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

Average treatment effect — where it appears

The mean causal effect over the whole population. It is the quantity most questions are about and it is not in general what an instrument identifies, since an instrument reports an average over the units it moved rather than over everybody.

Named by 6 essays across 4 fields — each of them below, with the objects they name alongside it.

A weight that balances, and one that unbalances. The standardised difference between the arms on each covariate, integrated over the population rather than counted in a sample. Unweighted, the arms differ by 0.8310 on the first covariate and 0.6015 on the second, which is what makes the raw difference of arm means 2.7102 against a true average effect of 1.0000. Weighting each unit by one over its own assignment probability removes both differences exactly — -2.78e-17 and -5.69e-19, which is machine precision and not a small number — because the weighted density of the treated arm is the population's own whatever the propensity is. Weighting by a score fitted without the second covariate balances the first to 0.0035 and pushes the second out to 0.7057, further apart than doing nothing.

A score that balances

Weighting each unit by one over its own assignment probability drives the standardised difference between the arms from 0.8310 to 2.8×10⁻¹⁷ — exactly, not nearly. A score fitted without the second covariate leaves that covariate at 0.7057, further apart than doing nothing at all.

weights · Weighting The estimator has no upper bound on what it costs. What a thinning overlap does to a stabilised inverse-probability estimate of an average effect of 1.0000, over 600 samples of 600 at each of six settings. The spread rises from 0.1965 to 0.8122 and the root mean square error from 0.1964 to 0.9219, so at the thin end the error is very nearly the whole of the quantity being estimated. The lower line is the share of the arm's weighted total the single largest observation owns, averaged over the same draws: 0.69% to 9.78%, and in the worst single draw of the sweep 82.75%. Coverage of the 95% interval goes from 93.7% to 55.5%.

The region with no comparison

A trimmed interval covers the average effect over everybody 90.8% of the time at six hundred rows and 41.0% at nine thousand six hundred, while covering the average effect over the units it kept 94.3% and 96.0% throughout. An interval that gets worse as the sample grows is an interval about something else.

weights · Weighting Four numbers, and only one of them is the question. Four quantities in a population where the effect is not the same for everybody: 40.0% compliers, 25.0% always-takers and 35.0% never-takers, with the compliers carrying an effect 1.0000 larger than everybody else's. The population average effect is 0.5000. The compliers' average effect is 1.1000. A valid instrument converges on 1.1000 — the second of those, not the first, and the two differ by 0.6000. Comparing the treated with the untreated as they stand gives 1.4182, wrong by 0.9182 in the same direction, because always-takers start 1.5000 above never-takers before any treatment happens. The instrument removes the selection and changes the question at the same time, and only one of those is reported.

Whose effect it is

With a perfectly valid instrument and no violation of anything, the estimate converges on 1.1000 where the population average effect is 0.5000. The gap is exactly θ(1 − p_c), the always-takers and never-takers cancel out of both halves of the ratio, and five per cent defiers move the answer to 1.2667.

instrument · Exclusion Either model is enough; neither is not. The bias of three estimators of an average effect of 1.0000, over 600 samples of 600 units with the assignment rule at strength 1, in each of the four cells made by getting each nuisance model right or wrong. The wrong model in both cases is one that omits the second covariate, which the outcome and the assignment both depend on. The outcome model alone is off by 0.8064 whenever it is the wrong one; weighting alone is off by 0.8190 whenever the propensity model is. The augmented estimator built from both is off by -0.0085, -0.0083 and -0.0016 in the three cells where at least one of them is right, and by 0.8118 in the fourth — which is between its two components rather than better than either.

Either model, but not neither

The augmented estimator's bias is −0.0085, −0.0083 and −0.0016 wherever one nuisance model is right, against components off by 0.8064 and 0.8190. One step past the overlap sweep it is the least biased estimator on the table at 0.0857 and the worst on it at 1.9265.

weights · Weighting An exact test rejecting a true hypothesis a fifth of the time. How often each analysis reports an effect when the average treatment effect is exactly zero and the effect varies between units, at 150 units with 25% treated. The permutation test on the difference in means reads 4.20% where the effect is constant — where the two nulls coincide and its exactness applies — and 22.93% where the effect varies with a standard deviation of 3. The same test on the studentised difference reads 6.27% there, and the ordinary large-sample t, which makes no exactness claim at all, reads 6.60%.

The null the exactness is for

A permutation test is exact under the hypothesis that the treatment changed nothing for anybody. Under the hypothesis it changed nothing on average, with a quarter of the units treated and the effect varying between them, it rejects a true null 22.93% of the time.

exact · Nuisance What the interval covers as the effect stops being one number. Coverage at 120 units under minimisation, 600 trials and 199 re-run allocations each, as the effect's slope on the prognosis rises from 0 to 1.5. the inverted test, unadjusted, for the population's mean: 96.7%, 96.2%, 95.7%, 95.7%, 94.8%. the same, for the sample's mean: 96.7%, 96.0%, 96.2%, 97.3%, 98.3%. the inverted test, adjusted, for the population's mean: 96.5%, 95.5%, 95.0%, 95.3%, 96.0%. the same, for the sample's mean: 96.5%, 95.7%, 96.5%, 98.0%, 99.5%. the regression with an interaction, for the population's mean: 96.2%, 95.5%, 94.0%, 91.0%, 84.0%. the same, for the sample's mean: 96.2%, 95.8%, 96.2%, 95.7%, 95.2%.

An effect that is not one number

A randomisation test becomes an interval for the effect by testing every constant effect in turn, so when the effect grows with the prognosis every hypothesis it tests is false. It does not fall apart. It covers the population's mean effect within a point of its level at every slope measured, and its width grows by exactly what the sample's own composition adds. A regression that models the interaction stays narrow at 0.72, covers the sample's mean at 95%, and covers the population's at 84%. The extra width was never a defect. It was the difference between two questions.

covadapt · Assignment

Named alongside it

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

Treatment effect heterogeneityConfidence intervalInverse-probability weightingPropensity scoreEstimandHeavy tailModel misspecificationMonte CarloNuisance modelRandomisation testRoot mean square errorStabilised weights

All concepts