A longitudinal Bayesian framework for estimating causal dose-response relationships

TL;DR

The paper tackles estimating causal dose–response relationships for continuous, time-varying exposures in longitudinal data with time-varying confounding. It develops a scalable, nonparametric Bayesian framework that embeds a Dirichlet process prior into generalized estimating equations and uses a generalized Bayesian bootstrap to infer the marginal APO $\mu(d)$. Key contributions include identifiability of $\mu(d)$ under standard causal assumptions, DP-based GPS integration into GEE-based estimands, simulation validation showing valid uncertainty quantification and extrapolation beyond observed doses, and a real-world application linking mass transit ridership to COVID-19 transmission with policy implications. This approach provides probabilistic, flexible dose–response inference for longitudinal continuous exposures, enabling robust decision-making in transportation and public health contexts.

Abstract

Existing causal methods for time-varying exposure and time-varying confounding focus on estimating the average causal effect of a time-varying binary treatment on an end-of-study outcome, offering limited tools for characterizing marginal causal dose-response relationships under continuous exposures. We propose a scalable, nonparametric Bayesian framework for estimating marginal longitudinal causal dose-response functions with repeated outcome measurements. Our approach targets the average potential outcome at any fixed dose level and accommodates time-varying confounding through the generalized propensity score. The proposed approach embeds a Dirichlet process specification within a generalized estimating equations structure, capturing temporal correlation while making minimal assumptions about the functional form of the continuous exposure. We apply the proposed methods to monthly metro ridership and COVID-19 case data from major international cities, identifying causal relationships and the dose-response patterns between higher ridership and increased case counts.

Figures (5)

• Figure 1: Directed acyclic graph for identification of the marginal dose--response function $\mu(d)=\mathbb{E}\{Y(d)\}$ under marginal ignorability. Observed covariates $X_t$ block all backdoor paths between exposure $D_t$ and outcome $Y_t$, while unobserved time-invariant heterogeneity $U$ is orthogonal to $X_t$.
• Figure 2: Example 1: Posterior predictive dose–response curves for the Gaussian outcome estimated using the Bayesian bootstrap, Dirichlet process, and Bayesian parametric MSM approaches.
• Figure 3: Example 2: Posterior predictive dose–response curves for the Poisson outcome estimated using the Bayesian bootstrap, Dirichlet process, and Bayesian parametric MSM approaches.
• Figure 4: Application: Posterior predictive for dose–response estimates for ridership ($\log$ scale) and COVID-19 cases. Panels labeled $\log(x+1)$ use Gaussian GEE models for log-transformed cases, while panels labeled $\log$ scale use Poisson GEE models with posterior predictive samples rescaled to the $\log$ scale. Results are shown for Bayesian bootstrap and DP inference under covariate-adjusted and weighted outcome regression specifications.
• Figure 5: Directed acyclic graph for a general longitudinal data-generating process with time-varying treatment $D_t$, time-varying confounders $X_t$, outcomes $Y_t$, and unobserved time-invariant heterogeneity $U$