@AndersHuitfeldt has helped to transform my understanding of effect measure collapsibility. I understand that one purpose of collapsibility to epidemiologists is to be to try to use the marginal RRs or ORs of a RCT from those observed in one population (e.g. in Norway) to estimate what might be observed in a different population (e.g. in Wales) where the conditional risks based on a covariate (e.g. males / females) are different. However this can be done using the collapsible property of natural frequencies and natural odds, which I have known as long as I have been using ‘P Maps’ to be always collapsible (although I did not use that word). This is explained in my previous post ( Should one derive risk difference from the odds ratio? - #540 by HuwLlewelyn ). This illustrates the point about nomenclature made by @ESMD and perhaps the reason that it has taken me so long to understand @Sander and others.
What interests me as a clinician however, is how to apply the result of a RCT to patients with the same disease but various degrees of disease severity (fundamental to clinical practice) and also to apply them to patients with different baseline risks for other reasons (e.g. the presence of multiple covariates that change the baseline risks). This can be helped when conditional and marginal effect measures such as RRs and ORs are identical as well as being natural frequency and natural odds collapsible.
The RR and OR can be very helpful for low risks encountered in epidemiology (when they can be very similar numerically). However the RR not only gives implausibly large RDs at high probabilities encountered in the clinical setting familiar to me but also probabilities > 1 if the treatment increases risk (a problem that can be prevented by the switch RR of course). The OR does not have these bad habits and allows the marginal OR to be transported plausibly to baseline probabilities that are higher and lower than the marginal probability in a convenient way. However the OR is not an accurate model because when the marginal and conditional odds ratios are the same, p(Z=1|X=0) ≠ p(Z=1|X=0).
Another approach is to fit logistic regression functions independently to both the control and treatment data, but this too can provide biases (e.g. due to choice of the training odds used). My solution is to use any of the above when convenient for practical reasons and then to calibrate the results by regarding the ‘calculated curves’ as being provisional. I explained how I do this in another recent post: Risk based treatment and the validity of scales of effect , that also explores a different aspect of causal inference when dealing with multiple covariates.