The extent to which the probability of the outcome differs between groups defined by baseline covariates such as obesity, is captured by the parameter for the covariate. There is no difference between the odds ratio, the risk difference and the risk ratio in terms of their ability to capture any baseline risk differences between groups due to the confounders.
If obesity was unobserved, and obesity was prognostic for the outcome, in this group the modeled effect of treatment should be assessed of less magnitude than in a group where obesity was measured and included in the model.
You need to be much clearer about the role of the parameter for baseline risk, the parameter for obesity, and the parameter for the intervention.
This is because the distribution of unmeasured obesity across treatment arms must influence our effect measures assessment of “how likely”.
In a randomized trial (and in the absence of confounding), the distribution of obesity will be independent of treatment arms. Avoiding confusion about this is one of the primary reasons that I insist on the need for using counterfactuals
So adding the additional variable obesity to the model should increase the estimated effect of treatment, via the effect measure, even if treatment and obesity were uncorrelated. In other words, the coefficient in the regression equation does not represent a sort of universal constant. If obesity (whether observed or not) is inherently involved in what the average causal effect of treatment in a population means, then the distribution of obesity in that population sets the context in which the effect of the treatment is realized
There are two separate questions to consider here:
- Is the prevalence of obesity independent of treatment arm?
- Assuming that the prevalence of obesity is independent of treatment arm, should the effect of treatment depend on the prevalence of obesity?
As explained above, I am going to assume we are in a situation where question one can be answered in the affirmative. This is expected to hold by design in an (infinitely large and perfect) randomized trial, but whether it is true in the data is not really relevant if we just define our effect measures in terms of counterfactuals.
For the second question, i agree that it is possible that the prevalence of obesity determines treatment effects. But in order to violate collapsibility, it must be the case that in the hypothetical situation where the odds ratio is equal between the group where obesity=0 (i.e. in a group where the prevalence of obesity is 0% in both the intervention group and the control group) and the group where obesity=1 (where the prevalence is 100%), then the pooled odds ratio (i.e. in a group where the prevalence of obesity is somewhere between 0 and 100%) is not also equal. You are going to have to find some way to explain this U-shape.
I also note that nobody is claiming that regression equations represent universal constants. I will however insist that the validity of a model depends on the extent to which the homogeneity assumptions that define it are to some approximation reflective of biological reality.