Inverse Probability Weighting (IPW)

IPW relies on the propensity score, defined as the conditional probability of receiving a treatment given observed covariates: e(X) = P(T=1 | X). Typically estimated via logistic regression or machine learning models, this score distills multi-dimensional confounding variables into a single balancing scalar. The weight assigned to each unit is the inverse of the probability of receiving the treatment they actually got: 1/e(X) for treated units and 1/(1-e(X)) for controls.

By applying weights, IPW creates a synthetic sample where the distribution of covariates is balanced between treatment and control groups. In this pseudo-population, treatment assignment becomes independent of the measured confounders, effectively mimicking a randomized controlled trial. This breaks the spurious association between the confounder and the outcome, allowing for an unbiased estimate of the Average Treatment Effect (ATE).

Standard IPW weights can exhibit extreme variability if some subjects have a very low probability of receiving their assigned treatment. Stabilized weights mitigate this by multiplying the inverse probability by the marginal probability of treatment: P(T=1)/e(X) for treated and P(T=0)/(1-e(X)) for controls. This reduces the variance of the estimator without sacrificing the bias correction, leading to narrower confidence intervals.

IPW requires the positivity assumption: every subject must have a non-zero probability of receiving either treatment level. Practical violations occur when certain covariate strata contain only treated or only control units. Trimming or truncating extreme weights (e.g., capping at the 1st and 99th percentiles) is a common heuristic to prevent a single observation from dominating the analysis, though it introduces a bias-variance trade-off.

IPW forms the foundation of Doubly Robust Estimation, which combines the propensity score model with an outcome regression model. The estimator remains consistent if either the propensity score model or the outcome model is correctly specified, but not necessarily both. This provides a crucial safety net against model misspecification, making it a preferred method in high-stakes causal inference tasks like policy evaluation.

Beyond treatment effects, IPW is critical for correcting informative censoring in survival analysis. When patients drop out of a clinical trial for reasons related to their prognosis, standard Kaplan-Meier curves become biased. By weighting uncensored observations by the inverse of their probability of remaining uncensored, analysts can recover the survival curve that would have been observed under independent censoring.