Defining and Estimating Causal Effects in Randomized Alternating Treatment Design for Single-Case Experiments: A Counterfactual Approach

Alternating Treatment Design (ATD) is a commonly used single-case experimental design (SCED) to evaluate the effectiveness of interventions or compare multiple treatments. Commonly used treatment effect estimators, such as simple mean differences and non-overlap indices, are not formally grounded in counterfactual logic and therefore lack a clear causal interpretation. This study addresses this gap by introducing a class of causal estimands in ATDs within a counterfactual framework and the inverse probability weighting (IPW) estimators and augmented IPW (AIPW) estimators for the average immediate treatment effect and the average lag-1 effect. Through Monte Carlo simulation studies, we evaluated the performance of IPW and AIPW estimators under conditions representative of psychological and behavioral research (e.g., short series lengths and restricted randomization). The main simulations compared these methods with commonly used estimators in terms of bias, type I error, and statistical power of randomization tests. The supplementary simulations further assessed the accuracy of standard error estimates and the coverage of corresponding confidence intervals. Results indicate that the AIPW estimator is a promising approach for quantifying the average immediate effect in randomized ATDs, as it produces accurate estimates even in the presence of residual effects and requires fewer assumptions than traditional methods. In addition, the AIPW estimator of the average lag-1 effect provides a practical tool for diagnosing potential multiple treatment interference. Finally, we illustrate the application of IPW and AIPW estimators using real data from an ATD study.