In the presence of qualitative effect modification, additive effect modification if and only if multiplicative effect modification? I am reading Robin's What If Causal Inference book Chapter 4 section 1. In the book, it says the following assertion where $V$ denotes subset of the groups. I could not figure out the reason why
"In the presence of qualitative effect modification, additive effect modification implies multiplicative effect modification, and vice versa."
"That is, there is qualitative effect modification because the average causal effects in the subsets $V = 1$ and $V = 0$ are in the opposite direction. In the presence of qualitative effect modification, additive effect modification implies multiplicative effect modification, and vice versa. In the absence of qualitative effect modification, however, one can find effect modification on one scale (e.g., multiplicative) but not on the other (e.g., additive)."
The statement sounds like stating a proposition without proof. Consider the following. Let $Y^1,Y^0$ denote (treatment and control) outcomes, $A$ treatment(=1 for treatment and =0 for control) and $V$ subgroups(=0 or =1).
Assume there is qualitative effect modification with $E[Y^1-Y^0|V=1]=\alpha$ and $E[Y^1-Y^0|V=0]=-\alpha$ with $\alpha\neq 0$. So there is additive effect modification. Denote $E[Y^1|V=1]=x_1,E[Y^1|V=0]=x_2$. Then $\frac{E[Y^1|V=1]}{E[Y^0|V=1]}=\frac{x_1}{x_1-\alpha}$ and $\frac{E[Y^1|V=0]}{E[Y^0|V=0]}=\frac{x_2}{x_2+\alpha}$. We can ask whether multiplicative effect the same here. Choose $x_2=\alpha$ and $x_1=-\alpha$. However, the statement says there will be multiplicative effect modification. I think I cooked up a counter example.
$Q1:$ What is the formal statement on effect modification here and what is the proof of the statement?
$Q2:$ This is not related to $Q1$. The book says in causal inference, we do not consider odds ratio in general. Why odds ratio is not big concerned here? This might have something to do with collapsibility. However, one might be still interested in odds ratios.

In response to Q2, the book "What If" indeed says that the odds ratio is not a parameter of interest in causal inference and does not really discuss effect modification on this scale. However see this paper that suggests that the OR scale is the only one on which effect modification should be assessed: Redefining effect modification
