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."

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

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

*Is this consistent with Pearl's DAGs?


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*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.
