“the important fact is that the treatment effect does not have the same interpretation, and hence there is no reason for it to have the same numerical value, under these groupings.”
-That statement needs to be refined along the following lines: It is a natural intuition to expect that a group effect measure will be an average of effects in its subgroups or among its individuals (subgroups of size 1). This intuitively natural property is obeyed by measures linear or loglinear in the outcome means, such as risk differences and ratios, but not by differences or ratios of odds or of hazard rates. Furthermore, risks typically provide a simpler and more accurate reflection of real-world cost functions (e.g., number of beds needed for patients) than do odds.
On the other hand, loglinear models for the odds or hazard rates provide the most well-behaved and easily computed smoothers for data analysis. They are thus most “natural” for statisticians, as reflected in the jargon in which log odds and log rates equal the “natural parameters” of binomial and Poisson outcomes, respectively. This creates a tension between statistics consumers who want a measure that behaves intuitively and tracks a realistic cost function, and on the other hand the technical suppliers of summary statistics who want a well-behaved easily computed data-processing algorithm.
Suppliers have at times been guilty of trying to sell their immediate product (i.e., odds or hazard ratios) as sufficient to meet consumer goals based on arguments that ignore consumer intuitions, needs, and costs. Yet quite early on, other statisticians (including Cornfield 1971 and Bishop, Fienberg & Holland 1975) recognized that the tension is resolvable by converting the statistically “natural” model outputs into the intuitively natural, collapsible measures needed for consumers to correctly grasp the outputs in context. I should hope we can agree that these long-standing approaches deserve to be implemented whenever we cannot assure that the odds or rate contrasts sufficiently approximate risk contrasts (as when the outcome may be “common” in some subgroups, or the sampling design has not forced an equality).
Here are a few of my articles on connecting “classical” acausal statistics for data processing (probability modeling) to subsequent consumer goals such as causal inference…
“Summarization, smoothing, and inference”, SJSM 1993:
https://journals.sagepub.com/doi/10.1177/140349489302100402
“Model-based estimation of relative risks and other epidemiologic measures in studies of common outcomes and in case-control studies”, AJE 2004:
https://academic.oup.com/aje/article/160/4/301/165918
“Smoothing observational data: a philosophy and implementation for the health sciences”, ISR 2006:
https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1751-5823.2006.tb00159.x