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Not really interested in correlations or random effects variances

Hi all,

let's hypothesize that I have a full-crossed experiment, that actually it
may be analysed with a standard ANOVA, but because I want to increase the
power of the analyses, and add as random slope the time passed
trail-by-trail since the beginning of the experiment, to take into account
for fatigue effects, I want to analyse these data with a multilevel linear
model.

Let's say that in this experiment I have two within-subjects factors that
are interacting. If I am not really interested neither in the variances
between the random slopes nor in the variances of the random effects, may I
totally avoid random slopes, using in their place random intercepts?

What I mean is, instead of this full slopes model: y~C1*C2+(C2*C2|ID)
can I use this "full intercept model":
y~C1*C2+(1|ID:C1:C2)+(1|ID:C1)+(1|ID:C2)+(1|ID)?

Is there any specific reason why it should not be recommended the second
model?

Thanks a lot for your help, best regards,

Michele Scandola