If the odds ratio is not constant that means there is an interaction between treatment and AER in the logistic model. The interaction needs to be taken into account when computing ARR or RR.
Thank you @f2harrell. If we have an RCT result where the proportion with an outcome on placebo is 0.4 and on treatment it is 0.2, the RR is 0.5 and the OR is 0.375. If an individual’s disease severity is S and the baseline risk at this disease severity is 0.8, then assuming a constant RR, the reduced risk on treatment is 0.8x0.5=0.4. However if the OR is constant then the reduced risk on treatment is 0.6.
If the likelihood of disease severity ‘S’ conditional on the outcome O is p(S|O)= 0.6, then the probability of someone in the trial having the severity S on placebo is
p(S) = p(O).p(S|O)/p(O|S) = 0.4x0.6/0.8 = 0.3.
If we assume a constant RR then the probability of someone in the trial having the severity S on treatment is also
p(S) = p(O).p(S|O)/p(O|S) = 0.2x0.6/0.4 = 0.3.
This is what we expect as randomisation should allow patients with similar degrees of severity to be present in both limbs of the RCT.
However, if we assume a constant OR then p(O|S) is now 0.6 so the probability of someone having the severity S on treatment is
p(S) = p(O).p(S|O)/p(O|S) = 0.2x0.6/0.6 = 0.2.
This means that the proportion of individuals with disease severity ‘S’ in the treatment limb is different from that in the control limb. This shows that when we use the OR model then we model interaction with disease severity (represented by the albumin excretion rate or AER in my example).
This is why I decided to fit a logistic regression functions independently to the treatment and placebo data. I was therefore able to illustrate the point that neither the RR nor OR are suitable for estimating the outcome probability of treatment from the baseline probability (e.g. on placebo).
The basis to emphasize (ARR, RR, OR) needs to come from an assimilation of hundreds of trials in the literature. Which measure is the most constant over subgroups? My money is on the OR
I agree that the OR is a better summary of efficacy than the RR. The OR and RR are of course similar at low probabilities but the RR gives bizarre results at higher probabilities. The OR should therefore give more consistent summary results across subgroups, as you say, especially if the average baseline risks vary between individual trials. However the OR as estimated in a RCT appears to be a summary of the range of different ORs at different baseline risks largely due to different severities of disease. I am very interested as a clinician in these degrees of severity and estimating accurately their effect on treatment efficacy in the form of absolute risk reduction, which is why I think it is more appropriate to fit logistic regression functions independently to the treatment and placebo data.