Using geometry to decide where to collect intervention data

Turning matching into geometry

Suppose researchers know a population’s outcomes under existing conditions very well, but have intervention outcomes for only a small sample. They want to estimate the intervention’s effect for people outside that sample.

One approach is matching: combine observed intervention outcomes using weights chosen so that the donors’ weighted characteristics reproduce those of a target. But several combinations can match the same target exactly. Which should researchers choose?

In Learning Where to Look: Delaunay Matching for Discovery and Data Collection, Opocher studies this question using a protective criterion: minimize the largest possible systematic estimation error across a specified class of outcome relationships. This is worst-case bias, not the error necessarily encountered in practice.

The proposed Delaunay Matching Estimator borrows a construction from computational geometry. With two characteristics, donor observations become points on a plane. A Delaunay triangulation connects them into triangles whose circumcircles contain no other donor point in their interiors.

For a target inside a triangle, the estimator combines intervention outcomes at its three corners. The weights reproduce the target’s position exactly. Subtracting the known status-quo outcome then gives an estimated intervention effect. Higher-dimensional versions use the corresponding geometric objects, called simplices.

What the guarantee requires

The paper proves that these weights minimize worst-case absolute conditional bias among nonnegative, exactly balanced matching rules whose weights sum to one. The guarantee applies to each target inside the donors’ convex hull—the region enclosed by their observations—with donor locations held fixed.

It rests on substantive assumptions. The expected intervention outcome must follow the same relationship with measured characteristics for sampled intervention units and the untreated target population. That relationship must be smooth, with a bound on its curvature. Status-quo outcomes must also be known, or estimated precisely enough to be treated as known.

Under these conditions, worst-case bias equals half the curvature bound multiplied by the weighted sum of squared distances from donors to the target. Delaunay weights minimize that geometric quantity. Their selection does not require knowing the curvature bound’s magnitude, although calculating a numerical bias bound does.

This is not a guarantee against unmeasured differences between populations. Nor does it justify predictions outside the donor convex hull.

Low bias is not the whole uncertainty story

Concentrating weight on a few nearby observations can reduce bias while increasing sampling uncertainty. The paper constructs finite-sample confidence intervals for bounded outcomes, combining a worst-case bias allowance with a sampling term derived from Hoeffding’s inequality.

The weights minimizing these intervals’ length need not be Delaunay weights. When the outcome relationship is linear, the optimum is a minimum-norm, exactly balanced synthetic-control rule, which spreads weight more broadly. As the allowed curvature grows, the optimum approaches Delaunay matching when the bias-minimizing weights are unique.

The distinction matters: minimizing systematic error does not automatically produce the most precise estimate. These intervals require known outcome bounds and a specified curvature bound.

Choosing the next observation

The same triangulation helps identify where new intervention evidence would reduce the largest worst-case bias across the existing convex hull.

For each simplex, consider the smallest ball enclosing its vertices. The largest squared radius across the triangulation determines the global worst-case bias, up to the common curvature factor.

The proposed collection plan numerically searches for an additional location that reduces this maximum most. Any strictly improving addition must fall inside the circumball of a simplex attaining the current maximum. This narrows the search; it does not mean every location there improves matters. The plan solves a one-addition problem and repeats it, rather than establishing a globally optimal allocation of an entire collection budget.

A deliberately constructed demonstration

The application uses data from an earlier biometric Smartcard rollout in Andhra Pradesh, India. Opocher retains empirical characteristics and constructs a status-quo outcome relationship, then imposes a known, nonlinear treatment-effect surface. It is therefore semi-synthetic, not a new measurement of Smartcard effectiveness.

Starting with 80 intervention donor locations, the comparison evaluates 922 grid targets inside their convex hull. The median pointwise worst-case bias ratio relative to Delaunay matching is about 7.1 for the specified synthetic-control benchmark, 8.0 for entropy balancing and 1.17 for adaptive nearest neighbors.

Sequential geometric collection reduces the maximum worst-case bias by 52.2% after one addition and 89.9% after twenty. None of 100 uniform-random collection paths reduces that maximum within twenty additions. These results concern this design and conservative error criterion, not a general failure of random sampling.

The arXiv preprint describes itself as work in progress. Its contribution is a conditional mathematical framework and an illustrative application—not evidence that those gains will transfer unchanged to other interventions.