Propensity Score Matching

Propensity score matching (PSM) creates pseudo-experimental conditions in observational studies by matching treated and control units with similar probabilities of receiving treatment.

Key Theorem

Assumptions

| Assumption | Meaning | Testable? |

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

| Unconfoundedness | No unobserved confounders | No |

| Overlap | for all X | Yes |

| SUTVA | No interference between units | Partially |

Estimation Steps

| Step | Action |

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

| 1 | Estimate propensity score (logistic regression, ML) |

| 2 | Check overlap (common support) |

| 3 | Match treated to control units |

| 4 | Assess covariate balance |

| 5 | Estimate treatment effect |

| 6 | Conduct sensitivity analysis

Matching Methods

| Method | Description |

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

| Nearest neighbor | Match each treated to closest control on propensity score |

| Caliper | Only match if propensity scores are within caliper distance |

| Full matching | Create matched sets that partition all units |

| Kernel matching | Weight all controls by kernel function of propensity score |

Covariate Balance

After matching, check that covariates are balanced between groups.

| SMD | Interpretation |

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

| < 0.1 | Excellent balance |

| 0.1 - 0.2 | Adequate balance |

| > 0.2 | Poor balance — matching failed |

ATT Estimation

Sensitivity Analysis

Python Implementation

import numpy as np

import pandas as pd

from sklearn.linear_model import LogisticRegression

from sklearn.neighbors import NearestNeighbors

import matplotlib.pyplot as plt



np.random.seed(42)



# Simulate observational data

n = 1000

X1 = np.random.randn(n)

X2 = np.random.binomial(1, 0.5, n)



# Propensity (confounded)

propensity = 1 / (1 + np.exp(-(0.5*X1 + 0.3*X2)))

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



# Outcome (true ATE = 2.0)

Y0 = 3*X1 + 2*X2 + np.random.randn(n)

Y1 = Y0 + 2.0

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



df = pd.DataFrame({'Y': Y, 'T': T, 'X1': X1, 'X2': X2})



# Estimate propensity score

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

df['ps'] = logit.predict_proba(df[['X1','X2']])[:, 1]



# Match

treated_idx = df[df['T']==1].index

control_idx = df[df['T']==0].index



nn = NearestNeighbors(n_neighbors=1, metric='euclidean')

nn.fit(df.loc[control_idx, ['ps']])

distances, matches = nn.kneighbors(df.loc[treated_idx, ['ps']])



# Balance check

for col in ['X1', 'X2']:

    before_smd = abs(df[df['T']==1][col].mean() - df[df['T']==0][col].mean()) / \

                 np.sqrt((df[df['T']==1][col].var() + df[df['T']==0][col].var())/2)

    

    matched_control = control_idx[matches.flatten()]

    after_smd = abs(df[df['T']==1][col].mean() - df.loc[matched_control, col].mean()) / \

                np.sqrt((df[df['T']==1][col].var() + df.loc[matched_control, col].var())/2)

    

    print(f"{col}: Before SMD={before_smd:.3f}, After SMD={after_smd:.3f}")



# ATT estimate

att = df[df['T']==1]['Y'].mean() - df.loc[matched_control, 'Y'].mean()

print(f"\nATT estimate: {att:.3f} (true: 2.0)")

Worked Example

Key Takeaways

Related Topics

• See Causal Inference for the potential outcomes framework

• See Randomized Controlled Trials for the gold standard

• See Missing Data for related topics on data quality