SPSS
This section closely follows Peugh and Enders (2005). It
demonstrates how to group-mean center level-1 covariates and
estimate multilevel models using SPSS syntax. Note that it is also
possible to use the Mixed Models option under the
Analyze pull-down menu (see Norusis 2005, pgs. 197-246).
However, length considerations limit the examples here to syntax.
The SPSS syntax editor can be accessed by going to File →
New → Syntax.
In the HSB data file, the student-level SES variable is in its
original metric (a standardized scale with a mean of zero).
Oftentimes researchers dealing with hierarchically structured data
wish to center a level-1 variable around the mean of all cases
within the same level-2 group in order to facilitate interpretation
of the intercept. To group-mean center a variable in SPSS, first use
the AGGREGATE command to estimate mean SES
scores by school. In this example, the syntax would be:
AGGREGATE OUTFILE=sesmeans.sav
/BREAK=id
/meanses=MEAN(ses)
The OUTFILE statement
specifies that the means are written out to the file sesmeans.sav in
the working directory. The BREAK subcommand
specifies the groups within which to estimate means. The final line
names the variable containing the school means meanses.
Next, the group means are sorted and merged with the original data
using the SORT CASES and MATCHFILES
commands. The centered variables are then created using
the COMPUTEcommand. (To grand mean
center a variable in SPSS requires only a single line of syntax. For
example, COMPUTE newvar = oldvar - mean(oldvar).) The
syntax for these steps would be:
SORT CASES BY id .
MATCH FILES
/TABLE=sesmeans.sav
/FILE=*
/BY id .
COMPUTE centses = ses - meanses .
EXECUTE .
The subcommands for MATCH FILES ask SPSS to take the data
file saved using the AGGREGATE command and merge it with the working data
(denoted by *). The matching variable is the school ID.
With the data prepared, the next step is to estimate the models of
interest. The following syntax corresponds to the empty model
(5):
MIXED mathach
/PRINT = SOLUTION TESTCOV
/FIXED = INTERCEPT
/RANDOM = INTERCEPT | SUBJECT(id) .
The command for estimating multilevel models
is MIXED followed immediately by the dependent
variable. PRINT = SOLUTION requests that SPSS
reports the fixed effects estimates and standard errors.
FIXED and RANDOM specify which
variables to treat as fixed and random effects, respectively. The
SUBJECT option following the vertical line |
identifies the grouping variable, in this case school ID.
The fixed and random effect estimates for this and subsequent models
are displayed in Table 1 at the bottom of the page. The intercept in the empty model
is equal to the overall average math achievement score, which for
this sample is 12.637. The variance component corresponding to the
random intercept is 8.614; for the level-1 error it is 39.1483.
Including the TESTCOVsubcommand requested that
SPSS report Wald-Z significance tests for the variance components,
equal to the estimate divided by its standard error. In this
example, the value of the Wald-Z statistic is 6.254, which is
significant (p<.001). Note, however, that these tests should not
be taken as conclusive. Singer (1998, pg. 351) writes,
``the validity of these tests has been called into question both because they rely on large sample approximations (not useful with the small sample sizes often analyzed using multilevel models) and because variance components are known to have skewed (and bounded) sampling distributions that render normal approximations such as these questionable.''
A more thorough test would thus estimate a second model
constraining the variance component to equal zero and compare the
two models using a likelihood ratio test.
The two variance components can be used to partition the variance
across levels according to equation 6 above. The intraclass
correlation coefficient for this example is equal to
=.1804, meaning that roughly 18% of
the variance is attributable to school traits. Because the
intraclass correlation coefficient shows a fair amount of variation
across schools, model 2 adds two school-level variables. These
variables are sector, defining whether a school
is private or public, and ttfamily meanses, which is the
average student socioeconomic status in the school. The SPSS syntax
to estimate this model is:
=.1804, meaning that roughly 18% of
the variance is attributable to school traits. Because the
intraclass correlation coefficient shows a fair amount of variation
across schools, model 2 adds two school-level variables. These
variables are sector, defining whether a school
is private or public, and ttfamily meanses, which is the
average student socioeconomic status in the school. The SPSS syntax
to estimate this model is:
MIXED mathach WITH meanses sector
/PRINT = SOLUTION TESTCOV
/FIXED = INTERCEPT meanses sector
/RANDOM = INTERCEPT | SUBJECT(id) .
The results, displayed in the second column
of Table 1, show that meanses and
sector significantly affect a school's average
math achievement score. The intercept, representing the expected
math achievement score for a student in a public school with average
SES, is equal to 12.1283. A one unit increase in average SES raises
the expected school mean by 5.5334. Private schools have expected
math achievement scores 1.2254 units higher than public schools. The
variance component corresponding to the random intercept has
decreased to 2.3140, demonstrating that the inclusion of the two
school-level variables has explained much of the level-2 variation.
However, the estimate is still more than twice the size of its
standard error, suggesting that there remains a significant amount
of unexplained school-level variance (though the same caution about
over-interpreting this test still applies).
A final model introduces a student-level covariate, the group-mean
centered SES variable centses. Because it is
possible that the effect of socioeconomic status may vary across
schools, SES is treated as a random effect. In addition, sector and meanses are included to
model the slope on the student-level SES variable. Modeling the
slope of a random effect is the same as specifying a cross-level
interaction, which can be specified in the FIXED
subcommand as in the following syntax:
MIXED mathach WITH meanses sector centses
/PRINT = SOLUTION TESTCOV
/FIXED = INTERCEPT meanses sector centses meanses*centses sector*centses
/RANDOM = INTERCEPT centses | SUBJECT(id) COVTYPE(UN) .
One important change over the previous models
is the addition of the COV(UN) option, which
specifies a structure for the level-2 covariance matrix. Only a
single school-level variance component was estimated in the previous
two models, thus it was unnecessary to deal with covariances. When
there is more than one level-2 variance component, SPSS will assume
a particular covariance structure. In many cross-sectional
applications of multilevel models, the researcher does not wish to
put any constraints on this covariance matrix. Thus the UN in the COV option specifies an unstructured matrix. In other contexts, the researcher may
wish to specify a first-order autoregressive (AR1), compound
symmetry (CS), identity (ID), or other structure. These alternatives
are more restrictive but may sometimes be appropriate.
The results from this final model appear in the last column of Table
1. The fixed effects are all significant. Given the
inclusion of the group-mean centered SES variable, the intercept is
interpreted as the expected math achievement in a public school with
average SES levels for a student at his or her school's average SES.
In this model, the expected outcome is 12.1279. Because there are
interactions in the model, the marginal fixed effects of each
variable will depend on the value of the other variable(s) involved
in the interaction. The marginal effect of a one-unit change in a
student's SES score on math achievement depends on whether a school
is public or private as well as on the school's average SES score.
For a public school (where sector=0), the
marginal effect of a one-unit change in the group-mean centered
student SES variable is equal to 
= γ10 + γ11(MEANSES) = 2.945041 + 1.039232(MEANSES).
For a private school (where sector=1), the
marginal effect of a one-unit change in student SES is equal to
= γ10 + γ11(MEANSES) + γ12 = 2.945041 + 1.039232(MEANSES) -
1.642674. When cross-level interactions are present, graphical
means may be appropriate for exploring the contingent nature of
marginal effects in greater detail (Raudenbush \& Bryk 2002;
Brambor, Clark, and Golder 2006). Here the simplest interpretation
is that the effect of student-level SES is significantly higher in
wealthier schools and significantly lower in private schools.
= γ10 + γ11(MEANSES) = 2.945041 + 1.039232(MEANSES).
For a private school (where sector=1), the
marginal effect of a one-unit change in student SES is equal to
= γ10 + γ11(MEANSES) + γ12 = 2.945041 + 1.039232(MEANSES) -
1.642674. When cross-level interactions are present, graphical
means may be appropriate for exploring the contingent nature of
marginal effects in greater detail (Raudenbush \& Bryk 2002;
Brambor, Clark, and Golder 2006). Here the simplest interpretation
is that the effect of student-level SES is significantly higher in
wealthier schools and significantly lower in private schools.
The variance component for the random intercept continues to be significant,
suggesting that there remains some variation in average school performance
not accounted for by the variables in the model. The variance component
for the random slope, however, is not significant. Thus the researcher
may be justified in estimating an alternative model that constrains
this variance component to equal zero.
