<s>
Multilevel	B-General_Concept
models	I-General_Concept
(	O
also	O
known	O
as	O
hierarchical	B-General_Concept
linear	I-General_Concept
models	I-General_Concept
,	O
linear	O
mixed-effect	O
model	O
,	O
mixed	B-General_Concept
models	I-General_Concept
,	O
nested	B-General_Concept
data	I-General_Concept
models	O
,	O
random	O
coefficient	O
,	O
random-effects	B-General_Concept
models	I-General_Concept
,	O
random	O
parameter	O
models	O
,	O
or	O
split-plot	B-General_Concept
designs	O
)	O
are	O
statistical	O
models	O
of	O
parameters	O
that	O
vary	O
at	O
more	O
than	O
one	O
level	O
.	O
</s>
<s>
These	O
models	O
can	O
be	O
seen	O
as	O
generalizations	O
of	O
linear	B-Algorithm
models	I-Algorithm
(	O
in	O
particular	O
,	O
linear	B-General_Concept
regression	I-General_Concept
)	O
,	O
although	O
they	O
can	O
also	O
extend	O
to	O
non-linear	O
models	O
.	O
</s>
<s>
Multilevel	B-General_Concept
models	I-General_Concept
are	O
particularly	O
appropriate	O
for	O
research	O
designs	O
where	O
data	O
for	O
participants	O
are	O
organized	O
at	O
more	O
than	O
one	O
level	O
(	O
i.e.	O
,	O
nested	B-General_Concept
data	I-General_Concept
)	O
.	O
</s>
<s>
While	O
the	O
lowest	O
level	O
of	O
data	O
in	O
multilevel	B-General_Concept
models	I-General_Concept
is	O
usually	O
an	O
individual	O
,	O
repeated	O
measurements	O
of	O
individuals	O
may	O
also	O
be	O
examined	O
.	O
</s>
<s>
As	O
such	O
,	O
multilevel	B-General_Concept
models	I-General_Concept
provide	O
an	O
alternative	O
type	O
of	O
analysis	O
for	O
univariate	O
or	O
multivariate	O
analysis	O
of	O
repeated	O
measures	O
.	O
</s>
<s>
Furthermore	O
,	O
multilevel	B-General_Concept
models	I-General_Concept
can	O
be	O
used	O
as	O
an	O
alternative	O
to	O
ANCOVA	B-General_Concept
,	O
where	O
scores	O
on	O
the	O
dependent	O
variable	O
are	O
adjusted	O
for	O
covariates	O
(	O
e.g.	O
</s>
<s>
Multilevel	B-General_Concept
models	I-General_Concept
are	O
able	O
to	O
analyze	O
these	O
experiments	O
without	O
the	O
assumptions	O
of	O
homogeneity-of-regression	O
slopes	O
that	O
is	O
required	O
by	O
ANCOVA	B-General_Concept
.	O
</s>
<s>
Multilevel	B-General_Concept
models	I-General_Concept
can	O
be	O
used	O
on	O
data	O
with	O
many	O
levels	O
,	O
although	O
2-level	O
models	O
are	O
the	O
most	O
common	O
and	O
the	O
rest	O
of	O
this	O
article	O
deals	O
only	O
with	O
these	O
.	O
</s>
<s>
refers	O
to	O
the	O
overall	O
regression	B-General_Concept
coefficient	I-General_Concept
,	O
or	O
the	O
slope	O
,	O
between	O
the	O
dependent	O
variable	O
and	O
the	O
Level	O
2	O
predictor	O
.	O
</s>
<s>
refers	O
to	O
the	O
overall	O
regression	B-General_Concept
coefficient	I-General_Concept
,	O
or	O
the	O
slope	O
,	O
between	O
the	O
dependent	O
variable	O
and	O
the	O
Level	O
1	O
predictor	O
.	O
</s>
<s>
Before	O
conducting	O
a	O
multilevel	B-General_Concept
model	I-General_Concept
analysis	O
,	O
a	O
researcher	O
must	O
decide	O
on	O
several	O
aspects	O
,	O
including	O
which	O
predictors	O
are	O
to	O
be	O
included	O
in	O
the	O
analysis	O
,	O
if	O
any	O
.	O
</s>
<s>
In	O
addition	O
,	O
this	O
model	O
provides	O
information	O
about	O
intraclass	O
correlations	O
,	O
which	O
are	O
helpful	O
in	O
determining	O
whether	O
multilevel	B-General_Concept
models	I-General_Concept
are	O
required	O
in	O
the	O
first	O
place	O
.	O
</s>
<s>
In	O
order	O
to	O
conduct	O
a	O
multilevel	B-General_Concept
model	I-General_Concept
analysis	O
,	O
one	O
would	O
start	O
with	O
fixed	O
coefficients	O
(	O
slopes	O
and	O
intercepts	O
)	O
.	O
</s>
<s>
One	O
such	O
statistic	O
is	O
the	O
chi-square	O
likelihood-ratio	B-General_Concept
test	I-General_Concept
,	O
which	O
assesses	O
the	O
difference	O
between	O
models	O
.	O
</s>
<s>
The	O
likelihood-ratio	B-General_Concept
test	I-General_Concept
can	O
be	O
employed	O
for	O
model	O
building	O
in	O
general	O
,	O
for	O
examining	O
what	O
happens	O
when	O
effects	O
in	O
a	O
model	O
are	O
allowed	O
to	O
vary	O
,	O
and	O
when	O
testing	O
a	O
dummy-coded	O
categorical	O
variable	O
as	O
a	O
single	O
effect	O
.	O
</s>
<s>
When	O
testing	O
non-nested	O
models	O
,	O
comparisons	O
between	O
models	O
can	O
be	O
made	O
using	O
the	O
Akaike	O
information	O
criterion	O
(	O
AIC	O
)	O
or	O
the	O
Bayesian	B-General_Concept
information	I-General_Concept
criterion	I-General_Concept
(	O
BIC	B-General_Concept
)	O
,	O
among	O
others	O
.	O
</s>
<s>
Multilevel	B-General_Concept
models	I-General_Concept
have	O
the	O
same	O
assumptions	O
as	O
other	O
major	O
general	O
linear	B-Algorithm
models	I-Algorithm
(	O
e.g.	O
,	O
ANOVA	B-General_Concept
,	O
regression	B-General_Concept
)	O
,	O
but	O
some	O
of	O
the	O
assumptions	O
are	O
modified	O
for	O
the	O
hierarchical	O
nature	O
of	O
the	O
design	O
(	O
i.e.	O
,	O
nested	B-General_Concept
data	I-General_Concept
)	O
.	O
</s>
<s>
Particularly	O
,	O
when	O
the	O
mean	O
part	O
of	O
the	O
level	O
1	O
regression	B-General_Concept
equation	O
is	O
replaced	O
with	O
a	O
non-linear	O
parametric	O
function	O
,	O
then	O
such	O
a	O
model	O
framework	O
is	O
widely	O
called	O
the	O
nonlinear	O
mixed-effects	O
model	O
.	O
</s>
<s>
The	O
multilevel	B-General_Concept
modelling	I-General_Concept
approach	O
can	O
be	O
used	O
for	O
all	O
forms	O
of	O
Generalized	O
Linear	B-Algorithm
models	I-Algorithm
.	O
</s>
<s>
The	O
assumption	O
of	O
homoscedasticity	B-General_Concept
,	O
also	O
known	O
as	O
homogeneity	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
,	O
assumes	O
equality	O
of	O
population	O
variances	O
.	O
</s>
<s>
Independence	O
is	O
an	O
assumption	O
of	O
general	O
linear	B-Algorithm
models	I-Algorithm
,	O
which	O
states	O
that	O
cases	O
are	O
random	O
samples	O
from	O
the	O
population	O
and	O
that	O
scores	O
on	O
the	O
dependent	O
variable	O
are	O
independent	O
of	O
each	O
other	O
.	O
</s>
<s>
One	O
of	O
the	O
main	O
purposes	O
of	O
multilevel	B-General_Concept
models	I-General_Concept
is	O
to	O
deal	O
with	O
cases	O
where	O
the	O
assumption	O
of	O
independence	O
is	O
violated	O
;	O
multilevel	B-General_Concept
models	I-General_Concept
do	O
,	O
however	O
,	O
assume	O
that	O
1	O
)	O
the	O
level	O
1	O
and	O
level	O
2	O
residuals	O
are	O
uncorrelated	O
and	O
2	O
)	O
The	O
errors	O
(	O
as	O
measured	O
by	O
the	O
residuals	O
)	O
at	O
the	O
highest	O
level	O
are	O
uncorrelated	O
.	O
</s>
<s>
The	O
regressors	O
must	O
not	O
correlate	O
with	O
the	O
random	B-General_Concept
effects	I-General_Concept
,	O
.	O
</s>
<s>
The	O
type	O
of	O
statistical	O
tests	O
that	O
are	O
employed	O
in	O
multilevel	B-General_Concept
models	I-General_Concept
depend	O
on	O
whether	O
one	O
is	O
examining	O
fixed	O
effects	O
or	O
variance	B-General_Concept
components	I-General_Concept
.	O
</s>
<s>
When	O
examining	O
fixed	O
effects	O
,	O
the	O
tests	O
are	O
compared	O
with	O
the	O
standard	O
error	O
of	O
the	O
fixed	O
effect	O
,	O
which	O
results	O
in	O
a	O
Z-test	B-General_Concept
.	O
</s>
<s>
A	O
t-test	B-General_Concept
can	O
also	O
be	O
computed	O
.	O
</s>
<s>
When	O
computing	O
a	O
t-test	B-General_Concept
,	O
it	O
is	O
important	O
to	O
keep	O
in	O
mind	O
the	O
degrees	O
of	O
freedom	O
,	O
which	O
will	O
depend	O
on	O
the	O
level	O
of	O
the	O
predictor	O
(	O
e.g.	O
,	O
level	O
1	O
predictor	O
or	O
level	O
2	O
predictor	O
)	O
.	O
</s>
<s>
Statistical	O
power	O
for	O
multilevel	B-General_Concept
models	I-General_Concept
differs	O
depending	O
on	O
whether	O
it	O
is	O
level	O
1	O
or	O
level	O
2	O
effects	O
that	O
are	O
being	O
examined	O
.	O
</s>
<s>
To	O
conduct	O
research	O
with	O
sufficient	O
power	O
,	O
large	O
sample	O
sizes	O
are	O
required	O
in	O
multilevel	B-General_Concept
models	I-General_Concept
.	O
</s>
<s>
In	O
order	O
to	O
detect	O
cross-level	O
interactions	O
,	O
given	O
that	O
the	O
group	O
sizes	O
are	O
not	O
too	O
small	O
,	O
recommendations	O
have	O
been	O
made	O
that	O
at	O
least	O
20	O
groups	O
are	O
needed	O
,	O
although	O
many	O
fewer	O
can	O
be	O
used	O
if	O
one	O
is	O
only	O
interested	O
in	O
inference	O
on	O
the	O
fixed	O
effects	O
and	O
the	O
random	B-General_Concept
effects	I-General_Concept
are	O
control	O
,	O
or	O
"	O
nuisance	O
"	O
,	O
variables	O
.	O
</s>
<s>
The	O
issue	O
of	O
statistical	O
power	O
in	O
multilevel	B-General_Concept
models	I-General_Concept
is	O
complicated	O
by	O
the	O
fact	O
that	O
power	O
varies	O
as	O
a	O
function	O
of	O
effect	O
size	O
and	O
intraclass	O
correlations	O
,	O
it	O
differs	O
for	O
fixed	O
effects	O
versus	O
random	B-General_Concept
effects	I-General_Concept
,	O
and	O
it	O
changes	O
depending	O
on	O
the	O
number	O
of	O
groups	O
and	O
the	O
number	O
of	O
individual	O
observations	O
per	O
group	O
.	O
</s>
<s>
As	O
a	O
simple	O
example	O
,	O
consider	O
a	O
basic	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
that	O
predicts	O
income	O
as	O
a	O
function	O
of	O
age	O
,	O
class	O
,	O
gender	O
and	O
race	O
.	O
</s>
<s>
A	O
simple	O
way	O
to	O
incorporate	O
this	O
into	O
the	O
regression	B-General_Concept
model	O
would	O
be	O
to	O
add	O
an	O
additional	O
independent	O
categorical	O
variable	O
to	O
account	O
for	O
the	O
location	O
(	O
i.e.	O
</s>
<s>
a	O
set	O
of	O
additional	O
binary	O
predictors	O
and	O
associated	O
regression	B-General_Concept
coefficients	I-General_Concept
,	O
one	O
per	O
location	O
)	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
a	O
simple	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
might	O
,	O
for	O
example	O
,	O
predict	O
that	O
a	O
given	O
randomly	O
sampled	O
person	O
in	O
Seattle	O
would	O
have	O
an	O
average	O
yearly	O
income	O
$	O
10,000	O
higher	O
than	O
a	O
similar	O
person	O
in	O
Mobile	O
,	O
Alabama	O
.	O
</s>
<s>
A	O
multilevel	B-General_Concept
model	I-General_Concept
,	O
however	O
,	O
would	O
allow	O
for	O
different	O
regression	B-General_Concept
coefficients	I-General_Concept
for	O
each	O
predictor	O
in	O
each	O
location	O
.	O
</s>
<s>
Essentially	O
,	O
it	O
would	O
assume	O
that	O
people	O
in	O
a	O
given	O
location	O
have	O
correlated	O
incomes	O
generated	O
by	O
a	O
single	O
set	O
of	O
regression	B-General_Concept
coefficients	I-General_Concept
,	O
whereas	O
people	O
in	O
another	O
location	O
have	O
incomes	O
generated	O
by	O
a	O
different	O
set	O
of	O
coefficients	O
.	O
</s>
<s>
Meanwhile	O
,	O
the	O
coefficients	O
themselves	O
are	O
assumed	O
to	O
be	O
correlated	O
and	O
generated	O
from	O
a	O
single	O
set	O
of	O
hyperparameters	B-General_Concept
.	O
</s>
<s>
Additional	O
levels	O
are	O
possible	O
:	O
For	O
example	O
,	O
people	O
might	O
be	O
grouped	O
by	O
cities	O
,	O
and	O
the	O
city-level	O
regression	B-General_Concept
coefficients	I-General_Concept
grouped	O
by	O
state	O
,	O
and	O
the	O
state-level	O
coefficients	O
generated	O
from	O
a	O
single	O
hyper-hyperparameter	O
.	O
</s>
<s>
Multilevel	B-General_Concept
models	I-General_Concept
are	O
a	O
subclass	O
of	O
hierarchical	O
Bayesian	O
models	O
,	O
which	O
are	O
general	O
models	O
with	O
multiple	O
levels	O
of	O
random	O
variables	O
and	O
arbitrary	O
relationships	O
among	O
the	O
different	O
variables	O
.	O
</s>
<s>
Multilevel	B-General_Concept
analysis	I-General_Concept
has	O
been	O
extended	O
to	O
include	O
multilevel	O
structural	O
equation	O
modeling	O
,	O
multilevel	O
latent	O
class	O
modeling	O
,	O
and	O
other	O
more	O
general	O
models	O
.	O
</s>
<s>
Multilevel	B-General_Concept
models	I-General_Concept
have	O
been	O
used	O
in	O
education	O
research	O
or	O
geographical	O
research	O
,	O
to	O
estimate	O
separately	O
the	O
variance	O
between	O
pupils	O
within	O
the	O
same	O
school	O
,	O
and	O
the	O
variance	O
between	O
schools	O
.	O
</s>
<s>
In	O
sociological	O
applications	O
,	O
multilevel	B-General_Concept
models	I-General_Concept
are	O
used	O
to	O
examine	O
individuals	O
embedded	O
within	O
regions	O
or	O
countries	O
.	O
</s>
<s>
They	O
are	O
often	O
used	O
in	O
ecological	O
research	O
as	O
well	O
under	O
the	O
more	O
general	O
term	O
mixed	B-General_Concept
models	I-General_Concept
.	O
</s>
<s>
This	O
model	O
assumes	O
that	O
each	O
group	O
has	O
a	O
different	O
regression	B-General_Concept
model	O
—	O
with	O
its	O
own	O
intercept	O
and	O
slope	O
.	O
</s>
<s>
Multilevel	B-General_Concept
models	I-General_Concept
have	O
two	O
error	O
terms	O
,	O
which	O
are	O
also	O
known	O
as	O
disturbances	O
.	O
</s>
<s>
However	O
,	O
variance	B-General_Concept
components	I-General_Concept
can	O
differ	O
,	O
as	O
some	O
groups	O
are	O
more	O
homogeneous	O
than	O
others	O
.	O
</s>
<s>
Multilevel	B-General_Concept
modeling	I-General_Concept
is	O
frequently	O
used	O
in	O
diverse	O
applications	O
and	O
it	O
can	O
be	O
formulated	O
by	O
the	O
Bayesian	O
framework	O
.	O
</s>
<s>
Particularly	O
,	O
Bayesian	O
nonlinear	O
mixed-effects	B-General_Concept
models	I-General_Concept
have	O
recently	O
received	O
significant	O
attention	O
.	O
</s>
<s>
A	O
basic	O
version	O
of	O
the	O
Bayesian	O
nonlinear	O
mixed-effects	B-General_Concept
models	I-General_Concept
is	O
represented	O
as	O
the	O
following	O
three-stage	O
:	O
</s>
