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Automating the Selection of Proxy Variables of Unmeasured Confounders
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Proposes Proxy-Rank (rank-deficiency of covariance submatrices) and Proxy-GIN (non-Gaussian GIN constraints) to automatically select valid negative-control proxy variables and estimate causal effects of multiple treatments under multiple unmeasured confounders in linear models.
Problem
Valid proxy variables (negative control exposures and outcomes) for unmeasured confounders are usually justified by background knowledge; the paper seeks data-driven selection of them for estimating causal effects of multiple treatments on an outcome when multiple unmeasured confounders exist in a l
Approach
Extends the single-confounder proxy estimator to q confounders via a determinant ratio of cross-covariance matrices, and identifies valid NCE/NCO sets either via rank constraints on second-order statistics (Proxy-Rank) or via GIN independence constraints exploiting non-Gaussian noise (Proxy-GIN), without prior knowledge of proxy validity.
Key results
- On synthetic Gaussian data, Proxy-Rank achieves little bias for all causal effects across sample sizes 1k, 3k, 5k, outperforming NAIVE and FindNC.
- In the non-Gaussian synthetic case, Proxy-GIN outperforms NAIVE and FindNC at all sample sizes, while Proxy-Rank fails for X2->Y and X5->Y because no valid NCE/NCO exist there.
- On the mouse obesity data, the method recovers findings such as Gstm2, Sirpa and 2010002N04Rik having positive effects on body weight and Dscam a negative effect, aligning with prior studies.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Proxy-Rank for causal-effect estimation | Synthetic data in the Gaussian case · Gaussian case; causal effects X2 → Y, X5 → Y, and X6 → Y | bias | Helped |
| Use Proxy-Rank for causal-effect estimation | Synthetic data in the non-Gaussian case · Non-Gaussian case; causal effect X6 → Y | bias | Helped |
| Use Proxy-Rank when no valid NCE and NCO exist | Synthetic data · Non-Gaussian case; causal effects X2 → Y and X5 → Y | bias | Hurt |
| Use Proxy-GIN for causal-effect estimation | Synthetic non-Gaussian data · Non-Gaussian case | bias | Helped |
| Use Proxy-Rank instead of NAIVE or FindNC | Synthetic data · Gaussian and non-Gaussian cases, where valid proxies exist | bias | Helped |
| Use the Extended Proxy Variables Estimator instead of the Traditional Proxy Variables Estimator | Synthetic data with multiple unmeasured confounders · q ≥ 2 | bias | Helped |
| Apply Proxy-Rank to estimate gene-expression effects | Mouse obesity dataset · Body weight; Gstm2, Sirpa, 2010002N04Rik, and Dscam | causal effect | Mixed |
Surprising findings
- Proxy-Rank is not consistent for X2->Y and X5->Y in the non-Gaussian setting because no valid proxies exist there, showing that rank constraints alone can fail to identify valid NCE/NCO under minimal assumptions.
Limitations
- Restricted to linear acyclic causal models; the number of unmeasured confounders q is assumed known.
- Requires that every treatment-confounder edge is non-zero (all entries of C non-zero) and rank-faithfulness.
- Worst-case complexity grows factorially with the number of treatments p.
Experimental setup
Datasets: mouse obesity dataset (F2 mice, 227 mice, 17 gene expressions)
Benchmarks: mouse obesity dataset
Metrics: bias of causal effect estimates
Baselines: NAIVE least-squares regression FindNC Traditional proxy variables estimator
Builds on: proximal causal learning negative control exposure and outcome LiNGAM Independent Noise constraint RFCI
Closest papers by meaning
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- Transferring Causal Effects using Proxies (NeurIPS 2025 · similarity 0.89)
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- Conditional Instrumental Variable Regression with Representation Learning for Causal Inference (ICLR 2024 · similarity 0.88)
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