Randomization Inference When N Equals One

TL;DR

This paper develops a new statistical inference framework for N-of-1 experiments with dynamic effects, combining causal inference and control theory to improve effect estimation in complex medical time series.

Contribution

It introduces a model for dynamic interference effects in linear systems and provides estimators with confidence intervals, enhancing N-of-1 experimental analysis.

Findings

Estimator is asymptotically normal under certain conditions

Provides formulas for higher moments of the estimator

Conditions identified for faster intervention effect estimation

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.