Consider a within-subject and within-item factorial design where the experimental treatment variable has two levels (conditions). Let
m1 be the maximal model and
m2 the no-random-correlations model.
m1: y ~ condition + (condition|subject) + (condition|item) m2: y ~ condition + (1|subject) + (0 + condition|subject) + (1|item) + (0 + condition|item)
Dale Barr states the following for this situation:
Edit (4/20/2018): As Jake Westfall pointed out, the following statements seem to refer to the datasets that are shown in Fig. 1 and 2 on this website only. However, the keynote remains the same.
In a deviation-coding representation (condition: -0.5 vs. 0.5)
m2 allows distributions, where subject's random intercepts are uncorrelated with subject's random slopes. Only a maximal model
m1 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?