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A Conditional Independence Test in the Presence of Discretization
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Proposes DCT, a conditional independence test for latent Gaussian (nonparanormal) variables observed only through discretized values, using bridge equations to recover covariances and nodewise regression to get precision coefficients with an asymptotic null distribution.
Problem
Conditional independence testing fails when the variables of interest are latent continuous variables observed only after discretization, so standard CI tests on the discretized data test the wrong CI relation.
Approach
Bridge equations link the proportion of discretized observations above their means to the latent Gaussian covariance (after binarization), and nodewise regression converts the recovered covariance into regression coefficients that serve as surrogates for precision entries, yielding an asymptotically normal test statistic under the CI null.
Key results
- Type I error of DCT stays near the nominal 0.05 across continuous, mixed and discrete data types and varying sample sizes, while Fisher-z and chi-square show inflated Type I error that grows with sample size and conditioning set size.
- DCT needs roughly n=1000 samples to reach satisfactory calibrated power, especially in mixed and continuous cases, and has lower power than baselines at small n.
- In PC-based causal discovery on synthetic data, DCT performs nearly on par with Fisher-z on original continuous data for small p and large n, and significantly outperforms discretized Fisher-z and chi-square.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use DCT instead of Fisher-Z or Chi-Square tests on discretized data | Conditional independence testing on synthetic data · Synthetic data; continuous, mixed, and discrete tested-variable pairs | Type I error | Helped |
| Use DCT instead of alternative tests | Conditional independence testing on synthetic data · Synthetic data; especially smaller sample sizes | Type II error | Hurt |
| Use DCT instead of Fisher-Z or Chi-Square tests on discretized data | PC causal discovery · Synthetic DAGs | F1 score | Helped |
| Use DCT instead of Fisher-Z or Chi-Square tests on discretized data | PC causal discovery · Synthetic DAGs; small p and large n | SHD | Helped |
| Use DCT instead of baseline tests | PC causal discovery · Synthetic DAGs | Recall | Hurt |
Surprising findings
- Baseline CI tests applied to discretized data have high recall and Type I error inflation, so their apparent power is partly spurious; DCT trades recall for calibrated error control.
Limitations
- Relies on the assumption that latent continuous variables are jointly Gaussian (nonparanormal); nonlinear dependence such as X1=X2^2 can be missed.
- Binarization discards information, so power is low at small sample sizes (needs about 1000 samples).
Experimental setup
Datasets: synthetic Gaussian data discretized into 2-12 levels
Benchmarks: synthetic data (Type I/II error) synthetic causal discovery skeleton and DAG recovery
Metrics: Type I error calibrated Type II error F1 precision recall SHD
Baselines: Fisher-Z test Chi-square test PC algorithm with Fisher-z on discretized data
Builds on: nodewise regression nonparanormal model PC algorithm Fisher-z test
Closest papers by meaning
- A Sample Efficient Conditional Independence Test in the Presence of Discretization (ICML 2025 · similarity 0.93)
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- PAIR-CI: Calibrated Conditional Independence Testing for Causal Discovery with Incomplete Data (NeurIPS 2026 · similarity 0.86)
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