Centering, More Random Coefficients Models
A Foreboding Example – The Heck Chapter 3 Math Data
Cue Ominous Music here
Heck et al. ch3multilevel.sav contains data on 8000+ students.
The interest of the investigators was on the relationship of math
achievement to SES level of the individual child.
The kids sampled were grouped into 400+ schools.
Key variables for our coverage of this example are
math
ses
schcode
public
Math Achievement scores of individual kids.
SES levels of the individual kids.
School code integer from 1 to 419.
Whether the school is public (=1) or private (=0).
The interest for this example is whether the relationship – intercept and
slope - of math to ses is the same for public schools as it is for private.
The Level 1 model is :
mathij = B0j + B1j*ses + eij
The Level 2 model of the intercept is: B0j = g00 + g01*publicj + u0j
In English: The overall Math Achievement level (intercept) is related
to whether the school is a public school or not.
The Level 2 model of the slope is:
B1j = g10 + g11*publicj + u1j
In English: The slope of the relationship of Math Achievement to
SES is related to whether the school is a public school or not.
This is so simple, I can almost do it in my sleep.
What could possibly be wrong with this?
Centering, More Random Coefficients models - 1
Here’s the Combined Model
B0j
B1j
mathij = g00 + g01*publicj + u0j + (g10 + g11*publicj + u1j)*ses + eij
mathij = g00 + g01*publicj + u0j + g10*ses + g11*publicj*ses + u1j*ses + eij
Analysis of the Heck Math Example in R
> modelA <- lme(fixed=math ~ ses + public + ses*public,random = ~ses|schcode,data=heckch3)
Error in lme.formula(fixed = math ~ ses + public + ses * public, random = ~ses |
nlminb problem, convergence error code = 1
message = iteration limit reached without convergence (10)
Argh!! The lme package won’t perform the analysis.
What’s going on?
Let’s try SPSS . . .
Centering, More Random Coefficients models - 2
:
An Analysis of the Heck Math Data in SPSS
Syntax
MIXED math WITH ses public
/FIXED=ses public ses*public | SSTYPE(3)
/METHOD=REML
/PRINT=G SOLUTION TESTCOV
/RANDOM=INTERCEPT ses | SUBJECT(schcode) COVTYPE(UN).
Results
Mixed Model Analysis
We got essentially the
same error message, but
SPSS prints out (possibly
erroneous) results.
Warnings
Iteration was terminated but convergence has not been achieved. The
MIXED procedure continues despite this warning. Subsequent results
produced are based on the last iteration. Validity of the model fit is
uncertain.
Fixed Effects
Estimates of Fixed Effectsa
Parameter
Estimate
Std. Error
Intercept
57.727385
.258527
ses
4.399813
.275427
public
-.014351
.302498
ses * public
-.634932
.320919
a. Dependent Variable: math.
df
338.581
357.776
342.999
335.813
t
223.294
15.975
-.047
-1.978
Sig.
.000
.000
.962
.049
95% Confidence Interval
Lower Bound Upper Bound
57.218865
58.235906
3.858154
4.941473
-.609335
.580633
-1.266198
-.003667
Covariance Parameters
Estimates of Covariance Parametersa
Parameter
Estimate Std. Error
Residual
61.074485 1.067696
Intercept + ses [subject UN (1,1) 3.686412
.598479
= schcode]
UN (2,1) -1.849761
.397749
UN (2,2)
.928426
.571506
a. Dependent Variable: math.
Wald Z
57.202
6.160
-4.651
1.625
Random Effect Covariance Structure (G)a
Intercept |
schcode
3.686412
-1.849761
Intercept | schcode
ses | schcode
Unstructured
a. Dependent Variable: math.
ses | schcode
-1.849761
.928426
Sig.
.000
.000
.000
.104
95% Confidence Interval
Lower
Upper
Bound
Bound
59.017285
63.203393
2.681721
5.067505
-2.629335
-1.070187
.277829
3.102533
-1.849761
r = -------------------------- = -.99986 !!!!!!
3.686412 * .9928426
All the output is here, but there is clearly a problem. The problem is that
the random intercepts are nearly perfectly negatively correlated with
the random slopes. The program cannot estimate them separately.
Centering, More Random Coefficients models - 3
Centering
One way of dealing with the correlated random Intercepts & Slopes
Problem
Consider the following situation.
The relationship of Y to X within each of 5 groups is examined.
Y is positively related to X within each group.
The Xs have mean = 5 within each group.
The only difference between the groups is that the slope of the relationship
changes.
Centering, More Random Coefficients models - 4
Here are scatterplots of the relationships within the 5 groups.
The Intercepts and slopes of
regressions within each group are
Intercept
-5.31
Slope
1.01
-9.91
1.94
-15.26
3.08
-19.89
3.98
-26.68
5.30
The relationship of intercept to
slopes across the 5 groups is
Note that the average height of the points in each scatterplot is the same.
So, if we’re using the intercepts as an indication of “how big the Y values
are, in general,” that’s not going to work.
Centering, More Random Coefficients models - 5
The intercepts in these scatterplots are suggesting that the Y values are
getting smaller as the slope gets larger.
Now here are the same data, except that the Xs have been centered so that
the mean of the Xs within each group is about 0.
The Intercepts and slopes of
regressions within each group are
Intercept
-.28
Slope
1.01
-.20
1.94
.13
3.08
.03
3.98
-.16
5.30
The relationship of intercept to
slopes across the 5 groups is
Note that when Xs are centered so that the group mean is 0 for each group,
there is essentially no relationship between slopes and intercepts.
Centering, More Random Coefficients models - 6
Another way of illustrating this is to plot all the groups on the same graph.
Here are the uncentered data. I’ve put a vertical line over X=0.
Neither centered data nor
uncentered
data
are
inherently “best”.
Here’s the centered data.
If there is a meaningful
X=0, then you should
respect it.
On the other hand, if the X
values are interval-scaled,
so that the 0 point is
arbitrary, then centering
may permit the program to
converge
or
prevent
outlandish estimates that
might occur for uncentered
data.
Generally, if you want the intercepts to represent “average” group
performance, you should center the data.
Centering, More Random Coefficients models - 7
The Effect of Centering on the Achieve Data
Recall the Achieve Data from the previous lecture
The file contains data on 10,320 students. Measures include demographic
information.
Of interest here are the following variables . . .
school
senroll
gevocab
geread
The number of the school (out of 160) attended by the student.
The size of the school attended by the student
A measure of student vocabulary.
A measure of reading ability.
Our Previous Application – a Random Coefficients Model
The Level 1 Model:
gereadij = B0j + B1j*gevocabij + eij
The Level 2 Intercept Model:
B0j = g00 + g01*senroll + u0j
In English: The overall level of Reading Ability (intercept) is related
to School Enrollment.
The Level 2 Slope Model:
B1j = g10 + g11*senroll + u1j
In English: The slope relating Reading Ability to Vocabulary Scores
depends on School Enrollment.
The Combined Model
gereadij = g00 + g01*senroll + u0j + (g10 + g11*senroll + u1j)*gevocabij + eij
gereadij = g00 + g01*senroll + u0j + g10*gevocabij + g11*senroll*gevocabij + u1j*gevocabij + eij
Centering, More Random Coefficients models - 8
Creating Centered Variables in R
R  Rcmdr  Import Data  Achieve.sav
> attach (achieve)
Error in attach(achieve) : object 'achieve' not found
> attach (Achieve)
> cgevocab = gevocab - mean(gevocab)
You cannot refer to variables
> csenroll = senroll - mean(senroll)
within a dataset at the R
> mean (gevocab)
[1] 4.493844
command prompt unless you
> mean (cgevocab)
first “attach” the dataset. You
[1] -4.112862e-16
> mean (senroll)
must also spell the name of the
[1] 533.4148
dataset correctly.
> mean (csenroll)
[1] 8.308599e-16
>
Centering, More Random Coefficients models - 9
Achieve data: Comparison of Uncentered vs Centered predictors in R
First, not centered.
> notcent <- lme(fixed = geread ~ gevocab + senroll +
gevocab*senroll,random = ~gevocab|school,data=Achieve)
> noncent
Error: object 'noncent' not found
> notcent
Linear mixed-effects model fit by REML
Data: Achieve
Log-restricted-likelihood: -21512.29
Fixed: geread ~ gevocab + senroll + gevocab * senroll
(Intercept)
gevocab
senroll gevocab:senroll
1.910363e+00
5.430546e-01
1.886407e-04
-4.523914e-05
Random effects:
Formula: ~gevocab | school
Structure: General positive-definite, Log-Cholesky parametrization
StdDev
Corr
(Intercept) 0.5327945 (Intr)
gevocab
0.1391362 -0.857
Residual
1.9146996
Number of Observations: 10320
Number of Groups: 160
Now, centered.
> cent <- lme(fixed = geread ~ cgevocab + csenroll +
cgevocab*csenroll,random = ~cgevocab|school,data=Achieve)
> cent
Linear mixed-effects model fit by REML
Data: Achieve
Log-restricted-likelihood: -21512.3
Fixed: geread ~ cgevocab + csenroll + cgevocab * csenroll
(Intercept)
cgevocab
csenroll cgevocab:csenroll
4.342972e+00
5.189304e-01
-1.464681e-05
-4.522354e-05
Random effects:
Formula: ~cgevocab | school
Structure: General positive-definite, Log-Cholesky parametrization
StdDev
Corr
Note that the correlation of random
(Intercept) 0.3220228 (Intr)
cgevocab
0.1391599 0.523
intercepts with random slopes has
Residual
1.9146941
Number of Observations: 10320
Number of Groups: 160
changed to a positive value. Generally,
zero or positive correlations are preferred.
Centering, More Random Coefficients models - 10
What about the Ch3multilevel Data???
Let’s try to fix the analysis of the Ch3 multilevel data using centering.
R  Rcmdr  load packages  Import Data . . .  Ch3multilevel
I’ll try a simple random coefficients model, without any predictor of the
slope.
Combined model is
mathij = g00 + u0j + g10*cses + u1j*cses + eij
> attach (ch3mult)
> mean (ses)
[1] 0.03190359
> cses <- ses - mean(ses)
> mean (cses)
[1] 3.148707e-18
> cent1 <- lme(fixed=math ~ cses, random = ~cses|schcode,data=ch3mult)
Error in lme.formula(fixed = math ~ cses, random = ~cses | schcode, data = ch3mult) :
nlminb problem, convergence error code = 1
message = iteration limit reached without convergence (10)
Really!!???
Centering was suppose to fix that problem. Let’s try SPSS.
Centering, More Random Coefficients models - 11
SPSS: Simple random coefficients model with centered predictor
Syntax
means ses.
compute cses = ses - .0319.
First, I used the MEANS command to get the
mean SES.
MIXED math WITH cses /FIXED=cses | SSTYPE(3) /METHOD=REML
/PRINT=G SOLUTION TESTCOV /RANDOM=INTERCEPT cses | SUBJECT(schcode)
COVTYPE(UNR).
Note that I used COVTYPE (UNR)
This requests SPSS to print the correlation between the random
intercepts and random slopes, so I don’t have to compute it by hand.
Output
Warnings
Iteration was terminated but convergence has not been achieved. The
MIXED procedure continues despite this warning. Subsequent results
produced are based on the last iteration. Validity of the model fit is
uncertain.
Estimates of Fixed Effectsa
Parameter
Estimate
Intercept
1294.869881
cses
3.878602
a. Dependent Variable: math.
Std. Error
43.610618
.136720
df
3885.026
3885.832
t
29.692
28.369
Sig.
.000
.000
95% Confidence Interval
Lower Bound
Upper Bound
1209.368003
1380.371758
3.610551
4.146652
Estimates of Covariance Parametersa
95% Confidence Interval
Lower
Bound
Upper Bound
60.893169
65.274099
.097123 120.020143
.
.
-1.000000
1.000000
Parameter
Estimate
Std. Error
Wald Z
Sig.
Residual
63.045592 1.117380
56.423
.000
Intercept + cses [subject Var(1)
3.414197 6.200918
.551
.582
= schcode]
Var(2)
1.891843E-7b
.000000
.
.
Corr(2,1)
.036367 12.010518
.003
.998
a. Dependent Variable: math.
b. This covariance parameter is redundant. The test statistic and confidence interval cannot be computed.
Random Effect Covariance Structure (G)a
Intercept |
schcode
3.414197
2.922730E-5
Intercept | schcode
cses | schcode
Unstructured Correlations
a. Dependent Variable: math.
cses | schcode
2.922730E-5
1.891843E-7
Whoa!! There is something wrong with this dataset. The correlation of
random intercepts and slopes is nearly zero. But there is still a problem.
I’m not sure what it is.
Centering, More Random Coefficients models - 12
Working around problems with complete random coefficients models
The initial problem was: Does the slope of the math-to-ses relationship
depend on whether the school is public or private?
The Level 1 model was:
mathij = B0j + B1j*ses + eij
The Level 2 model of the intercept is: B0j = g00 + g01*publicj + u0j
In English: The overall Math Achievement level (intercept) is related
to whether the school is a public school or not.
The Level 2 model of the slope is:
B1j = g10 + g11*publicj + u1j
In English: The slope of the relationship of Math Achievement to
SES is related to whether the school is a public school or not.
We’ve discovered we cannot estimate the variance of u1j along with the
variance of u0j using R. SPSS will estimate it but warns us of possible
problems.
What to do???
Let’s try dropping the assumption that u1j varies from group to group.
Let’s make the Level 2 model of the slope: B1j = g10 + g11*publicj
It kind of violates the whole notion of random coefficients, but we’ll at
least get a vague idea of whether or not slopes vary systematically from
public to private schools.
Centering, More Random Coefficients models - 13
The (semi-) random coefficients model of Ch3multilevel in R
> cent2 <- lme(fixed=math ~ cses + public + cses*public, random = ~1|schcode,data=ch3mult)
> summary (cent2)
Linear mixed-effects model fit by REML
Data: ch3mult
AIC
BIC
logLik
48225.35 48266.35 -24106.67
Random effects:
Formula: ~1 | schcode
(Intercept) Residual
StdDev:
1.867186 7.923816
Fixed effects: math ~ cses + public + cses * public
Value Std.Error
DF
t-value p-value
(Intercept) 57.77746 0.2557442 6449 225.91894 0.0000
cses
4.27266 0.2627613 6449 16.26063 0.0000
public
-0.06969 0.2988403 6449 -0.23320 0.8156
cses:public -0.54758 0.3076524 6449 -1.77987 0.0751
SPSS Syntax
MIXED math WITH cses public /FIXED=cses public cses*public| SSTYPE(3) /METHOD=REML
/PRINT=G SOLUTION TESTCOV /RANDOM=INTERCEPT | SUBJECT(schcode) COVTYPE(UNR).
SPSS Output excerpts
Estimates of Fixed Effectsa
95% Confidence Interval
Parameter
Estimate
Std. Error
df
t
Sig.
Lower Bound
Upper Bound
Intercept
57.777447
.255743
382.622
225.920
.000
57.274608
58.280285
cses
4.272668
.262761
4203.178
16.261
.000
3.757517
4.787818
public
-.069690
.298839
385.813
-.233
.816
-.657247
.517868
cses * public
-.547581
.307652
4102.788
-1.780
.075
-1.150746
.055584
a. Dependent Variable: math.
Estimates of Covariance Parametersa
95% Confidence Interval
Parameter
Estimate
Std. Error
Residual
62.786894
1.108639
56.634
3.486328
.541221
6.442
Intercept [subject =
Variance
Wald Z
Sig.
Lower Bound
Upper Bound
.000
60.651170
64.997823
.000
2.571739
4.726173
schcode]
a. Dependent Variable: math.
The program ran without an error message. The Relationship of math to
ses is NOT officially moderated by whether or not the school is a public or
private school.
Centering, More Random Coefficients models - 14
Even though public does NOT moderate the math <- ses relationship,
The p-value is nearly <= .05. On the assumption that it might be
significant, if we could perform the appropriate analysis . . .
Graphical description of the almost significant interaction of public
and ses . . .
Private
Public
Centering, More Random Coefficients models - 15
If there were random variability in the slopes
Centering, More Random Coefficients models - 16
Example – The PUPCROSS data
The data are on 1000 pupils from different school systems.
Variables are
achiev
pupses
pupsex
pschool
pdenom
Individual student achievement score:the outcome variable
Individual student SES level
0=boy; 1=girl
Code representing the primary school
Denomination of the primary school (1=Prot; 0=non)
What would be the research question here?
Oh, I don’t know. How about, “Is Achievement related to SES?”
That would make the Level 1 model: achievij = B0j + B1j*sesij + eij
Suppose we were interested in the effect of school denomination (pdenom)
on the relationship of achievement to ses.
Specifically, suppose we were interested in the effect of denomination on
both the intercept and the slope of the achieve  ses relationship.
Level 2 model of intercept:
B0j = g00 + g01*pdenom + u0j
Level 2 model of slope:
B1j = g10 + g11*pdenom + u1j
(Let’s hope this dataset will allow estimation of both random
intercepts and random slopes.)
Centering, More Random Coefficients models - 17
Rules for specifying models to R and SPSS
1. All Level 1 predictors are entered in the fixed specification.
2. All Level 2 predictors of intercepts are entered in the fixed
specification.
3. All Level 2 predictors of slopes are entered both individually in the
fixed specification and as interactions with Level 1 predictors (L2
Predictor * L1 Predictor) in the fixed specification.
4. Specify whether intercepts are random and whether level 1 slopes are
random.
Here’s how this applies to this problem
Level 1 Model:
achievij = B0j + B1j*sesij + eij
Level 2 Intercept Model:
B0j = g00 + g01*pdenom + u0j
Level 2 Slope Model:
B1j = g10 + g11*pdenom + u1j
Predictors entered in the fixed specification: ses, pdenom, pdenom*ses
So, in R
Lme(fixed = achiev ~ ses + pdenom + pdenom*ses,
random = ~ses|pschool,data=???)
In SPSS
MIXED achieve with ses pdenom /fixed = ses pdenom pdenom*ses
/random = intercept ses |pschool.
Centering, More Random Coefficients models - 18
PUPCROSS Analysis in R
R  rcmdr  nlme  Import Data  pupcross.sav
> attach (pupcross)
> cpupses = pupses - mean(pupses)
> mean (cpupses)
[1] 1.347746e-16
> mean (pupses)
[1] 4.098
> res1 <- lme(fixed = achiev ~ pupses pdenom cpupses cpupses*pdenom,random=
~cpupses|pschool,data=pupcross)
Error: unexpected symbol in "res1 <- lme(fixed = achiev ~ pupses pdenom"
Duh! I forgot to put + between predictors.
Centering, More Random Coefficients models - 19
> res3 <- lme(fixed = achiev ~ cpupses + pdenom + cpupses*pdenom,
random = ~cpupses|pschool,data=pupcross)
> summary (res3)
Linear mixed-effects model fit by REML
Data: pupcross
AIC
BIC
logLik
2353.62 2392.85 -1168.81
Random effects:
Formula: ~cpupses | pschool
Structure: General positive-definite, Log-Cholesky parametrization
StdDev
Corr
(Intercept) 0.4056005 (Intr)
cpupses
0.1085470 0.537
Residual
0.7281753
Fixed effects: achiev ~ cpupses + pdenom + cpupses * pdenom
Value Std.Error DF t-value p-value
(Intercept)
6.240868 0.09772229 948 63.86330 0.0000
cpupses
0.119281 0.03641824 948 3.27531 0.0011
pdenom
0.199885 0.12650984 48 1.57999 0.1207
cpupses:pdenom -0.006046 0.04692968 948 -0.12883 0.8975
SPSS –
MIXED achiev WITH cpupses pdenom /FIXED=cpupses pdenom cpupses*pdenom | SSTYPE(3)
/METHOD=REML /PRINT=G SOLUTION TESTCOV
/RANDOM=INTERCEPT cpupses | SUBJECT(pschool)COVTYPE(UNR).
Estimates of Fixed Effectsa
Parameter
Estimate
Std. Error
df
t
Sig.
6.241107
Intercept
.097746
47.604
63.850
.000
.119281
cpupses
.036418
49.783
3.275
.002
.199873
pdenom
.126541
48.109
1.580
.121
-.006046
cpupses * pdenom
.046930
49.896
-.129
.898
a. Dependent Variable: achiev Outcome: achievement score in secundary school.
95% Confidence Interval
Lower Bound
Upper Bound
6.044532
6.437681
.046125
.192437
-.054540
.454285
-.100312
.088220
Estimates of Covariance Parametersa
95% Confidence Interval
Estimate Std. Error Wald Z
Sig.
Lower Bound Upper Bound
.530239
.000
.024875
21.316
.483659
.581306
.164606
.000
Var(1)
.039156
4.204
.103268
.262379
.011782
.022
Var(2)
.005126
2.298
.005022
.027643
.537477
.005
Corr(2,1)
.193083
2.784
.068325
.811965
a. Dependent Variable: achiev Outcome: achievement score in secundary school.
Parameter
Residual
Intercept + cpupses
[subject = pschool]
Achiev is positively related to SES.
Achiev is not related to whether or not the school is denominational.
The slope of the achiev  ses relationship is the same across
denominations.
Variances of u0j and u1j are significantly larger than 0.
Correlation of u0j with u1j is positive: r= .537.
Centering, More Random Coefficients models - 20
One Final Example from Heck Ch3multilevel
This is Model 5 described beginning on page 115
The data are 8000+ students from 400+ school.
Key variables are
math
ses
public
per4yrc
ses_mean
Math achievement scores of individual students
SES levels of individual students
Whether the school is a public school or not
The proportion of students in the school going on to 4 yr
college
Mean SES scores of students in each school
The Level 1 model is mathij = B0j + B1j*sesj + eij
The authors investigated the effect of the 3 school level variables – public,
per4yrc, and ses_mean – on the intercepts and slopes of the math  ses
relationship.
We’ll do the same.
The intercept model: B0j =
The slope model:
g00
+ g01*public
+ g02*per4yrc
+ g03*ses_mean
+ u0j
B1j =
g10
+ g11*public
+ g12*per4yrc
+g13*ses_mean
+ u1j
Note that we will NOT estimate a random slope component.
Centering, More Random Coefficients models - 21
The combined model
mathij =
B0j
B1j*ses
g00
+ g01*public
+ g02*per4yrc
+ g03*ses_mean
+ u0j
+ g10*ses
+ g11*public*ses
+g12*per4yrc*ses
+g13*ses_mean*ses
+eij
From the rules for specifying an analysis
1. All Level 1 predictors are entered in the fixed specification.
2. All Level 2 intercept predictors variables are entered into the fixed
specification.
3. All Level 2 slope model predictors are entered individually and as
products of the Level 1 predictor(s) into the fixed specification.
4. Appropriate random effects are specified
So
Rule 1:
Rule 2:
Rule 3:
Rule 4:
ses
public, per4yrc, ses_mean
public*ses, per4yrc*ses, ses_mean*ses
Intercept only since this dataset won’t allow estimation of
both random intercepts and random slopes
Centering, More Random Coefficients models - 22
Heck Model 5 in R
> res4 <- lme(fixed=math~ses+public+per4yrc+ses_mean+public*ses+per4yrc*ses+ses_mean*ses,
random=~1|schcode,data=ch3mult)
> summary res4
Error: unexpected symbol in "summary res4"
> summary (res4)
Linear mixed-effects model fit by REML
Data: ch3mult
AIC
BIC
logLik
48144.92 48213.26 -24062.46
Random effects:
Formula: ~1 | schcode
(Intercept) Residual
StdDev:
1.540142 7.913786
Fixed effects: math ~ ses + public + per4yrc + ses_mean + public * ses + per4yrc *
ses + ses_mean * ses
Value Std.Error
DF
t-value p-value
(Intercept) 56.48396 0.4904389 6447 115.17024 0.0000
ses
3.77109 0.5612092 6447
6.71958 0.0000
public
-0.16420 0.2757663 6447 -0.59544 0.5516
per4yrc
1.41812 0.4848464 416
2.92489 0.0036
ses_mean
2.51952 0.3189236 416
7.90007 0.0000
ses:public
-0.61861 0.3034719 6447 -2.03844 0.0415
ses:per4yrc -0.15683 0.5465211 6447 -0.28696 0.7742
ses:ses_mean -0.11516 0.2703184 6447 -0.42600 0.6701
Correlation:
(Intr) ses
public pr4yrc ses_mn ss:pbl ss:pr4
ses
0.130
public
-0.421 0.043
per4yrc
-0.869 -0.172 0.013
ses_mean
0.263 -0.179 -0.054 -0.255
ses:public
0.044 -0.436 0.003 -0.046 0.030
ses:per4yrc -0.177 -0.875 -0.052 0.224 0.026 0.048
ses:ses_mean -0.128 0.320 0.019 0.007 -0.271 -0.123 -0.294
Standardized Within-Group Residuals:
Min
Q1
Med
Q3
-3.7697122 -0.5528381 0.1317089 0.6563923
Max
5.7714576
Number of Observations: 6871
Number of Groups: 419
Centering, More Random Coefficients models - 23
SPSS Results
Syntax
MIXED math WITH ses ses_mean public per4yrc
/FIXED=ses ses_mean public per4yrc ses*ses_mean ses*public ses*per4yrc | SSTYPE(3)
/METHOD=REML /PRINT=G SOLUTION TESTCOV
/RANDOM=INTERCEPT | SUBJECT(schcode) COVTYPE(UNR).
Estimates of Fixed Effectsa
95% Confidence Interval
Parameter
Estimate
Std. Error
df
t
Sig.
Lower Bound
Upper Bound
Intercept
56.483964
.490436
460.529
115.171
.000
55.520194
57.447735
ses
3.771100
.561209
5039.371
6.720
.000
2.670886
4.871313
ses_mean
2.519516
.318922
765.571
7.900
.000
1.893450
3.145582
public
-.164201
.275765
405.815
-.595
.552
-.706307
.377904
per4yrc
1.418119
.484844
452.165
2.925
.004
.465293
2.370946
ses * ses_mean
-.115155
.270318
1728.651
-.426
.670
-.645339
.415029
ses * public
-.618609
.303472
3848.811
-2.038
.042
-1.213590
-.023629
ses * per4yrc
-.156836
.546521
4965.697
-.287
.774
-1.228258
.914586
a. Dependent Variable: math.
Estimates of Covariance Parametersa
95% Confidence Interval
Parameter
Estimate
Std. Error
Residual
62.628058
1.103449
56.757
2.371965
.444817
5.332
Intercept [subject =
Variance
Wald Z
Sig.
Lower Bound
Upper Bound
.000
60.502253
64.828556
.000
1.642411
3.425586
schcode]
a. Dependent Variable: math.
Math achievement scores were positively related to individual student ses,
to the average ses in the student’s school, and to the proportion of students
in the student’s school going on to 4 year colleges.
Students in public schools were not significantly different on average from
those in private schools.
However, the relationship of math to ses was shallower for public schools.
There was significant variability about predicted individual values and of
the general level of students in the schools (intercept variation).
Centering, More Random Coefficients models - 24