Causal Inference

Overview

Causal inference determines cause-and-effect relationships and estimates treatment effects, going beyond correlation to understand what causes what.

When to Use

• Evaluating the impact of policy interventions or business decisions
• Estimating treatment effects when randomized experiments aren't feasible
• Controlling for confounding variables in observational data
• Determining if a marketing campaign or product change caused an outcome
• Analyzing heterogeneous treatment effects across different user segments
• Making causal claims from non-experimental data using propensity scores or instrumental variables

Key Concepts

• Treatment: Intervention or exposure
• Outcome: Result or consequence
• Confounding: Variables affecting both treatment and outcome
• Causal Graph: Visual representation of relationships
• Treatment Effect: Impact of intervention
• Selection Bias: Non-random treatment assignment

Causal Methods

• Randomized Controlled Trials (RCT): Gold standard
• Propensity Score Matching: Balance treatment/control
• Difference-in-Differences: Before/after comparison
• Instrumental Variables: Handle endogeneity
• Causal Forests: Heterogeneous treatment effects

Implementation with Python

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
from sklearn.linear_model import LinearRegression, LogisticRegression
from sklearn.preprocessing import StandardScaler
from scipy import stats

# Generate observational data with confounding
np.random.seed(42)

n = 1000

# Confounder: Age (affects both treatment and outcome)
age = np.random.uniform(25, 75, n)

# Treatment: Training program (more likely for younger people)
treatment_prob = 0.3 + 0.3 * (75 - age) / 50  # Inverse relationship with age
treatment = (np.random.uniform(0, 1, n) < treatment_prob).astype(int)

# Outcome: Salary (affected by both treatment and age)
# True causal effect of treatment: +$5000
salary = 40000 + 500 * age + 5000 * treatment + np.random.normal(0, 10000, n)

df = pd.DataFrame({
    'age': age,
    'treatment': treatment,
    'salary': salary,
})

print("Observational Data Summary:")
print(df.describe())
print(f"\nTreatment Rate: {df['treatment'].mean():.1%}")
print(f"Average Salary (Control): ${df[df['treatment']==0]['salary'].mean():.0f}")
print(f"Average Salary (Treatment): ${df[df['treatment']==1]['salary'].mean():.0f}")

# 1. Naive Comparison (BIASED - ignores confounding)
naive_effect = df[df['treatment']==1]['salary'].mean() - df[df['treatment']==0]['salary'].mean()
print(f"\n1. Naive Comparison: ${naive_effect:.0f} (BIASED)")

# 2. Regression Adjustment (Covariate Adjustment)
X = df[['treatment', 'age']]
y = df['salary']
model = LinearRegression()
model.fit(X, y)
regression_effect = model.coef_[0]

print(f"\n2. Regression Adjustment: ${regression_effect:.0f}")

# 3. Propensity Score Matching
# Estimate probability of treatment given covariates
ps_model = LogisticRegression()
ps_model.fit(df[['age']], df['treatment'])
df['propensity_score'] = ps_model.predict_proba(df[['age']])[:, 1]

print(f"\n3. Propensity Score Matching:")
print(f"PS range: [{df['propensity_score'].min():.3f}, {df['propensity_score'].max():.3f}]")

# Matching: find control for each treated unit
matched_pairs = []
treated_units = df[df['treatment'] == 1].index
for treated_idx in treated_units:
    treated_ps = df.loc[treated_idx, 'propensity_score']
    treated_age = df.loc[treated_idx, 'age']

    # Find closest control unit
    control_units = df[(df['treatment'] == 0) &
                      (df['propensity_score'] >= treated_ps - 0.1) &
                      (df['propensity_score'] <= treated_ps + 0.1)].index

    if len(control_units) > 0:
        closest_control = min(control_units,
                             key=lambda x: abs(df.loc[x, 'propensity_score'] - treated_ps))
        matched_pairs.append({
            'treated_idx': treated_idx,
            'control_idx': closest_control,
            'treated_salary': df.loc[treated_idx, 'salary'],
            'control_salary': df.loc[closest_control, 'salary'],
        })

matched_df = pd.DataFrame(matched_pairs)
psm_effect = (matched_df['treated_salary'] - matched_df['control_salary']).mean(
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