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?

  • $\begingroup$ 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/… $\endgroup$ – Jeremy Miles Aug 14 '17 at 20:24
  • $\begingroup$ 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). $\endgroup$ – Carlos Cinelli Nov 29 '17 at 4:20

I assumed you are asking about what happens when you have a latent confounder measured by error-prone indicators. There have been a few approaches to solving this problem, but the current best practice is to use SEM to generate factor scores for the latent variables and then use those scores in a propensity score analysis as usual with your observed variables. That is, simply replace the latent variables with their factor score estimates. Though not perfect, some research has found this to be effective (Jakuboski, 2015; Raykov, 2012). More research will come out soon on this topic (including my own). Graphically, the causal model is depicted identically to how it would be if the variables were observed, except that the observed indicators are consequences of the latent variables.

If you are considering a latent treatment variable, then I'm not familiar with any solutions. It's hard to know what a treatment effect is when you don't know people's treatment condition.

It's important to note that the confounders are not the indicators of some latent "propensity to treatment". They are the causes of propensity to treatment. It's very important to distinguish between causal indicators and measurement indicators in latent variable models.

  • $\begingroup$ I was considering exposure/treatment to be measured whereas other confounders were latent. "Pseudotreatment" is a nod to quasiexperimental design: basically, observational research. I think this answers my question, and I have seen some other research which supports this approach. Especially +1 for the last remark. Indicators of a latent confounder are very different from observed confounders, and this suggests that poorly designed (in terms of not directly measuring confounders) studies cannot avail much methodology. $\endgroup$ – AdamO Dec 11 '17 at 22:22

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