WHOMP: Optimizing Randomized Controlled Trials via Wasserstein Homogeneity

TL;DR

A novel partitioning method called the Wasserstein Homogeneity Partition (WHOMP), which optimally minimizes type I and type II errors that often result from imbalanced group splitting or partitioning, commonly referred to as accidental bias, in comparative and controlled trials is introduced.

Abstract

We investigate methods for partitioning datasets into subgroups that maximize diversity within each subgroup while minimizing dissimilarity across subgroups. We introduce a novel partitioning method called the $\textit{Wasserstein Homogeneity Partition}$ (WHOMP), which optimally minimizes type I and type II errors that often result from imbalanced group splitting or partitioning, commonly referred to as accidental bias, in comparative and controlled trials. We conduct an analytical comparison of WHOMP against existing partitioning methods, such as random subsampling, covariate-adaptive randomization, rerandomization, and anti-clustering, demonstrating its advantages. Moreover, we characterize the optimal solutions to the WHOMP problem and reveal an inherent trade-off between the stability of subgroup means and variances among these solutions. Based on our theoretical insights, we design algorithms that not only obtain these optimal solutions but also equip practitioners with tools to select the desired trade-off. Finally, we validate the effectiveness of WHOMP through numerical experiments, highlighting its superiority over traditional methods.

Figures (3)

• Figure 1: This example illustrates that anti-clustering (left) tends to produce subgroups at different scales: compare the size of the larger triangle formed by $x_1, x_5, x_9$ with the size of the smaller triangles formed by $x_2, x_6, x_7$ and $x_3, x_4, x_8$, respectively. In comparison, WHOMP Matching (right) leads to the desired subgroup partition at the same scale.
• Figure 2: It is evident from the frequency plot above that the worst-case Wasserstein distances resulting from WHOMP solutions are almost the best-case Wasserstein distance resulting from the random partition or Pocock $\&$ Simon's covariate-adaptive randomization.
• Figure 3: It is evident from the frequency plot above that the worse-case Wasserstein distances resulting from WHOMP solutions are significantly better than the worst-case Wasserstein distance resulting from the random partition or Pocock $\&$ Simon's covariate-adaptive randomization. In the case of 2 subgroups, the 90th percentile worst-case distance in WHOMP solutions is equivalent to the 50th percentile worst-case distance in the other two methods.

Theorems & Definitions (25)

• Definition 2.1: Wasserstein Homogeneity Partition
• Definition 2.2: Balanced K-means Clustering
• Theorem 2.1: No type I or type II error due to subgroup
• Example 2.1: Theorem \ref{['th:no_type_error']} in Hypothesis Testing
• Lemma 2.1: $\hat{\tau}$ is an unbiased estimator of $\tau$
• Corollary 2.1: Zero-One p-value
• Corollary 2.2: Lipschitz Statistics Error Bound
• Theorem 2.2: Variance Bound for Average Treatment Effect Estimator
• Lemma 3.1: Subsample Wasserstein Deviation Lower Bound
• Corollary 3.1: Distribution of Subsample Wasserstein Deviation Lower Bounds