Research map / Learning theory and algorithms / Prediction and decision losses
Unit Selection with Nonbinary Treatment and Effect
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Extends Li and Pearl's binary unit selection benefit function to nonbinary treatment and effect, giving an identifiability test and bounds computed from experimental and observational data.
Problem
Unit selection (identifying individuals who would respond in a desired counterfactual pattern) has only been solved for binary treatment and binary effect; the benefit function must be generalized to multi-valued treatments and outcomes.
Approach
Generalizes the benefit function as a linear combination of nm response-type probabilities of causation over m-valued treatment and n-valued effect, provides an algorithm to test identifiability from experimental data, and an algorithm that bounds the function by combining Li-Pearl theorems for nonbinary probabilities of causation over equivalent rewritings.
Key results
- In the vaccine example, bounds for the benefit function are -0.228 to -0.107 (Task 1), while a re-weighted benefit vector is identifiable with value -0.167 (Task 2).
- Simulations with m=2, n=3 show bounds from Theorem 2 closely bracket the real benefit; average bound gaps are 0.330, 0.6520, 0.3284, and 0.3266 for the four benefit vectors (largest possible gaps 6, 9, 3, 3).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| extend benefit function from binary to nonbinary treatment and effect with bounds (Algorithm 2) | unit selection benefit function · Vaccine clinical study | bounds on benefit function | Helped |
| reweighting benefit vector to 2/-2 on severe-condition terms makes benefit function identifiable from experimental data | benefit function with nonbinary vaccine example · Vaccine clinical study | identifiability and point value | Mixed |
| bounds from Theorem 2 across four common benefit vectors on simulated populations | nonbinary benefit function bounds · Simulated populations (m=2, n=3, 1000 populations) | average bound gap | Helped |
Surprising findings
- Identifiability for nonbinary benefit functions can hold for specific benefit vectors even when individual probabilities of causation are not identifiable.
Limitations
- Algorithms have exponential run time in the number of treatment and outcome values.
- Bounds are only evaluated on simulations and one illustrative example; statistical tightness not analyzed.
- Assumes treatment does not affect the population-specific characteristics c.
Experimental setup
Datasets: simulated populations with experimental and observational samples clinical vaccine study example data (illustrative tables)
Benchmarks: simulated populations with four common benefit vectors (1000 populations each)
Metrics: lower bound upper bound midpoint average bound gap real benefit comparison
Baselines: Li-Pearl binary benefit function bounds Balke linear programming approach
Builds on: Li and Pearl unit selection benefit function Li and Pearl nonbinary probabilities of causation bounds Balke linear programming bounds
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