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Causal Attribution Analysis for Continuous Outcomes

ICML 2025 · Probabilistic and scientific modeling › Treatment effects · approach: Proximal and deep causal effect estimation

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Extends posterior causal attribution (postTCE, postNDE, postNIE) from binary to continuous outcomes using quantile-based counterfactual mappings under sequential ignorability, monotonicity and perfect positive rank, with a two-step estimator.

Problem

Retrospectively attributing multiple correlated binary causes to an observed continuous outcome; binarizing the outcome loses information and can bias conclusions, and prior posterior-causal-effect work covers only binary outcomes.

Approach

Define posterior intervention effects, total effects, and natural direct/indirect effects conditioned on an observed continuous outcome interval; identify individual counterfactual mappings via quantile matching under a perfect positive rank assumption, then nonparametrically estimate the posterior estimands from these mappings.

Key results

• On the hypertension example with evidence X=(1,1,1,1,1), Y>140, postTCE for heart disease (X4) is 17.023 versus 10.628 for exercise (X1), with postNDE of X4 equal to its postTCE.
• Simulations with n=1000 to 10000 (500 repetitions) show decreasing bias and standard errors as sample size grows; estimates approach true posterior ICE values (e.g., n=10000 estimates close to true values such as -19 vs -19.036).
• Under evidence X=(1,1,1,0,1), Y>140, the paper reports postTCE of X4 = 10.483 while the binary method of Lu et al. (2023) gives zero, showing the continuous framework can detect a cause the binarized analysis misses.

Measured effects reported in the paper

Surprising findings

• Under the continuous framework, heart disease (X4) receives a nonzero posterior total effect where the binarized method of Lu et al. (2023) assigns zero, because the perfect rank assumption does not guarantee the unique-risk-factor property.

Limitations

• Requires sequential ignorability, monotonicity among causes, and perfect positive rank, which cannot be directly tested.
• Current framework handles binary causes and a single continuous outcome; continuous causes and multiple outcomes are left for future work.
• Assumes known compact support of the outcome; trimming is needed to relax the interior-density assumption.

Experimental setup

Datasets: simulated hypertension causal network (Lu et al. 2023 structure) NTP developmental toxicology TCPP mouse study (NTP 2023)

Benchmarks: simulated hypertension example NTP developmental toxicity dataset

Metrics: posterior intervention effect posterior total causal effect posterior natural direct effect posterior natural indirect effect bias standard error

Baselines: Lu et al. 2023 binary posterior TCE (postTCE*) probability of necessity PN*

Builds on: posterior total causal effect (Lu et al. 2023) rank preservation (Heckman et al. 1997) quantile regression (Koenker & Bassett 1978) counterfactual mapping (Vuong & Xu 2017; Feng et al. 2020) probability of causation (Dawid et al. 2014)

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