Under the
proportional odds assumption, the relative odds that ,
associated with a unit increase in the predictor, does not
depend on m.
Here
is a "threshold" that separates categories m
1 and m. For example, when M = 4,
Unrestricted model
This
model is appropriate when the aim of the study is to collect
T observations per aprticipant according to a fixed
design. However, one or more observations may be missing at
random. We assume a constant but otherwise arbitrary T
x T covariance matrix for each person's "complete
data".
The level1
model relates the observed data, Y, to the complete
data, :
where is
the rth outcome for person i associated with
time h. Here is
the value that person i would have displayed if that
person had been observed at time t, and is
an indicator variable taking on a value of 1 if the hth
measurement for person i did occur at time t,
0 if not. Thus, ,
t = 1, ..., T, represent the complete data for
person i while ,
h = 1, ..., A are
the observed data, and the indicators tell
us the pattern of missing data for person i.
Homogeneous model
Under the
special case in which the withinperson design is fixed ,
with T observations per person and randomly missing
time points, the twolevel HLM can be derived from the unrestricted
model by imposing restrictions on the covariance matrix, A.
(Note: regressors having varying designs may be included
in the level1 model, but coefficients associated with such
values must not have random effects at level 2). The most
frequently used assumption in the standard HLM is that the
withinperson residuals are independent with a constant variance,
.
The level1
model has a similar form to that in the case of the unrestricted
model
with .
Heterogeneous model
One can
model heterogeneity of level1 variance as a function of the
occasion of measurement. Such a model is suitable when we
suspect that the level1 residual variance varies across
occasions. The models that can be estimated are a subset of
the models that can be estimated within the standard HLM2.
The level1 model is the same as in the case of homogenous
variances, except that now
that is,
is
now diagonal with elements ,
the variance associated with occasion t, t =
1, …, T.
The number
of parameters estimated is .
Now r must be no larger than .
When ,
the results will duplicate those based on the unrestricted
model.
Loglinear model
The model
with varying level1 variance, described above, assumes a
unique level1 variance for every occasion. A more parsimonious
model would specify a functional relationship between aspects
of the occasion (e.g. time or age) and the variance. We would
again have ,
but now
Thus, the
natural log of the level1 variance may be a linear or quadratic
function of age. If the explanatory variables are
dummy
variables, each indicating the occasion of measurement, the
results will duplicate those of the previous section.
The number
of parameters estimated is now .
Again, r must be no larger than and
H must be no larger than .
When this
option is selected, the Predictors of level1 variance
button is activated. Click on this button to open a dialog
box that can be used to select the predictors of the level1
variance.
1storder autoregressive model
This model
allows the level1 residuals to be correlated under Markov
assumptions (a level1 residual depends on previous level1
residuals only through the immediately preceding level1 residuals).
This leads to the level1 covariance structure
Thus, the
variance at each time point is
and each correlation diminishes with the distance between
time points, so that the correlations are as
the distance between occasions is 1, 2, 3, ....
The number
of parameters estimated is now .
r must be no larger than .
Note that
level1 predictors are assumed to have the same values for
all level2 units of the complete data. This assumption can
be relaxed. However, if the design for varies
over i, its coefficient cannot vary randomly at level
2. In this regard, the standard 2level model (HLM2) is more
flexible than HMLM.
Cross classified random effects models
A general
random crossclassified model consists of two submodels:
level1 or withincell and level2 or betweencell models.
The cells refer to the crossclassifications by the two higherlevel
units. For example, if the research problem consists of data
on students crossclassified by schools and neighborhoods,
the level1 or withincell model will represent the relationships
among the studentlevel variables, the level2 or betweencell
model will capture the influence of school and neighborhoodlevel
factors. Formally, there are
level1 units (e.g., students) nested within cells crossclassified
by = 1,..., J first level2 units (e.g., neighborhoods),
designated as rows, and k = 1,..., K second level2
units (e.g., schools), designated as columns.
We represent
in the level1 or withincell model the outcome for case
i in individual cells crossclassified by level2 units
j and k.
where

()
are level1 coefficients,

is the level1 predictor p for case i
in cell jk,

is the level1 or withincell random effect, and

is the variance of ,
that is the level1 or withincell variance.
Here we assume that the random term .
Level2
or "betweencell" model
Each of
the
coefficients in the level1 or withincell model becomes
an outcome variable in the level2 or betweencell model:
where
 is
the model intercept, the expected value of when
all explanatory variables are set to zero;

are the fixed effects of columnspecific predictors
;
 are
the random effects associated with columnspecific predictors .
They vary randomly over rows j = 1,..., J;

are the fixed effects of rowspecific predictors
;

are the random effects associated with rowspecific predictors
. They vary randomly over columns k = 1,…, K;

are the fixed effects of cellspecific predictors
,
which are the interaction terms created as the products
of and
, s = 1, …,
and ;
and
 ,
,
and
are residual row, column, and cellspecific random effects,
respectively, on ,
after taking into account ,
,
and .
We assume that ,
,
and that the effects are independent of each other.
Vknown models
The Vknown
option in HLM2 is a general routine that can be used for applications
where the level1 variances (and covariances) are known. Included
here are problem of metaanalysis and a wide range of other
possible uses as discussed in Chapter 7 of Hierarchical
Linear Models. The program input consists of Q
random level1 statistics for each group and their associated
error variances and covariances. While this model is usually
only run in interactive/batch mode, a Vknown analysis can
be eprformed using the Windows interface when Q = 1.
Analysis of multiply imputed data/plausible
values
Users can
apply HLM2 and HLM3 to mutlipleimputed data to produce appropriate
estimates that incorporate the uncertainty resulting from
imputation. HLM has two methods to analyze multipleimputed
data. They both use the same equations to compute the averages,
so the method chosen depends on the data being analyzed.
 Plausible values: is usually preferable
for data sets that have only one variable (outcome or predictor)
for whivch you have several plausible values. In this case,
one MDM file, containing all of the plausible values
plus any other variables of interest, is required as input.
 Multiple imputation: is necessary when
there is more than one variable for which multiplyimputed
data are available. In this case, as many MDM files as there
are plausible values are required as input for HLM.
Latent variable regression
Researchers
may be interested in studying the randomly varying coefficients
not only as outcomes, but as predictors as well. Treating
these coefficients as latent variables, the HLM2, HLM3 and
HMLM modules allow researchers to study direct as well indirect
effects among them and to assess their impacts on coefficients
associated with observed covariates in the model. Furthermore,
using HMLM with unrestricted covariance structures, one may
use latent variable analysis to run regressions with missing
data.
The graph
suggests that there is significant variation in the rate of
acceleration in vocabulary growth in children during the second
year of life. For instance, the confidence intervals of the
EB estimates of the AGE12S coefficients for the last four
children from the left did not overlap with those of the first
eleven children.
Users can
look at the actual empirical Bayes estimates and their 95%
confidence intervals of individual level2 units by clicking
on the confidence interval plots, and can also choose to include
a level2 classification variable when examining the confidence
interval plots.