[R-sig-ME] Why does treatment coding always result in a correlation between random slope and intercept?

[Maarten Jung]

Dear list,

Consider a within-subject and within-item factorial design where the
experimental treatment variable has two levels (conditions). Let m1 be the
maximal model, m2 the no-within-unit-intercepts model and m3 the
no-random-correlation model:

m1: y ~ condition + (condition|subject) + (condition|item)
m2: y ~ condition + (0 + condition|subject) + (0 + condition|item)
m3: y ~ condition + (1|subject) + (0 + condition|subject) + (1|item) + (0 +
condition|item)

Dale Barr states the following for this situation [1]:
In a deviation-coding representation (condition: -0.5 vs. 0.5) both models,
m1 and m2, allow distributions, where subject's random intercepts are
uncorrelated with subject's random slopes. Only a maximal model allows
distributions, where the two are correlated.

In the treatment-coding representation (condition: 0 vs. 1) these
distributions, where subject's random intercepts are uncorrelated with
subject's random slopes, cannot be fitted using the no-random-correlations
model, *since in each case there is a correlation between random slope and
intercept in the treatment-coding representation.*

Why does treatment coding always result in a correlation between random
slope and intercept?

Please note that I asked the question on Stack Exchange [2] about 10 days
ago.

Regards,
Maarten

[1] http://talklab.psy.gla.ac.uk/simgen/rsonly.html
[2]
https://stats.stackexchange.com/questions/337158/why-does-treatment-coding-always-result-in-a-correlation-between-random-slope-an

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