# Connection with propensity score adjustment and latent variable modeling

We often use graphical models to represent causal networks. In a model considering the relation between an exposure and an outcome, a number of confounding factors may be at play. For instance: in relating smoking to cancer risk, age, gender, alcohol use, diet, and exercise are causally related to smoking and increase cancer risk. In a logistic regression model, we fit a multivariate model with effects for smoking as well as separate effects for these confounding factors. A methodologically consistent approach to this is adjusting for the propensity score as a covariate (Austin Madama 2006). The propensity score is a predictor of likelihood of receipt of exposure, conceiving of the exposure as a pseudo-treatment (in this case: smoking). The propensity score, then is a linear combination of confounding variables which predicts the exposure and blocks the effects of confounding.

In structural equation modeling, it is possible to fit regression models using a mixture of latent and observed variables. If we accept the propensity adjustment approach, we might depict it graphically by showing a "likelihood of receipt of (pseudo)treatment" as a latent variable with manifest variables being the confounders (age, sex, alcohol, diet, and so on). Is this approach correct? If so, would the latent/propensity variable have a path leading to only the exposure, only the outcome, or both as a means of obtaining approximately similar inference?

• This sounds a bit like a Complier Average Causal Effect (CACE) model. Here are some slides: statmodel.com/bmuthen/Ed255C/Lecture/Lecture%206/… and a paper: journals.sagepub.com/doi/abs/10.3102/… – Jeremy Miles Aug 14 '17 at 20:24
• Do you mean the usual representation of Z --> X being replaced by Z ---> p(x|z) --> X where Z is the confounder, X the treatment and p(x|z) the propensity score? If that's the case, then the ps intercepts the path of the confounder to the exposure only. You can also do a different model to interecept the path from confounder to outcome, but that's a disease risk score (modeling outcome), not a propensity score (modeling treatment). – Carlos Cinelli Nov 29 '17 at 4:20