HLM in Stata
HLM in Stata
HLM in Stata
SAS
Jeremy J. Albright
May 11, 2007
Multilevel data are pervasive in the social sciences. Students may be nested within schools,
voters within districts, or workers within firms, to name a few examples. Statistical methods that explicitly take into account hierarchically structured data have gained popularity
in recent years, and there now exist several special-purpose statistical programs designed
specifically for estimating multilevel models (e.g. HLM, MLwiN). In addition, the increasing
use of of multilevel models also known as hierarchical linear and mixed effects models
has led general purpose packages such as SPSS, Stata, and SAS to introduce their own
procedures for handling nested data.
Nonetheless, researchers may face two challenges when attempting to determine the appropriate syntax for estimating multilevel/mixed models using general purpose software.
First, many users from the social sciences come to multilevel modeling with a background
in regression models, whereas much of the software documentation utilizes examples from
experimental disciplines [due to the fact that multilevel modeling methodology evolved out
of ANOVA methods for analyzing experiments with random effects (Searle, Casella, and McCulloch, 1992)]. Second, notation for multilevel models is often inconsistent across disciplines
(Ferron 1997).
The purpose of this document is to demonstrate how to estimate multilevel models using
SPSS, Stata, and SAS. It first seeks to clarify the vocabulary of multilevel models by defining
what is meant by fixed effects, random effects, and variance components. It then compares
the model building notation frequently employed in applications from the social sciences with
1
the more general matrix notation found in much of the software documentation. The syntax
for centering variables and estimating multilevel models is then presented for each package.
Models for multilevel data have developed out of methods for analyzing experiments with
random effects. Thus it is important for those interested in using hierarchical linear models to
have a minimal understanding of the language experimental researchers use to differentiate
between effects considered to be random or fixed. In an ideal experiment, the researcher
is interested in whether the presence or absence of one factor affects scores on an outcome
variable.1 Does a particular pill reduce cholesterol more than a placebo? Can behavioral
modification reduce a particular phobia better than psychoanalysis or no treatment? The
factors in these experiments are said to be fixed because the same, fixed levels would be
included in replications of the study (Maxwell and Delaney, pg. 469). That is, the researcher
is only interested in the exact categories of the factor that appear in the experiment. The
typical model for a one-factor experiment is:
yij = + j + eij
(1)
where the score on the dependent variable for individual i is equal to the grand mean of the
sample (), the effect of receiving treatment j, and an individual error term eij . In general,
some kind of constraint is put on the alpha values, such as that they sum to zero, so that the
model is identified. In addition, it is assumed that the errors are independent and normally
distributed with constant variance.
In some experiments, however, a particular factor may not be fixed and perfectly replicable across experiments. Instead, the distinct categories present in the experiment represent
a random sample from a larger population. For example, different nurses may administer
1
In the parlance of experiments, a factor is a categorical variable. The term covariate refers to continuous
independent variables.
an experimental drug to subjects. Usually the effect of a specific nurse is not of theoretical interest, but the researcher will want to control for the possibility that an independent
caregiver effect is present beyond the fixed drug effect being investigated. In such cases the
researcher may add a term to control for the random effect:
(2)
where represents the effect of the kth level of the random effect, and represents the
interaction between the random and fixed effects. A model that contains only fixed effects
and no random effects, such as equation 1, is known as a fixed effects model. One that
includes only random effects and no fixed effects is termed a random effects model. Equation
2 is actually an example of a mixed effects model because it contains both random and fixed
effects.
While the notation in equation 2 for the random effect is the same as for the fixed effect
(that is, both are denoted by subscripted Greek letters), an important difference exists in
the tests for the drug and nurse factors. For the fixed effect, the researcher is interested in
only those levels included in the experiment, and the null hypothesis is that there are no
differences in the means of each treatment group:
H0 : 1 = 2 = ... = j
H1 : j 6= j 0
For the random effect in the drug example, the researcher is not interested in the particular
nurses per se but instead wishes to generalize about the potential effects of drawing different
nurses from the larger population. The null hypothesis for the random effect is therefore that
H0 : 2 = 0
H1 : 2 > 0
mentation for statistical applications capable of estimating mixed models. Political scientists
or sociologists, for example, come to utilize mixed models because they recognize that hierarchically structured data violate standard linear regression assumptions. However, because
mixed models developed out of methods for evaluating experiments, much of the documentation for packages like SPSS, SAS, and Stata is made up of experimental examples. Hence
it is important to recognize the connection between random effects ANOVA and hierarchical
linear models.
Note that the motivation for utilizing mixed models for multilevel data does not rest in
the different number of observations at each level, as any model including a dummy variable
involves nesting (e.g. survey respondents are nested within gender). The justification instead
lies in the fact that the errors within each randomly sampled level-2 unit are likely correlated,
necessitating the estimation of a random effects model. Once the researcher has accounted
for error non-independence it is possible to make more accurate inferences about the fixed
effects of interest.
Even if one is comfortable distinguishing between fixed and random effects, additional confusion may emerge when trying to make sense of the notation used to describe multilevel
models. In non-experimental disciplines, researchers tend to use the notation of Raudenbush
and Bryk (2002) that explicitly models the nested structure of the data. Unfortunately his
approach can be rather messy, and software documentation typically relies instead on matrix
notation. Both approaches are detailed in this section.
In the archetypical cross-sectional example, a researcher is interested in predicting test
performance as a function of student-level and school-level characteristics. Using the modelbuilding notation, an empty (i.e. lacking predictors) student-level model is specified first:
Yij = 0j + rij
5
(3)
The outcome variable Y for individual i nested in school j is equal to the average outcome in
unit j plus an individual-level error rij . Because there may also be an effect that is common
to all students within the same school, it is necessary to add a school-level error term. This
is done by specifying a separate equation for the intercept:
0j = 00 + u0j
(4)
where 00 is the average outcome for the population and u0j is a school-specific effect. Combining equations 3 and 4 yields:
(5)
Denoting the variance of rij as 2 and the variance of u0j as oo , the percentage of
observed variation in the dependent variable attributable to school-level characteristics is
found by dividing 00 by the total variance:
00
00 + 2
(6)
(7)
(8)
If the researcher wishes to treat student SES as a random effect (that is, the researcher
feels the effect of a students SES status varies between schools), he can do so by specifying
an equation for the slope in the same manner as was previously done with the intercept
equation:
1j = 10 + u1j
(9)
Finally, it is possible that the effect of a level-1 variable changes across scores on a level-2
variable. The effect of a students SES status may be less important in a private rather
than a public school, or a students individual SES status may be more important in schools
with higher average SES scores. To test these possibilities, one can add the MEANSES and
SECTOR variables to equation 9.
(10)
(11)
This approach of building a multilevel model through the specification and combination
7
of different level-1 and level-2 models makes clear the nested structure of the data. However,
it is long and messy, and what is more, it is inconsistent with the notation used in much
of the documentation for general statistical packages. Instead of the step-by-step approach
taken above, the pithier, and more general, matrix notation is often used:
y = X + Zu +
(12)
(13)
Note that it is possible for a variable to appear as both a fixed effect and a random effect
(appearing in both X and Z from 12). In this example, estimating 13 would yield both fixed
effect and random effect estimates for the student-level SES variable. The fixed effect would
refer to the overall expected effect of a students socioeconomic status on test scores; the
random effect gives information on whether or not this effect differs between schools.
Estimation
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 COMPUTE
command.2 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
2
To grand mean center a variable in SPSS requires only a single line of syntax. For example, COMPUTE
newvar = oldvar - mean(oldvar).
10
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. 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 TESTCOV subcommand
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.
11
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
8.614
8.614+39.1483
= .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 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 schools 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
12
Y
CEN T SES
10 + 11 (M EAN SES) = 2.945041 + 1.039232(M EAN SES). For a private school (where
sector=1), the marginal effect of a one-unit change in student SES is equal to
Y
CEN T SES
10 + 11 (M EAN SES) + 12 = 2.945041 + 1.039232(M EAN SES) 1.642674. When crosslevel 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.
13
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.
[Table 1 about here.]
Stata
This section discusses how to center variables and estimate multilevel models using Stata.
A fuller treatment is available in Rabe-Hesketh and Skrondal (2005) and in the Stata documentation. Since release 9, Stata includes the command .xtmixed to estimate multilevel
models. The .xt prefix signifies that the command belongs to the larger class of commands
used to estimate models for longitudinal data. This reflects the fact that panel data can be
thought of as multilevel data in which observations at multiple time points are nested within
an individual. However, the command is appropriate for mixed model estimation in general,
including cross-sectional applications.
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. Group-mean centering can be accomplished by using one of two commands in Stata.
The first is to use the .collapse command, which creates a new data file consisting of
summary statistics for groups (analogous to the AGGREGATE command in SPSS). Following
the .collapse command, the desired summary statistic is listed in parentheses followed by a
list of variables for which the corresponding statistic(s) will be estimated. The resulting data
file can then be sorted, saved, and merged with the original data. The .generate command
completes the creation of a group-centered variable.
14
With the original data file (hsb.dta) as the working data, the full syntax to group-mean
center the SES variable is:
.collapse (mean) meanses=ses, by(id)
.sort id
.save sesmeans, replace
.use hsb.dta, replace
.sort id
.merge id using sesmeans
.gen centses=ses-meanses
Here meanses=ses tells Stata to name the summary variable meanses. The .merge
command combines two data files.
Alternatively, the .statsby command can also estimate and temporarily store summary
statistics for groups. The syntax for this approach is:
.statsby meanses=r(mean), by(id) saving(sesmeans, replace):
.sort id
.merge id using sesmeans
summarize ses
The .statsby command stores temporary variables created by a particular command for
the groups specified in the by option. In this example, Stata saves the group means as the
variable sesmeans in a new data file. By default, the new data file replaces the working data.
The saving option instead saves the data to a file, here named sesmeans, in the working
directory. The .sort and .merge commands sort the working data and merge it with the
newly created file.3
The syntax for estimating multilevel models in Stata begins with the .xtmixed command
followed by the dependent variable and a list of independent variables. The last independent
variable is followed by double vertical lines ||, after which the grouping variable and random
effects are specified. .xtmixed will automatically specify the intercept to be random. A
list of variables whose slopes are to be treated as random follows the colon. Note that, by
default, Stata reports variance components as standard deviations (equal to the square root
of the variance components). To get Stata to report variances instead, add the var option.
3
To grand mean center in Stata requires two commands. First, .quietly summarize oldvar. Then,
.gen newvar=oldvar-r(mean).
15
, var
The results are displayed in Table 2. The average test score across schools, reflected in the
intercept term, is 12.63697. The variance component corresponding to the random intercept
is 8.61403. Because this estimate is substantially larger than its standard error, there appears
to be significant variation in school means.
The two variance components can be used to partition the variance across levels. The
intraclass correlation coefficient is equal to
8.61403
39.14832+8.61403
, var
The intercept, which now corresponds to the expected math achievement score in a public
school with average SES scores, is 12.12824. Moving to a private school bumps the expected
score by 1.2254 points. In addition, a one-unit increase in the average SES score is associated
with an expected increased in math achievement of 5.3328. These estimates are all significant.
The variance component corresponding to the random intercept has decreased to 2.313986,
reflecting the fact that the inclusion of the level-2 variables has accounted for some of the
variance in the dependent variable. Nonetheless, the estimate is still more than twice the
size of its standard error, suggesting that there remains variance unaccounted for.
A final model introduces the student socioeconomic status variable. Because it is possible
that the effect of individual SES status varies across schools, this slope is treated as random.
In addition, a schools average SES score and its sector (public or private) may interact with
student-level SES, accounting for some of the variance in the slope. In order to include these
cross-level interactions in the model, however, it is necessary to first explicitly create the
16
centses,
The results are displayed in the final column of Table 2. The intercept is 12.12793,
which here is the expected math achievement score in a public school with average SES
scores for a student at his or her schools average SES level. Because there are interactions
in the model, the marginal fixed effects of each variable now depend on the value of the
other variable(s) involved in the interaction. The marginal effect of a one-unit change in
students SES on math achievement will depend on whether a school is public or private
as well as on the average SES score for the school. For a public school (where sector=0),
the marginal effect of a one-unit change in the group-mean centered SES variable is equal
to
Y
CEN T SES
school (where sector=1), the marginal effect of a one-unit change in student SES is equal to
Y
CEN T SES
When cross-level interactions are present, graphical means may be appropriate for exploring
the contingent nature of marginal effects in greater detail. Here the simplest interpretation
of the interaction coefficients 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 is 2.379597, which is still large relative
to its standard error of 0.3714584. Thus there remains some school-level variance unaccounted
for in the model. The variance component corresponding to the slope, however, is quite
17
small relative to its standard error. This suggests that the researcher may be justified in
constraining the effect to be fixed.
By default, Stata does not report model fit statistics such as the AIC or BIC. These
can be requested, however, by using the postestimation command .estat ic. This displays
the log-likelihood, which can be converted to Deviance according to the formula 2 log
likelihood. It also displays the AIC and BIC statistics in smaller-is-better form. Comparing
both the AIC and BIC statistics in Table 2 it is clear that the final model is preferable to
the first two models.
[Table 2 about here.]
SAS
This section follows Singer (1998); a thorough treatment is available from Littell et al. (2006).
The SAS procedure for estimating multilevel models is PROC MIXED.
In the HSB data file the student-level SES variable is in its original metric (a standardized scale with a mean of zero). Oftentimes, the researcher will prefer to center a variable
around the mean of all observations within the same group. Group-mean centering in SAS
is accomplished using the SQL procedure. The following commands create a new data file,
HSB2, in the Work library that includes two additional variables: the group means for the
SES variable (saved as the variable sesmeans) and the group-mean centered SES variable
cses.4
PROC SQL;
CREATE TABLE hsb2 AS
SELECT *, mean(ses) as meanses,
ses-mean(ses) AS cses
FROM hsb
GROUP BY id;
QUIT;
4
Grand-mean centering also uses PROC SQL. Excluding the GROUP BY statement causes the mean(ses)
function to estimate the grand mean for the ses variable. The ses-mean(ses) statement then creates the
grand-mean centered variable.
18
8.6097
8.6097+39.1487
school characteristics.
In order to explain some of the school-level variation in math achievement scores it is
possible to incorporate school-level predictors into the model. For example, the average
socioeconomic status of a schools students may affect performance. In addition, whether a
school is public or private may also make a difference. The SAS program for a model with
two school level predictors is the following:
PROC MIXED COVTEST DATA=hsb2;
CLASS id;
MODEL mathach = meanses sector /SOLUTION;
RANDOM intercept/SUBJECT=id;
19
RUN;
The MODEL statement now includes the two school-level predictors following the equals
sign. Nothing else is changed from the previous program.
The results are displayed in the second column of Table 3. The intercept is 12.1282, which
now corresponds to the expected math achievement score for a student in a public school
at that schools average SES level. A one-unit increase in the schools average SES score
is associated with a 5.3328-unit increase in expected math achievement, and moving from
a public to a private school is associated with an expected improvement of 1.2254. These
estimates are all significant.
The variance component corresponding to the random intercept has now dropped to
2.3139, demonstrating that the inclusion of the average SES and school sector variables
explains a good deal of the school-level variance. Still, the estimate remains more than twice
the size of its standard error of 0.3700, suggesting that some of the school-level variance
remains unexplained.
A final model adds a student-level covariate, the group-mean centered SES variable.
Because it is possible that the effect of a students SES may vary across schools, the final
model treats the slope as random. Additionally, because the slope may vary according to
school-level characteristics such as average SES and sector (private versus public), the final
model also incorporates cross-level interactions.
The syntax for this last model is the following:
PROC MIXED COVTEST DATA=hsb2;
CLASS id;
MODEL mathach = meanses sector cses meanses*cses sector*cses/solution;
RANDOM intercept cses / TYPE=UN SUB=id;
RUN;
The MODEL statement adds the cses variable along with the cross-level interactions between csesat the student level and sector and meansesat the school level. CSES is also
added to the RANDOM statement. The TYPE=UN option specifies an unstructured covariance
matrix for the random effects.
20
The results are displayed in the final column of Table 3. The intercept of 12.1279 now
refers to the expected math achievement score in a public school with average SES scores
for a student at his or her schools average SES level. Because there are interactions in the
model, the marginal fixed effects of each variable depend on the value of the other variable(s)
involved in the interaction. The marginal effect of a one-unit change in a students SES
score on math achievement will depend on whether a school is public or private as well as
on the schools 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
Y
CEN T SES
(where sector=1), the marginal effect of a one-unit change in a students SES is equal to
Y
CEN T SES
cross-level interactions are present, graphical means may be appropriate for exploring the
contingent nature of marginal effects in greater detail. Here the simplest interpretation of
the interaction coefficients is that the effect of student-level SES is significantly higher in
wealthier schools and significantly lower in private schools.
The variance component corresponding to the random intercept is 2.3794, which remains
much larger than its standard error of .3714. Thus there is most likely additional school-level
variation unaccounted for in the model. The variance component for the random slope is
smaller than its standard error, however, suggesting that the model picks up most of the
variance in this slope that exists across schools.
[Table 3 about here.]
References
[1] Brambor, T., Clark, W. R., & Golder, M. (2006). Understanding interaction models: Improving
emprical analysis. Political Analysis, 14, 63-82.
21
[2] Ferron, J. (1997). Moving between hierarchical modeling notations. Journal of Educational and
Behavioral Statistics, 22, 119-12.
[3] Hayes, W. L. (1973). Statistics for the Social Sciences. New York: Holt, Rinehart, & Winston
[4] Henderson, C. R. (1953). Estimation of variance and covariance components. Biometrics, 9,
226-252.
[5] Hox, J. J. (1994). Multilevel analysis methods. Sociological Methods & Research, 22, 283-299.
[6] Littell, R. C., Milliken, G. A., Stroup, W. A., Wolfinger, R. D., & Schabenberger, O. (2006).
SAS for Mixed Models, Second Edition. Cary, NC: SAS Institute.
[7] Maxwell, S. E. & Delaney, H. D. (2004). Designing Experiments and Analyzing Data: A Model
Comparison Perspective, Second Edition. Mahwah, NJ: Lawrence Erlbaum Associates, Inc.
[8] Norusis, M. J. (2005). SPSS 14.0 Advanced Statistical Procedures Companion. Upper Saddle,
NJ: Prentice Hall.
[9] Peugh, J. L. & Enders, C. K. (2005). Using the SPSS mixed procedure to fit cross-sectional
and longitudinal multilevel models. Educational and Psychological Measurement, 65, 714-741.
[10] Raudenbush, S.W. & Bryk, A. S. (2002). Hierarchical Linear Models: Applications and Data
Analysis Methods, Second Edition. Newbury Park, CA: Sage.
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and Nonlinear Modeling. Lincolnwood, IL: Scientific Software International.
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and individual growth models. Journal of Educational and Behavioral Statistics, 24, 323-355.
22
[15] Snijders, T. A. B., & Bosker, R. J. (1999). Multilevel Analysis: An Introduction to Basic and
Advanced Multilevel Modeling. Thousand Oaks, CA: Sage.
[16] StataCorp. (2005). Stata Longitudinal/Panel Data Reference Manual, Release 9. College Station, TX: StataCorp LP.
23
Model 1
12.636974
(0.244394)
Model 2
12.128236
(0.199193)
5.332838
(0.368623)
1.225400
(0.3058)
Model 1
8.614025
(1.078804)
Model 2
2.313989
(0.370011)
39.148322
(0.660645)
Model 1
47116.8
47122.79
47143.43
39.161358
(0.660933)
Model 2
46946.5
46956.5
46990.9
MEANSES 01
SECTOR 02
CENTSES 10
MEANSES*CENTSES 11
SECTOR*CENTSES 12
Random Effects
Intercept 00
CENTSES 11
Residual 2
Model Fit Statistics
Deviance
AIC
BIC
24
Model 3
12.127931
(0.199290)
5.3329
(0.369164)
1.226579
(0.306269)
2.945041
(0.155599)
1.039232
(0.298894)
1.642674
(0.239778)
Model 3
2.379579
(0.371456)
0.101216
(0.213792)
36.721155
(0.626134)
Model 3
46714.24
46726.23
46767.51
Model 1
12.63697
(0.2443937)
Model 2
12.12824
(0.1992)
5.332838
(0.3686225)
1.2254
(0.3058)
Model 1
8.614034
(1.078805)
Model 2
2.313986
(0.3700)
39.14832
(0.6606446)
Model 1
47116.8
47120.8
47126.9
39.16136
(0.6609331)
Model 2
46946.5
46950.5
46956.6
MEANSES 01
SECTOR 02
CENTSES 10
MEANSES*CENTSES 11
SECTOR*CENTSES 12
Random Effects
Intercept 00
CENTSES 11
Residual 2
Model Fit Statistics
Deviance
AIC
BIC
25
Model 3
12.12793
(0.1992901)
5.332876
(0.3691648)
1.226578
(0.3062703)
2.94504
(0.1555962)
1.039237
(0.2988895)
1.642675
(0.2397734)
Model 3
2.379597
(0.3714584)
0.1012
(0.2138)
36.7212
(0.660)
Model 3
46503.7
46511.7
46524.0
Model 1
12.6370
(0.2443)
Model 2
12.1282
(0.1992)
5.3328
(0.3686)
1.2254
(0.3058)
Model 1
8.6097
(1.0778)
Model 2
2.3139
(0.3700)
39.1487
(0.6607)
Model 1
47116.8
47120.8
47126.9
39.1614
(0.6609)
Model 2
46946.5
46950.5
46956.6
MEANSES 01
SECTOR 02
CENTSES 10
MEANSES*CENTSES 11
SECTOR*CENTSES 12
Random Effects
Intercept 00
CENTSES 11
Residual 2
Model Fit Statistics
Deviance
AIC
BIC
26
Model 3
12.1279
(0.1993)
5.3329
(0.3692)
1.2266
(0.3063)
2.9450
(0.1556)
1.0392
(0.2989)
1.6427
(0.2398)
Model 3
2.3794
(0.3714)
0.1012
(0.2138)
36.7212
(0.6261)
Model 3
46503.7
46511.7
46524.0