Optimal refinement of strata to balance covariates

Area of Science:

• Statistics
• Observational Studies
• Biostatistics

Background:

• Stratification is crucial for reducing covariate imbalance in observational studies.
• Existing methods may not optimally split strata to minimize within-stratum imbalance across multiple covariates.
• Integer programming offers a theoretical solution but is computationally intractable for large datasets.

Purpose of the Study:

• To develop an optimal method for splitting a single stratum into two to maximally reduce within-stratum covariate imbalance.
• To approximate the integer programming solution using a computationally feasible linear programming relaxation and randomized rounding.
• To demonstrate the method's effectiveness in improving covariate balance in an observational study.

Main Methods:

• Formulated the stratum splitting problem as an integer program.
• Approximated the integer program solution using a linear programming relaxation and randomized rounding.
• Analyzed the theoretical properties of randomized rounding, showing its close resemblance to the linear programming relaxation under certain conditions.

Main Results:

• The randomized rounding approach provides a usable approximation to the optimal integer programming solution.
• When stratum size is large relative to the number of covariates, randomized rounding closely approximates the linear programming relaxation.
• Applied to an observational study, refining 5 propensity score strata into 10 achieved excellent covariate balance while retaining all patients.

Conclusions:

• The proposed method effectively refines strata to achieve superior covariate balance in observational studies.
• Randomized rounding of linear programs offers a practical and theoretically sound approach to solving complex stratification problems.
• An R package, optrefine, is available to implement this method.