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

Stratification — where it appears

Dividing the units into cells by their baseline characteristics and randomising separately inside each, so the cells' counts stay even. It balances the cells it was told about and says nothing about a covariate it was not, which is the same limitation every balancing rule has.

Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.

What a median split can see. A standard normal covariate with its median marked, and the mean of each category as a vertical rule: -0.7979, 0.7979. A rule that balances the categories is balancing those numbers and nothing else, so the part of the covariate it can act on is the variance between them — 0.6366 of the total, which at two categories is exactly 2/π because the two half-normal means are ±√(2/π). The rest, 0.3634, is variation inside the categories that the rule cannot see and does not touch: the assignment within a category is still a coin. Everything the next figure measures is a consequence of this one, and it is available before any unit has arrived.

A covariate with no levels

Every balancing rule on this site reads a level. Age and blood pressure have none, so somebody cuts them into categories — and a median split can see exactly 2/π of a normal covariate, whatever the rule does with the halves.

continuous · Assignment 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.

Balanced on the wrong function

A rule that reads a covariate's numbers halves the variance of the treatment estimate, if the covariate enters the outcome as a straight line. If it enters as a threshold the rule is worth a fifth of that, and if it enters as a curve every rule here is worse than a coin.

shape · Blocking A threshold in the tail is a threshold nothing balances. 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.

A threshold in the tail

How much of a threshold's imbalance a balanced covariate removes is a correlation, and the correlation is a closed form. At the median it is exactly 2/π — the same 2/π a median split throws away — and two standard deviations out it is an eighth.

shape · Blocking One odds ratio in every stratum, and a different one marginally. Five strata with baseline risks from 5% to 85%, a treatment allocated by a coin in each, and a conditional odds ratio of exactly 2.5 throughout. The marginal odds ratio is 1.789. Nothing is confounded; the odds ratio is simply not a weighted average of odds ratios.

The change that is not confounding

Five strata, a treatment allocated by a coin in every one, and an odds ratio of exactly 2.5 in all five. The odds ratio computed on the pooled table is 1.789. Nothing is confounded — an odds ratio is not a weighted average of odds ratios, and the risk difference, on the same table, is exactly its own stratum value.

reversal · Simpson A cut at a quantile, and a cut at a value. Two rules that read identically in a protocol. One splits each covariate at its median; the other splits it at 1 on the covariate's own scale — a dose, a temperature, a clinical threshold. At a correlation of 0.5 the first removes exactly nothing of the interaction between its own two splits, under every marginal here, because a median split is a function of the sign of the latent normal whatever the marginal is. The second removes what the bars show, and it does so on a normal covariate too: the threshold sits at 1.000 on the latent scale rather than at zero, so it is 59.4% odd and 40.6% even. The exact zero was never about the cut; it was about the cut being at the median.

The cut that is not a quantile

A protocol that says split the covariate at a threshold and one that says split it at the median read the same and are different rules. One has an exact guarantee under every marginal and the other has none under any.

skew · Criterion What stratifying on a mediator estimates, and what it does not. The direct effect stays at -0.109 throughout, because it is defined with the mediator held fixed. The total effect runs from -0.218 to 0.027 and crosses zero at 1.5. On 15% of the sweep the two have opposite signs, and both are correct answers.

Conditioning on what the treatment caused

When the grouping variable lies on the path from treatment to outcome, the stratified answer is the direct effect and the aggregate is the total effect. Both are correct. Over 15% of a sweep of the indirect path they have opposite signs, and no arithmetic on the table says which question was being asked.

reversal · Simpson What balancing several numbers at once costs each of them. The criterion generalises without a word changing — the covariate imbalance becomes a vector and the correction a quadratic form — so the question is what it is worth rather than whether it can be done. At n = 200 with 200 trials per point, a rule balancing one covariate leaves 12.7% of a coin's imbalance in it; balancing eight leaves 23.2% in each. The assignment has a fixed amount of freedom and every covariate added takes a share of it. The rule degrades rather than failing: at eight covariates it is still four times better balanced than a coin, and the eight are being held simultaneously rather than in turn.

Balancing more than one number

The criterion generalises to several covariates without a word changing, which makes the question what it is worth rather than whether it can be done. Each one added takes a share of the assignment's freedom, and the imbalance left in every one of them rises.

continuous · Blocking

Named alongside it

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

RandomisationContinuous covariateCovariate balanceThresholdAggregationAllocation ruleBlockingClosed formConfoundingCovariate-adaptive randomisationCovariate imbalanceEfficiency

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