Adjusting for unconfounding in DAG context? I am reading Cox's Case-Control Studies(2014) pg 6 of Preamble chapter. The book uses DAGs but it seems that it deviates from DAGs. I am not sure whether the following statement should hold in DAGs causal inference context.
Consider $X$ treatment and $Y$ outcome with arrow $X\to Y$ with observed confounding $W_O$ associated to arrow $W_O\to X,W_O\to Y$. Suppose there is an unobserved intrinsic variable $W_U$ yielding arrow $W_U\to Y$. In this case, there is no confounding besides $W_O$. Thus to estimate causal effect of $X$ on $Y$, it suffices to condition on $W_O$ without controlling $W_U$. However, the book says the following. Furthermore, discussion holds even for $W_O$ being empty. The book says this holds for $Y$ being binary.
"If the arrow from $W_U$ to $X$ is missing but there remains an arrow from $W_U$ to $Y$ then there is no confounding but we may still face bias if $W_U$ is ignored in some special situations,
notably when the association between X and Y is measured using an odds ratio."

*

*Is the statement true for $Y$ non-binary?

*What is the reason behind $Y$ binary resulting bias?

*Is this consistent with Pearl's DAGs?

 A: Preliminary remarks:
After the passage you cited, the book states, "This relates to the discussion around Figure 0.3(a)". There (p.4 in my copy) they point out that they are referring to the issue of non-collapsibility. Indeed, collapsibility is concerned with whether some functionals of your probability densities like risk difference or odds-ratio are the same with and without controlling for some covariate like $W_U$.
Your first question:
The odds ratio is an example of a functional that is in general non-collapsible. And since the odds ratio is the standard measure used for binary outcomes, they say that non-collapsibility is a problem with binary outcomes. They do not, however, say, that for non-binary outcomes all functionals are collapsible, which is indeed not the case. So, yes, the statement (of non-collapsibility) can also be true for non-binary outcomes.
Your second question:
As stated above non-collapsibility does also occur with non-binary outcomes. However, for odds ratio for binary outcomes, they give an example in the book that shows non-collapsibility, see p.52 "Example 2.2 Non-collapsibility of odds ratios". I am not sure what you mean by "the reason", but I think this example is sufficiently instructive to get the intuition across. After all, for showing that we don't always have collapsibility it is sufficient to find a counterexample which is provided by this example.
Your third question:
The notion of non-collapsibility exists completely independent of Pearl's causal graphs. But those DAGs help understand non-collapsibility better. In fact, in his book:

Pearl, Judea. Causality. Cambridge university press, 2009

Pearl investigates the notion of collapsibility quite thoroughly, mainly in section 6.5.
Final remark:
Two important points Pearl shows in his book and that are relevant w.r.t. your questions, too, are:

*

*Noncollapsibility and confounding are in general two distinct notions; neither implies the other.

*Confounding and non-collapsibility need not correspond even in linear functionals.

A: *

*The statement is also true for non-binary variables. If there is a single confounder, adjusting for it will recover the causal effect no matter the type of variable, provided the model is correctly specified.


*From the quote, I think the "bias" that is being referred to would be an odds ratio that may not reflect the causal relationship between the variables. For example, the $W_U \to Y$ could make one of the outcomes way more likely than it otherwise would be. An odds ratio between $X$ and $Y$ would know nothing about this. This is not because of the binary nature of the variable but simply a property of the odds ratio. I think the book is simply trying to say that there are some measures of association which are not easily interpretable in a causal sense even in a simple DAG without confounding.


*All the statements about confounding and the interpretation of different measures are consistent with Pearl's theory of causal DAGs.
