IP Weighting Fundamentals

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What is IP Weighting?

Inverse Probability (IP) Weighting is a powerful method for estimating causal effects that creates a pseudo-population where treatment is independent of measured confounders. This approach is fundamental to marginal structural models and has broad applications in causal inference.

Inverse Probability Weighting: A method that weights each observation by the inverse of its probability of receiving the treatment it actually received, thereby creating a pseudo-population where treatment assignment is independent of measured confounders.

The key intuition:

In observational data, treated and untreated groups differ systematically. IP weighting "re-weights" the data so that each treatment group represents what the full population would look like.

A treated patient who was unlikely to be treated (low PS) gets high weight
A treated patient who was likely to be treated (high PS) gets low weight
The reverse applies to untreated patients

Example: Smoking cessation study
Consider a person with characteristics that make quitting very unlikely (young, heavy smoker, no health concerns)—say PS = 0.05. If they did quit, their weight would be 1/0.05 = 20. This person represents many similar people who did not quit.

Conversely, someone very likely to quit (older, light smoker, health-motivated)—say PS = 0.80—gets weight 1/0.80 = 1.25 if they quit. They represent fewer people since most similar individuals also quit.

Why IP weighting creates balance:

After weighting, the distribution of confounders becomes the same in treated and untreated groups. Mathematically, in the pseudo-population:

$P^*(L|A=1) = P^*(L|A=0) = P(L)$

where $P^*$ denotes probability in the weighted pseudo-population.

Reference: Hernan MA, Robins JM. What If (2020), Chapter 12: "IP Weighting and Marginal Structural Models" provides the theoretical foundation for IP weighting.

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