7 Example 2: Using SAS and SPSS to Conduct Three-Level Multivariate Analysis
Tải bản đầy đủ
596
â†œæ¸€å±®
â†œæ¸€å±®
Multivariate Multilevel Modeling
within-school dependence into account by adding a third level—the school level—to
the multilevel model. Further, instead of including only the treatment variable in the
model, we include other explanatory variables, including student gender, student
pretest knowledge, a school average of these pretest scores, and a treatment-gender
productÂ€term.
There is one primary hypothesis underlying these analyses. That is, while treatment
effects are expected to be present for intention and knowledge for both boys and girls,
boys are expected to derive greater benefit from the computer-based instruction. The
reason for this extra impact of the intervention, we assume, is that fifth-grade boys will
enjoy playing the instructional video game more than girls. As a result, the impact that
the experimental program has for intention and knowledge will be greater for boys
than girls. Thus, the investigators hypothesize the presence of a treatment-by-gender
interaction for both outcomes, where the intervention will have stronger effects on
intention and knowledge for boys than for girls.
In addition, because the cluster randomized trial with this limited number of schools (i.e.,
40) does not generally provide for great statistical power, knowledge pretest scores were
collected from all students. These scores are expected to be fairly strongly associated with
both outcomes. Further, because associations may be stronger at the school level than at the
student level, the researchers computed school averages of the knowledge pretest scores
and plan to include this variable in the model to provide for increased power.
Three MVMM analyses are illustrated next. The first analysis includes the treatment variable as the sole explanatory variable. The purpose of this analysis is to obtain a preliminary
estimate of the treatment effect for each outcome. The second analysis includes all of the
explanatory variables as well as the treatment-by-gender interaction. The primary purpose
of this analysis is to test the hypothesized interactions. If the multivariate test for the interaction is significant, the analysis will focus on examining the treatment-by-gender interaction for each outcome, and if significant, describing the nature of any interactions obtained.
The third analysis will illustrate a multivariate test for multiple variance and covariance
elements. Often, in practice, it is not clear if, for example, the association between a student
explanatory variable and outcome is the same or varies across schools. Researchers may
then rely on empirical evidence (e.g., a statistical test result) to address this issue.
14.7.1â•‡ AÂ€Three-Level Model for Treatment Effects
For this first analysis, EquationÂ€1, which had previously been the level-1 model, needs
to be modified slightly in order to acknowledge the inclusion of the school level. The
level-1 model nowÂ€is:
Yijk = π1 jk a1 jk + π 2 jk a2 jk ,
(6)
which is identical to EquationÂ€1 except that subscript k has been added. Thus, π1jk and
π2jk represent the intention and knowledge posttest scores, respectively, for a given
Chapter 14
â†œæ¸€å±®
â†œæ¸€å±®
student j who is attending a given school k. The second- or student-level of the model,
with no explanatory variables included, isÂ€then
π1 jk = β10 k + r1 jk (7)
π 2 jk = β 20 k + r2 jk ,
(8)
where β10k and β20k represent the mean for a given school k for intention and knowledge, respectively. The student-level or within-school residual terms (r1jk and r2jk) are
assumed to follow a bivariate normal distribution, with an expected mean of zero, variances (τπ1 and τπ2), and covariance (τπ12). Since treatment assignment varies across and
not within schools, the treatment indicator variable (coded −.5 and .5 for control and
experimental schools, respectively) appears in the school-level model. This third- or
school-level modelÂ€is
β10 k = γ 100 + γ 101Treatk + u10 k
(9)
β 20 k = γ 200 + γ 201Treatk + u20 k , (10)
where γ100 and γ200 represent the overall average for intention and knowledge, respectively. The key parameters are γ101 and γ201, which represent the differences in
means between the experimental and control groups for intention and knowledge. The
school-level residual terms are u10k and u20k, which are assumed to follow a bivariate
normal distribution with an expected mean of zero and constant variances (τβ11 and
τβ22), and covariance (τβ12).
The software commands for reorganizing a data set given in TableÂ€14.4 can be used
here to change the data set from the wide to the needed long format. Note that the Keep
commands in TableÂ€14.4 should be modified to also include variables gender, pretest,
meanpretest, and TXG, which are used in subsequent analyses. TableÂ€14.10 shows
some cases for the reorganized data set that is needed for this section.
The variables in this data set include a school and student id, the index variable identifying the response as Y1 or Y2, response containing the scores for the outcomes,
treatment (with −.5 for the control group and .5 for the experimental group), gender (with −.5 indicating female and .5 male), pretest knowledge, meanpretest, and a
treatment-by-gender product variable (denoted TXG), which is needed to model the
interaction of interest. To ensure that the output you obtain will correspond to that in
the text, all variables except the index and id variables should appear as continuous
variables in the dataÂ€set.
The model described in EquationsÂ€ 6–10 has four fixed effects (the four γs) and six
variance-covariance elements, for a total of 10 parameters. As shown in TableÂ€14.11,
we build upon the SAS and SPSS commands for the two-level models to estimate
these parameters and present selected results in TableÂ€14.12. Note that the R matrix is
597
598
â†œæ¸€å±®
â†œæ¸€å±®
Multivariate Multilevel Modeling
Table 14.10:â•‡ Selected Cases Showing Variables in Long Format for Three-Level Models
Mean
Record School Student Index1 Response Treat Gender Pretest Pretest TXG
1
2
3
4
5
6
7
8
1
1
1
1
1
1
1
1
1
1
2
2
3
3
4
4
1
2
1
2
1
2
1
2
29
47
52
50
42
36
47
64
−.50
−.50
−.50
−.50
−.50
−.50
−.50
−.50
−.50
−.50
.50
.50
−.50
−.50
.50
.50
48
48
52
52
41
41
63
63
46
46
46
46
46
46
46
46
.25
.25
−.25
−.25
.25
.25
−.25
−.25
1599
1600
40
40
800
800
1
2
66
50
.50
.50
.50
.50
41
41
53
53
.25
.25
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
the variance-covariance matrix for the student-level residuals, and the G matrix is the
variance-covariance matrix for the school-level residuals.
In TableÂ€14.12, the outputs (in the fixed effects tables) show that students scored higher,
on average, for both intentions (γ101Â€=Â€8.72, p < .05) and knowledge (γ201Â€=Â€8.63, p <
.05) when they were exposed to the experimental nutritional educational program. In
addition, after taking treatment membership into account, most of the posttest score
variability is within schools, as the proportion of remaining variability that is between
schools for intention is about .03 (i.e., 3.31 / (3.31 + 97.62)) and .08 (i.e., 8.74 / (8.74
+ 100.56)) for knowledge. Note that estimates for the variances and covariances appear
in the covariance parameter tables and in the R and G matrices of TableÂ€14.12. The
correlation among the residuals, which can be calculated manually, at the student level
is .40 and at the school level is .38. Note that if desired, an empty model omitting the
treatment variable from EquationsÂ€9 and 10 could be estimated prior to this model. If
that were done, model deviances could be compared as in sectionsÂ€14.6.1 and 14.6.2
to test the overall multivariate null hypothesis of no treatment effect. AÂ€test of model
deviances will be used to provide a multivariate test of the interaction of interest.
14.7.2â•‡ AÂ€Three-Level Model With Multiple Predictors
In the second analysis, all explanatory variables are included and the multivariate null
hypothesis of no treatment-by-gender interaction for any of the outcomes is tested.
For this analysis, EquationÂ€6 remains the level-1 model. The student-level model
is modified to include gender (coded −.5 for females and .5 for males) and pretest,
which is group-mean centered. For the remaining models in this chapter, variable
names, instead of symbols, are used to ease understanding of the models. Thus, the
student-level modelÂ€is
Table 14.11:â•‡ SAS and SPSS Control Lines for Estimating the Three-Level Model With
Treatment Effects
SAS
SPSS
PROC MIXED DATA=LONG METHOD=ML
COVTEST;
CLASS INDEX1 STUDENT SCHOOL;
(1)
MODEL RESPONSEÂ€=Â€INDEX1
TREAT*INDEX1 / NOINT SOLUTION;
Â�
RANDOM INDEX1 / SUBJECT=SCHOOL
(2)
TYPE=UN G;
REPEATED INDEX1 / SUBJECTÂ€=Â€
(3)
STUDENT(SCHOOL) TYPE=UN R;
MIXED RESPONSE BY INDEX1 WITH
TREAT/
FIXED=INDEX1 TREAT*INDEX1 |
NOINT/
METHOD=ML/
PRINT=G R SOLUTION TESTCOV/
RANDOM=INDEX1 | SUBJECT
(4)
(SCHOOL) COVTYPE(UN)/
(5)
REPEATED=INDEX1 | SUBJECT
(SCHOOL*STUDENT)
COVTYPE(UN).
(1)â•‡ We add the level-3 unit identifier (SCHOOL) as a CLASS variable.
(2)â•‡The RANDOM statement estimates separate random effects for INDEX1 (i.e., Y1 and Y2) at the SCHOOL
level and displays the corresponding variance-covariance matrix, G matrix.
(3)â•‡ The nesting of level-2 units within level-3 units appears as STUDENT(SCHOOL). The R matrix is the
person-level variance covariance matrix.
(4)â•‡The RANDOM subcommand specifies random effects for Y1 and Y2 at the school level and requests an
unstructured school-level variance-covariance matrix, G matrix.
(5)â•‡ SCHOOL*STUDENT refers to the nesting of level-2 units within level-3 units and specifies an unstructured matrix for the student-level variance-covariance matrix, which is the R matrix.
Table 14.12:â•‡ Selected Output for the Three-Level Model With Treatment Effects
SAS
Solution for Fixed Effects
Effect
Index1
Index1
Treat*Index1
Treat*Index1
Index1
Estimate
Standard
Error
DF
tÂ€Value
Pr > |t|
1
2
1
2
50.0548
50.2909
8.7233
8.6257
0.4525
0.5867
0.905
1.1734
76
76
1520
1520
110.62
85.72
9.64
7.35
<.0001
<.0001
<.0001
<.0001
Estimated R Matrix for Student (School)
Row
Col1
Col2
1
2
97.6186
40.0357
40.0357
100.56
(Continued)
Covariance Parameter Estimates
Cov Parm
Subject
Estimate
Standard Error
Z Value
UN(1,1)
UN(2,1)
UN(2,2)
UN(1,1)
UN(2,1)
UN(2,2)
School
School
School
Student(School)
Student(School)
Student(School)
3.3094
2.0222
8.7416
97.6186
40.0357
100.56
1.8484
1.806
3.0898
5.0077
3.8763
5.1586
1.79
1.12
2.83
19.49
10.33
19.49
Pr Z
0.0367
0.2629
0.0023
<.0001
<.0001
<.0001
Fit Statistics
-2 Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
11813.4
11833.4
11833.5
11850.3
Estimated G Matrix
Row
Effect
Index1
School
Col1
Col2
1
2
Index1
Index1
1
2
1
1
3.3094
2.0222
2.0222
8.7416
SPSS
Estimates of Fixed Effectsa
95% Confidence
Interval
Parameter
Estimate
Std. Error
Df
t
[Index1=1]
[Index1=2]
[Index1=1] * Treat
[Index1=2] * Treat
50.054814
50.290916
8.723254
8.625689
0.452502
0.58672
0.905004
1.17344
40
40
40
40
110.618
85.715
9.639
7.351
a
Lower
Sig. Bound
49.140274
49.105111
6.894173
6.254078
Upper
Bound
50.969355
51.476721
10.552335
10.997299
Dependent Variable: Response.
Estimates of Covariance Parametersa
95% Confidence
Interval
Parameter
Repeated
Measures
Estimate
UN (1,1)
UN (2,1)
97.618558
40.035727
Std. Error
5.007726
3.876274
Wald Z Sig.
19.494
10.328
Lower
Bound
Upper
Bound
88.280884
32.438371
107.943901
47.633084
Table 14.12:â•‡ (Continued)
Estimates of Covariance Parametersa
95% Confidence
Interval
Parameter
Index1
[subject =
School]
a
Estimate
UN (2,2)
UN (1,1)
UN (2,1)
UN (2,2)
Std. Error
100.55996
3.309392
2.022168
8.741618
Wald Z Sig.
5.158614 19.494
1.848448 1.79
1.806048 1.12
3.089764 2.829
0.073
0.263
0.005
Lower
Bound
Upper
Bound
90.940926
1.107421
-1.517621
4.37251
111.196421
9.889709
5.561956
17.476435
Dependent Variable: Response.
Random Effect Covariance Structure (G)a
[Index1=1] | School
[Index1=2] | School
[Index1=1] | School
[Index1=2] | School
3.309392
2.022168
2.022168
8.741618
Unstructured
a
Dependent Variable: Response.
Information Criteriaa
-2 Log Likelihood
Akaike’s Information Criterion (AIC)
Hurvich and Tsai’s Criterion (AICC)
Bozdogan’s Criterion (CAIC)
Schwarz’s Bayesian Criterion (BIC)
11813.380
11833.38
11833.518
11897.157
11887.157
The information criteria are displayed in smaller-is-better forms.
a
Dependent Variable: Response.
Residual Covariance (R) Matrixa
[Index1 = 1]
[Index1 = 2]
Unstructured
a
Dependent Variable: Response.
[Index1 = 1]
[Index1 = 2]
97.618558
40.035727
40.035727
100.55996
602
â†œæ¸€å±®
â†œæ¸€å±®
Multivariate Multilevel Modeling
π1 jk = β10 k + β11k Genderjk + β12 k Pretest jk + r1 jk (11)
π 2 jk = β 20 k + β 21k Genderjk + β 22 k Pretest jk + r2 jk . (12)
The student-level or within-school residual terms (r1jk and r2jk) are assumed to follow
a bivariate normal distribution, with an expected mean of zero, some variance, and a
covariance.
At the school level, each of the regression coefficients in EquationsÂ€11 and 12 may
be considered as outcomes to be modeled. However, the investigators assume that the
association between the pretest and each of the outcomes is the same across schools,
so β12k and β22k are modeled as fixed effects in the school-level model. Also, in order
to model the treatment-by-gender interaction, the treatment variable needs to be added
in the model for β11k and β21k. Further, meanpretest, which is grand-mean centered, is
included in the model for β10k and β20k so that it may serve as a covariate for each outcome. This school-level modelÂ€is
β10 k = γ 100 + γ 101Treatk + γ 102 MeanPretestk + u10 k
(13)
β11k = γ 110 + γ 111Treatk
(14)
β12 k = γ 120
(15)
β 20 k = γ 200 + γ 201Treatk + γ 202 MeanPretestk + u20 k
(16)
β 21k = γ 210 + γ 211Treatk (17)
β 22 k = γ 220 . (18)
Note that there are no residual terms included in the EquationsÂ€14 and 17, which suggests that any systematic between-school variability in male-female performance is due
to the treatment. This assumption is tested in the third analysis. Thus, EquationsÂ€13–18
have two residual terms, u10k and u20k, which are assumed to follow a bivariate normal
distribution with an expected mean of zero and constant variance and covariance.
The focus of this analysis is on the interaction between treatment and gender. Perhaps the best way to recognize which coefficients represent this interaction is to form
equations for Y1 (intention) and Y2 (knowledge), separately. Recall that Y1 is the same
as π1jk in EquationÂ€11, and Y2 is the same as π2jk in EquationÂ€12. Therefore, separate
equations for the outcomes can be formed by replacing each of the β terms on the right
hand side of EquationsÂ€11 and 12 with the expressions for these coefficients found in
EquationsÂ€13–18. Thus, the equations for Y1 and Y2 may be expressedÂ€as
Y 1 = γ 100 + γ 101Treatk + γ 102 MeanPretestk + γ 110 Genderjk + γ 111TXG jk
+ γ 120 Pretestk + u10 k + r1 jk
(19)
Chapter 14
â†œæ¸€å±®
â†œæ¸€å±®
Y 2 = γ 200 + γ 201Treatk + γ 202 MeanPretestk + γ 210Genderjk + γ 211TXG jk
+ γ 220 Pretestk + u20 k + r2 jk .
(20)
From EquationsÂ€19 and 20, the treatment-by-gender product variable (TXG) is readily
recognizable, and the absence of other product terms indicates that no other interactions are included in the model. Thus, γ111 and γ211 represent the treatment-by-gender
interactions (cross-level interactions) for intention and knowledge. Note that while
some software programs (e.g., HLM) would include these cross-level interaction terms
without a user needing to enter the specific product variable, the SAS and SPSS programs require a user to enter this productÂ€term.
Note that in this data set, the number of girls and boys is the same in each of the 40
schools (which is not a requirement of the model). As a result, the use of the coding −.5
and .5 for females and males effectively makes gender a centered variable (centered
within schools). Such centering is useful here because it (1) results in parameters β11k
and β21k of EquationsÂ€11 and 12 reflecting only within-school gender differences on
the outcomes and (2) reduces multicollinearity, given that the product of gender and
treatment appears in the model. Similarly, pretest is also centered within-schools, so
that (1) parameters β12k and β22k of EquationsÂ€11 and 12 represent the within-school
associations of pretest and each of the outcomes and (2) parameters γ102 and γ202 of
EquationsÂ€13 and 16 represent the between-school associations between meanpretest
and each of the outcomes. We also center meanpretest in EquationsÂ€13 and 16, which
while not necessary, is done here so that the intercepts of these equations continue to
represent the means for Y1 and Y2. TableÂ€14.13 shows the SAS and SPSS commands
that can be used to create a group-mean centered student pretest variable, called pretest_cen, and a centered school pretest variable, called meanpretest_cen.
Note that in this model, there are 12 fixed effects, six γs in each of the equations for
Y1 and Y2 and six variance-covariance elements, with three such terms at each of the
student and school levels. To estimate these parameters, we insert additional terms
into the SAS and SPSS commands from TableÂ€14.11. These additions are shown in
TableÂ€14.14, and selected results are presented in TableÂ€14.15.
The multivariate hypothesis of no interaction for the two outcomes can be conducted
by comparing the deviance from the current model to the deviance from the model
that omits the TXG variable from EquationsÂ€19 and 20. Although the results from the
model where both interactions are constrained to be zero (i.e., γ111Â€=Â€γ211Â€=Â€0) are not
shown here, we estimated that model, and its deviance is 11,358.5. Note that this no
interaction model has 16 parameters estimated (i.e., two fewer than the current model
with the removal of TXG from EquationsÂ€19 and 20). As shown in the SAS and SPSS
outputs in TableÂ€14.15, the deviance from the current model is 11,338.9, and there are
18 parameters estimated. The difference in model fit, as reflected by the difference in
model deviances, is then 11,358.5 − 11,338.9Â€=Â€19.6, which is statistically significant
as it exceeds the chi-square critical value of 5.99 (αÂ€=Â€.05, dfÂ€=Â€2). Thus, the statistically
603
604
â†œæ¸€å±®
â†œæ¸€å±®
Multivariate Multilevel Modeling
Table 14.13:â•‡ SAS and SPSS Control Lines for Creating Centered Student Pretest and
School Pretest Variables
SAS
SPSS
CREATING CENTERED PRETEST VARIABLE
DATA LONG; SET LONG;
(1) PRETEST_CEN=PRETESTMEANPRETEST;
OMPUTE PRETEST_CEN=PRETEST
(1) C
— MEANPRETEST.
EXECUTE.
CREATING CENTERED MEANPRETEST VARIABLE
(2) PROC SQL;
(3) CREATE TABLE LONG2 AS
(4)
SELECT *, MEAN
(MEANPRETEST),
MEANPRETEST — MEAN
(5)
MEANPRETEST) as
MEANPRETEST_CEN
(6) FROM LONG
(7) QUIT;
(8) A
GGREGATE/ MEANPRETEST_
MEAN=MEAN(MEANPRETEST).
(5)
COMPUTE MEANPRETEST_
CEN=MEANPRETEST — MEANPRETEST_MEAN.
EXECUTE.
(1)â•‡ We create the group-mean centered variable (PRETEST_CEN) by subtracting the respective school’s
mean (MEANPRETEST) from each student’s pretest score (PRETEST).
(2)â•‡The SQL procedure is just one way to centerÂ€data.
(3)â•‡ The general form for the CREATE statement is CREATE TABLE name of new datasetÂ€AS.
(4)â•‡The SELECT statement includes a SELECT clause and a FROM clause. The * selects all the columns
from the dataset specified in (6) below. The MEAN function calculates the mean of the scores for the variable
within the parentheses (here, school pretest scores or MEANPRETEST).
(5)â•‡ We create a centered variable (MEANPRETEST_CEN) by subtracting the grand mean from each
school’sÂ€mean.
(6)â•‡ The name of the original dataset appears in the FROM clause.
(7)â•‡ QUIT terminates PROCÂ€SQL.
(8)â•‡ We use the AGGREGATE and following subcommand to create a MEANPRETEST score
(MEANPRETEST_MEAN) created for each record.
significant improvement in fit obtained by allowing for treatment-by-gender interactions for both outcomes suggests the presence of a treatment-by-gender interaction for
at least one of the dependent variables.
Examining the outputs for the estimates of the treatment-by-gender interaction for
each outcome in TableÂ€ 14.15 (in the fixed effects tables) shows that the point estimates of the interaction for intention is 4.791 (SEÂ€=Â€1.247) and for knowledge is 3.293
(SEÂ€=Â€1.116). The corresponding t ratios, 3.84 and 2.95, and p values (each smaller
than .05) suggest that the treatment-by-gender interaction is significant for each outcome. To better understand these interactions, we use the LSMEANS statement in
Chapter 14
â†œæ¸€å±®
â†œæ¸€å±®
Table 14.14:â•‡ SAS and SPSS Control Lines for Estimating the Three-Level Model With
All Explanatory Variables and a Treatment-by-Gender Interaction
SAS
PROC MIXED DATA=LONG METHOD=ML
COVTEST;
CLASS INDEX1 STUDENT
SCHOOL;
MODEL RESPONSEÂ€=Â€INDEX1
(1)
TREAT*INDEX1 GENDER*INDEX1
Â�PRETEST_CEN*INDEX1
MEANPRETEST_CEN*INDEX1 TXG*INDEX1 / NOINT SOLUTION;
RANDOM INDEX1 / SUBJECT=
SCHOOL TYPE=UN G;
REPEATED INDEX1 /
SUBJECTÂ€=Â€STUDENT(SCHOOL)
TYPE=UN R;
SPSS
(2) M
IXED RESPONSE BY
INDEX1Â€WITH TREAT GENDER
PRETEST_CEN
MEANPRETEST_CEN TXG /
(1)
FIXED=INDEX1 TREAT*INDEX1
GENDER*INDEX1 PRETEST_
CEN*INDEX1
MEANPRETEST_CEN*INDEX1
TXG*INDEX1 | NOINT/
METHOD=ML/
PRINT=G R SOLUTION
TESTCOV/
RANDOM=INDEX1 | SUBJECT
(SCHOOL) COVTYPE(UN)/
REPEATED=INDEX1 |
SUBJECT(SCHOOL*
STUDENT) COVTYPE(UN).
(1)â•‡ We add GENDER*INDEX1, PRETEST_CEN*INDEX1, MEANPRETEST_CEN*INDEX1, and
TXG*INDEX1 as fixed effects.
(2)â•‡ We include GENDER, PRETEST_CEN, MEANPRETEST_CEN, and TXG as covariates.
SAS and the EMMEANS subcommand in SPSS to obtain the experimental and control
group means for males and females, holding constant the values of the other explanatory variables at their means, as well as tests of the simple effects of the treatment.
TableÂ€14.16 shows the commands from TableÂ€14.14 along with the changes required
for the LSMEANS and EMMEANS commands. Selected output is summarized in
TableÂ€14.17.
For intention, the differences in means between the experimental and control groups
shown in TableÂ€14.17 suggest that the intervention has positive effects for both males
and females, but that this impact is greater for males. Specifically, the treatment effect
for males is 11.00 points and for females is 6.21 points. The extra impact the treatment
provides to males then is 11.00 − 6.21 or 4.79, which is equal to γ111 in EquationÂ€19,
with this additional impact being statistically significant as shown in TableÂ€ 14.15.
As described earlier, while the computer-based intervention is hypothesized to have
greater effects for boys than girls, the investigators also hypothesized that the intervention will have positive effects for both boys and girls. The p-values for the tests of
these simple effects, shown in TableÂ€14.17, suggest that the intervention has a positive impact on intention for both groups. Note that SAS also provides the associated
605
Table 14.15:â•‡ Selected Output for the Three-Level Model With a Treatment-By-Gender
Interaction
SAS
Solution for Fixed Effects
Effect
Index1
Estimate
Standard
Error
DF
tÂ€Value
Pr > |t|
Index1
Index1
Treat*Index1
Treat*Index1
Gender*
Index1
Gender*
Index1
Pretest_
cen*Index1
Pretest_
cen*Index1
MeanPretest_
c*Index1
MeanPretest_
c*Index1
TXG*Index1
TXG*Index1
1
2
1
2
1
50.0548
50.2909
8.6057
8.2513
3.6551
0.4269
0.3409
0.8555
0.6832
0.6243
74
74
1514
1514
1514
117.24
147.51
10.06
12.08
5.85
<.0001
<.0001
<.0001
<.0001
<.0001
2
2.3247
0.5589
1514
4.16
<.0001
1
0.3981
0.03232
1514
12.32
<.0001
2
0.6123
0.02893
1514
21.16
<.0001
1
0.2854
0.1285
1514
2.22
0.0265
2
0.9091
0.1026
1514
8.86
<.0001
1
2
4.7911
3.2932
1.247
1.1163
1514
1514
3.84
2.95
0.0001
0.0032
Covariance Parameter Estimates
Cov Parm
Subject
Estimate
Standard Error
Z Value
Pr Z
UN(1,1)
UN(2,1)
UN(2,2)
UN(1,1)
UN(2,1)
UN(2,2)
School
School
School
Student(School)
Student(School)
Student(School)
3.4041
0.4806
1.5342
77.7445
13.6003
62.3026
1.6425
0.9475
1.0518
3.9882
2.5723
3.1961
2.07
0.51
1.46
19.49
5.29
19.49
0.0191
0.612
0.0723
<.0001
<.0001
<.0001
Estimated G Matrix
Row
Effect
Index1
School
Col1
Col2
1
2
Index1
Index1
1
2
1
1
3.4041
0.4806
0.4806
1.5342
(Continued)