Summary: Propensity score matching (PSM) estimates the causal effect of a binary “treatment” (feature rollout) by pairing treated and control units with similar probabilities of treatment given covariates, thereby reducing confounding from observed covariates.Assumptions:- Conditional ignorability (unconfoundedness): given covariates X, treatment is independent of potential outcomes (no unobserved confounders).- Positivity / overlap: 0 < P(T=1 | X) < 1 for all X (common support).- Correct specification of propensity model (or use flexible methods).Implementation steps:1. Define covariates X that predict both treatment assignment and outcome.2. Estimate propensity scores p(X) using logistic regression, random forest, or gradient boosting.3. Match treated to control (nearest‑neighbor, caliper, 1:k, with/without replacement, or optimal).4. Check balance and overlap; adjust matching parameters if needed.5. Estimate average treatment effect on treated (ATT) by comparing outcomes within matched pairs (simple difference, paired regression, or weighted estimator).6. Compute robust SEs (bootstrap or analytic formulas accounting for matching).Minimal Python example:python
# estimate propensity
from sklearn.linear_model import LogisticRegression
import numpy as np
lr = LogisticRegression(max_iter=1000)
lr.fit(X, treat)
ps = lr.predict_proba(X)[:,1]
# nearest neighbor matching (1:1, caliper optional)
from sklearn.neighbors import NearestNeighbors
nn = NearestNeighbors(n_neighbors=1)
nn.fit(ps[control_idx].reshape(-1,1))
dist, idx = nn.kneighbors(ps[treated_idx].reshape(-1,1))
