Ron Heck, Fall 2011
1
EDEP 768E: Seminar in Multilevel Modeling
Oct. 24
,
2011
Week 10
:
Notes
Covariance Matrices in IBM SPSS
One of the decisions that must be made
about developing
two

and three

level models is the
covariance structure at each level. This concerns what variances and covariances for outcomes
will be reported at
each level. Are there any particular rules about defining the covariance
structure in a multilevel model? I really don't have a definitive answer for this question
—
which
reflects my thinking that it sort of "depends" on the situation and the goals of your
research.
IBM
SPSS provides very flexible specification of the covariance matrix at both level 1 and level 2

but I don't think there is any one "best" solution. Here are a few thoughts.
1.
If you have a categorical outcome (like dichotomous, count, or ordin
al) it does not matter
at level 1, since the variance is scaled to 1.0 anyway because there is no separate error
term when the outcome is categorical. At level 2 you typically have some choices. If
there is one random effect (e.g., a random intercept) it w
ould be a identity covariance
matrix. That just means there is one variance in the diagonal (and no covariance since
there is only one random effect). With two random effects, the default would be diagonal,
unless you have some interest in the covariance (
or a correlation) between the random
effects. If you use the default specification (VC, or “variance components”) it will default
to a diagonal matrix if there is more than one random effect. For one random effect, VC
will default to an identity matrix.
I
n the case you wish to look at covariances between random effects, you can define the
matrix as "unstructured." Note, however, that specifying “unstructured” can result in
more models that do not converge. By converge, we mean reach a solution where
everyt
hing is estimated. Sometimes when the program fails to reach a proper solution, it
is because the covariance is near zero and therefore cannot be estimated accurately. In
this case you will likely see zeros in the covariance parameter table. So you may hav
e to
go back to using a diagonal covariance matrix.
2.
If the outcome is continuous, it sort of depends on other factors about the set up of the
model. If there is only one outcome and the model is cross

sectional, the level

1 matrix
will just be an identity
matrix. There is just one residual variance. At level 2, if there is
one random effect, it will be an identity matrix, and for a two, using the default VC, the
covariance matrix will be diagonal. If you have some reason to want to examine all the
possibl
e variances and covariances between the random effects, you might define the
level 2 matrix as "unstructured"; however, as I mentioned previously, this is typically
harder to get to converge on a solution.
(EDEP 768) Week 10
: Notes
on Covariance Matrices
2
So I usually start with a simple structure (ident
ity, or diagonal) and proceed from there.
3.
I have found through experience that, at least in most cases, fully defining the covariance
matrix at level 1 does not produce any substantive change in the model's fixed effects.
4.
For longitudinal models, again at
level 1 there are several choices. People may use VC
and identify a diagonal matrix. This will provide a variance for each occasion of
measurement, but this does not take into consideration that the successive measurements
may be substantially correlated.
Usually with a longitudinal model, using an unstructured
matrix is going "too far." It is not necessary. A good solution is an autoregressive matrix,
of which there are a couple of types available in
IBM
SPSS. I usually use the
abbreviation AR1, which cap
tures the covariance structure of the repeated measures in an
"assumed" diagonal structure with a correlation (defined as rho in the output)
representing the average correlation between successive measurements. It is a reasonable
compromise; that is, it is
generally preferable to a diagonal matrix since it considers the
correlations between measures, but not as complicated as a full unstructured matrix
where each variance and covariance is separately measured. Another possible structure
sometimes used for l
ongitudinal models is “compound symmetry” (CS).
There are occasions where the variances are almost the same between occasions where I
have just used a simple "identity" matrix as well (just capturing the average variance at
level 1).
5.
As a conclusion:
In
most cases, it does not too much matter in terms of the fixed effects. Raudenbush and
Bryk always say to look at the correlation between an intercept and slope, but I find that
in many situations the model with a fully unstructured covariance matrix will n
ot
converge. So as I mentioned, I usually start from a simple structure and go from there. To
me, it depends on whether you have some theoretical reason for looking in detail at the
covariance structure (like a particular reason for examining the correlati
on between the
slope and intercept, etc.).
That said, in few cases you will find that a particular type of covariance matrix may not
converge. In that case, you may have to try one or two others as a type of compromise in
order to get the model to converg
e. To me, convergence is the prime consideration. You
cannot accept a model that does not converge. After that, then the covariance structure
should be based on practical (ease) and theoretically relevant criteria depending on the
particular study.
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