Multilevel models are statistical models of parameters that vary at more than one level.
These models can be seen as generalizations of linear models, although they can also extend
to non-linear models. These models became much more popular after sufficient computing
power and software became available. Multilevel models are particularly appropriate
for research designs where data for participants are organized at more than one level. The
units of analysis are usually individuals who are nested within contextual/aggregate
units. While the lowest level of data in multilevel models is usually an individual, repeated
measurements of individuals may also be examined. As such, multilevel models provide an alternative
type of analysis for univariate or multivariate analysis of repeated measures. Individual
differences in growth curves may be examined. Furthermore, multilevel models can be used
as an alternative to ANCOVA, where scores on the dependent variable are adjusted for
covariates before testing treatment differences. Multilevel models are able to analyze these
experiments without the assumptions of homogeneity-of-regression slopes that is required by ANCOVA.
Multilevel models can be used on data with many levels, although 2-level models are the
most common. The dependent variable must be examined at the lowest level of analysis.
Level 1 regression equation Yij = β0j + β1j(Xij) + eij
Yij refers to the score on the dependent variable for an individual observation at Level 1.
Xij refers to the Level 1 predictor. β0j refers to the intercept of the dependent
variable in group j. β1j refers to the slope for the relationship
in group j between the Level 1 predictor and the dependent variable.
eij refers to the random errors of prediction for the Level 1 equation. At Level 1, both
the intercepts and slopes in the groups can be either fixed, non-randomly varying, or
randomly varying. Level 2 regression equation
The dependent variables are the intercepts and the slopes for the independent variables
at Level 1 in the groups of Level 2. β0j = γ00 + γ01Wj +u0j
β1j = γ10 + u1j γ00 refers to the overall intercept. This
is the grand mean of the scores on the dependent variable across all the groups when all the
predictors are equal to 0. Wj refers to the Level 2 predictor.
γ01 refers to the overall regression coefficient, or the slope, between the dependent variable
and the Level 2 predictor. u0j refers to the random error component for
the deviation of the intercept of a group from the overall intercept.
γ10 refers to the overall regression coefficient, or the slope, between the dependent variable
and the Level 1 predictor. u1j refers to the error component for the
slope. Types of models
Before conducting a multilevel model analysis, a researcher must decide on several aspects,
including which predictors are to be included in the analysis, if any. Second, the researcher
must decide whether parameter values will be fixed or random. Fixed parameters are composed
of a constant over all the groups, whereas a random parameter has a different value for
each of the groups. Additionally, the researcher must decide whether to employ a maximum likelihood
estimation or a restricted maximum likelihood estimation type.
Random intercepts model A random intercepts model is a model in which
intercepts are allowed to vary, and therefore, the scores on the dependent variable for each
individual observation are predicted by the intercept that varies across groups. This
model assumes that slopes are fixed. In addition, this model provides information about intraclass
correlations, which are helpful in determining whether multilevel models are required in
the first place. Random slopes model
A random slopes model is a model in which slopes are allowed to vary, and therefore,
the slopes are different across groups. This model assumes that intercepts are fixed.
Random intercepts and slopes model A model that includes both random intercepts
and random slopes is likely the most realistic type of model, although it is also the most
complex. In this model, both intercepts and slopes are allowed to vary across groups,
meaning that they are different in different contexts.
Developing a multilevel model In order to conduct a multilevel model analysis,
one would start with fixed coefficients. One aspect would be allowed to vary at a time,
and compared with the previous model in order to assess better model fit. Thus, there are
three different questions that a researcher would ask in assessing a model. First, is
it a good model? Second, is a more complex model better? Third, what contribution do
individual predictors make to the model? In order to assess models, different model fit
statistics would be examined. One such statistic is the chi-square likelihood-ratio test, which
assesses the difference between models. The chi-square test can be employed for model
building in general, for examining what happens when effects in a model are allowed to vary,
and when testing a dummy-coded categorical variable as a single effect. However, this
test can only be used when models are nested. When testing non-nested models, comparisons
between models can be made using Akaike's Information Criterion, Hurvich and Tsai's
Criterion, Bozdogan's Criterion or Schwarz's Bayesian Criterion.
Assumptions Multilevel models have the same assumptions
as other major general linear models, but some of the assumptions are modified for the
hierarchical nature of the design. Linearity
The assumption of linearity states that there is a rectilinear relationship between variables.
Normality The assumption of normality states that the
error terms at every level of the model are normally distributed.
Homoscedasticity The assumption of homoscedasticity, also known
as homogeneity of variance, assumes equality of population variances.
Independence of observations Independence is an assumption of general linear
models, which states that cases are random samples from the population and that scores
on the dependent variable are independent of each other.
Modification of assumptions The assumptions of linearity and normality
do not pose any problems for multilevel models, and are thus retained. However, the assumptions
of homoscedasticity and independence of observations must be adapted in order to proceed with this
type of analysis. The latter two assumptions pose several problems. First, units of observations
in the same group are more similar than those in different groups. Second, while groups
are independent of each other, observations within a group share values on variables,
and thus, they are not independent. However, one advantage of employing multilevel models
over other types of analyses is that independence is not required, because it is violated at
each level of the analysis. Additionally, multilevel models are designed to deal with
this intraclass correlation, which assumes that data from the same context are more similar
than data from different contexts. Multilevel modelling analysis serves to measure the variability
within contexts. Statistical tests
The type of statistical tests that are employed in multilevel models depend on whether one
is examining fixed effects or variance components. When examining fixed effects, the tests are
compared with the standard error of the fixed effect, which results in a Z-test. A t-test
can also be computed. When computing a t-test, it is important to keep in mind the degrees
of freedom, which will depend on the level of the predictor. For a level 1 predictor,
the degrees of freedom are based on the number of level 1 predictors, the number of groups
and the number of individual observations. For a level 2 predictor, the degrees of freedom
are based on the number of level 2 predictors and the number of groups.
Statistical power Statistical power for multilevel models differs
depending on whether it is level 1 or level 2 effects that are being examined. Power for
level 1 effects is dependent upon the number of individual observations, whereas the power
for level 2 effects is dependent upon the number of groups. To conduct research with
sufficient power, large sample sizes are required in multilevel models. However, the number
of individual observations in groups is not as important as the number of groups in a
study. In order to detect cross-level interactions, given that the group sizes are not too small,
recommendations have been made that at least 20 groups are needed. The issue of statistical
power in multilevel models is complicated by the fact that power varies as a function
of effect size and intraclass correlations, it differs for fixed effects versus random
effects, and it changes depending on the number of groups and the number of individual observations
per group. Applications of multilevel models
Level The concept of level is the keystone of this
approach. In an educational research example, the levels might be:
pupil class
school district
The researcher must establish for each variable the level at which it was measured. In this
example "test score" might be measured at pupil level, "teacher experience" at class
level, "school funding" at school level, and "urban" at district level.
Example As a simple example, consider a basic linear
regression model that predicts income as a function of age, class, gender and race. It
might then be observed that income levels also vary depending on the city and state
of residence. A simple way to incorporate this into the regression model would be to
add an additional independent categorical variable to account for the location. This
would have the effect of shifting the mean income up or down — but it would still assume,
for example, that the effect of race and gender on income is the same everywhere. In reality,
this is unlikely to be the case — different local laws, different retirement policies,
differences in level of racial prejudice, etc. are likely to cause all of the predictors
to have different sorts of effects in different locales.
In other words, a simple linear regression model might, for example, predict that a given
randomly sampled person in Seattle would have an average yearly income $10,000 higher than
a similar person in Mobile, Alabama. However, it would also predict, for example, that a
white person might have an average income $7,000 above a black person, and a 65-year-old
might have an income $3,000 below a 45-year-old, in both cases regardless of location. A multilevel
model, however, would allow for different regression coefficients for each predictor
in each location. Essentially, it would assume that people in a given location have correlated
incomes generated by a single set of regression coefficients, whereas people in another location
have incomes generated by a different set of coefficients. Meanwhile, the coefficients
themselves are assumed to be correlated and generated from a single set of hyperparameters.
Additional levels are possible: For example, people might be grouped by cities, and the
city-level regression coefficients grouped by state, and the state-level coefficients
generated from a single hyper-hyperparameter. Multilevel models are a subclass of hierarchical
Bayesian models, which are general models with multiple levels of random variables and
arbitrary relationships among the different variables. Multilevel analysis has been extended
to include multilevel structural equation modeling, multilevel latent class modeling,
and other more general models. Uses of multilevel models
Multilevel models have been used in education research or geographical research, to estimate
separately the variance between pupils within the same school, and the variance between
schools. In psychological applications, the multiple levels are items in an instrument,
individuals, and families. In sociological applications, multilevel models are used to
examine individuals embedded within regions or countries. In organizational psychology
research, data from individuals must often be nested within teams or other functional
units. Different covariables may be relevant on different
levels. They can be used for longitudinal studies, as with growth studies, to separate
changes within one individual and differences between individuals.
Cross-level interactions may also be of substantive interest; for example, when a slope is allowed
to vary randomly, a level-2 predictor may be included in the slope formula for the level-1
covariate. For example, one may estimate the interaction of race and neighborhood so that
an estimate of the interaction between an individual's characteristics and the context.
Applications to longitudinal data
Alternative ways of analyzing hierarchical data
There are several alternative ways of analyzing hierarchical data, although most of them have
some problems. First, traditional statistical techniques can be used. One could disaggregate
higher-order variables to the individual level, and thus conduct an analysis on this individual
level. The problem with this approach is that it would violate the assumption of independence,
and thus could bias our results. This is known as atomistic fallacy. Another way to analyze
the data using traditional statistical approaches is to aggregate individual level variables
to higher-order variables and then to conduct an analysis on this higher level. The problem
with this approach is that it discards all within-group information. As much as 80-90%
of the variance could be wasted, and the relationship between aggregated variables is inflated,
and thus distorted. This is known as ecological fallacy, and statistically, this type of analysis
results in decreased power in addition to the loss of information.
Another way to analyze hierarchical data would be through a random-coefficients model. This
model assumes that each group has a different regression model - with its own intercept
and slope. Because groups are sampled, the model assumes that the intercepts and slopes
are also randomly sampled from a population of group intercepts and slopes. This allows
for an analysis in which one can assume that slopes are fixed but intercepts are allowed
to vary. However this presents a problem, as individual components are independent but
group components are independent between groups, but dependent within groups. This also allows
for an analysis in which the slopes are random; however, the correlations of the error terms
are dependent on the values of the individual-level variables. Thus, the problem with using a
random-coefficients model in order to analyze hierarchical data is that is still not possible
to incorporate higher order variables. Error terms
Multilevel models have two error terms, which are also known as disturbances. The individual
components are all independent, but there are also group components, which are independent
between groups but correlated within groups. However, variance components can differ, as
some groups are more homogeneous than others. See also
Hyperparameter Mixed-design analysis of variance
Restricted randomization References
Books Goldstein, H.. Multilevel Statistical Models.
4th ed. London: Wiley. Hox, J. J.. Multilevel analysis: Techniques
and applications. 2nd ed. Hogrefe and Huber. Raudenbush, S. W. and Bryk, A. S.. Hierarchical
linear models: Applications and data analysis methods. 2nd ed. Thousand Oaks, CA: Sage.
Snijders, T. A. B. and Bosker, R. J.. Multilevel Analysis: an Introduction to Basic and Advanced
Multilevel Modeling. 2nd ed. London: Sage. External links
Software Specialised multilevel statistical software
aML HLM
Latent GOLD MLwiN
Mplus Don Hedeker: Free Multilevel Software
Standard statistical software R
lme4 package nlme package
SAS using the MIXED procedure SPSS using the Advanced Statistics product
Stata gllamm user written command
runmlwin user written command xtmixed command
xtmelogit command xtmepoisson command
WinBUGS JAGS
STAN Other resources
Centre for Multilevel Modelling Tom Snijders' Multilevel Analysis Page