I have been asked to write about noncollapsibility by the editor of a medical journal and obviously (given the discussions previously on this blog) this will be the first paper that will re-define this a good and desirable property of an effect measure.

However in the process of creating some examples in the paper I decided to go for a non-confounding prognostic (for the outcome) third variable to demonstrate the change in effect expected. Then I realized that a third variable could be:
a) non-prognostic for the outcome
b) prognostic only
c) prognostic and confounding
d) prognostic and effect modifying
e) prognostic, confounding and effect modifying

Of course adjusting for a) adds nothing so no issues here

Q1. Adjusting for b) leads to a change in effect for the main effect but also adds an effect for the prognostic variable so we can assess a phenomenon Frank calls risk magnification - and to quote âit has nothing to do with HTEâ with which I agree but it does change the baseline risk by level of that variable and thus will impact the response to the âfixedâ effect of treatment - is this why it was discussed under HTE and would I be wrong to call this a variant of HTE?

Q2. Change in main effect with variable type b) can only be away from the null but with type c) there is no point in talking about noncollapsibility at all since the confounding effect is indeed a synonym for the noncollapsibility effect since it is also a consequence of prognosis but this time the change in effect can be in both directions - is that right?

Q3. For variables type d) and e) there is mostly just random variation due to artifacts of the sample and a true effect modification is so rare that we should not even discuss it and therefore d) should be handled as b) and e) as c) above - right? That means its only important to discuss a) , b) and c).

Q4. The implication is that the bulk of the epidemiology of precision medicine is risk magnification - right?

[please lets not discuss collapsible measures here ]

Very interesting and the last related link took me here

The analysis of the GUSTO-I data was very interesting so I tried it out on the same 30,510 patients this time adjusting for age (rcs with three knots) and Killip class. McFaddenâs R^2 went up from 0.0007 to 0.1450 suggesting that the age and Killip class account for the bulk of the impact on day 30 mortality. The main effect of treatment strengthened to OR=0.81 and would I be right in saying that this is the closest empiric treatment effect for the individual patient (not OR=0.85)?
Interestingly, within strata of Killip the age-adjusted main effect was
OR=0.80
OR=0.84
OR=0.72
OR=1.33
We can safely conclude that these variations relate to artifacts of the sample and 0.81 only is correct?

0.81 is correct. I donât know how you get different ORs by Killip class if you adjust for all the other variables. There is no evidence for a treatment by Killip class interaction.

Agree - I only adjusted for age and stratified by Killip class so the four analyses had 34825, 5141, 551 and 313 participants so indeed the differences just represent increasing random error not a treatment by Killip class interaction. That is what I was trying to illustrate to justify dropping d) and e) in the first post

I revisited this paper from Steyerbergâs group which we have discussed in a different thread and references this post referenced above as ref 10.

They propose three regression based methods of predictive approaches to HTE that aim to provide âindividualizedâ treatment effect estimates from group-based data:

Risk-based methods

Treatment effect modeling methods

Optimal treatment regime methods

We are discussing 1) here and the authors say: âRisk-based approaches use baseline risk determined by a multivariate equation to define the reference class of a patient as the basis for predicting HTE. Two distinct approaches were identified: 1.1) risk magnification assumes constant relative treatment effect across all patient subgroups, while 1.2) risk stratification analyzes treatment effects within strata of predicted risk. This approach is straightforward to implement, and may provide adequate assessment of HTE in the absence of strong prior evidence for potential effect modification. The approach might better be labeled âbenefit magnificationâ, since benefit increases by higher baseline risk and a constant relative risk.â

1.1 is what we are discussing here since 1.2 is back to risk-by-treatment interactions and as seen above these are unreliable and more likely artifacts of the sample than true risk-by-treatment interactions (I am using Killip class to stratify patients)