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Chapter 12 of Hierarchical
Linear Models discusses two applications of a cross-classified
random effects model . The first is from a study of neighborhood
and school effects on educational attainment in Scotland
(Garner & Raudenbush, 1991).
As there were students who resided in a specific neighborhood
and enrolled in schools located in a different neighborhood,
the data collected did not have a strictly hierarchical data
structure. Here the attainment data of students were cross-classified
by neighborhoods and schools. The second case is an assessment
of the effects of classrooms on children's cognitive growth
during the primary years. This latter example is from the
Immersion Study. As there were changes in classroom memberships
among the students during the course of the investigation,
the achievement data were cross-classified by students and
classrooms.
A general random cross-classified
model consists of two sub-models: level-1 or within-cell and
level-2 or between-cell models. The cells refer to the cross-classifications
by the two higher-level units. For example, if the research
problem consists of data on students cross-classified by schools
and neighborhoods, the level-1 or within-cell model will represent
the relationships among the student-level variables, the level-2
or between-cell model will capture the influence of school-
and neighborhood-level factors. Formally, there are
level-1 units (e.g., students)
nested within cells cross-classified by = 1,...,
J first level-2 units (e.g.,
neighborhoods), designated as rows, and k
= 1,..., K second level-2 units (e.g., schools),
designated as columns.
Level-1 or "within-cell" model
We represent in the level-1
or within-cell model the outcome for case i
in individual cells cross-classified by level-2 units
j and k.
where
( 
) are level-1 coefficients,
is the level-1 predictor p for
case i in cell
jk,
is the level-1 or within-cell random effect,
and
is the variance of
, that is the level-1 or within-cell
variance. Here we assume that the random term
.
Level-2
or "between-cell" model
Each of the
coefficients in the level-1 or within-cell model
becomes an outcome variable in the level-2 or between-cell
model :
where
is the model intercept, the expected
value of
when all explanatory variables
are set to zero;
are the fixed effects of column-specific
predictors
 ;
are the random effects associated
with column-specific predictors
 .
They vary randomly over rows j = 1,...,
J;
are the fixed effects of row-specific
predictors
 ;
are the random effects associated
with row-specific predictors 
. They vary randomly over columns k = 1,.,
K;
are the fixed effects of cell-specific
predictors 
, which are the interaction terms created as the products
of
and 
, s = 1, .,
and
; and
 ,
and
are residual row, column, and cell-specific random
effects, respectively, on
, after taking into account ,
,
and  .
We assume that
 ,
 ,
and that the effects are independent of each other.
However, the vector containing
elements  is
assumed multivariate normal with a mean zero and a full covariance
matrix 
. Similarly the vector with elements 
is assumed multivariate
normal with mean vector zero and full covariance matrix
 .
Parameter
estimation
Three kinds of parameter estimates
are available in HCM2:
empirical Bayes estimates
of randomly varying effects of level-1 or within-cell
and row- and column-specific coefficients; maximum-likelihood
estimates
of the level-2 or row-, column- and cell-specific
coefficients; and maximum likelihood estimate of the level-1
or within-cell and level-2 or between-cell variance-covariance
components. The estimation procedure uses a full maximum likelihood
approach (Kang, 1992; Raudenbush,
1993).
Hypothesis
testing
As in the case of HLM2,
HCM2 routinely
prints standard errors and t-tests for each
of the level-2 coefficients (the "fixed effects") as well
as a chi-square test of homogeneity for each random effect.
In addition, optional "multivariate hypothesis tests" are
available in HCM2.
Multivariate tests for the level-2 coefficients enable both
omnibus tests and specific comparisons of the parameter estimates.
These, along with multivariate tests regarding alternative
variance-covariance structures at level 2 proceed just as
described for HLM2 models.
 
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