Chapter 3b Example: Means by session for RT outcome

The MEANS Procedure

Analysis Variable : rt rt: Response Time in Milliseconds
N Obs Mean Std Error Minimum Maximum
606 1770.70 20.0709378 917.6666667 4159.14

Analysis Variable : rt rt: Response Time in Milliseconds
session: Occasion
(1-6)		 N Obs Mean Std Error Minimum Maximum
1 101 1961.89 54.6804707 1055.40 4159.14
2 101 1815.17 50.6541433 991.2857143 3954.37
3 101 1750.03 48.0684335 992.9259259 4086.14
4 101 1717.80 46.4101014 930.4615385 3611.80
5 101 1707.18 45.8254952 982.7666667 3290.98
6 101 1672.14 44.1344897 917.6666667 3226.00


Eq 3b.10: Between-Person Independent ANOVA

The Mixed Procedure

Model Information
Data Set WORK.CHAPTER3B
Dependent Variable rt
Covariance Structure Variance Components
Subject Effect PersonID
Estimation Method REML
Residual Variance Method Parameter
Fixed Effects SE Method Model-Based
Degrees of Freedom Method Satterthwaite

Dimensions
Covariance Parameters 1
Columns in X 7
Columns in Z 0
Subjects 101
Max Obs Per Subject 6

Number of Observations
Number of Observations Read 606
Number of Observations Used 606
Number of Observations Not Used 0

Iteration History
Iteration Evaluations -2 Res Log Like Criterion
0 1 9155.43252939  
1 1 9155.43252939 0.00000000

Convergence criteria met.

Estimated R Matrix for PersonID 101
Row Col1 Col2 Col3 Col4 Col5 Col6
1 236813          
2   236813        
3     236813      
4       236813    
5         236813  
6           236813

Estimated R Correlation Matrix for PersonID 101
Row Col1 Col2 Col3 Col4 Col5 Col6
1 1.0000          
2   1.0000        
3     1.0000      
4       1.0000    
5         1.0000  
6           1.0000

Covariance Parameter Estimates
Cov Parm Subject Estimate Standard Error Z Value Pr > Z
session PersonID 236813 13672 17.32 <.0001

Fit Statistics
-2 Res Log Likelihood 9155.4
AIC (smaller is better) 9157.4
AICC (smaller is better) 9157.4
BIC (smaller is better) 9160.0

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

Information Criteria
Neg2LogLike Parms AIC AICC HQIC BIC CAIC
9155.4 1 9157.4 9157.4 9158.5 9160.0 9161.0

Solution for Fixed Effects
Effect session: Occasion
(1-6)		 Estimate Standard Error DF t Value Pr > |t| Alpha Lower Upper
Intercept   1672.14 48.4219 600 34.53 <.0001 0.05 1577.04 1767.23
session 1 289.76 68.4789 600 4.23 <.0001 0.05 155.27 424.24
session 2 143.04 68.4789 600 2.09 0.0371 0.05 8.5488 277.52
session 3 77.8986 68.4789 600 1.14 0.2558 0.05 -56.5889 212.39
session 4 45.6604 68.4789 600 0.67 0.5052 0.05 -88.8271 180.15
session 5 35.0397 68.4789 600 0.51 0.6091 0.05 -99.4478 169.53
session 6 0 . . . . . . .

Type 3 Tests of Fixed Effects
Effect Num DF Den DF Chi-Square F Value Pr > ChiSq Pr > F
session 5 600 23.67 4.73 0.0003 0.0003

Least Squares Means
Effect session: Occasion
(1-6)		 Estimate Standard Error DF t Value Pr > |t| Alpha Lower Upper
session 1 1961.89 48.4219 600 40.52 <.0001 0.05 1866.80 2056.99
session 2 1815.17 48.4219 600 37.49 <.0001 0.05 1720.08 1910.27
session 3 1750.03 48.4219 600 36.14 <.0001 0.05 1654.94 1845.13
session 4 1717.80 48.4219 600 35.48 <.0001 0.05 1622.70 1812.89
session 5 1707.18 48.4219 600 35.26 <.0001 0.05 1612.08 1802.27
session 6 1672.14 48.4219 600 34.53 <.0001 0.05 1577.04 1767.23

Differences of Least Squares Means
Effect session: Occasion
(1-6)		 session: Occasion
(1-6)		 Estimate Standard Error DF t Value Pr > |t| Alpha Lower Upper
session 1 2 146.72 68.4789 600 2.14 0.0325 0.05 12.2334 281.21
session 1 3 211.86 68.4789 600 3.09 0.0021 0.05 77.3712 346.35
session 1 4 244.10 68.4789 600 3.56 0.0004 0.05 109.61 378.58
session 1 5 254.72 68.4789 600 3.72 0.0002 0.05 120.23 389.21
session 1 6 289.76 68.4789 600 4.23 <.0001 0.05 155.27 424.24
session 2 3 65.1377 68.4789 600 0.95 0.3419 0.05 -69.3498 199.63
session 2 4 97.3759 68.4789 600 1.42 0.1556 0.05 -37.1116 231.86
session 2 5 108.00 68.4789 600 1.58 0.1153 0.05 -26.4909 242.48
session 2 6 143.04 68.4789 600 2.09 0.0371 0.05 8.5488 277.52
session 3 4 32.2382 68.4789 600 0.47 0.6380 0.05 -102.25 166.73
session 3 5 42.8589 68.4789 600 0.63 0.5316 0.05 -91.6286 177.35
session 3 6 77.8986 68.4789 600 1.14 0.2558 0.05 -56.5889 212.39
session 4 5 10.6207 68.4789 600 0.16 0.8768 0.05 -123.87 145.11
session 4 6 45.6604 68.4789 600 0.67 0.5052 0.05 -88.8271 180.15
session 5 6 35.0397 68.4789 600 0.51 0.6091 0.05 -99.4478 169.53


Eq 3b.10: Univariate Repeated Measures ANOVA

The Mixed Procedure

Model Information
Data Set WORK.CHAPTER3B
Dependent Variable rt
Covariance Structure Compound Symmetry
Subject Effect PersonID
Estimation Method REML
Residual Variance Method Profile
Fixed Effects SE Method Model-Based
Degrees of Freedom Method Satterthwaite

Dimensions
Covariance Parameters 2
Columns in X 7
Columns in Z 0
Subjects 101
Max Obs Per Subject 6

Number of Observations
Number of Observations Read 606
Number of Observations Used 606
Number of Observations Not Used 0

Iteration History
Iteration Evaluations -2 Res Log Like Criterion
0 1 9155.43252939  
1 1 8353.35510068 0.00000000

Convergence criteria met.

Estimated R Matrix for PersonID 101
Row Col1 Col2 Col3 Col4 Col5 Col6
1 236813 202677 202677 202677 202677 202677
2 202677 236813 202677 202677 202677 202677
3 202677 202677 236813 202677 202677 202677
4 202677 202677 202677 236813 202677 202677
5 202677 202677 202677 202677 236813 202677
6 202677 202677 202677 202677 202677 236813

Estimated R Correlation Matrix for PersonID 101
Row Col1 Col2 Col3 Col4 Col5 Col6
1 1.0000 0.8559 0.8559 0.8559 0.8559 0.8559
2 0.8559 1.0000 0.8559 0.8559 0.8559 0.8559
3 0.8559 0.8559 1.0000 0.8559 0.8559 0.8559
4 0.8559 0.8559 0.8559 1.0000 0.8559 0.8559
5 0.8559 0.8559 0.8559 0.8559 1.0000 0.8559
6 0.8559 0.8559 0.8559 0.8559 0.8559 1.0000

Covariance Parameter Estimates
Cov Parm Subject Estimate Standard Error Z Value Pr Z
CS PersonID 202677 29470 6.88 <.0001
Residual   34136 2158.96 15.81 <.0001

Fit Statistics
-2 Res Log Likelihood 8353.4
AIC (smaller is better) 8357.4
AICC (smaller is better) 8357.4
BIC (smaller is better) 8362.6

Null Model Likelihood Ratio Test
DF Chi-Square Pr > ChiSq
1 802.08 <.0001

Information Criteria
Neg2LogLike Parms AIC AICC HQIC BIC CAIC
8353.4 2 8357.4 8357.4 8359.5 8362.6 8364.6

Solution for Fixed Effects
Effect session: Occasion
(1-6)		 Estimate Standard Error DF t Value Pr > |t| Alpha Lower Upper
Intercept   1672.14 48.4219 129 34.53 <.0001 0.05 1576.33 1767.94
session 1 289.76 25.9993 500 11.14 <.0001 0.05 238.68 340.84
session 2 143.04 25.9993 500 5.50 <.0001 0.05 91.9550 194.12
session 3 77.8986 25.9993 500 3.00 0.0029 0.05 26.8173 128.98
session 4 45.6604 25.9993 500 1.76 0.0797 0.05 -5.4209 96.7418
session 5 35.0397 25.9993 500 1.35 0.1784 0.05 -16.0417 86.1211
session 6 0 . . . . . . .

Type 3 Tests of Fixed Effects
Effect Num DF Den DF Chi-Square F Value Pr > ChiSq Pr > F
session 5 500 164.24 32.85 <.0001 <.0001

Least Squares Means
Effect session: Occasion
(1-6)		 Estimate Standard Error DF t Value Pr > |t| Alpha Lower Upper
session 1 1961.89 48.4219 129 40.52 <.0001 0.05 1866.09 2057.70
session 2 1815.17 48.4219 129 37.49 <.0001 0.05 1719.37 1910.98
session 3 1750.03 48.4219 129 36.14 <.0001 0.05 1654.23 1845.84
session 4 1717.80 48.4219 129 35.48 <.0001 0.05 1621.99 1813.60
session 5 1707.18 48.4219 129 35.26 <.0001 0.05 1611.37 1802.98
session 6 1672.14 48.4219 129 34.53 <.0001 0.05 1576.33 1767.94

Differences of Least Squares Means
Effect session: Occasion
(1-6)		 session: Occasion
(1-6)		 Estimate Standard Error DF t Value Pr > |t| Alpha Lower Upper
session 1 2 146.72 25.9993 500 5.64 <.0001 0.05 95.6396 197.80
session 1 3 211.86 25.9993 500 8.15 <.0001 0.05 160.78 262.94
session 1 4 244.10 25.9993 500 9.39 <.0001 0.05 193.02 295.18
session 1 5 254.72 25.9993 500 9.80 <.0001 0.05 203.64 305.80
session 1 6 289.76 25.9993 500 11.14 <.0001 0.05 238.68 340.84
session 2 3 65.1377 25.9993 500 2.51 0.0125 0.05 14.0564 116.22
session 2 4 97.3759 25.9993 500 3.75 0.0002 0.05 46.2945 148.46
session 2 5 108.00 25.9993 500 4.15 <.0001 0.05 56.9153 159.08
session 2 6 143.04 25.9993 500 5.50 <.0001 0.05 91.9550 194.12
session 3 4 32.2382 25.9993 500 1.24 0.2156 0.05 -18.8432 83.3196
session 3 5 42.8589 25.9993 500 1.65 0.0999 0.05 -8.2225 93.9403
session 3 6 77.8986 25.9993 500 3.00 0.0029 0.05 26.8173 128.98
session 4 5 10.6207 25.9993 500 0.41 0.6831 0.05 -40.4606 61.7021
session 4 6 45.6604 25.9993 500 1.76 0.0797 0.05 -5.4209 96.7418
session 5 6 35.0397 25.9993 500 1.35 0.1784 0.05 -16.0417 86.1211


Likelihood Ratio Test for FitBPANOVA vs. FitUnivANOVA

Name Neg2LogLike Parms AIC BIC DevDiff DFdiff Pvalue
FitBPANOVA 9155.4 1 9157.4 9160.0 . . .
FitUnivANOVA 8353.4 2 8357.4 8362.6 802.077 1 0


Eq 3b.10: Multivariate Repeated Measures ANOVA

The Mixed Procedure

Model Information
Data Set WORK.CHAPTER3B
Dependent Variable rt
Covariance Structure Unstructured
Subject Effect PersonID
Estimation Method REML
Residual Variance Method None
Fixed Effects SE Method Model-Based
Degrees of Freedom Method Satterthwaite

Dimensions
Covariance Parameters 21
Columns in X 7
Columns in Z 0
Subjects 101
Max Obs Per Subject 6

Number of Observations
Number of Observations Read 606
Number of Observations Used 606
Number of Observations Not Used 0

Iteration History
Iteration Evaluations -2 Res Log Like Criterion
0 1 9155.43252939  
1 1 8229.78846855 0.00000000

Convergence criteria met.

Estimated R Matrix for PersonID 101
Row Col1 Col2 Col3 Col4 Col5 Col6
1 301985 235659 217994 202607 192154 195360
2 235659 259150 230217 213232 202092 193268
3 217994 230217 233368 205209 196919 188604
4 202607 213232 205209 217544 193676 185321
5 192154 202092 196919 193676 212098 187840
6 195360 193268 188604 185321 187840 196733

Estimated R Correlation Matrix for PersonID 101
Row Col1 Col2 Col3 Col4 Col5 Col6
1 1.0000 0.8424 0.8212 0.7905 0.7593 0.8015
2 0.8424 1.0000 0.9361 0.8981 0.8620 0.8559
3 0.8212 0.9361 1.0000 0.9108 0.8851 0.8802
4 0.7905 0.8981 0.9108 1.0000 0.9016 0.8958
5 0.7593 0.8620 0.8851 0.9016 1.0000 0.9196
6 0.8015 0.8559 0.8802 0.8958 0.9196 1.0000

Covariance Parameter Estimates
Cov Parm Subject Estimate Standard Error Z Value Pr Z
UN(1,1) PersonID 301985 42707 7.07 <.0001
UN(2,1) PersonID 235659 36578 6.44 <.0001
UN(2,2) PersonID 259150 36649 7.07 <.0001
UN(3,1) PersonID 217994 34350 6.35 <.0001
UN(3,2) PersonID 230217 33686 6.83 <.0001
UN(3,3) PersonID 233368 33003 7.07 <.0001
UN(4,1) PersonID 202607 32672 6.20 <.0001
UN(4,2) PersonID 213232 31913 6.68 <.0001
UN(4,3) PersonID 205209 30476 6.73 <.0001
UN(4,4) PersonID 217544 30765 7.07 <.0001
UN(5,1) PersonID 192154 31776 6.05 <.0001
UN(5,2) PersonID 202092 30953 6.53 <.0001
UN(5,3) PersonID 196919 29711 6.63 <.0001
UN(5,4) PersonID 193676 28922 6.70 <.0001
UN(5,5) PersonID 212098 29995 7.07 <.0001
UN(6,1) PersonID 195360 31237 6.25 <.0001
UN(6,2) PersonID 193268 29721 6.50 <.0001
UN(6,3) PersonID 188604 28545 6.61 <.0001
UN(6,4) PersonID 185321 27774 6.67 <.0001
UN(6,5) PersonID 187840 27751 6.77 <.0001
UN(6,6) PersonID 196733 27822 7.07 <.0001

Fit Statistics
-2 Res Log Likelihood 8229.8
AIC (smaller is better) 8271.8
AICC (smaller is better) 8273.4
BIC (smaller is better) 8326.7

Null Model Likelihood Ratio Test
DF Chi-Square Pr > ChiSq
20 925.64 <.0001

Information Criteria
Neg2LogLike Parms AIC AICC HQIC BIC CAIC
8229.8 21 8271.8 8273.4 8294.0 8326.7 8347.7

Solution for Fixed Effects
Effect session: Occasion
(1-6)		 Estimate Standard Error DF t Value Pr > |t| Alpha Lower Upper
Intercept   1672.14 44.1345 100 37.89 <.0001 0.05 1584.57 1759.70
session 1 289.76 32.7000 100 8.86 <.0001 0.05 224.88 354.63
session 2 143.04 26.2031 100 5.46 <.0001 0.05 91.0502 195.02
session 3 77.8986 22.8842 100 3.40 0.0010 0.05 32.4970 123.30
session 4 45.6604 20.7853 100 2.20 0.0303 0.05 4.4230 86.8979
session 5 35.0397 18.1168 100 1.93 0.0559 0.05 -0.9035 70.9829
session 6 0 . . . . . . .

Type 3 Tests of Fixed Effects
Effect Num DF Den DF Chi-Square F Value Pr > ChiSq Pr > F
session 5 100 83.60 16.72 <.0001 <.0001

Least Squares Means
Effect session: Occasion
(1-6)		 Estimate Standard Error DF t Value Pr > |t| Alpha Lower Upper
session 1 1961.89 54.6805 100 35.88 <.0001 0.05 1853.41 2070.38
session 2 1815.17 50.6541 100 35.83 <.0001 0.05 1714.68 1915.67
session 3 1750.03 48.0684 100 36.41 <.0001 0.05 1654.67 1845.40
session 4 1717.80 46.4101 100 37.01 <.0001 0.05 1625.72 1809.87
session 5 1707.18 45.8255 100 37.25 <.0001 0.05 1616.26 1798.09
session 6 1672.14 44.1345 100 37.89 <.0001 0.05 1584.57 1759.70

Differences of Least Squares Means
Effect session: Occasion
(1-6)		 session: Occasion
(1-6)		 Estimate Standard Error DF t Value Pr > |t| Alpha Lower Upper
session 1 2 146.72 29.8209 100 4.92 <.0001 0.05 87.5572 205.88
session 1 3 211.86 31.3658 100 6.75 <.0001 0.05 149.63 274.09
session 1 4 244.10 33.6427 100 7.26 <.0001 0.05 177.35 310.84
session 1 5 254.72 35.8456 100 7.11 <.0001 0.05 183.60 325.83
session 1 6 289.76 32.7000 100 8.86 <.0001 0.05 224.88 354.63
session 2 3 65.1377 17.8233 100 3.65 0.0004 0.05 29.7769 100.50
session 2 4 97.3759 22.3009 100 4.37 <.0001 0.05 53.1316 141.62
session 2 5 108.00 25.7681 100 4.19 <.0001 0.05 56.8734 159.12
session 2 6 143.04 26.2031 100 5.46 <.0001 0.05 91.0502 195.02
session 3 4 32.2382 20.0232 100 1.61 0.1105 0.05 -7.4872 71.9636
session 3 5 42.8589 22.6091 100 1.90 0.0609 0.05 -1.9969 87.7147
session 3 6 77.8986 22.8842 100 3.40 0.0010 0.05 32.4970 123.30
session 4 5 10.6207 20.4625 100 0.52 0.6049 0.05 -29.9763 51.2177
session 4 6 45.6604 20.7853 100 2.20 0.0303 0.05 4.4230 86.8979
session 5 6 35.0397 18.1168 100 1.93 0.0559 0.05 -0.9035 70.9829


Likelihood Ratio Test for FitUnivANOVA vs. FitMultivANOVA

Name Neg2LogLike Parms AIC BIC DevDiff DFdiff Pvalue
FitUnivANOVA 8353.4 2 8357.4 8362.6 . . .
FitMultivANOVA 8229.8 21 8271.8 8326.7 123.567 19 0