According to the references below, except Begg et al, this is interpreted
as the effect of xbar_i when xbar_i changes between clusters. However, Begg
et al argue that the estimate of b1 is not estimating the direct effect of
the cluster-level mean.
The equation
Y_ij = a + (Xbar_i) b11 + v_i + e_ij (2)
estimates b11 as the same value as b1 in equation (1)
Therefore, b1 in equation 1 does not seem to be adjusting for anything,
especially the change in Xij that can occur with the change in Xbar_i .
The coefficient b12 seems to give the true effect of a change in cluster
mean, in that
b12 = b1 - b2
- xtreg y x_ij,i(cluster) be -
gives a coefficient that is equal to b1 and b11.
Any comments or clarifications would be much appreciated.
The above has cited
1. Stata reference manual xtreg,
2. http://www.stata.com/support/faqs/stat/xt.html
3. Neuhaus JM and Kalbfleisch DD (1998) Between- and within-cluster
covariate effects in the analysis of clustered data. Biometrics 54:638-645
4. Begg MD and Parides MK (2003) Separation of individual-level and
cluster-level covariate effects in regression analysis of correlated data
