Type
and nature of variables
Variables can be defined at any level of the hierarchy
in the multi-level analysis. At each level in the hierarchy, we may have
several types of variables. In multilevel data, there is not one ‘proper’ level
at which the data should be analysed. Rather, all levels present in the data
are important in their own way. This becomes clear when we investigate cross-level
hypotheses, or multilevel problems [4].
The multilevel models assume hierarchical data, in
which the dependent variable is measured at the lowest level and the
independent variables are measured at all available levels [5]. Structural data
are hierarchical or nested data are likely to be correlated [6]. Clustered data
(nested data) are measurements are taken on subjects that share a common
category that leads to correlation. Clustered longitudinal data outcome is
measured repeatedly for the same subject over time, and subjects are clustered
within some unit. In a clustered data, each “level” represents a factor that
can be thought of as a random sample from a larger population, otherwise
classification [7].
The fixed effects are regression coefficients; in
which values of interest are all represented in the dataset while random
effects are variations of regression coefficients between levels (variance
components) ever-existing natural heterogeneity among subjects. Cross-level
interaction effects are fixed effects of the joint effect of variables at level
one in conjunction with variables at level two. Mixed-effects models combine
both factor(s) [8].
Types
of multilevel models
Random
intercepts model: In this model, the
intercepts are allowed to vary, so the dependent variable for each individual
observation will be predicted by the intercept that varies across groups [9].
The model also assumes that slopes are fixed (the same across different
contexts). In addition, this model provides information about intra class
correlations, which are helpful in determining whether multilevel models are
required in the first place [10].
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 (the same across different
contexts) [10].
Random
intercepts and slopes model: In this model, both
intercepts and slopes are allowed to vary across groups, meaning that they are
different in different contexts [10].
Statistical
model analysis
The multilevel model can be conceptualized as a
two-stage equation.
Step 1: Estimates separate regression equations within
units. Relationships within units (intercept and slopes).
Step 2: Uses “summaries” of between unit relationships
as group variables.
Mathematically
Level 1: Regression lines estimated separately for
each unit or within unit relationships for each unit. Analyse the model with no
explanatory variables. The intercept only model is given by the model of
equations (1).
Y = ß 0j + ß 1j X ij
+ ? ij (1)
Where: ß 0j and ß 1j =
regression coefficients
Level 2: Variance in intercepts and slopes are
predicted by between unit variables. Level 2 models variance in level-1
parameters (intercepts and slopes) with between unit variables. Analyse the
model with all lower level explanatory variables.
ß 0j = ?00 + ?01
(Group j) + U0j (2)
ß 1j = ?10 + ?11
(Group j) + U1j (3)
Adding cross level interaction between explanatory
group level variables and those individual level explanatory variables leads a
full model formulated in equation (4). So, substitute equations (2) and (3) in
equation [1]:
Y ij = ?00 + ?01g
j + ?10Ii j + ?11g j Iij + U0j
+ U1j Iij + ?ij (4)
Where:
?00 is variances of group intercepts (over
all intercept)
?10 is regression coefficient (slop of
individual variable)
?11 is slop of interaction (how large Z
affect X)
?01 is variances of group slopes (group
regression coefficient)
U0j and U 1j are errors in the group-level
equations;
U0j is group level deviation of each
intercept from all intercept (?00?00)
The full multilevel regression model assumes that
their hierarchal data set, with one single dependent variable at all existing
levels. For example, Joop Hox’s multilevel analysis [4] assume data from J
classes, with a different number of pupils nj in each class. On the pupil
level, the outcome variable ‘popularity’ (Y), measured by a self-rating scale
that ranges from 0 (very unpopular) to 10 (very popular). They have two
explanatory variables on the pupil level: pupil gender (X1: 0 = boy, 1 = girl)
and pupil extraversion (X2, measured on a self-rating scale ranging from 1 to
10), and one class level explanatory variable teacher experience (Z: in years,
ranging from 2 to 25). Using variable labels the equation is:
Popularityij = ?0j + ?1j
genderij + ?2j extraversionij + eij (5)
In this regression equation, ?0j is the intercept; ?1j
is the regression coefficient (regression slope) for the dichotomous explanatory
variable gender, ?2j is the regression coefficient (slope) for the continuous
explanatory variable (extraversion), and the usual residual error term is eij.
The subscript j is for the classes (j = 1 . . . J) and the subscript i is for
individual pupils (i = 1 . . . nj).
The next step in the hierarchical regression model is
to explain the variation of the regression coefficients ?j introducing
explanatory variables at the class level:
?0j = ?00 + ?01Zj
+ u0j (6)
?1j = ?10 + ?11Zj
+ u1j
(7)
?2j = ?20 + ?21Zj
+ u2j (8)
The model with two pupil-level and one class-level
explanatory variable can be written as a single complex regression equation by
substituting equations (6), (7) and (8) into equation (5) gives:
Popularityij = ?00 +
?10 genderij + ?20 extraversionij +
?01 experiencej + ?11 genderij ×
experiencej + ?21 extraversionij × experiencej
+ u1j genderij + u2j extraversionij
+ u0j + eij (9)
Sample
size determination
It is generally accepted that increasing sample sizes
at all levels estimates and standard errors improve. To be statistically safe,
as “rule of thumb”, researchers should use ‘30/30’ rule, a sample of at least
30 groups with at least 30 individuals per group. On the other hand, the
numbers should be modified as if there is strong interest in cross-level
interactions, the number of groups should be larger, (a 50/20 rule-50 groups
with 20 individuals/ group); if there is stronger interest in the random part,
or in the variance and/ or covariance components, the number of involving groups
should be larger, leading to a 100/10 rule (100 groups with 10
individuals/group). One should take into account the costs attached to data
collection, so if the number of groups is increased, than the number of
individuals per group might decreases [5].
Assumptions
of multilevel modeling
Multilevel models have the same assumptions as other
major general linear models (e.g., Anova, regression), but some of the
assumptions are modified for the hierarchical nature of the design (i.e.,
nested data). Accordingly, checking and improving the specification of a
multilevel model in many cases can be carried out while staying within the
assumption of the multilevel model.
Linearity
The assumption of linearity states that there is a
rectilinear (straight-line, as opposed to non-linear or U-shaped) relationship
between variables [11]. However, the model can be extended to nonlinear
relationships [12]. A regression analysis expected to fit the best rectilinear
line that explains the most data given your set of parameters. Therefore, the
base models rely on the assumption that the data follow a straight line (though
the models can be expanded to handle curvilinear data). Graphically, by
plotting the model residuals (the difference between the observed value and the
model-estimated value) versus the predictor, linearity can be tested. If a
pattern emerges, a higher-order term may need to be included or you may need to
mathematically transform a predictor/response [13].
Normality
The assumption of normality states that the error
terms at every level of the model are normally distributed [11]. QQ plots which
are obtained in standard regression modeling in R can provide an estimation of
where the standardized residuals lie with respect to normal quintiles. Strong
deviation from the provided line indicates that the residuals themselves are
not normally distributed [13].
Homoscedasticity
The assumption of homoscedasticity or homogeneity of
variance assumes equality of population variances. However, different
variance-correlation matrix can be specified and the heterogeneity of variance
can itself be modelled [11]. In R, we extract the residuals from the model,
place them in our original table, take their absolute value, and then square
them (for a more robust analysis with respect to issues of normality. Finally, take
a look at the ANOVA of the between-subjects residuals [13].
Independence
of observations
Independence of observation is an assumption which
states that cases are random samples from the population and that scores on the
dependent variable are independent of each other [11].
Model
fitness
One way of assessing model fit is the chi-square
likelihood-ratio test, which assesses the difference between models. The
likelihood-ratio test can only be used when models are nested. It can be used
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,
when testing non-nested models, comparisons between models can be made using
the Akaike information criterion or the Bayesian information criterion, among
others [10,14].
Statistical
tests and power
The types of statistical tests in multilevel models
depend on whether one is examining fixed effects or variance components. When
examining fixed effects, the tests should be 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 consider degrees of freedom. For a
level one predictor, the degrees of freedom are based on the number of level
one predictor, the number of groups and the number of individual observations.
For a level two predictor, the degrees of freedom are based on the number of
level two predictors and the number of groups [10].
Statistical power for multilevel models differs
depending on whether it is level one or level two effects that are being
examined. Power for level one effect is dependent upon the number of individual
observations, whereas the power for level two effects is dependent upon the
number of groups [14].
Benefits
of multilevel analysis
The multilevel approach offers several advantages.
First, the result can generalize to a wider population. Second, fewer
parameters are needed. The dummy variables approach would require 25 additional
parameters. In the handling of more complex models and a limited amount of
data, a reduction in the number of parameters is important. Third, information
can be shared between groups. This is due to assuming that the random effects
resulted from a common distribution. As a result, the precision of predictions
for groups that have relatively little data improves. Finally, it can deal with
data in which the times of the measurements vary from subject to subject [2].
Limitation
of multilevel analysis
Analysing variables from different levels at one
single common level leads to two distinct types of problems. The first problem
is statistical. If data are aggregated, the result is that different data
values from many sub-units are combined into fewer values for fewer higher-level
units. As a result, statistical analysis loses power due to information loss.
On the other hand, for a larger number of sub-units, few data values from a
small number of super-units will ‘blown up’ into many more values, if data are
disaggregated. The second problem is conceptual. That is analysing the data at
one level and formulating conclusions at another level [4].
Acknowledgements
We are thankful to
Wollo University for providing the necessary facilities to conduct this review.
Declaration of Interest
The author has no
relevant affiliations or financial involvement with a financial interest in or financial with the subject matter or materials discussed in
the manuscript.
