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Bayesian Nonparametrics for Principal Stratification with Continuous Post-Treatment Variables

Dafne Zorzetto, Antonio Canale, Fabrizia Mealli, Francesca Dominici, Falco J. Bargagli-Stoffi

TL;DR

This paper tackles causal inference under principal stratification when the post-treatment variable is continuous. It introduces CASBAH, a Confounders-Aware SHared-atoms Bayesian mixture model that uses a dependent Dirichlet process with shared atoms to model potential post-treatment variables while incorporating covariate-dependent weights, enabling data-adaptive discovery of principal strata and full uncertainty quantification of stratum membership. Through extensive simulations, CASBAH outperforms existing methods in identifying principal strata and estimating stratum-specific causal effects. The method is applied to the 2005 National Ambient Air Quality Standards revision, revealing three strata with distinct patterns in PM$_{2.5}$ changes and mortality, and demonstrating the practical value of stratum-aware causal analysis for environmental policy evaluation.

Abstract

Principal stratification provides a causal inference framework for investigating treatment effects in the presence of a post-treatment variable. Principal strata play a key role in characterizing the treatment effect by identifying groups of units with the same or similar values for the potential post-treatment variable at all treatment levels. The literature has focused mainly on binary post-treatment variables. Few papers considered continuous post-treatment variables. In the presence of a continuous post-treatment, a challenge is how to identify and characterize meaningful coarsening of the latent principal strata that lead to interpretable principal causal effects. This paper introduces the Confounders-Aware SHared atoms BAyesian mixture (CASBAH), a novel approach for principal stratification with binary treatment and continuous post-treatment variables. CASBAH leverages Bayesian nonparametric priors with an innovative hierarchical structure for the potential post-treatment outcomes that overcomes some of the limitations of previous works. Specifically, the novel features of our method allow for (i) identifying coarsened principal strata through a data-adaptive approach and (ii) providing a comprehensive quantification of the uncertainty surrounding stratum membership. Through Monte Carlo simulations, we show that the proposed methodology performs better than existing methods in characterizing the principal strata and estimating principal effects of the treatment. Finally, CASBAH is applied to a case study in which we estimate the causal effects of US national air quality regulations on pollution levels and health outcomes.

Bayesian Nonparametrics for Principal Stratification with Continuous Post-Treatment Variables

TL;DR

This paper tackles causal inference under principal stratification when the post-treatment variable is continuous. It introduces CASBAH, a Confounders-Aware SHared-atoms Bayesian mixture model that uses a dependent Dirichlet process with shared atoms to model potential post-treatment variables while incorporating covariate-dependent weights, enabling data-adaptive discovery of principal strata and full uncertainty quantification of stratum membership. Through extensive simulations, CASBAH outperforms existing methods in identifying principal strata and estimating stratum-specific causal effects. The method is applied to the 2005 National Ambient Air Quality Standards revision, revealing three strata with distinct patterns in PM changes and mortality, and demonstrating the practical value of stratum-aware causal analysis for environmental policy evaluation.

Abstract

Principal stratification provides a causal inference framework for investigating treatment effects in the presence of a post-treatment variable. Principal strata play a key role in characterizing the treatment effect by identifying groups of units with the same or similar values for the potential post-treatment variable at all treatment levels. The literature has focused mainly on binary post-treatment variables. Few papers considered continuous post-treatment variables. In the presence of a continuous post-treatment, a challenge is how to identify and characterize meaningful coarsening of the latent principal strata that lead to interpretable principal causal effects. This paper introduces the Confounders-Aware SHared atoms BAyesian mixture (CASBAH), a novel approach for principal stratification with binary treatment and continuous post-treatment variables. CASBAH leverages Bayesian nonparametric priors with an innovative hierarchical structure for the potential post-treatment outcomes that overcomes some of the limitations of previous works. Specifically, the novel features of our method allow for (i) identifying coarsened principal strata through a data-adaptive approach and (ii) providing a comprehensive quantification of the uncertainty surrounding stratum membership. Through Monte Carlo simulations, we show that the proposed methodology performs better than existing methods in characterizing the principal strata and estimating principal effects of the treatment. Finally, CASBAH is applied to a case study in which we estimate the causal effects of US national air quality regulations on pollution levels and health outcomes.
Paper Structure (18 sections, 2 theorems, 43 equations, 7 figures, 3 tables, 1 algorithm)

This paper contains 18 sections, 2 theorems, 43 equations, 7 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Causal Inference Setup
  3. Model
  4. Confounders-Aware Shared-atoms Bayesian Mixture Model
  5. Causal Estimands
  6. Discovery of Principal Strata
  7. Posterior Inference
  8. Simulation Study
  9. Assessing the Impact of Environmental Policies on Particulate Pollution and Health
  10. Data and Study Design
  11. Results
  12. Discussion
  13. Conjugate prior distribution for probit regression parameters
  14. Proof Theorem 1
  15. Proof Dissociative Stratum Probability
  16. ...and 3 more sections

Key Result

Theorem 1

Invoking assumptions ass:sutva and ass:sita, the expected value of each potential outcome can be rewritten as the following where the inner expectations $\mathbb{E}\{Y_i \mid T_i=t, X_i=x, P_i(1)=p_1,P_i(0)=p_0 \}$ is estimate with the outcome model $Y_i \mid T_i, X_i, P_i(1),P_i(0)$, as well the probability $\hbox{pr}(P_i(1)=p_1,P_i(0)=p_0\mid T_i=t,S_{i}^{(1)}=s_1, S_{i}^{(0)}=s_0, X_i=x )$,

Figures (7)

  • Figure 1: Probability to belong to the dissociative stratum. Function varies according to different values of $\alpha(x)=\beta_0+\beta_1x$.
  • Figure 2: Posterior medians and $90\%$ credible intervals of the three identified strata for (left) the conditional average of the difference of the post-treatment variables, and (right) the expected associative/dissociative effects. The x-axes indicate (left) the pm$_{2.5}$ variation in $\mu g/m^3$ and (right) the mortality variation in ${}^\text{o}\mkern-5mu/\mkern-3mu_\text{oooo}$. The light-blue vertical lines show the value zero, identifying the null effects.
  • Figure 3: Representation of the characteristics of the identified strata. Each spider plot reports in the colored area the strata-specific characteristics (the mean of the analyzed covariates) and in the gray area the collective characteristics (the mean of the covariates among all the analyzed counties in the Eastern U.S.). We can consider the gray area as the benchmark to understand how the characteristics of each stratum differ from the collective characteristics of the analyzed population.
  • Figure 4: Distributions of (right) the post-treatment variables and (left) the outcomes, for the five simulated scenarios. In light blue the distribution of the variables given the treatment and in yellow given the control.
  • Figure 5: Representations of the five simulated scenarios. (Left) The expected value of the difference of the post-treatment variables under treatment and control given the strata allocation. (Right) The expected associative/dissociative causal effects. In green and indicated with $V=-1$, the associative negative stratum, i.e. corresponding to $S_i^{(1)} \prec S_i^{(0)}$, in yellow and indicated with $V=0$, the dissociative stratum, i.e., $S_i^{(1)} = S_i (0)$, and in red and indicated with $V=+1$ the associative positive stratum, i.e., $S_i^{(1)} \succ S_i^{(0)}$.
  • ...and 2 more figures

Theorems & Definitions (3)

  • Definition 1: Causal Estimands
  • Theorem 1
  • Theorem 2