variable-and person-centered approaches and multilevel models

Following up on an unanswered (I think) old post:

I had not heard the term "person-centred approach" before, but after
searching a bit, it seems to refer to focus on clustering under a
specific grouping term instead of a specific co-variate. I think this
distinction is somewhat blurred in mixed-effects models. While the fixed
effects are more or less what is called "variable centered" (indeed, the
canonical examples in the psych literature are things like linear
regression and ANOVA), the random effects allow for variation to follow
the grouping variables, which is more like (although maybe not exactly
the same) as the "person-centred" approach. The covariance matrix / the
correlations between random effects is more difficult to comprehend, but
is actually the part that perhaps best captures "person centering", as
it shows the relationship between the by-person regression slopes.

As a side note, I'm not sure it makes sense to include the personal mean
as a separate covariate, as depending on the exact structure of your
data, it may already be captured in random intercept by participant.

Phillip