Propensity Score Matching
Estimate treatment effects when randomization isn't possible. Match treated and control subjects with similar propensity scores to reduce selection bias.
📊 The Selection Bias Problem
When treatment isn't random, treated and control groups differ systematically. Naive comparison conflates treatment effect with selection effect.
Example
VIP program members have higher LTV. But is it the VIP perks, or were they already high-value customers before joining?
The PSM Solution
1. Estimate P(treatment | covariates)
2. Match treated to controls with similar P
3. Compare matched pairs only
"Compare apples to apples" by matching on treatment propensity.
Parameters
📊 Effect Estimates
Propensity Score Distribution
Treated
Control
Matching Process
Estimate Propensity
Logistic regression: P(T|X)
Match
Pair treated with similar controls
Compare
ATT = mean(Y_T - Y_C) for pairs
🎰 Betting Applications
VIP Effect
Match VIPs to non-VIPs with similar prior activity.
Compare LTV post-enrollment.
Promo Impact
Match promo recipients to non-recipients by propensity.
Estimate true promo ROI.
Feature Adoption
Match users who adopted new feature vs those who didn't.
Measure engagement lift.
R Code Equivalent
# Propensity score matching
library(MatchIt)
# Simulate data
set.seed(42)
n <- 200
x <- rnorm(n, 50, 15)
propensity <- plogis(-2 + 5 * 0.01 * x)
treated <- rbinom(n, 1, propensity)
y <- 100 + 0.5 * x + 10 * treated + rnorm(n, 0, 10)
df <- data.frame(x, treated, y)
# Naive comparison
naive <- mean(df$y[df$treated == 1]) - mean(df$y[df$treated == 0])
cat("Naive effect:", naive, "\n")
# Propensity score matching
m.out <- matchit(treated ~ x, data = df, method = "nearest")
m.data <- match.data(m.out)
# Matched comparison
matched_effect <- mean(m.data$y[m.data$treated == 1]) -
mean(m.data$y[m.data$treated == 0])
cat("Matched effect:", matched_effect, "\n")
# Check balance
summary(m.out)✅ Key Takeaways
- • PSM reduces selection bias in observational data
- • Match on P(treatment | covariates)
- • Check for common support (overlap)
- • ATT: Average Treatment effect on Treated
- • Requires all confounders measured
- • Balance checks essential post-matching