Causal Inference — Potential Outcomes Framework

Statistics

The Gold Standard for Answering Causal Questions

The potential outcomes framework (Rubin Causal Model) defines causal effects as comparisons between what happened and what would have happened. It provides the conceptual foundation for all modern causal inference methods.

Medical Research — Define treatment effects rigorously in clinical trial analysis
Policy Evaluation — Measure program impacts by comparing actual and counterfactual outcomes
Social Science — Establish clear criteria for when causal claims are justified

The fundamental problem — we never observe both potential outcomes — drives all of causal inference.

Causal inference asks: What would have happened if a different treatment had been applied? The potential outcomes framework (Rubin Causal Model) provides a rigorous framework for answering this question.

Potential Outcomes

Fundamental Problem of Causal Inference

We can only estimate population-level effects (average treatment effects).

Average Treatment Effects

SUTVA

The Stable Unit Treatment Value Assumption has two parts:

| Component | Meaning |

|-----------|---------|

| No interference | One unit's treatment does not affect another unit's outcome |

| Treatment variation irrelevance | There is only one version of each treatment level |

Selection Bias

The naive comparison of treated and control groups may be biased because treatment assignment is often not random.

Causal Identifying Assumptions

| Assumption | Description | Violation Consequence |

|-----------|------------|----------------------|

| Unconfoundedness | | Selection bias |

| Overlap | for all X | Cannot estimate effects for some subgroups |

| SUTVA | No interference; one treatment version | Spillover effects bias estimates |

Estimation Under Unconfoundedness

When treatment is unconfounded given covariates , we can use matching, weighting, or regression adjustment.

Python Implementation


import numpy as np

import pandas as pd

from sklearn.linear_model import LogisticRegression

import matplotlib.pyplot as plt



np.random.seed(42)



# Simulate data with confounding

n = 1000

X = np.random.randn(n, 3)

propensity = 1 / (1 + np.exp(-(0.5*X[:,0] + 0.3*X[:,1] - 0.2*X[:,2])))

T = np.random.binomial(1, propensity)

Y0 = 2*X[:,0] + X[:,1] + np.random.randn(n)

Y1 = Y0 + 1.5  # True ATE = 1.5

Y = T * Y1 + (1 - T) * Y0



df = pd.DataFrame({'Y': Y, 'T': T, 'X1': X[:,0], 'X2': X[:,1], 'X3': X[:,2]})



# Naive comparison (biased)

naive_diff = df[df['T']==1]['Y'].mean() - df[df['T']==0]['Y'].mean()

print(f"Naive difference: {naive_diff:.3f}")



# IPW estimator

prop_model = LogisticRegression().fit(df[['X1','X2','X3']], df['T'])

e_hat = prop_model.predict_proba(df[['X1','X2','X3']])[:, 1]



ipw = np.mean(T * Y / e_hat - (1 - T) * Y / (1 - e_hat))

print(f"IPW estimate: {ipw:.3f}")

print(f"True ATE: 1.500")

Causal Inference — Potential Outcomes Framework

Causal Inference — Potential Outcomes Framework

The Gold Standard for Answering Causal Questions

Potential Outcomes

Fundamental Problem of Causal Inference

Average Treatment Effects

SUTVA

Selection Bias

Causal Identifying Assumptions

Estimation Under Unconfoundedness

Python Implementation

Worked Example

Key Takeaways

Related Topics

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