Multilevel models deal
with the analysis of data where observations are nested
within groups. Social, behavioral and even economic data
often have a hierarchical structure. A frequently cited
example is in education, where students are grouped in classes.
Classes are grouped in schools, schools in school districts,
etc. We thus have variables describing individuals, but
the individuals may be grouped into larger or higher-order
units. In the case of repeated measurement data, the individual
serves as the group, with multiple measurements nested within
the individual.
Traditionally, fixed
parameter linear regression models are used for the analysis
of such data, and statistical inference is based on the
assumptions of linearity, normality, homoscedasticity, and
independence. Ideally, only the first of these assumptions
should be used. It has been shown by Aitkin & Longford
(1986), that the aggregation over individual observations
may lead to misleading results. Aggregation of, for example,
student characteristics over classes facilitate a class
analysis, but in the process all individual information
is lost. As within-group variation frequently accounts for
most of the total variation in the outcome, this loss of
information can have an adverse effect on the analysis and
lead to distortion of relationships between variables. The
alternative, disaggregation, implies the assignation of
all class, school, and higher-level characteristics to the
individual students. In the process, the assumption of independent
observations no longer holds. Both the aggregation of individual
variables to a higher level of observation and the disaggregation
of higher order variables to an individual level have been
somewhat discredited (Bryk & Raudenbush, 1992). It has
also been pointed out by Holt, Scott and Ewings, (1980),
that serious inferential errors may result from the analysis
of complex survey data if it is assumed that the data have
been obtained under a simple random sampling scheme.
In hierarchical data,
individuals in the same group are also likely to be more
similar than individuals in different groups. Due to this,
the variations in outcome may be due to differences between
groups, and to individual differences within a group. Thus,
variance component models, where disturbance may have both
a group and an individual component, can be of help in analyzing
data of this nature. Within these models, individual components
are independent, but while group components are independent
within groups, they are perfectly correlated within the
groups. Random regression models have been developed to
model continuous data (Bock, 1983), and also dichotomous
repeated measurement data (Gibbons & Bock, 1987) where
certain characteristics of the data preclude the use of
traditional ANOVA models. In random regression models, however,
there is still no possibility of including higher level
variables. In order to accommodate both random coefficients
and higher order variables, multilevel models should be
used.
Multilevel analysis
allows characteristics of different groups to be included
in models of individual behavior. Most analyses of social
sciences data entail the analysis of data with built-in
hierarchies, usually obtained as a sequence of complex sampling
methods. Thus, the scope for application of multilevel models
is very wide. The formulation of such models and estimation
procedures may be seen as an effort to develop a new family
of analytical tools that correspond to the classical experimental
designs. These models are much more flexible in that they
are capable of handling unbalanced data, the analysis of
variance-covariance components and the analysis of both
continuous and discrete response variables. As the characteristics
of individual groups are incorporated into the multilevel
model, the hierarchical structure of the data is taken into
account and correct estimates of standard errors are obtained. The exploration of variation between groups, which may be
of interest in its own right, is facilitated. Valid tests
and confidence intervals can also be constructed and stratification
variables used in the sample design can be incorporated
into the model.
The use of multilevel
models has been hampered in the past by the fact that closed
form mathematical formulas to estimate the variance and
covariance components have only been available for perfectly
balanced designs. Iterative numerical procedures must be
used to obtain efficient estimates for unbalanced designs. Among the procedures suggested are full maximum likelihood
(Goldstein, 1986, and Longford, 1987), and restricted maximum
likelihood estimation as proposed by Mason et. al. (1983)
and Bryk and Raudenbush (1986). Another approach is the
procedure of Bayes estimation (Dempster et.al., 1981). Some
other procedures include the use of Iteratively Reweighted
Generalized Least Squares (Goldstein, 1986), and a Fisher
scoring algorithm (Longford, 1987). Increased interest in
these models, which are known by various names in the literature
(hierarchical linear models, multilevel models, mixed-effects
models, random-effects models, random coefficient regression
models, covariance components models) have led to new developments
in this field in recent years.
An interesting development
was in terms of the type of outcome variables considered:
where previously interest was confined to continuous outcome
variables, statistical theory has been extended and implemented
in software such as HLM to appropriately handle binary outcomes,
ordered categorical outcomes, and multi-category nominal
scale outcomes within the hierarchical framework. Goldstein
(1991) and Longford (1993) both developed software that
allowed the use of several types of discrete outcomes for
two- and three-level models, followed by an improved second-order
approximation by Goldstein in 1995. Hedeker and Gibbons
(1993) and Pinheiro and Bates (1995) contributed accurate
approximations to maximum likelihood using Gauss-Hermite
quadrature that has since been implemented in software such
as MIXOR and SAS PROC NLMIXED. The use of a high-order Laplace
transform as an alternative approximation was introduced
by Raudenbush, Yang & Yosef in 2000 and is implemented in
the HLM program.
In the preceding discussion,
the assumption was that each lower-level unit, for example
an individual student, was nested within an unique higher-level
unit such as a school. In other words, a one to one relationship
was assumed to define the nesting of units within groups.
Excluded from this were hierarchies in which cross-classification
occur, for example where students from multiple neighborhoods
may end up going to multiple schools; a situation where
students are "cross-classified" by neighborhoods
and schools. To address this situation and allow for the
inclusion of predictors for more than one "classification"
variable where the coefficients of a individual level model
describing the association between individual-level variables
and the outcome for groups defined by the "classification"
variables, cross-classified random-effect models were developed
(Raudenbush, 1993; Goldstein, 1995). A two-level cross-classified
random-effect model has been implemented in HLM 6.
Due to increased interest in multivariate outcome models,
such as repeated measurement data, contributions by Jennrich
& Schluchter (1986), and Goldstein (1995) led to the
inclusion of multivariate models in most of the available
hierarchical linear modeling programs. These models allow
the researcher to study cases where the variance at the
lowest level of the hierarchy can assume a variety of forms/structures. The approach also provides the researcher with the opportunity
to fit latent variable models (Raudenbush & Bryk, 2002),
with the first level of the hierarchy representing associations
between fallible, observed data and latent, "true"
data. An application that has received attention in this
regard recently is the analysis of item response models,
where an individuals "ability" or "latent
trait" is based on the probability of a given response
as a function of characteristics of items presented to an
individual.
