🚀 go-pugleaf

Dear Colleagues, I have a question about between group
and within group correlation. Having two vars x, y, and
a third grouping var, g, that stands for 20 groups with
equal group size, is the following true: (1) Between-group
correlation of x and y is the correlation between the
group means of x and group means of y. (2) Within-group
correlation between x and y is the correlation between
deviations of x (from group mean) and deviations of y.
(3) Do (1), (2) still hold if group size is not equal?
If not, how to compute between and within correlations
when group size is unequal?
   An extension of the above would be multiple regression
where there are more than one predictor variable. If there
are y, x1, x2, and 20 equal sized groups, can I
use the group means and group deviations of the
y, x1, x2 to conduct two sets of regression and
claim that the one using the group means is the
between-group regressions of y on x1 and x2, and
the set using group deviation scores represent the
within-group regression of the y on x1 and x2.
    Thank you very so much for any comments.
    ---chang---