Experimental Design When N Equals One

2026-06-26

AI Summary

This study addresses the lack of theoretical foundations for optimal treatment assignment designs in N-of-1 trials. The authors propose a unified experimental design framework based on Markov transition matrices, which minimizes ordinary least squares estimation error by strategically controlling the temporal dependence of treatment assignments under a finite-order impulse response model. They develop large-sample asymptotic theory for two structured design classes—stochastic and cyclic switching—and uncover intrinsic connections between optimal designs and target effects, such as cumulative or lagged responses. The analysis further demonstrates the robustness of i.i.d. Bernoulli designs. Simulations confirm that the proposed approach substantially improves estimation accuracy and provides clear guidance on optimal switching structures.

Abstract

N-of-1 trials, or time-series experiments, are widely used in clinical research and online platforms. Yet the theoretically optimal design for estimating many treatment effects remains unclear. We propose a simple Markovian framework for experimental design in which the treatment assignment process is governed by possibly time-varying transition matrices. This formulation encompasses many existing N-of-1 designs and provides a principled way to control temporal dependence in treatment assignment through Markov transition probabilities. Under a finite-order impulse-response model, we formulate the design objective as minimizing the estimation error of ordinary least squares estimators for target treatment effects, and propose practical design optimization procedures. To characterize the optimal temporal structure, we focus on two structured design classes, random-switch and cycle-switch designs, and establish a complete large-$T$ asymptotic theory for the optimal designs in both classes. Our results justify the robustness of i.i.d. Bernoulli designs in N-of-1 trials and quantify how the optimal design depends on the target estimand, including cumulative and lag-specific treatment effects. Simulations demonstrate the effectiveness and robustness of the proposed designs across multiple scenarios.

Problem

• N-of-1 trials
• experimental design
• treatment effect estimation
• time-series experiments
• optimal design

Innovation

• N-of-1 trials
• Markovian experimental design
• temporal dependence
• optimal design
• impulse-response model