Causal Inference¶

← Constraints | Semi-Supervised →

Causal inference is the study of how one variable (the treatment \(T\)) affects another (the outcome \(Y\)). Unlike standard regression which measures correlation, causal inference seeks to estimate the Average Treatment Effect (ATE) and Conditional Average Treatment Effect (CATE) after adjusting for confounding variables \(X\).

The Fundamental Problem¶

In observational data, we only ever see one outcome for each individual: either the treated outcome \(Y(1)\) or the control outcome \(Y(0)\). We never observe the counterfactual \(Y(1-T)\).

Causal inference methods in torchregress use doubly-robust estimation [1, 4] to overcome this missing data problem.

Why Use Causal Inference?¶

Core Methods¶

1. Doubly Robust ATE (dr_ate)¶

The Doubly Robust (DR) estimator [1] combines an outcome model \(\hat{\mu}(x, t)\) and a propensity model \(\hat{e}(x) = P(T=1 \mid x)\) to estimate the average treatment effect across the entire population. Implementations in torchregress cross-fit these nuisance models on held-out folds rather than accepting precomputed \(\hat{\mu}\) or \(\hat{e}\).

from sklearn.linear_model import LinearRegression, LogisticRegression
from torchregress.causal import dr_ate

result = dr_ate(
    x, t, y,
    outcome_model=LinearRegression,
    propensity_model=LogisticRegression(max_iter=1000),
    folds=2,
    alpha=0.05,
)
ate = result["estimate"]
ci = (result["ci_lower"], result["ci_upper"])
overlap = result["diagnostics"]  # overlap / ESS checks

2. Conditional ATE (dr_cate)¶

Estimates the treatment effect as a function of the covariates \(X\). This is crucial for personalised medicine or targeted marketing, where we want to know \(\tau(x) = \mathbb{E}[Y(1) - Y(0) \mid x]\).

from sklearn.linear_model import LinearRegression, LogisticRegression
from torchregress.causal import dr_cate

result = dr_cate(
    x, t, y,
    cate_model=LinearRegression,
    outcome_model=LinearRegression,
    propensity_model=LogisticRegression(max_iter=1000),
    folds=2,
)
cate_hat = result["cate_hat"]  # per-sample CATE estimates

3. Policy Evaluation (dr_policy_value)¶

Estimates the expected outcome if we were to apply a specific treatment policy \(\pi(x)\) to the entire population.

from sklearn.linear_model import LinearRegression, LogisticRegression
from torchregress.causal import dr_policy_value

# What happens if we treat only if feature 0 > 0.5?
policy = (x[:, 0] > 0.5).float()
result = dr_policy_value(
    x, t, y,
    policy=policy,
    outcome_model=LinearRegression,
    propensity_model=LogisticRegression(max_iter=1000),
)
value = result["estimate"]

Method Comparison¶

Requirements: The Three Pillars¶

For causal estimates to be valid, three assumptions must hold:

1. Unconfoundedness: No unmeasured variables influence both \(T\) and \(Y\).
2. Positivity: Every individual has a non-zero probability of being treated or not (\(0 < \hat{e}(x) < 1\)).
3. Exchangeability: The treated and control groups are comparable after adjusting for \(X\).

References¶

Limitations¶

1. Unconfoundedness is untestable: All causal estimates assume no unmeasured confounders — variables that affect both treatment assignment \(T\) and outcome \(Y\). This assumption cannot be verified from observed data alone. Sensitivity analyses are essential.
2. Positivity violations: If any subpopulation has \(\hat{e}(x) \approx 0\) or \(\hat{e}(x) \approx 1\), the doubly-robust estimator becomes unstable. Always run causal_overlap_report before interpreting results.
3. Cross-fitting dependency: dr_ate and dr_cate cross-fit nuisance models on held-out folds. With small \(n\) or few folds (\(\le 2\)), the nuisance models may be poorly estimated, degrading the doubly-robust property.
4. CATE is harder than ATE: Conditional treatment effect estimates require more data and stronger modelling assumptions than population-average effects. CATE estimates are inherently higher-variance.
5. Not for dynamic treatments: The current API handles binary, static treatments only. Time-varying treatments, sequential interventions, and instrumental variable methods are not supported.

Recommendations¶

• Default workflow: Run dr_ate (population effect) → causal_overlap_report (positivity check) → dr_cate (heterogeneity). All three together provide a complete causal picture.
• Diagnose before interpreting: Always inspect the diagnostics dict from dr_ate/dr_cate — it contains effective sample size (ESS) and overlap warnings.
• Propensity clipping: If propensity scores are extreme (\(< 0.05\) or \(> 0.95\)), consider clipping or trimming before estimation. See Propensity utilities.
• Complement with PPI: For causal quantities that can be framed as population means, PPI inference can leverage unlabeled data for tighter confidence intervals.
• Example pipelines: See Causal DR uplift comparison (--dataset synthetic or --dataset diabetes for real covariates).

Next Steps¶

• Learn about Propensity Weighted Losses
• View the Causal DR Uplift Comparison
• Explore Inference Methods Detail