Concept

Nuisance model — where it appears

A model fitted only so that some other quantity can be estimated, and whose own coefficients nobody reports. A propensity model and an outcome model are both nuisance models for a treatment effect, and being wrong in one of the two is survivable.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as unconfoundedness — the same set of essays touches all of them, so they are one junction rather than several.

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

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

Named alongside it

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

Average treatment effect Inverse-probability weighting Model misspecification Propensity score Unconfoundedness Augmented estimator Balancing score Confidence interval Covariate balance Covariate imbalance Doubly robust Heavy tail

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