# Causal Analysis in Theory and Practice

## November 10, 2009

### The Intuition Behind Inverse Probability Weighting

Filed under: Discussion,Intuition,Marginal structural models — moderator @ 11:00 pm

### Michael Foster from University of North Carolina writes:

I’m an economist here in the UNC school of public health and trying to work on the intuition of MSM for my non-methodologists collaborators. My bios and epi colleagues can give me mechanical answers but are short on intuition at times. Here are two questions:

1. Consider a regressor that is a confounding variable but that is also a victim of unobserved confounding itself. Why does weighting with this troublesome covariate not cause bias that regression causes (collider bias)? In this case, I’m principally thinking about past exposures and how to handle them in an analysis of dynamic treatment. Marginal structural models (MSM) including them in calculating the weights; Robins suggests that including them as covariates in the outcome equation produces the “null paradox”.

Here’s my answer. A confounding variable has two characteristics–it is related to the exposure and to the outcome. When we weight with that variable, we break the link between the exposure and that variable. However, other than the portion due to the exposure, we do not eliminate the relationship between the covariate and the outcome. In that way (by not breaking both links), we avoid the bias created by the collider issue.

1. How do I know what variables to include in the numerator of the MSM weight?

Here’s my answer: I would include in the weights those variables that will be included in the analysis of the outcome. Their presence in the denominator of the weight is essentially duplicative–we’re accounting for them there and in the outcome model.

### Judea Pearl replies (updated 11/19/2009):

Your question deals with the intuition behind “Inverse Probability Weighting” (IPW), an estimation technique used in several frameworks, among them Marginal Structural Models (MSM). However, the division by the propensity score P(X=1| Z=z) or the probability of treatment X = 1 given observed covariates Z = z, is more than a step taken by one estimation technique; it is dictated by the very definition of “causal effect,” and appears therefore, in various guises, in every method of effect estimation — it is a property of Nature, not of our efforts to unveil the secrets of Nature.

Let us first see how this probability ends up in the denominator of the effect estimand, and then deal with the specifics of your question, dynamic treatment and unobserved confounders.

As always, we welcome your views on this topic. To continue the discussion, please use the comment link below to add your thoughts. You can also suggest a new topic of discussion using our submission form by clicking here.

P
(
Z
1
, X
1
=
x
1
, Z
2
, X
2
=
x
2
, Y
) =
P
(
Z
1
, X
1
=
x
1
, Z
2
, X
2
=
x
2
, Y
)
P
(
X
1
=
x
1
|
Z
1
)
P
(
X
2
=
x
2
|
Z
2
, X
1
=
x
1
)