Don't Compromise Control Groups

Randomized, double-blind, controlled trials (RCTs) are widely considered the gold standard for modern intervention-based clinical research. While the importance of control groups is well recognized, their impact on statistical efficiency is often overlooked. A common practice, for example, is to allocate more patients to experimental groups than to control groups, which can ultimately compromise the efficiency of statistical analyses.

Consider a clinical trial with one control group and k experimental groups, where the primary endpoint is a continuous variable with constant variance (σ², assumed known for simplicity) across all groups. Suppose all experimental groups are of equal importance and the primary interest lies in the pairwise comparisons between each experimental group and the control group - a typical objective in clinical studies. A natural question arises: what is the most efficient sample size allocation (n₀, n₁, ..., nₖ) given a fixed total sample size n? Here, n₀ through nₖ denote the sample sizes for the control and experimental groups, respectively. A reasonable approach is to determine the allocation that minimizes the total (or equivalently, average) variance of the k pairwise comparisons (i.e., A-optimal design). Under a one-way analysis of variance model, this can be formally stated as the following optimization problem subject to the restriction that n₀ + n₁, ..., + nₖ = n:

Applying the method of Lagrange multipliers, the optimal solution is

Therefore, the optimal allocation ratio between each experiment group and the control group is 1: √k. Note that, in practice, an exact ratio can be calculated only when k is a perfect square.

The table below shows the efficiency of alternative designs relative to this optimal allocation, specifically for scenarios where the ratio between each experimental group and the control group is 1:1 and 2:1, respectively.

These results suggest that the control group should be at least as large as the experimental groups, as deviations from the optimal allocation can result in substantial losses in statistical efficiency. Importantly, this principle often applies beyond the specific scenario discussed here. For example, in dose-response testing under a monotonic relationship (e.g., a simple linear regression model), statistical power is often enhanced by allocating more patients to both the placebo group and the highest dose group. Since a control group serves as the benchmark (or standard) for evaluating treatment effects, reliable statistical inference is nearly impossible without adequate information about the control effect.