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 level-1
model relates the observed data, Y, to the complete
data, 
:
where 
is
the r-th 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 h-th
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 within-person design is fixed ,
with T observations per person and randomly missing
time points, the two-level 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 level-1 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
within-person residuals are independent with a constant variance,
.
The level-1
model has a similar form to that in the case of the unrestricted
model
with 
.
Heterogeneous model
One can
model heterogeneity of level-1 variance as a function of the
occasion of measurement. Such a model is suitable when we
suspect that the level-1 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 level-1 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.
Log-linear model
The model
with varying level-1 variance, described above, assumes a
unique level-1 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 level-1 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 level-1 variance
button is activated. Click on this button to open a dialog
box that can be used to select the predictors of the level-1
variance.
1st-order auto-regressive model
This model
allows the level-1 residuals to be correlated under Markov
assumptions (a level-1 residual depends on previous level-1
residuals only through the immediately preceding level-1 residuals).
This leads to the level-1 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
level-1 predictors are assumed to have the same values for
all level-2 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 2-level model (HLM2) is more
flexible than HMLM.
Cross classified random effects models
A general
random cross-classified model consists of two sub-models:
level-1 or within-cell and level-2 or between-cell models.
The cells refer to the cross-classifications by the two higher-level
units. For example, if the research problem consists of data
on students cross-classified by schools and neighborhoods,
the level-1 or within-cell model will represent the relationships
among the student-level variables, the level-2 or between-cell
model will capture the influence of school- and neighborhood-level
factors. Formally, there are 
level-1 units (e.g., students) nested within cells cross-classified
by = 1,..., J first level-2 units (e.g., neighborhoods),
designated as rows, and k = 1,..., K second level-2
units (e.g., schools), designated as columns.
We represent
in the level-1 or within-cell model the outcome for case
i in individual cells cross-classified by level-2 units
j and k.
where
(
)
are level-1 coefficients, 
is the level-1 predictor p for case i
in cell jk, 
is the level-1 or within-cell random effect, and
is the variance of ,
that is the level-1 or within-cell variance.
Here we assume that the random term 
.
Level-2
or "between-cell" model
Each of
the
coefficients in the level-1 or within-cell model becomes
an outcome variable in the level-2 or between-cell model:
where
is
the model intercept, the expected value of when
all explanatory variables are set to zero; 
are the fixed effects of column-specific predictors 
; 
are
the random effects associated with column-specific predictors 
.
They vary randomly over rows j = 1,..., J;
are the fixed effects of row-specific predictors 
; 
are the random effects associated with row-specific predictors 
. They vary randomly over columns k = 1,…, K;
are the fixed effects of cell-specific predictors
,
which are the interaction terms created as the products
of and
, s = 1, …,
and 
;
and 
,
,
and 
are residual row, column, and cell-specific random effects,
respectively, on ,
after taking into account ,
,
and .
We assume that 
,
,
and that the effects are independent of each other.
V-known models
The V-known
option in HLM2 is a general routine that can be used for applications
where the level-1 variances (and covariances) are known. Included
here are problem of meta-analysis and a wide range of other
possible uses as discussed in Chapter 7 of Hierarchical
Linear Models. The program input consists of Q
random level-1 statistics for each group and their associated
error variances and covariances. While this model is usually
only run in interactive/batch mode, a V-known 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 mutliple-imputed data to produce appropriate
estimates that incorporate the uncertainty resulting from
imputation. HLM has two methods to analyze multiple-imputed
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 multiply-imputed
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 level-2 units by clicking
on the confidence interval plots, and can also choose to include
a level-2 classification variable when examining the confidence
interval plots.
Types
of models available in HLM
, to 
to insure that the predictions are constrained to lie within
a given interval. Such a transformation is called a link function.
In the normal case, no transformation is necessary. However,
this decision not to transform may be made explicit by writing
,
is 0.5, the odds of success is 1.0 and the log-odds or "logit"
is zero. When the probability of success is less than 0.5,
the odds are less than one and the logit is negative; when
the probability is greater than 0.5, the odds are greater
than unity and the logit is positive. Thus, while 
be the number of "successes" in 
trials. Then we write that 
.
According to the Poisson distribution, the expected value
and variance of are then 
as the log-odds of falling into category m relative
to that of falling into category M. Specifically
