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7 Example 2: Using SAS and SPSS to Conduct Three-Level Multivariate Analysis

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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

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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

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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

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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)

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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

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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

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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)

## 2016 keenan a pituch, james p stevens applied multivariate statistics for the social sciences analyses with SAS and IBMs SPSS routledge (2015)

## 2 Type I Error, Type II Error, and Power

## 2 Addition, Subtraction, and Multiplication of a Matrix by a Scalar

## K-Group MANOVA: A Priori and Post Hoc Procedures

## 14 Power Analysis—A Priori Determination of Sample Size

## 9 MANCOVA—Several Dependent Variables and Several Covariates

## 10 McFadden’s Pseudo R-Square for Strength of Association

## 9 Sample Size for Power = .80 in Single-Sample Case

## 5 Example 1: Examining School Differences in Mathematics

## 8 Example 2: Evaluating the Efficacy of a Treatment

## 6 Example 1: Using SAS and SPSS to Conduct Two-Level Multivariate Analysis

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