time*treatment vs time + time:treatment in RCTs

Hi. In medicine's RCTs, with 3 or more time-points, whenever LMMs are used
and the code is available, a variation of  y ~ time*treatment + (1 | ID)
*(M1)* is always used (from what I have seen).

Recently I came across the model  time + time:treatment + (1 | ID)* (M2)*
in Solomun Kurz's blog and in the book of Galecki (LMMs using R).

Questions:
*1)* Are there any modelling reasons for M2 to be less used in medicine's
RCTs?

It depends a bit on what `y` is: change from baseline or the 'raw'
measure. If it's the raw measure, then (M2) doesn't include a
description of differences at baseline between the groups.

Perhaps most importantly though: (M2) violates the principle of
marginality discussed e.g. in Venables' Exegeses on Linear Models
(https://www.stats.ox.ac.uk/pub/MASS3/Exegeses.pdf)

*2)* Can anyone explain, in layman terms, what is the estimand in M2? I
still struggle to understand what model is really measuring.

Approximately the same thing as M1, except that the "overall" effect of
treatment is assumed to be zero. "Overall" is a bit vague because it
depends on the contrast coding used for time and treatment.

You can see this for yourself. M1 can also be written as:

y ~ time + time:treatment + treatment + (1|ID).

If you force the coefficient on treatment to be zero, then you have M2.

*3)* On a general basis, in a RCT with 3 time points (baseline, 3-month and
4-month), would you tend to gravitate more towards model 1 or 2?

Definitely (1).

PS: When referencing a blog entry, please provide a link to it. :)

Thank you
Jorge

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