear 1.4% on values and 28.1% on ranks; split 58.7% on values and 25.0% on ranks. A rank's hold on the split does not move with the marginal.