You're on the right track! Propensity Score Matching (PSM) is a powerful quasi-experimental method to balance observed covariates between treatment and control groups, thereby reducing confounding and allowing for more robust estimation of treatment effects.

Let's break down the process step-by-step, starting with your requested first step: Estimating Propensity Scores using Logistic Regression.

Propensity Score Matching (PSM) Guide

Goal: To create a matched sample where treated and control units are similar on observed covariates, allowing for an unbiased estimate of the treatment effect.

Assumptions for PSM:

• Conditional Independence (CIA) / Ignorability: All confounding variables that influence both treatment assignment and the outcome are observed and included in the propensity score model. This is untestable.


• Common Support / Overlap: For every treated unit, there is a control unit with a similar propensity score, and vice-versa.


• Stable Unit Treatment Value Assumption (SUTVA): Treatment assignment for one unit does not affect the outcome of another unit, and there are no different versions of the treatment.

Step 1: Estimate Propensity Scores using Logistic Regression

The propensity score ($e(X)$) is the probability of receiving the treatment, conditional on observed baseline covariates ($X$).
$e(X) = P(Treatment=1 | X)$

Logistic regression is the standard method for estimating these probabilities because the outcome (treatment assignment) is binary (0 = control, 1 = treated).

Key Considerations for Covariate Selection:

• **Include all pre-treatment confounders:** These are variables that are related to both the treatment assignment and the outcome.


• Do NOT include variables that are:


• Only related to the outcome, but not treatment (they don't confound).


• Only related to treatment, but not outcome (they don't confound).


• Mediators (on the causal path between treatment and outcome). Including them would block part of the treatment effect.


• Colliders (variables affected by both treatment and outcome).


• Balance vs. Bias: The goal of the propensity score model is to achieve covariate balance, not necessarily to predict treatment perfectly. A high R-squared isn't the primary goal.


• Flexibility: Consider including squared terms or interaction terms if non-linear relationships or interactions are theoretically justified. However, this can make common support harder to achieve.


• Sample Size: Ensure enough units to estimate the logistic regression model robustly, especially with many covariates or interactions. A general rule of thumb is 10-20 events (treated units) per covariate.

Practical Implementation (R & Python Examples):

Let's assume you have a dataset df with:

• treatment: Binary variable (1 for treated, 0 for control)


• outcome: Your continuous or binary outcome variable


• X1, X2, X3, ...: Your pre-treatment confounding covariates

Example in R:

# Install and load necessary packages
# install.packages("glm2") # For robust logistic regression if issues
# install.packages("dplyr") # For data manipulation
library(glm2)
library(dplyr)
# --- 1. Load/Prepare Data (Example Data) ---
# Let's create some dummy data for demonstration
set.seed(123)
n <- 1000
df <- data.frame(
  X1 = rnorm(n, 50, 10),
  X2 = rbinom(n, 1, 0.4),
  X3 = runif(n, 0, 100)
)
# Simulate treatment assignment: correlated with X1, X2
df$treatment <- rbinom(n, 1, plogis(-1 + 0.05 * df$X1 + 0.8 * df$X2 + 0.01 * df$X3))
# Simulate outcome: correlated with X1, X2, X3 and treatment
df$outcome <- 10 + 0.5 * df$X1 + 2 * df$X2 - 0.1 * df$X3 + 5 * df$treatment + rnorm(n, 0, 5)
# --- 2. Estimate Propensity Scores using Logistic Regression ---
# Define your covariates
covariates <- c("X1", "X2", "X3")
# Formula for logistic regression
fmla <- as.formula(paste("treatment ~", paste(covariates, collapse = " + ")))
# Fit the logistic regression model
# Use glm() with family = binomial(link = "logit")
# You might use glm2::glm.fit2 for more robust fitting in case of convergence issues
prop_model <- glm(fmla, data = df, family = binomial(link = "logit"))
# Summarize the model (optional, but good for understanding)
summary(prop_model)
# Predict propensity scores (probabilities)
# type = "response" gives you the probabilities P(Treatment=1|X)
df$prop_score <- predict(prop_model, type = "response")
# --- 3. Inspect Propensity Scores (Preliminary Check) ---
# View the range of propensity scores
summary(df$prop_score)
# Check for common support visually (density plots are great)
# This will be revisited in Step 3, but a quick look is good
hist(df$prop_score[df$treatment == 1], main = "Treated Propensity Scores", xlab = "Propensity Score", breaks = 30)
hist(df$prop_score[df$treatment == 0], main = "Control Propensity Scores", xlab = "Propensity Score", breaks = 30)
# Or using a density plot (better for comparison)
plot(density(df$prop_score[df$treatment == 1]), col = "blue", lwd = 2,
     main = "Propensity Score Distribution by Treatment Group",
     xlab = "Propensity Score")
lines(density(df$prop_score[df$treatment == 0]), col = "red", lwd = 2)
legend("topright", legend = c("Treated", "Control"), col = c("blue", "red"), lwd = 2)
# Save the updated data frame for the next steps
# write.csv(df, "data_with_prop_scores.csv", row.names = FALSE)

Example in Python:

import pandas as pd
import numpy as np
import statsmodels.api as sm
from sklearn.linear_model import LogisticRegression
import matplotlib.pyplot as plt
import seaborn as sns
# --- 1. Load/Prepare Data (Example Data) ---
# Let's create some dummy data for demonstration
np.random.seed(123)
n = 1000
data = {
    'X1': np.random.normal(50, 10, n),
    'X2': np.randint(0, 2, n),
    'X3': np.random.uniform(0, 100, n)
}
df = pd.DataFrame(data)
# Simulate treatment assignment: correlated with X1, X2
prob_treatment = 1 / (1 + np.exp(-(-1 + 0.05 * df['X1'] + 0.8 * df['X2'] + 0.01 * df['X3'])))
df['treatment'] = np.random.binomial(1, prob_treatment, n)
# Simulate outcome: correlated with X1, X2, X3 and treatment
df['outcome'] = 10 + 0.5 * df['X1'] + 2 * df['X2'] - 0.1 * df['X3'] + 5 * df['treatment'] + np.random.normal(0, 5, n)
# --- 2. Estimate Propensity Scores using Logistic Regression ---
# Define your covariates
covariates = ['X1', 'X2', 'X3']
# Method 1: using statsmodels (provides more detailed statistical output)
X = df[covariates]
X = sm.add_constant(X) # Add an intercept to the model
y = df['treatment']
logit_model = sm.Logit(y, X)
result = logit_model.fit()
print(result.summary())
df['prop_score_sm'] = result.predict(X)
# Method 2: using scikit-learn (often preferred for prediction/ML pipelines)
# Note: scikit-learn's LogisticRegression by default includes an intercept.
# Scaling features is often good practice for sklearn models, but not strictly required for Logit.
# from sklearn.preprocessing import StandardScaler
# scaler = StandardScaler()
# X_scaled = scaler.fit_transform(df[covariates])
# lr_model = LogisticRegression(solver='liblinear') # 'liblinear' is a good default for small datasets
# lr_model.fit(X_scaled, df['treatment'])
# df['prop_score_sk'] = lr_model.predict_proba(X_scaled)[:, 1] # Get probability of the positive class (1)
# Let's stick with statsmodels for clearer statistical interpretation
df['prop_score'] = df['prop_score_sm']
# --- 3. Inspect Propensity Scores (Preliminary Check) ---
# View the range of propensity scores
print("\nPropensity Score Summary:")
print(df['prop_score'].describe())
# Check for common support visually (density plots)
plt.figure(figsize=(10, 6))
sns.histplot(df[df['treatment'] == 1]['prop_score'], color='blue', label='Treated', kde=True, stat='density', alpha=0.5)
sns.histplot(df[df['treatment'] == 0]['prop_score'], color='red', label='Control', kde=True, stat='density', alpha=0.5)
plt.title('Propensity Score Distribution by Treatment Group')
plt.xlabel('Propensity Score')
plt.ylabel('Density')
plt.legend()
plt.show()
# Save the updated data frame for the next steps
# df.to_csv("data_with_prop_scores.csv", index=False)

Next Steps (Brief Overview):

After you have estimated the propensity scores for each unit in your dataset, the PSM process continues:

Step 2: Matching

• Method Selection: Choose a matching algorithm (e.g., Nearest Neighbor, Caliper, Optimal, Kernel).


• Ratio: Decide on the number of control units to match to each treated unit (e.g., 1:1, 1:n).


• Replacement: Decide whether control units can be used more than once.


• Pruning: Remove units that fall outside the common support region.

• Crucial Step! After matching, you must check if the covariates are actually balanced between the matched treatment and control groups.


• Methods: Standardized Mean Differences (SMD), variance ratios, QQ plots, Love plots (covariate balance plots), histograms of propensity scores.


• Iterative Process: If balance is not achieved, you may need to go back to Step 1 (re-specify the logistic regression model, e.g., add interactions, polynomial terms), or adjust matching parameters in Step 2.

• Analyze Matched Data: Perform your outcome analysis only on the matched sample.


• Regression: Use regression (e.g., OLS) with treatment as a predictor on the matched data.


• Standard Errors: Be mindful of standard errors. If matching with replacement, or with 1:N matching, the observations are not independent. Clustered standard errors (by the matched pairs/sets) are often recommended.

• Assess how robust your findings are to the potential influence of unobserved confounders.

I recommend starting with Step 1 as detailed above. Once you have your propensity scores, you can move on to the matching phase! Let me know if you want to dive into the next step.