Trends for many units

Wouldn't you expect two grades in one school to be more similar than two
grades in different schools?  And would not slopes for different grades
in one school be more similar than across schools?  Those translate into
dependence.

For an lm-type model you can circumvent this by treating all the
school-grade combinations as fixed effects.  Thus

(r)lm(Pct.Excl ~ Unit*Grade*Year)

(and I would centre Year on 1998) fits 3520 lines with a common assumed
error variance. That's a lot of parameters to fit in one go, and you will
probably find lmList in package nlme helpful.  But my suggestion for a
model is

i Unit
j Grade
t Year

y_{ijt} = mu + beta_j + gamma * t + eta_i + zeta_{ij} + epsilon_{ijt}

eta, zeta, epsilon iid with common variances in each group.

that is fixed effects for Grade, random effects for Unit and Unit | Grade.
You may or may not need additional random effects

lambda_i * Year + kappa_{ij} * Year

As set up here, independence of all the rvs is plausible, but lme does not
require it.  The predict.lme will give you BLUP lines for each Unit-Grade
combination, and they will not be the fitted values in the fixed-effects
model.  Most social statisticians I know (and we have have some local
stars) think that the second is more valuable, and routinely use it.

Snijders, T.A.B. and Bosker, R.J. (1999) Multilevel Analysis. Sage.

have an example of IQ tests adminstered to students in classes in schools
done in exactly this way.  (And that was the ref the experts recommended
for social applications.)

Sent on 08-Jan-2001 at 09:49:27 with xfmail