Randomization Inference When N Equals One
Tengyuan Liang, Benjamin Recht
TL;DR
The paper addresses inference for N-of-1 trials where treatment effects propagate over time, violating SUTVA due to temporal interference. It develops a linear time-invariant (LTI) framework with impulse response g and uses a generalized estimator $\widehat{\tau}^{\mathcal{L}}(q)$ based on random treatment schedules to recover linear estimands $\tau^{\mathcal{L}}(q) = \langle D_T q, g \rangle$, along with a circular-convolution analogue $\widehat{\tau}(q)$. The authors prove asymptotic normality under decay conditions on $g$ and $q$, derive higher-moment formulas, and propose a plug-in variance estimator for valid confidence intervals, showing that circular and linear models are asymptotically equivalent. They also demonstrate a rapid interleaving design that improves signal-to-noise and enables estimation of a broad class of estimands beyond standard N-of-1 designs. The work connects dynamic causal inference, system identification, and N-of-1 experimentation, and outlines directions for optimal design and extensions to nonstationary or nonlinear dynamics.$
Abstract
N-of-1 experiments, where a unit serves as its own control and treatment in different time windows, have been used in certain medical contexts for decades. However, due to effects that accumulate over long time windows and interventions that have complex evolution, a lack of robust inference tools has limited the widespread applicability of such N-of-1 designs. This work combines techniques from experiment design in causal inference and system identification from control theory to provide such an inference framework. We derive a model of the dynamic interference effect that arises in linear time-invariant dynamical systems. We show that a family of causal estimands analogous to those studied in potential outcomes are estimable via a standard estimator derived from the method of moments. We derive formulae for higher moments of this estimator and describe conditions under which N-of-1 designs may provide faster ways to estimate the effects of interventions in dynamical systems. We also provide conditions under which our estimator is asymptotically normal and derive valid confidence intervals for this setting.