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Primary Submission Category: Propensity Scores

Optimal Refinement of Strata to Balance Covariates

Authors: Katherine Brumberg, Dylan Small, Paul Rosenbaum,

Presenting Author: Katherine Brumberg*

What is the best way to split one stratum into two if the goal is to maximally reduce the imbalance in many covariates? We formulate this problem as an integer program and show how to nearly solve it by randomized rounding of a linear program. A linear program may assign a fraction of a person to one refined stratum and the remainder to the other. Randomized rounding views fractional people as probabilities, assigning intact people to strata using biased coins. Randomized rounding of a linear program is a well-studied technique for approximating the optimal solution of classes of insoluble but amenable integer programs. When the number of people in a stratum is large relative to the number of covariates, we prove the following new results: (i) randomized rounding to split a stratum does very little randomizing, so it closely resembles the unusable linear programming solution that splits intact people, (ii) the unusable linear programming solution and the randomly rounded solution place lower and upper bounds on the unattainable integer programming solution, and because of (i), these bounds are often close, ratifying the usable randomly rounded solution. We illustrate using an observational study that balanced many covariates by forming 1008 matched pairs from 2016 patients selected from 5735 using a propensity score. Instead, we form five strata using the propensity score and refine them into ten, obtaining excellent covariate balance while retaining all 5735 patients.