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.