Doubly Robust Estimation

By the end of this lesson, you should be able to: explain how a doubly robust estimator combines an outcome model with a propensity model, say when it's right and when it isn't, and explain why it doesn't solve unmeasured confounding.

Two models, two ways to be wrong

The last three lessons used two different kinds of model.

An outcome model (lesson 5) predicts each user's outcome with and without treatment from their covariates. Regression adjustment relies on it being right.
A propensity model (lessons 6 and 7) predicts each user's chance of treatment. Weighting relies on it being right.

Each can be wrong: a missed curve, a missed interaction, the wrong functional form. A doubly robust estimator uses both, and gives the right answer if either model is right.

How it works

The most common doubly robust estimator is augmented inverse-probability weighting (AIPW). For each user, start from the outcome model's prediction of their effect, then correct it with the outcome model's error, weighted by the propensity score:

\hat{\tau} = \frac{1}{n}\sum_i \left[\hat{m}_1(x_i) - \hat{m}_0(x_i) + \frac{T_i\,(Y_i - \hat{m}_1(x_i))}{\hat{e}(x_i)} - \frac{(1-T_i)\,(Y_i - \hat{m}_0(x_i))}{1 - \hat{e}(x_i)}\right]

Here $\hat{m}_1$ and $\hat{m}_0$ are the predicted outcomes with and without treatment, and $\hat{e}$ is the propensity score.

Why that works, in two sentences:

If the outcome model is right, its errors average out to zero, so the correction terms vanish and you're left with the model's correct prediction.
If the propensity model is right, the weighted correction exactly repairs whatever the outcome model got wrong.

Facebook's study used a version of this called IPWRA (inverse-probability-weighted regression adjustment), which also combines a weighting model with an outcome regression.