# 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.

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1. Isn’t there a factor of N_z missing on the right hand side of the second displayed equation?

Comment by Cosma Shalizi — November 11, 2009 @ 6:45 am

2. Yes,
add Nz to the right hand side of th second displayed
equation.
Likewise,add Z1 to the secon term in the denominator
of the 3rd displayed equation.
Also, change P(X=1, Z=z) into P(X=1 | Z=z).

Comment by judea — November 11, 2009 @ 11:54 pm

3. A comment regarding the term “Dynamic Treatment”.
A colleague alerted me to the fact that “Dynamic treatment” has two interpretations. 1. A sequence of
actions do(X1 =x1), do(X2=x2)….do(Xk = xk) and,
2. a sequence of conditional actions, for example,
do(X1=x1 if Z1=z1, do X1=0 otherwise), do(X2=x2 if Z2….) etc…or, in general,do (Xi=gi(z) whenever Z=z), i = 1,2,3….

The second interpreation poses no special problem to
those concerned with the denominator of the post-treatment expression. Again, the rule is: decompose into a product and replace each pre-treatment mechanism by
its post-treatment expression. Thus, whereas in the case
of fixed treatment we had P'(X=1|Z=z)= 1, this term will become P*(X=1|Z=z), where P* is the specified treatment selection strategy in the post-treatment
world. This means that every treatment term in the original distribution will give rise to a fraction
P(X=1|Z=z)/P*(X=1|Z=z) in the denominator of the post-treatment distribution.
An example taken from process control is given on page 75
of Causality (2009). Again, the content of all denominators is dictated by quantity we wish to estimate,
and not by the technique we adapt to estimate it.

Identification conditions for partially observed confounders are discussed on page 354 of Causality.

Request to future bloggers: please use the term
“conditional plan” or “conditional treatments” when referring to this interpretation
of “dynamic treatment”.

Comment by judea — November 12, 2009 @ 12:46 am

4. […] causality-blog has been enriched with two recent discussions: ” The intuition behind ‘Inverse Probability Weighting’” and “Accounting for measurement cost and estimators […]

Pingback by Causal Analysis in Theory and Practice » Message from Judea Pearl — December 9, 2009 @ 3:19 am

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