SAS Model 2.1: Empty Model

The Mixed Procedure

Model Information
Data Set WORK.EX2_2
Dependent Variable y_raw
Covariance Structure Unstructured
Subject Effects ID, ID*target*setsize
Estimation Method ML
Residual Variance Method None
Fixed Effects SE Method Model-Based
Degrees of Freedom Method Satterthwaite

Dimensions
Covariance Parameters 6
Columns in X 2
Columns in Z Per Subject 2
Subjects 148
Max Obs Per Subject 12

Number of Observations
Number of Observations Read 1776
Number of Observations Used 1776
Number of Observations Not Used 0

Estimated R Correlation Matrix
for ID*target*setsize 201
1 3
Row Col1 Col2
1 1.0000 0.4193
2 0.4193 1.0000

Estimated G Correlation Matrix
Row Effect Participant ID Col1 Col2
1 dv1 201 1.0000 0.05010
2 dv2 201 0.05010 1.0000

Covariance Parameter Estimates
Cov Parm Subject Estimate Standard Error Z Value Pr Z
UN(1,1) ID 0.03312 0.004340 7.63 <.0001
UN(2,1) ID 0.000705 0.001493 0.47 0.6369
UN(2,2) ID 0.005973 0.001016 5.88 <.0001
UN(1,1) ID*target*setsize 0.02500 0.001299 19.24 <.0001
UN(2,1) ID*target*setsize 0.008409 0.000799 10.52 <.0001
UN(2,2) ID*target*setsize 0.01609 0.000837 19.24 <.0001

Fit Statistics
-2 Log Likelihood -1550.2
AIC (smaller is better) -1534.2
AICC (smaller is better) -1534.1
BIC (smaller is better) -1510.2

Null Model Likelihood Ratio Test
DF Chi-Square Pr > ChiSq
5 877.53 <.0001

Solution for Fixed Effects
Effect Estimate Standard Error DF t Value Pr > |t|
dv1 6.7317 0.01587 148 424.11 <.0001
dv2 0.1720 0.007647 148 22.49 <.0001

Type 3 Tests of Fixed Effects
Effect Num DF Den DF F Value Pr > F
dv1 1 148 179867 <.0001
dv2 1 148 505.90 <.0001



SAS Model 2.2: Main Effects Model

The Mixed Procedure

Model Information
Data Set WORK.EX2_2
Dependent Variable y_raw
Covariance Structure Unstructured
Subject Effects ID, ID*target*setsize
Estimation Method ML
Residual Variance Method None
Fixed Effects SE Method Model-Based
Degrees of Freedom Method Satterthwaite

Dimensions
Covariance Parameters 24
Columns in X 8
Columns in Z Per Subject 6
Subjects 148
Max Obs Per Subject 12

Number of Observations
Number of Observations Read 1776
Number of Observations Used 1776
Number of Observations Not Used 0

Estimated R Correlation Matrix
for ID*target*setsize 201
1 3
Row Col1 Col2
1 1.0000 0.01598
2 0.01598 1.0000

Estimated G Correlation Matrix
Row Effect Participant ID Col1 Col2 Col3 Col4 Col5 Col6
1 dv1 201 1.0000 0.1589 0.6651 0.2351 -0.09823 -0.08470
2 dv2 201 0.1589 1.0000 -0.3305 0.4972 -0.1646 -0.1837
3 dv1*c_size 201 0.6651 -0.3305 1.0000 0.2361 0.4283 -0.09030
4 c_size*dv2 201 0.2351 0.4972 0.2361 1.0000 0.1411 -0.2428
5 dv1*c_targ 201 -0.09823 -0.1646 0.4283 0.1411 1.0000 0.1191
6 c_targ*dv2 201 -0.08470 -0.1837 -0.09030 -0.2428 0.1191 1.0000

Covariance Parameter Estimates
Cov Parm Subject Estimate Standard Error Z Value Pr Z
UN(1,1) ID 0.03317 0.004242 7.82 <.0001
UN(2,1) ID 0.002416 0.001586 1.52 0.1276
UN(2,2) ID 0.006976 0.001175 5.94 <.0001
UN(3,1) ID 0.001339 0.000334 4.01 <.0001
UN(3,2) ID -0.00031 0.000166 -1.84 0.0659
UN(3,3) ID 0.000122 0.000049 2.48 0.0065
UN(4,1) ID 0.000399 0.000287 1.39 0.1649
UN(4,2) ID 0.000387 0.000152 2.54 0.0110
UN(4,3) ID 0.000024 0.000032 0.76 0.4486
UN(4,4) ID 0.000087 0.000042 2.09 0.0185
UN(5,1) ID -0.00175 0.002049 -0.85 0.3942
UN(5,2) ID -0.00134 0.001056 -1.27 0.2040
UN(5,3) ID 0.000462 0.000210 2.20 0.0277
UN(5,4) ID 0.000128 0.000189 0.68 0.4978
UN(5,5) ID 0.009523 0.001914 4.97 <.0001
UN(6,1) ID -0.00100 0.001587 -0.63 0.5301
UN(6,2) ID -0.00099 0.000904 -1.10 0.2730
UN(6,3) ID -0.00006 0.000163 -0.40 0.6927
UN(6,4) ID -0.00015 0.000150 -0.98 0.3291
UN(6,5) ID 0.000751 0.001083 0.69 0.4884
UN(6,6) ID 0.004172 0.001227 3.40 0.0003
UN(1,1) ID*target*setsize 0.009764 0.000655 14.90 <.0001
UN(2,1) ID*target*setsize 0.000148 0.000439 0.34 0.7363
UN(2,2) ID*target*setsize 0.008746 0.000587 14.90 <.0001

Fit Statistics
-2 Log Likelihood -2307.6
AIC (smaller is better) -2243.6
AICC (smaller is better) -2242.4
BIC (smaller is better) -2147.7

Null Model Likelihood Ratio Test
DF Chi-Square Pr > ChiSq
23 1230.78 <.0001

Solution for Fixed Effects
Effect Estimate Standard Error DF t Value Pr > |t|
dv1 6.6896 0.01571 148 425.79 <.0001
dv1*c_size 0.03645 0.001630 148 22.36 <.0001
dv1*c_targ 0.08018 0.01041 148 7.70 <.0001
dv1*c_age 0.006925 0.002947 148 2.35 0.0201
dv2 0.1515 0.008187 146 18.50 <.0001
c_size*dv2 0.02639 0.001493 148 17.68 <.0001
c_targ*dv2 0.03889 0.008221 148 4.73 <.0001
c_age*dv2 0.003615 0.001523 148 2.37 0.0189

Type 3 Tests of Fixed Effects
Effect Num DF Den DF F Value Pr > F
dv1 1 148 181299 <.0001
dv1*c_size 1 148 499.91 <.0001
dv1*c_targ 1 148 59.34 <.0001
dv1*c_age 1 148 5.52 0.0201
dv2 1 146 342.39 <.0001
c_size*dv2 1 148 312.44 <.0001
c_targ*dv2 1 148 22.38 <.0001
c_age*dv2 1 148 5.64 0.0189



SAS Model 2.3a: All Interactions Model

The Mixed Procedure

Model Information
Data Set WORK.EX2_2
Dependent Variable y_raw
Covariance Structure Unstructured
Subject Effects ID, ID*target*setsize
Estimation Method ML
Residual Variance Method None
Fixed Effects SE Method Model-Based
Degrees of Freedom Method Satterthwaite

Dimensions
Covariance Parameters 24
Columns in X 16
Columns in Z Per Subject 6
Subjects 148
Max Obs Per Subject 12

Number of Observations
Number of Observations Read 1776
Number of Observations Used 1776
Number of Observations Not Used 0

Estimated R Correlation Matrix
for ID*target*setsize 201
1 3
Row Col1 Col2
1 1.0000 0.002027
2 0.002027 1.0000

Estimated G Correlation Matrix
Row Effect Participant ID Col1 Col2 Col3 Col4 Col5 Col6
1 dv1 201 1.0000 0.1655 0.6525 0.2386 -0.1096 -0.09451
2 dv2 201 0.1655 1.0000 -0.2965 0.4818 -0.1544 -0.1707
3 dv1*c_size 201 0.6525 -0.2965 1.0000 0.3138 0.3835 -0.1460
4 c_size*dv2 201 0.2386 0.4818 0.3138 1.0000 0.1597 -0.2179
5 dv1*c_targ 201 -0.1096 -0.1544 0.3835 0.1597 1.0000 0.1088
6 c_targ*dv2 201 -0.09451 -0.1707 -0.1460 -0.2179 0.1088 1.0000

Covariance Parameter Estimates
Cov Parm Subject Estimate Standard Error Z Value Pr Z
UN(1,1) ID 0.03328 0.004236 7.86 <.0001
UN(2,1) ID 0.002510 0.001573 1.60 0.1104
UN(2,2) ID 0.006914 0.001156 5.98 <.0001
UN(3,1) ID 0.001321 0.000325 4.06 <.0001
UN(3,2) ID -0.00027 0.000161 -1.70 0.0887
UN(3,3) ID 0.000123 0.000048 2.58 0.0049
UN(4,1) ID 0.000407 0.000286 1.42 0.1550
UN(4,2) ID 0.000374 0.000151 2.49 0.0129
UN(4,3) ID 0.000033 0.000032 1.03 0.3025
UN(4,4) ID 0.000087 0.000041 2.11 0.0175
UN(5,1) ID -0.00196 0.002029 -0.97 0.3334
UN(5,2) ID -0.00126 0.001040 -1.21 0.2256
UN(5,3) ID 0.000418 0.000205 2.04 0.0418
UN(5,4) ID 0.000146 0.000188 0.78 0.4355
UN(5,5) ID 0.009640 0.001890 5.10 <.0001
UN(6,1) ID -0.00110 0.001567 -0.70 0.4816
UN(6,2) ID -0.00091 0.000888 -1.02 0.3067
UN(6,3) ID -0.00010 0.000160 -0.65 0.5170
UN(6,4) ID -0.00013 0.000148 -0.88 0.3800
UN(6,5) ID 0.000683 0.001068 0.64 0.5228
UN(6,6) ID 0.004089 0.001211 3.38 0.0004
UN(1,1) ID*target*setsize 0.009327 0.000626 14.90 <.0001
UN(2,1) ID*target*setsize 0.000018 0.000427 0.04 0.9659
UN(2,2) ID*target*setsize 0.008666 0.000582 14.90 <.0001

Fit Statistics
-2 Log Likelihood -2340.2
AIC (smaller is better) -2260.2
AICC (smaller is better) -2258.3
BIC (smaller is better) -2140.3

Null Model Likelihood Ratio Test
DF Chi-Square Pr > ChiSq
23 1251.83 <.0001

Solution for Fixed Effects
Effect Estimate Standard Error DF t Value Pr > |t|
dv1 6.6893 0.01571 148 425.77 <.0001
dv1*c_size 0.03035 0.002085 361 14.55 <.0001
dv1*c_targ 0.07936 0.01037 148 7.65 <.0001
dv1*c_age 0.008067 0.003318 148 2.43 0.0162
dv1*c_size*c_targ 0.01179 0.002651 444 4.45 <.0001
dv1*c_size*c_age 0.000504 0.000440 361 1.15 0.2528
dv1*c_targ*c_age 0.002796 0.002190 148 1.28 0.2037
dv1*c_si*c_tar*c_age 0.000408 0.000560 444 0.73 0.4663
dv2 0.1521 0.008154 148 18.65 <.0001
c_size*dv2 0.02431 0.001964 377 12.38 <.0001
c_targ*dv2 0.03817 0.008180 148 4.67 <.0001
c_age*dv2 0.001618 0.001722 148 0.94 0.3489
c_size*c_targ*dv2 0.004322 0.002556 444 1.69 0.0915
c_size*c_age*dv2 0.000040 0.000415 377 0.10 0.9238
c_targ*c_age*dv2 0.002470 0.001727 148 1.43 0.1549
c_si*c_tar*c_age*dv2 -0.00066 0.000540 444 -1.22 0.2218

Type 3 Tests of Fixed Effects
Effect Num DF Den DF F Value Pr > F
dv1 1 148 181280 <.0001
dv1*c_size 1 361 211.79 <.0001
dv1*c_targ 1 148 58.56 <.0001
dv1*c_age 1 148 5.91 0.0162
dv1*c_size*c_targ 1 444 19.78 <.0001
dv1*c_size*c_age 1 361 1.31 0.2528
dv1*c_targ*c_age 1 148 1.63 0.2037
dv1*c_si*c_tar*c_age 1 444 0.53 0.4663
dv2 1 148 347.87 <.0001
c_size*dv2 1 377 153.25 <.0001
c_targ*dv2 1 148 21.77 <.0001
c_age*dv2 1 148 0.88 0.3489
c_size*c_targ*dv2 1 444 2.86 0.0915
c_size*c_age*dv2 1 377 0.01 0.9238
c_targ*c_age*dv2 1 148 2.04 0.1549
c_si*c_tar*c_age*dv2 1 444 1.50 0.2218



SAS revised Model 2.3b: Significant Interactions Only Model

The Mixed Procedure

Model Information
Data Set WORK.EX2_2
Dependent Variable y_raw
Covariance Structure Unstructured
Subject Effects ID, ID*target*setsize
Estimation Method ML
Residual Variance Method None
Fixed Effects SE Method Model-Based
Degrees of Freedom Method Satterthwaite

Dimensions
Covariance Parameters 24
Columns in X 10
Columns in Z Per Subject 6
Subjects 148
Max Obs Per Subject 12

Number of Observations
Number of Observations Read 1776
Number of Observations Used 1776
Number of Observations Not Used 0

Estimated R Correlation Matrix
for ID*target*setsize 201
1 3
Row Col1 Col2
1 1.0000 0.000017
2 0.000017 1.0000

Estimated G Correlation Matrix
Row Effect Participant ID Col1 Col2 Col3 Col4 Col5 Col6
1 dv1 201 1.0000 0.1686 0.6525 0.2446 -0.1135 -0.1010
2 dv2 201 0.1686 1.0000 -0.2972 0.4939 -0.1642 -0.1783
3 dv1*c_size 201 0.6525 -0.2972 1.0000 0.3184 0.3818 -0.1431
4 c_size*dv2 201 0.2446 0.4939 0.3184 1.0000 0.1390 -0.2428
5 dv1*c_targ 201 -0.1135 -0.1642 0.3818 0.1390 1.0000 0.1327
6 c_targ*dv2 201 -0.1010 -0.1783 -0.1431 -0.2428 0.1327 1.0000

Covariance Parameter Estimates
Cov Parm Subject Estimate Standard Error Z Value Pr Z
UN(1,1) ID 0.03333 0.004247 7.85 <.0001
UN(2,1) ID 0.002563 0.001582 1.62 0.1052
UN(2,2) ID 0.006931 0.001165 5.95 <.0001
UN(3,1) ID 0.001320 0.000326 4.05 <.0001
UN(3,2) ID -0.00027 0.000161 -1.70 0.0894
UN(3,3) ID 0.000123 0.000048 2.57 0.0050
UN(4,1) ID 0.000416 0.000288 1.45 0.1481
UN(4,2) ID 0.000383 0.000152 2.53 0.0116
UN(4,3) ID 0.000033 0.000032 1.04 0.3000
UN(4,4) ID 0.000087 0.000042 2.09 0.0185
UN(5,1) ID -0.00205 0.002053 -1.00 0.3174
UN(5,2) ID -0.00135 0.001053 -1.29 0.1986
UN(5,3) ID 0.000419 0.000207 2.02 0.0434
UN(5,4) ID 0.000128 0.000189 0.68 0.4978
UN(5,5) ID 0.009807 0.001910 5.13 <.0001
UN(6,1) ID -0.00119 0.001589 -0.75 0.4537
UN(6,2) ID -0.00096 0.000901 -1.06 0.2876
UN(6,3) ID -0.00010 0.000162 -0.63 0.5272
UN(6,4) ID -0.00015 0.000150 -0.98 0.3291
UN(6,5) ID 0.000849 0.001082 0.78 0.4326
UN(6,6) ID 0.004172 0.001227 3.40 0.0003
UN(1,1) ID*target*setsize 0.009338 0.000627 14.90 <.0001
UN(2,1) ID*target*setsize 1.577E-7 0.000430 0.00 0.9997
UN(2,2) ID*target*setsize 0.008746 0.000587 14.90 <.0001

Fit Statistics
-2 Log Likelihood -2331.3
AIC (smaller is better) -2263.3
AICC (smaller is better) -2262.0
BIC (smaller is better) -2161.4

Null Model Likelihood Ratio Test
DF Chi-Square Pr > ChiSq
23 1246.73 <.0001

Solution for Fixed Effects
Effect Estimate Standard Error DF t Value Pr > |t|
dv1 6.6889 0.01572 148 425.50 <.0001
dv1*c_size 0.03030 0.002084 361 14.54 <.0001
dv1*c_targ 0.08018 0.01041 148 7.70 <.0001
dv1*c_age 0.009558 0.003211 148 2.98 0.0034
dv1*c_size*c_targ 0.01191 0.002648 444 4.50 <.0001
dv1*c_size*c_age 0.000688 0.000333 148 2.06 0.0408
dv2 0.1517 0.008169 147 18.57 <.0001
c_size*dv2 0.02639 0.001493 148 17.68 <.0001
c_targ*dv2 0.03889 0.008221 148 4.73 <.0001
c_age*dv2 0.003026 0.001549 148 1.95 0.0527

Type 3 Tests of Fixed Effects
Effect Num DF Den DF F Value Pr > F
dv1 1 148 181054 <.0001
dv1*c_size 1 361 211.28 <.0001
dv1*c_targ 1 148 59.34 <.0001
dv1*c_age 1 148 8.86 0.0034
dv1*c_size*c_targ 1 444 20.23 <.0001
dv1*c_size*c_age 1 148 4.26 0.0408
dv2 1 147 344.70 <.0001
c_size*dv2 1 148 312.44 <.0001
c_targ*dv2 1 148 22.38 <.0001
c_age*dv2 1 148 3.81 0.0527



SAS Re-Parameterized Model 2.4: Standardized Response Variables

The Mixed Procedure

Model Information
Data Set WORK.EX2_2
Dependent Variable y_SD
Covariance Structure Unstructured
Subject Effects ID, ID*target*setsize
Estimation Method ML
Residual Variance Method None
Fixed Effects SE Method Model-Based
Degrees of Freedom Method Satterthwaite

Dimensions
Covariance Parameters 24
Columns in X 12
Columns in Z Per Subject 6
Subjects 148
Max Obs Per Subject 12

Number of Observations
Number of Observations Read 1776
Number of Observations Used 1776
Number of Observations Not Used 0

Estimated R Correlation Matrix
for ID*target*setsize 201
1 3
Row Col1 Col2
1 1.0000 0.000017
2 0.000017 1.0000

Estimated G Correlation Matrix
Row Effect Participant ID Col1 Col2 Col3 Col4 Col5 Col6
1 dv1 201 1.0000 0.1674 0.6524 0.2397 -0.1122 -0.09881
2 dv2 201 0.1674 1.0000 -0.2976 0.4836 -0.1620 -0.1777
3 c_size*dv1 201 0.6524 -0.2976 1.0000 0.3204 0.3832 -0.1407
4 c_size*dv2 201 0.2397 0.4836 0.3204 1.0000 0.1586 -0.2165
5 c_targ*dv1 201 -0.1122 -0.1620 0.3832 0.1586 1.0000 0.1322
6 c_targ*dv2 201 -0.09881 -0.1777 -0.1407 -0.2165 0.1322 1.0000

Covariance Parameter Estimates
Cov Parm Subject Estimate Standard Error Z Value Pr Z
UN(1,1) ID 0.5725 0.07291 7.85 <.0001
UN(2,1) ID 0.07096 0.04403 1.61 0.1071
UN(2,2) ID 0.3138 0.05260 5.97 <.0001
UN(3,1) ID 0.02268 0.005593 4.06 <.0001
UN(3,2) ID -0.00766 0.004496 -1.70 0.0884
UN(3,3) ID 0.002111 0.000820 2.57 0.0050
UN(4,1) ID 0.01135 0.007982 1.42 0.1552
UN(4,2) ID 0.01695 0.006835 2.48 0.0132
UN(4,3) ID 0.000921 0.000880 1.05 0.2954
UN(4,4) ID 0.003914 0.001876 2.09 0.0185
UN(5,1) ID -0.03486 0.03526 -0.99 0.3229
UN(5,2) ID -0.03726 0.02935 -1.27 0.2043
UN(5,3) ID 0.007227 0.003566 2.03 0.0427
UN(5,4) ID 0.004075 0.005297 0.77 0.4418
UN(5,5) ID 0.1686 0.03283 5.13 <.0001
UN(6,1) ID -0.03262 0.04431 -0.74 0.4616
UN(6,2) ID -0.04344 0.04076 -1.07 0.2865
UN(6,3) ID -0.00282 0.004519 -0.62 0.5326
UN(6,4) ID -0.00591 0.006804 -0.87 0.3851
UN(6,5) ID 0.02368 0.03017 0.79 0.4324
UN(6,6) ID 0.1904 0.05551 3.43 0.0003
UN(1,1) ID*target*setsize 0.1605 0.01077 14.90 <.0001
UN(2,1) ID*target*setsize 4.375E-6 0.01193 0.00 0.9997
UN(2,2) ID*target*setsize 0.3936 0.02642 14.90 <.0001

Fit Statistics
-2 Log Likelihood 3576.5
AIC (smaller is better) 3648.5
AICC (smaller is better) 3650.1
BIC (smaller is better) 3756.4

Null Model Likelihood Ratio Test
DF Chi-Square Pr > ChiSq
23 1042.30 <.0001

Solution for Fixed Effects
Effect Estimate Standard Error DF t Value Pr > |t|
Intercept -0.1773 0.06515 148 -2.72 0.0073
c_size 0.1256 0.008641 361 14.54 <.0001
c_targ 0.3324 0.04315 148 7.70 <.0001
c_age 0.03803 0.01343 148 2.83 0.0053
c_size*c_targ 0.04937 0.01098 444 4.50 <.0001
c_size*c_age 0.002752 0.001386 148 1.99 0.0489
dv2 0.04110 0.07937 148 0.52 0.6054
c_size*dv2 0.03859 0.01539 391 2.51 0.0125
c_targ*dv2 -0.07073 0.06783 148 -1.04 0.2988
c_age*dv2 -0.01963 0.01624 148 -1.21 0.2286
c_size*c_targ*dv2 -0.02159 0.02040 444 -1.06 0.2903
c_size*c_age*dv2 -0.00466 0.002430 148 -1.92 0.0570

Type 3 Tests of Fixed Effects
Effect Num DF Den DF F Value Pr > F
c_size 1 361 211.36 <.0001
c_targ 1 148 59.34 <.0001
c_age 1 148 8.02 0.0053
c_size*c_targ 1 444 20.23 <.0001
c_size*c_age 1 148 3.94 0.0489
dv2 1 148 0.27 0.6054
c_size*dv2 1 391 6.29 0.0125
c_targ*dv2 1 148 1.09 0.2988
c_age*dv2 1 148 1.46 0.2286
c_size*c_targ*dv2 1 444 1.12 0.2903
c_size*c_age*dv2 1 148 3.68 0.0570