Chapter 3a Example: Means by group and time for learning outcome

The MEANS Procedure

Analysis Variable : outcome outcome: Learning Outcome
N Obs Mean Std Error Minimum Maximum
100 53.3395829 0.6351006 37.5335041 68.6163028

Analysis Variable : outcome outcome: Learning Outcome
time: Occasion (1=pre-test,
2=post-test)		 N Obs Mean Std Error Minimum Maximum
1 50 49.9177687 0.7297690 37.5335041 62.1309134
2 50 56.7613972 0.7870204 44.5615778 68.6163028

Analysis Variable : outcome outcome: Learning Outcome
group: Treatment
Group (1=control,
2=treatment)		 N Obs Mean Std Error Minimum Maximum
1 50 51.9879804 0.8943692 37.5335041 67.1060321
2 50 54.6911855 0.8691455 40.5321932 68.6163028


Eq 3a.1: Empty Between-Person Model

The Mixed Procedure

Model Information
Data Set WORK.CHAPTER3A
Dependent Variable outcome
Covariance Structure Variance Components
Subject Effect PersonID
Estimation Method REML
Residual Variance Method Parameter
Fixed Effects SE Method Model-Based
Degrees of Freedom Method Between-Within

Dimensions
Covariance Parameters 1
Columns in X 1
Columns in Z 0
Subjects 50
Max Obs Per Subject 2

Number of Observations
Number of Observations Read 100
Number of Observations Used 100
Number of Observations Not Used 0

Iteration History
Iteration Evaluations -2 Res Log Like Criterion
0 1 651.58043111  
1 1 651.58043111 0.00000000

Convergence criteria met.

Estimated R Matrix for PersonID 1
Row Col1 Col2
1 40.3353  
2   40.3353

Estimated R Correlation Matrix
for PersonID 1
Row Col1 Col2
1 1.0000  
2   1.0000

Covariance Parameter Estimates
Cov Parm Subject Estimate Standard Error Z Value Pr > Z
time PersonID 40.3353 5.7330 7.04 <.0001

Fit Statistics
-2 Res Log Likelihood 651.6
AIC (smaller is better) 653.6
AICC (smaller is better) 653.6
BIC (smaller is better) 655.5

Null Model Likelihood Ratio Test
DF Chi-Square Pr > ChiSq
0 0.00 1.0000

Information Criteria
Neg2LogLike Parms AIC AICC HQIC BIC CAIC
651.6 1 653.6 653.6 654.3 655.5 656.5

Solution for Fixed Effects
Effect Estimate Standard Error DF t Value Pr > |t| Alpha Lower Upper
Intercept 53.3396 0.6351 49 83.99 <.0001 0.05 52.0633 54.6159


Eq 3a.2: Empty Within-Person Model

The Mixed Procedure

Model Information
Data Set WORK.CHAPTER3A
Dependent Variable outcome
Covariance Structure Compound Symmetry
Subject Effect PersonID
Estimation Method REML
Residual Variance Method Profile
Fixed Effects SE Method Model-Based
Degrees of Freedom Method Between-Within

Dimensions
Covariance Parameters 2
Columns in X 1
Columns in Z 0
Subjects 50
Max Obs Per Subject 2

Number of Observations
Number of Observations Read 100
Number of Observations Used 100
Number of Observations Not Used 0

Iteration History
Iteration Evaluations -2 Res Log Like Criterion
0 1 651.58043111  
1 1 646.80947282 0.00000000

Convergence criteria met.

Estimated R Matrix for PersonID 1
Row Col1 Col2
1 40.4590 12.2526
2 12.2526 40.4590

Estimated R Correlation Matrix
for PersonID 1
Row Col1 Col2
1 1.0000 0.3028
2 0.3028 1.0000

Covariance Parameter Estimates
Cov Parm Subject Estimate Standard Error Z Value Pr Z
CS PersonID 12.2526 6.0256 2.03 0.0420
Residual   28.2064 5.6413 5.00 <.0001

Fit Statistics
-2 Res Log Likelihood 646.8
AIC (smaller is better) 650.8
AICC (smaller is better) 650.9
BIC (smaller is better) 654.6

Null Model Likelihood Ratio Test
DF Chi-Square Pr > ChiSq
1 4.77 0.0289

Information Criteria
Neg2LogLike Parms AIC AICC HQIC BIC CAIC
646.8 2 650.8 650.9 652.3 654.6 656.6

Solution for Fixed Effects
Effect Estimate Standard Error DF t Value Pr > |t| Alpha Lower Upper
Intercept 53.3396 0.7260 49 73.47 <.0001 0.05 51.8806 54.7986


Eq 3a.7: Conditional Between-Person Model (top of Eq. 3.7)
Manual Contrasts for Time and Group

The Mixed Procedure

Model Information
Data Set WORK.CHAPTER3A
Dependent Variable outcome
Covariance Structure Variance Components
Subject Effect PersonID
Estimation Method REML
Residual Variance Method Parameter
Fixed Effects SE Method Model-Based
Degrees of Freedom Method Between-Within

Dimensions
Covariance Parameters 1
Columns in X 4
Columns in Z 0
Subjects 50
Max Obs Per Subject 2

Number of Observations
Number of Observations Read 100
Number of Observations Used 100
Number of Observations Not Used 0

Iteration History
Iteration Evaluations -2 Res Log Like Criterion
0 1 602.50704558  
1 1 602.50704558 0.00000000

Convergence criteria met.

Estimated R Matrix for PersonID 1
Row Col1 Col2
1 27.2245  
2   27.2245

Estimated R Correlation Matrix
for PersonID 1
Row Col1 Col2
1 1.0000  
2   1.0000

Covariance Parameter Estimates
Cov Parm Subject Estimate Standard Error Z Value Pr > Z
time PersonID 27.2245 3.9295 6.93 <.0001

Fit Statistics
-2 Res Log Likelihood 602.5
AIC (smaller is better) 604.5
AICC (smaller is better) 604.5
BIC (smaller is better) 606.4

Null Model Likelihood Ratio Test
DF Chi-Square Pr > ChiSq
0 0.00 1.0000

Information Criteria
Neg2LogLike Parms AIC AICC HQIC BIC CAIC
602.5 1 604.5 604.5 605.2 606.4 607.4

Solution for Fixed Effects
Effect Estimate Standard Error DF t Value Pr > |t| Alpha Lower Upper
Intercept 49.0768 1.0435 48 47.03 <.0001 0.05 46.9786 51.1750
time1 5.8224 1.4758 48 3.95 0.0003 0.05 2.8551 8.7896
treat 1.6819 1.4758 48 1.14 0.2601 0.05 -1.2853 4.6492
time1*treat 2.0425 2.0871 48 0.98 0.3327 0.05 -2.1538 6.2389

Type 3 Tests of Fixed Effects
Effect Num DF Den DF Chi-Square F Value Pr > ChiSq Pr > F
time1 1 48 15.56 15.56 <.0001 0.0003
treat 1 48 1.30 1.30 0.2544 0.2601
time1*treat 1 48 0.96 0.96 0.3278 0.3327

Estimates
Label Estimate Standard Error DF t Value Pr > |t| Alpha Lower Upper
Mean: Control Group at Pre-Test 49.0768 1.0435 48 47.03 <.0001 0.05 46.9786 51.1750
Mean: Control Group at Post-Test 54.8992 1.0435 48 52.61 <.0001 0.05 52.8010 56.9973
Mean: Treatment Group at Pre-Test 50.7587 1.0435 48 48.64 <.0001 0.05 48.6606 52.8569
Mean: Treatment Group at Post-Test 58.6236 1.0435 48 56.18 <.0001 0.05 56.5254 60.7218
Time Effect for Control Group 5.8224 1.4758 48 3.95 0.0003 0.05 2.8551 8.7896
Time Effect for Treatment Group 7.8649 1.4758 48 5.33 <.0001 0.05 4.8976 10.8322
Group Effect at Pre-Test 1.6819 1.4758 48 1.14 0.2601 0.05 -1.2853 4.6492
Group Effect at Post-Test 3.7245 1.4758 48 2.52 0.0150 0.05 0.7572 6.6917


Eq 3a.7: Conditional Within-Person Model (bottom of Eq. 3.7)
Manual Contrasts for Time and Group

The Mixed Procedure

Model Information
Data Set WORK.CHAPTER3A
Dependent Variable outcome
Covariance Structure Compound Symmetry
Subject Effect PersonID
Estimation Method REML
Residual Variance Method Profile
Fixed Effects SE Method Model-Based
Degrees of Freedom Method Between-Within

Dimensions
Covariance Parameters 2
Columns in X 4
Columns in Z 0
Subjects 50
Max Obs Per Subject 2

Number of Observations
Number of Observations Read 100
Number of Observations Used 100
Number of Observations Not Used 0

Iteration History
Iteration Evaluations -2 Res Log Like Criterion
0 1 602.50704558  
1 1 544.69899365 0.00000000

Convergence criteria met.

Estimated R Matrix for PersonID 1
Row Col1 Col2
1 27.2245 22.7794
2 22.7794 27.2245

Estimated R Correlation Matrix
for PersonID 1
Row Col1 Col2
1 1.0000 0.8367
2 0.8367 1.0000

Covariance Parameter Estimates
Cov Parm Subject Estimate Standard Error Z Value Pr Z
CS PersonID 22.7794 5.1236 4.45 <.0001
Residual   4.4451 0.9073 4.90 <.0001

Fit Statistics
-2 Res Log Likelihood 544.7
AIC (smaller is better) 548.7
AICC (smaller is better) 548.8
BIC (smaller is better) 552.5

Null Model Likelihood Ratio Test
DF Chi-Square Pr > ChiSq
1 57.81 <.0001

Information Criteria
Neg2LogLike Parms AIC AICC HQIC BIC CAIC
544.7 2 548.7 548.8 550.2 552.5 554.5

Solution for Fixed Effects
Effect Estimate Standard Error DF t Value Pr > |t| Alpha Lower Upper
Intercept 49.0768 1.0435 48 47.03 <.0001 0.05 46.9786 51.1750
time1 5.8224 0.5963 48 9.76 <.0001 0.05 4.6234 7.0214
treat 1.6819 1.4758 48 1.14 0.2601 0.05 -1.2853 4.6492
time1*treat 2.0425 0.8433 48 2.42 0.0193 0.05 0.3469 3.7382

Type 3 Tests of Fixed Effects
Effect Num DF Den DF Chi-Square F Value Pr > ChiSq Pr > F
time1 1 48 95.33 95.33 <.0001 <.0001
treat 1 48 1.30 1.30 0.2544 0.2601
time1*treat 1 48 5.87 5.87 0.0154 0.0193

Estimates
Label Estimate Standard Error DF t Value Pr > |t| Alpha Lower Upper
Mean: Control Group at Pre-Test 49.0768 1.0435 48 47.03 <.0001 0.05 46.9786 51.1750
Mean: Control Group at Post-Test 54.8992 1.0435 48 52.61 <.0001 0.05 52.8010 56.9973
Mean: Treatment Group at Pre-Test 50.7587 1.0435 48 48.64 <.0001 0.05 48.6606 52.8569
Mean: Treatment Group at Post-Test 58.6236 1.0435 48 56.18 <.0001 0.05 56.5254 60.7218
Time Effect for Control Group 5.8224 0.5963 48 9.76 <.0001 0.05 4.6234 7.0214
Time Effect for Treatment Group 7.8649 0.5963 48 13.19 <.0001 0.05 6.6659 9.0639
Group Effect at Pre-Test 1.6819 1.4758 48 1.14 0.2601 0.05 -1.2853 4.6492
Group Effect at Post-Test 3.7245 1.4758 48 2.52 0.0150 0.05 0.7572 6.6917


Eq 3a.7: Conditional Within-Person Model (bottom of Eq. 3.7)
Categorical Predictors for Time and Group

The Mixed Procedure

Model Information
Data Set WORK.CHAPTER3A
Dependent Variable outcome
Covariance Structure Compound Symmetry
Subject Effect PersonID
Estimation Method REML
Residual Variance Method Profile
Fixed Effects SE Method Model-Based
Degrees of Freedom Method Between-Within

Dimensions
Covariance Parameters 2
Columns in X 9
Columns in Z 0
Subjects 50
Max Obs Per Subject 2

Number of Observations
Number of Observations Read 100
Number of Observations Used 100
Number of Observations Not Used 0

Iteration History
Iteration Evaluations -2 Res Log Like Criterion
0 1 602.50704558  
1 1 544.69899365 0.00000000

Convergence criteria met.

Estimated R Matrix for PersonID 1
Row Col1 Col2
1 27.2245 22.7794
2 22.7794 27.2245

Estimated R Correlation Matrix
for PersonID 1
Row Col1 Col2
1 1.0000 0.8367
2 0.8367 1.0000

Covariance Parameter Estimates
Cov Parm Subject Estimate Standard Error Z Value Pr Z
CS PersonID 22.7794 5.1236 4.45 <.0001
Residual   4.4451 0.9073 4.90 <.0001

Fit Statistics
-2 Res Log Likelihood 544.7
AIC (smaller is better) 548.7
AICC (smaller is better) 548.8
BIC (smaller is better) 552.5

Null Model Likelihood Ratio Test
DF Chi-Square Pr > ChiSq
1 57.81 <.0001

Information Criteria
Neg2LogLike Parms AIC AICC HQIC BIC CAIC
544.7 2 548.7 548.8 550.2 552.5 554.5

Solution for Fixed Effects
Effect time1:
Time (0=pre-test,
1=post-test)		 treat:
Treatment
Group
(0=control,
1=treatment)		 Estimate Standard Error DF t Value Pr > |t| Alpha Lower Upper
Intercept     58.6236 1.0435 48 56.18 <.0001 0.05 56.5254 60.7218
time1 0   -7.8649 0.5963 48 -13.19 <.0001 0.05 -9.0639 -6.6659
time1 1   0 . . . . . . .
treat   0 -3.7245 1.4758 48 -2.52 0.0150 0.05 -6.6917 -0.7572
treat   1 0 . . . . . . .
time1*treat 0 0 2.0425 0.8433 48 2.42 0.0193 0.05 0.3469 3.7382
time1*treat 0 1 0 . . . . . . .
time1*treat 1 0 0 . . . . . . .
time1*treat 1 1 0 . . . . . . .

Type 3 Tests of Fixed Effects
Effect Num DF Den DF Chi-Square F Value Pr > ChiSq Pr > F
time1 1 48 263.41 263.41 <.0001 <.0001
treat 1 48 3.65 3.65 0.0560 0.0619
time1*treat 1 48 5.87 5.87 0.0154 0.0193

Least Squares Means
Effect time1:
Time (0=pre-test,
1=post-test)		 treat:
Treatment
Group
(0=control,
1=treatment)		 Estimate Standard Error DF t Value Pr > |t| Alpha Lower Upper
time1*treat 0 0 49.0768 1.0435 48 47.03 <.0001 0.05 46.9786 51.1750
time1*treat 0 1 50.7587 1.0435 48 48.64 <.0001 0.05 48.6606 52.8569
time1*treat 1 0 54.8992 1.0435 48 52.61 <.0001 0.05 52.8010 56.9973
time1*treat 1 1 58.6236 1.0435 48 56.18 <.0001 0.05 56.5254 60.7218

Differences of Least Squares Means
Effect time1:
Time (0=pre-test,
1=post-test)		 treat:
Treatment
Group
(0=control,
1=treatment)		 time1:
Time (0=pre-test,
1=post-test)		 treat:
Treatment
Group
(0=control,
1=treatment)		 Estimate Standard Error DF t Value Pr > |t| Alpha Lower Upper
time1*treat 0 0 0 1 -1.6819 1.4758 48 -1.14 0.2601 0.05 -4.6492 1.2853
time1*treat 0 0 1 0 -5.8224 0.5963 48 -9.76 <.0001 0.05 -7.0214 -4.6234
time1*treat 0 0 1 1 -9.5468 1.4758 48 -6.47 <.0001 0.05 -12.5141 -6.5796
time1*treat 0 1 1 0 -4.1404 1.4758 48 -2.81 0.0072 0.05 -7.1077 -1.1731
time1*treat 0 1 1 1 -7.8649 0.5963 48 -13.19 <.0001 0.05 -9.0639 -6.6659
time1*treat 1 0 1 1 -3.7245 1.4758 48 -2.52 0.0150 0.05 -6.6917 -0.7572

Tests of Effect Slices
Effect time1:
Time (0=pre-test,
1=post-test)		 treat:
Treatment
Group
(0=control,
1=treatment)		 Num DF Den DF F Value Pr > F
time1*treat 0   1 48 1.30 0.2601
time1*treat 1   1 48 6.37 0.0150
time1*treat   0 1 48 95.33 <.0001
time1*treat   1 1 48 173.95 <.0001