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When using IPW with a binary outcome, the results after IPW are not bound between 0 and 1.

I am using IPW to get doubly robust scores. $$ \mu + \frac{D_i(y_i - \mu)}{ps_i} $$ where $\mu$ is the predicted outcome, $D$ is the treatment indicator, $y$ the observed outcome, and $ps$ the propensity score.

What are approaches to bound the outcomes?

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    $\begingroup$ Could you please explain how you are using IPW? What are you estimating and what is your estimator? Can you give a simple example of how it produces erroneous, strange, or invalid results? $\endgroup$
    – whuber ♦
    Commented Sep 30, 2022 at 14:26
  • $\begingroup$ @whuber: the doubly robust score with the ipw gives values outside of 0 and 1 for the individuals that actually received the treatment. $\endgroup$
    – jkortner
    Commented Oct 1, 2022 at 7:42
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    $\begingroup$ Looks like you're going for Augmented IPW (AIPW). First, just verifying that $ps_i=\Pr[A=a_i|X_i]$ (the inverse probability weight) and not $\Pr[A=1|X_i]$ (as these are sometimes get confused). Second, as Noah wrote below, AIPW, just like IPW, is not well-defined at the individual level, but rather only after averaging over the dataset. There do exist other doubly-robust methods that support individual prediction, like TMLE. $\endgroup$
    – ehudk
    Commented Oct 19, 2022 at 19:07

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If you use the Hajek estimator, the most commonly used estimator for IPW, the expected potential outcomes are bounded between 0 and 1 as long as the weights are non-negative, which they will be in most applications. The Hajek estimator of a counterfactual mean is computed as $$ E[Y^a]=\frac{\sum_{i=1}^n{I(A_i=a)w_i Y_i}}{\sum_{i=1}^n{I(A_i=a)w_i} } $$ or just the weighted mean of the outcome $Y$ in the treatment group $A=a$. If all the $Y_i$s are 0 or 1 and all the $w_i$s are nonnegative, there is no way for this estimator to yield a weighted mean outside 0 to 1. This is not a property of IPW but of all weighted means. When you use an unadjusted weighted linear regression of the outcome on the treatment, the resulting coefficient on the treatment is equal to the Hajek estimator of the treatment effect.

Things change when your weights are negative, which can be the case when the outcome regression includes other covariates, especially those that are not balanced. See Chattopadhyay and Zubizarreta (2022) for more details on negative weights implied by a linear regression.

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  • $\begingroup$ I see that the mean will yield a value between 0 and 1, but the individual expected potential outcomes are not bound as I understand it. $\endgroup$
    – jkortner
    Commented Oct 1, 2022 at 7:44
  • $\begingroup$ IPW doesn't estimate the individual potential outcomes. It uses the raw outcomes themselves, which are just 0s and 1s. If you are using a DR method, you can just use a method of estimating the potential outcomes that bounds them between 0 and 1, like logistic regression or extracting the predicted probability from a classification algorithm. That has nothing to do with IPW, though. $\endgroup$
    – Noah
    Commented Oct 1, 2022 at 15:51

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