A sensitivity analysis approach to principal stratification with a continuous longitudinal intermediate outcome: Applications to a cohort stepped wedge trial

Causal inference in the presence of intermediate variables is a challenging problem in many applications. Principal stratification (PS) provides a framework to estimate principal causal effects (PCE) in such settings. However, existing PS methods primarily focus on settings with binary intermediate variables. We propose a novel approach to estimate PCE with continuous intermediate variables in the context of stepped wedge cluster randomized trials (SW-CRTs). Our method leverages the time-varying treatment assignment in SW-CRTs to calibrate sensitivity parameters and identify the PCE under realistic assumptions. We demonstrate the application of our approach using data from a cohort SW-CRT evaluating the effect of a crowdsourcing intervention on HIV testing uptake among men who have sex with men in China, with social norms as a continuous intermediate variable. The proposed methodology expands the scope of PS to accommodate continuous variables and provides a practical tool for causal inference in SW-CRTs.

💡 Research Summary

This paper introduces a novel sensitivity‑analysis framework for principal stratification (PS) when the intermediate variable is continuous and measured longitudinally within a stepped‑wedge cluster randomized trial (SW‑CRT). Traditional PS methods focus on binary intermediates and rely on strong structural assumptions such as monotonicity and exclusion restriction, which are often untenable in real‑world studies, especially when baseline covariates are limited. Leveraging the staggered rollout inherent to SW‑CRTs, the authors exploit the fact that both treatment and control periods are observed for each cluster at different time points, allowing direct calibration of sensitivity parameters from the data.

The methodological contributions are threefold. First, a copula‑based assumption is introduced to model the joint distribution of the potential intermediate outcomes under treatment and control. By separating marginal distributions from the dependence structure, the approach accommodates non‑normality, clustering, and complex correlation patterns without discretizing the continuous mediator. Second, a marginal structural assumption links potential outcomes to potential intermediates, replacing the untestable principal ignorability assumption with a set of more plausible, data‑driven constraints. Third, a Bayesian hierarchical model incorporating random cluster and time effects is developed. This model naturally handles monotone missingness (common in longitudinal cohorts) and yields posterior distributions for the principal causal effects (PCEs) as well as for the sensitivity parameters themselves.

A calibration procedure is described whereby observed intermediate and outcome data across pre‑ and post‑intervention periods are used to estimate the copula correlation and the parameters governing the marginal structural relationship. This grounding of sensitivity analysis in observed data mitigates reliance on arbitrary prior specifications and enhances interpretability.

The authors apply the method to a closed‑cohort SW‑CRT evaluating a crowdsourcing HIV‑testing intervention among men who have sex with men in eight Chinese cities. The continuous intermediate is a social‑norms score (six‑item Likert scale). By defining principal strata based on changes in social norms (e.g., “norms increase” vs. “norms stable/decrease”), they estimate short‑ and long‑term PCEs for HIV‑testing uptake. Results indicate a substantial positive effect (≈12 percentage‑point increase) among participants whose perceived norms rose, while the effect is negligible or slightly negative for those whose norms did not improve. Sensitivity analyses across a range of copula correlations (0.3–0.7) show the conclusions are robust.

Simulation studies demonstrate that, compared with methods that impose monotonicity and exclusion restrictions, the proposed approach yields lower bias and appropriate coverage even when those assumptions are violated. The paper also discusses limitations, including dependence on the chosen copula family and the need for monotone missingness assumptions, and outlines future extensions to non‑Gaussian copulas, non‑linear structural models, and multiple continuous intermediates.

In sum, this work expands principal stratification to continuous longitudinal mediators within stepped‑wedge designs, provides a practical data‑driven sensitivity‑analysis toolkit, and delivers actionable insights for public‑health interventions where mediator heterogeneity drives effect modification.