time*treatment vs time + time:treatment in RCTs

Hmm, the automatic HTML-to-plaintext conversion didn?t work too well. 
Here?s a plaintext version:

No, you actually get equivalent results for your two models (which is 
really one model, just parametrised differently). The likelihood for 
the two models should be identical, and the P-value for testing 
whether there is an interaction should be identical. Here?s a simple 
simulation for data very similar to the ones you have:

library(lmerTest)

d_temp = expand.grid(group = c("PLA", "FUT"),
???????????????????? time = c("Baseline", "3month", "4month"))
n_varcombo = nrow(d_temp)
d_temp$exp = c(29, 30, 24.5, 28, 24, 27)
n_ind = 30
dat_long = d_temp[rep(1:n_varcombo, each = n_ind), ]
dat_long$ID = rep(1:n_ind, each = n_varcombo)
set.seed(6)
dat_long$ID_effect = rep(rnorm(n_ind, sd = 3), each = n_varcombo)
dat_long$vo2 = dat_long$exp + rnorm(n_varcombo * n_ind, sd = 3)

# Model without interaction
res0 <- lmer(vo2 ~ group + time + (1 | ID), data = dat_long)

# Two (equivalent) models with interaction
res1 <- lmer(vo2 ~? group * time + (1? | ID), data = dat_long)
summary(res1)
#> Fixed effects:
#>???????????????????? Estimate Std. Error????? df t value Pr(>|t|)
#> (Intercept)????????? 28.6258???? 0.5439 24.0000? 52.635 < 2e-16 ***
#> groupFUT????????????? 1.2876???? 0.7691 24.0000?? 1.674 0.1071
#> time3month?????????? -4.8032???? 0.7691 24.0000? -6.245 1.87e-06 ***
#> time4month?????????? -4.8628???? 0.7691 24.0000? -6.322 1.55e-06 ***
#> groupFUT:time3month?? 2.7573???? 1.0877 24.0000?? 2.535 0.0182 *
#> groupFUT:time4month?? 1.5326???? 1.0877 24.0000?? 1.409 0.1717
#>

res2 <- lmer(vo2 ~? time + group:time + (1 | ID), data = dat_long)
summary(res2)
#> Fixed effects:
#>?????????????????????? Estimate Std. Error????? df t value Pr(>|t|)
#> (Intercept)??????????? 28.6258???? 0.5439 24.0000? 52.635 < 2e-16 ***
#> time3month???????????? -4.8032???? 0.7691 24.0000? -6.245 1.87e-06 ***
#> time4month???????????? -4.8628???? 0.7691 24.0000? -6.322 1.55e-06 ***
#> timeBaseline:groupFUT?? 1.2876???? 0.7691 24.0000?? 1.674 0.10710
#> time3month:groupFUT???? 4.0449???? 0.7691 24.0000?? 5.259 2.16e-05 ***
#> time4month:groupFUT???? 2.8202???? 0.7691 24.0000?? 3.667 0.00122 **

# The models have the same log likelihood (and degrees of freedom)
logLik(res1)
#> 'log Lik.' -442.2284 (df=8)
logLik(res2)
#> 'log Lik.' -442.2284 (df=8)

# And the P-values for the interaction are exactly the same
anova(res0, res1)
#> refitting model(s) with ML (instead of REML)
#> Data: dat_long
#> Models:
#> res0: vo2 ~ group + time + (1 | ID)
#> res1: vo2 ~ group * time + (1 | ID)
#>????? npar??? AIC??? BIC? logLik deviance? Chisq Df Pr(>Chisq)
#> res0??? 6 906.62 925.78 -447.31 894.62
#> res1??? 8 903.79 929.33 -443.89?? 887.79 6.8379? 2 0.03275 *
#> ---
#> Signif. codes:? 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

anova(res0, res2)
#> refitting model(s) with ML (instead of REML)
#> Data: dat_long
#> Models:
#> res0: vo2 ~ group + time + (1 | ID)
#> res2: vo2 ~ time + group:time + (1 | ID)
#>????? npar??? AIC??? BIC? logLik deviance? Chisq Df Pr(>Chisq)
#> res0??? 6 906.62 925.78 -447.31 894.62
#> res2??? 8 903.79 929.33 -443.89?? 887.79 6.8379? 2 0.03275 *
#> ---
#> Signif. codes:? 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


As you can see, the two models are actually equivalent, and any 
inference based on the models should give the same results.


Karl Ove Hufthammer


Karl Ove Hufthammer skreiv 29.08.2022 20:56:

No, you actually get equivalent results for your two models (which is
really one model, just parametrised differently). The likelihood for the
two models should be identical, and the P-value for testing whether
there is an interaction should be identical. Here?s a simple simulation
for data very similar to the ones you have:

|library(lmerTest)
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-13>d_temp 

= expand.grid(group = c("PLA", "FUT"),
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-14> 

time = c("Baseline", "3month", "4month"))
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-15>n_varcombo 

= nrow(d_temp)
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-16>d_temp$exp 

= c(29, 30, 24.5, 28, 24, 27)
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-17>n_ind 

= 30
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-18>dat_long 

= d_temp[rep(1:n_varcombo, each = n_ind), ]
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-19>dat_long$ID 

= rep(1:n_ind, each = n_varcombo)
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-20>set.seed(6) 

<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-21>dat_long$ID_effect 

= rep(rnorm(n_ind, sd = 3), each = n_varcombo)
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-22>dat_long$vo2 

= dat_long$exp + rnorm(n_varcombo * n_ind, sd = 3)
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-23> 

# Model without interaction
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-25>res0 

<- lmer(vo2 ~ group + time + (1 | ID), data = dat_long)
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-26> 

# Two (equivalent) models with interaction
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-28>res1 

<- lmer(vo2 ~ group * time + (1 | ID), data = dat_long)
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-29>summary(res1) 

#> Fixed effects:
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-48>#> 

Estimate Std. Error df t value Pr(>|t|)
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-49>#> 

(Intercept) 28.6258 0.5439 24.0000 52.635 < 2e-16 ***
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-50>#> 

groupFUT 1.2876 0.7691 24.0000 1.674 0.1071
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-51>#> 

time3month -4.8032 0.7691 24.0000 -6.245 1.87e-06 ***
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-52>#> 

time4month -4.8628 0.7691 24.0000 -6.322 1.55e-06 ***
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-53>#> 

groupFUT:time3month 2.7573 1.0877 24.0000 2.535 0.0182 *
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-54>#> 

groupFUT:time4month 1.5326 1.0877 24.0000 1.409 0.1717 #>
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-65>res2 

<- lmer(vo2 ~ time + group:time + (1 | ID), data = dat_long)
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-66>summary(res2) 

#> Fixed effects:
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-85>#> 

Estimate Std. Error df t value Pr(>|t|)
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-86>#> 

(Intercept) 28.6258 0.5439 24.0000 52.635 < 2e-16 ***
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-87>#> 

time3month -4.8032 0.7691 24.0000 -6.245 1.87e-06 ***
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-88>#> 

time4month -4.8628 0.7691 24.0000 -6.322 1.55e-06 ***
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-89>#> 

timeBaseline:groupFUT 1.2876 0.7691 24.0000 1.674 0.10710
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-90>#> 

time3month:groupFUT 4.0449 0.7691 24.0000 5.259 2.16e-05 ***
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-91>#> 

time4month:groupFUT 2.8202 0.7691 24.0000 3.667 0.00122 ** # The models
have the same log likelihood (and degrees of freedom)
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-104>logLik(res1) 

<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-105>#> 

'log Lik.' -442.2284 (df=8)
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-106>logLik(res2) 

<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-107>#> 

'log Lik.' -442.2284 (df=8)
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-108> 

# And the P-values for the interaction are exactly the same
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-110>anova(res0, 

res1)
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-111>#> 

refitting model(s) with ML (instead of REML)
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-112>#> 

Data: dat_long
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-113>#> 

Models:
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-114>#> 

res0: vo2 ~ group + time + (1 | ID)
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-115>#> 

res1: vo2 ~ group * time + (1 | ID)
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-116>#> 

npar AIC BIC logLik deviance Chisq Df Pr(>Chisq)
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-117>#> 

res0 6 906.62 925.78 -447.31 894.62 #> res1 8 903.79 929.33 -443.89
887.79 6.8379 2 0.03275 *
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-119>#> 

---
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-120>#> 

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-121>anova(res0, 

res2)
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-122>#> 

refitting model(s) with ML (instead of REML)
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-123>#> 

Data: dat_long
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-124>#> 

Models:
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-125>#> 

res0: vo2 ~ group + time + (1 | ID)
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-126>#> 

res2: vo2 ~ time + group:time + (1 | ID)
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-127>#> 

npar AIC BIC logLik deviance Chisq Df Pr(>Chisq)
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-128>#> 

res0 6 906.62 925.78 -447.31 894.62 #> res2 8 903.79 929.33 -443.89
887.79 6.8379 2 0.03275 *
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-130>#> 

---
<http://localhost:28019/session/grave-cobra_reprex_preview.html?viewer_pane=1&capabilities=1&host=http%3A%2F%2F127.0.0.1%3A55034#cb1-131>#> 

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1|


As you can see, the two models are actually equivalent, and any
inference based on the models should give the same results.


Karl Ove Hufthammer

Jorge Teixeira skreiv 29.08.2022 20:20:

Thank you all for the replies. Still processing them...

Indeed, Wolfgang, I was mainly thinking of time as a factor. 
Although, I
welcome comments as if it was numeric as well.

Your reply is surprising to me, because in my data I get different 
results.
The ES and p-values are very different regarding the interactions at 
3 and
4-months, which are the relevant data to me. My df has 3 time points.

?? res1 <- lmer(vo2 ~? group*time + ( 1? | ID? ), data = dat_long )

???? res2 <- lmer(vo2 ~? time + group:time + ( 1 | ID? ), data =? 
dat_long )


*res1:*
Fixed effects:
???????????????????????? Estimate Std. Error????? df t value Pr(>|t|)
(Intercept)???????????? 29.0705???? 0.9998 61.4510? 29.076 < 2e-16 ***
groupFUT????????????? 1.0395???? 1.4140 61.4510?? 0.735 0.465036
time3month????????????? -4.4917???? 1.0918 64.1740? -4.114 0.000113 ***
time4month????????????? -5.0305???? 1.0622 63.8295? -4.736 1.26e-05 ***
*groupFUT:time3month *? 2.5467???? 1.4396 61.8093?? 1.769 0.081822 .
*groupFUT:time4month*?? 1.7643???? 1.4424 61.8409?? 1.223 0.225909


*res2:*
Fixed effects:
???????????????????????? Estimate Std. Error????? df t value Pr(>|t|)
(Intercept)???????????? 29.0705???? 0.9998 61.4510? 29.076 < 2e-16 ***
time3month????????????? -4.4917???? 1.0918 64.1740? -4.114 0.000113 ***
time4month????????????? -5.0305???? 1.0622 63.8295? -4.736 1.26e-05 ***
time0month:groupFUT?? 1.0395???? 1.4140 61.4510?? 0.735 0.465036
*time3month:groupFUT*?? 3.5862???? 1.5402 73.4895?? 2.328 0.022643 *
*time4month:groupFUT*? 2.8038???? 1.5428 73.7427?? 1.817 0.073226 .



Viechtbauer, Wolfgang 
(NP)<wolfgang.viechtbauer at maastrichtuniversity.nl>
escreveu no dia segunda, 29/08/2022 ?(s) 16:11:

I strongly suspect that 'time' is treated as a factor in the examples
Jorge is referring to. In this case, the two formulations are just
different parameterizations of the same model. We can use the 
'Orthodont'
data to illustrate this. Think of 'age' as the time variable (as a
four-level factor) and 'Sex' as the treatment variable (as a two-level
factor). In fact, I will throw in a third parameterization, which I 
think
is even more intuitive.

library(lme4)

data("Orthodont", package="nlme")

Orthodont$age <- factor(Orthodont$age)

res1 <- lmer(distance ~ age*Sex + (1 | Subject), data=Orthodont)
summary(res1)

res2 <- lmer(distance ~ age + age:Sex + (1 | Subject), data=Orthodont)
summary(res2)

res3 <- lmer(distance ~ 0 + age + age:Sex + (1 | Subject), 
data=Orthodont)
summary(res3)

logLik(res1)
logLik(res2)
logLik(res3)

The fit is identical.

In 'res3', we get the estimated intercepts (means) of the reference 
group
(in this case for 'Male') at all 4 timepoints and the age:Sex 
coefficients
are the difference between the Female and Male groups at those 4 
timepoints.

Since these are just all different parameterizations of the same 
model,
there is no reasons for preferring one over the other.

One has to be careful though when using anova() on those models, 
esp. with
respect to the age:Sex test. In anova(res1), the test examines if the
difference between males and females is the same at all 4 
timepoints, while
in anova(res2) and anova(res3) the test examines if the difference 
is 0 at
all 4 timepoints. However, one could get either test out of all three
parameterizations, by forming appropriate contrasts. So again, no 
reason to
prefer one over the other (except maybe convenience depending on 
what one
would like to test).

Best,
Wolfgang

-----Original Message-----
From: R-sig-mixed-models 
[mailto:r-sig-mixed-models-bounces at r-project.org]

On

Behalf Of Douglas Bates
Sent: Monday, 29 August, 2022 16:14
To: Phillip Alday
Cc: R-mixed models mailing list; Jorge Teixeira
Subject: Re: [R-sig-ME] time*treatment vs time + time:treatment in 
RCTs

M2 is an appropriate model if time corresponds to "time on 
treatment" or

in

general if the covariate over which the measurements are repeated 
has a
scale where 0 is meaningful.? I think of it as the "zero dose" model
because zero dose of treatment 1 is the same as zero dose of 
treatment 2

is

the same as zero dose of the placebo.? Similarly zero time on 
treatment is
the same for any of the treatments or the placebo.

In those cases we would not expect a main effect for treatment 
because

that

corresponds to systematic differences before the study begins (or 
at zero
dose), but we would expect an interaction of time (or dose) with

treatment.

On Mon, Aug 29, 2022 at 8:28 AM Phillip Alday<me at phillipalday.com>

wrote:

On 8/29/22 05:53, Jorge Teixeira wrote:

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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