[R-sig-ME] Equivalence of (0 + factor|group) and (1|group) + (1|group:factor) random effect specifications in case of compound symmetry

[Maarten Jung]

Hello everyone,

I have a question regarding the equivalence of the following models:

m1 <- lmer(y ~ factor + (0 + factor|group))
m2 <- lmer(y ~ factor + (1|group) + (1|group:factor))

Douglas Bates states (slide 91 in this presentation [1])  that these models
are equivalent in case of compound symmetry.

1. I realized that I don't really understand the random slope by factor
model (m1) and espacially why it reduces to m2 given compound symmetry.
Also, why is there no random intercept in m1?
Can anyone explain the difference between the models and how m1 reduces to
m2 in an intuitive way.

2. If m1 is a special case of m2 – this could be an interesting option for
model reduction but I’ve never seen something like m2 in papers. Instead
they suggest dropping the random slope and thus the interaction completely
(e.g. Matuschek et al. 2017 [3]).
What do you think about it?

Please note that I asked the question on Stack Exchange [2] but some
consider it as off-topic. So, I hope one of you can help me out.


Best regards,
Maarten

[1] http://www.stat.wisc.edu/~bates/UseR2008/WorkshopD.pdf
[2] https://stats.stackexchange.com/q/304374/136579
[3] https://doi.org/10.1016/j.jml.2017.01.001

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