Applied Longitudinal Data Analysis: Modeling Change and Event Occurrenceby Judith D. Singer and John B. Willett Chapter 7: Examining the multilevel model s error covariance structure

Table 7.2 on page 246.

title1 'Table 7.2: Conventional multilevel model for change: Opposite naming data';
title2 'Restricted ML';
proc mixed data='c:\alda\opposites_pp' noclprint noinfo covtest noitprint method=reml;
  class id;
  model opp=time ccog time*ccog / solution ddfm=bw;
  random intercept time / sub=id type=un;
run;

Table 7.2: Conventional multilevel model for change: Opposite naming data
Restricted ML

The Mixed Procedure

Covariance Parameter Estimates

Standard Z Cov Parm Subject Estimate Error Value Pr Z UN(1,1) ID 1236.41 332.40 3.72 <.0001 UN(2,1) ID -178.23 85.4298 -2.09 0.0370 UN(2,2) ID 107.25 34.6767 3.09 0.0010 Residual 159.48 26.9566 5.92 <.0001

           Fit Statistics
-2 Res Log Likelihood          1260.3
AIC (smaller is better)        1268.3
AICC (smaller is better)       1268.6
BIC (smaller is better)        1274.5

  Null Model Likelihood Ratio Test
    DF    Chi-Square      Pr > ChiSq
     3        120.72          <.0001

Solution for Fixed Effects

Standard Effect Estimate Error DF t Value Pr > |t| Intercept 164.37 6.2061 33 26.49 <.0001 TIME 26.9600 1.9939 103 13.52 <.0001 CCOG -0.1136 0.5040 33 -0.23 0.8231 TIME*CCOG 0.4329 0.1619 103 2.67 0.0087

Type 3 Tests of Fixed Effects

Num Den Effect DF DF F Value Pr > F TIME 1 103 182.83 <.0001 CCOG 1 33 0.05 0.8231 TIME*CCOG 1 103 7.15 0.0087

Table 7.3 on pages 258-259.

title1 'Table 7.3: Conventional multilevel model for change: Opposite naming data';
proc mixed data='c:\alda\opposites_pp' noclprint noinfo covtest noitprint method=reml;
  title2 'Comparing alternative error structures at level-1, with none at level-2 (REML)';
  title3 'completely unstructured';
  class id wave;
  model opp=time ccog time*ccog / solution ddfm=bw notest;
  repeated wave / subject=id type=un r;
run;
 
proc mixed data='c:\alda\opposites_pp' noclprint noinfo covtest noitprint method=reml;
  title3 'compound symmetric';
  class id wave;
  model opp=time ccog time*ccog / solution ddfm=bw notest;
  repeated wave / subject=id type=cs r;
run;
 
proc mixed data='c:\alda\opposites_pp' noclprint noinfo covtest noitprint method=reml;
  title3 'compound symmetric heterogeneous CSH';
  class id wave;
  model opp=time ccog time*ccog / solution ddfm=bw notest;
  repeated wave / subject=id type=csh r;
run;
 
proc mixed data='c:\alda\opposites_pp' noclprint noinfo covtest noitprint method=reml;
  title3 'ar(1)';
  class id wave;
  model opp=time ccog time*ccog / solution ddfm=bw notest;
  repeated wave / subject=id type=ar(1) r;
run;
 
proc mixed data='c:\alda\opposites_pp' noclprint noinfo covtest noitprint method=reml;
  title3 'ar(1)heterogeneous';
  class id wave;
  model opp=time ccog time*ccog / solution ddfm=bw notest;
  repeated wave / subject=id type=arh(1) r;
run;
 
proc mixed data='c:\alda\opposites_pp' noclprint noinfo covtest noitprint method=reml;
  title3 'Toeplitz';
  class id wave;
  model opp=time ccog time*ccog / solution ddfm=bw notest;
  repeated wave / subject=id type=toep r;
run;

Table 7.3: Conventional multilevel model for change: Opposite naming data
Comparing alternative error structures at level-1, with none at level-2 (REML)
completely unstructured

The Mixed Procedure

             Estimated R Matrix for ID 1

 Row        Col1        Col2        Col3        Col4

   1     1344.84     1005.57      946.05      583.12
   2     1005.57     1150.30     1028.43      846.49
   3      946.05     1028.43     1235.65      969.24
   4      583.12      846.49      969.24     1205.89

                 Covariance Parameter Estimates
                                   Standard         Z
Cov Parm    Subject    Estimate       Error     Value        Pr Z
UN(1,1)     ID          1344.84      330.16      4.07      <.0001
UN(2,1)     ID          1005.57      277.88      3.62      0.0003
UN(2,2)     ID          1150.30      282.13      4.08      <.0001
UN(3,1)     ID           946.05      276.91      3.42      0.0006
UN(3,2)     ID          1028.43      272.75      3.77      0.0002
UN(3,3)     ID          1235.65      301.84      4.09      <.0001
UN(4,1)     ID           583.12      243.58      2.39      0.0167
UN(4,2)     ID           846.49      252.08      3.36      0.0008
UN(4,3)     ID           969.24      270.78      3.58      0.0003
UN(4,4)     ID          1205.89      296.70      4.06      <.0001

           Fit Statistics
-2 Res Log Likelihood          1255.8
AIC (smaller is better)        1275.8
AICC (smaller is better)       1277.5
BIC (smaller is better)        1291.3

Null Model Likelihood Ratio Test DF Chi-Square Pr > ChiSq 9 125.22 <.0001

Solution for Fixed Effects

Standard Effect Estimate Error DF t Value Pr > |t| Intercept 165.83 5.9523 33 27.86 <.0001 TIME 26.5843 1.9257 33 13.80 <.0001 CCOG -0.07408 0.4834 33 -0.15 0.8791 TIME*CCOG 0.4583 0.1564 33 2.93 0.0061

Table 7.3: Conventional multilevel model for change: Opposite naming data
Comparing alternative error structures at level-1, with none at level-2 (REML)
compound symmetric

The Mixed Procedure

Estimated R Matrix for ID 1

Row Col1 Col2 Col3 Col4 1 1231.36 900.07 900.07 900.07 2 900.07 1231.36 900.07 900.07 3 900.07 900.07 1231.36 900.07 4 900.07 900.07 900.07 1231.36

Covariance Parameter Estimates

Standard Z Cov Parm Subject Estimate Error Value Pr Z CS ID 900.07 242.25 3.72 0.0002 Residual 331.28 46.1633 7.18 <.0001

           Fit Statistics
-2 Res Log Likelihood          1287.0
AIC (smaller is better)        1291.0
AICC (smaller is better)       1291.1
BIC (smaller is better)        1294.2

  Null Model Likelihood Ratio Test
    DF    Chi-Square      Pr > ChiSq
     1         93.96          <.0001

Solution for Fixed Effects

Standard Effect Estimate Error DF t Value Pr > |t| Intercept 164.37 5.6870 33 28.90 <.0001 TIME 26.9600 1.3759 103 19.59 <.0001 CCOG -0.1136 0.4619 33 -0.25 0.8073 TIME*CCOG 0.4329 0.1117 103 3.87 0.0002

Table 7.3: Conventional multilevel model for change: Opposite naming data
Comparing alternative error structures at level-1, with none at level-2 (REML)
compound symmetric heterogeneous CSH

The Mixed Procedure

Estimated R Matrix for ID 1

Row Col1 Col2 Col3 Col4 1 1438.04 912.85 946.52 1009.46 2 912.85 1067.67 815.57 869.80 3 946.52 815.57 1147.87 901.88 4 1009.46 869.80 901.88 1305.61

                 Covariance Parameter Estimates
Cov                               Standard         Z
Parm       Subject    Estimate       Error     Value        Pr Z
Var(1)     ID          1438.04      354.99      4.05      <.0001
Var(2)     ID          1067.67      255.23      4.18      <.0001
Var(3)     ID          1147.87      273.55      4.20      <.0001
Var(4)     ID          1305.61      323.74      4.03      <.0001
CSH        ID           0.7367     0.05971     12.34      <.0001

           Fit Statistics
-2 Res Log Likelihood          1285.0
AIC (smaller is better)        1295.0
AICC (smaller is better)       1295.4
BIC (smaller is better)        1302.7

  Null Model Likelihood Ratio Test
    DF    Chi-Square      Pr > ChiSq
     4         96.06          <.0001

Solution for Fixed Effects

Standard Effect Estimate Error DF t Value Pr > |t| Intercept 164.32 5.6936 33 28.86 <.0001 TIME 26.9260 1.4192 103 18.97 <.0001 CCOG -0.1937 0.4624 33 -0.42 0.6780 TIME*CCOG 0.4397 0.1153 103 3.81 0.0002

Table 7.3: Conventional multilevel model for change: Opposite naming data
Comparing alternative error structures at level-1, with none at level-2 (REML)
ar(1)

The Mixed Procedure

Estimated R Matrix for ID 1

Row Col1 Col2 Col3 Col4 1 1256.72 1037.24 856.08 706.57 2 1037.24 1256.72 1037.24 856.08 3 856.08 1037.24 1256.72 1037.24 4 706.57 856.08 1037.24 1256.72

Covariance Parameter Estimates

Standard Z Cov Parm Subject Estimate Error Value Pr Z AR(1) ID 0.8253 0.03948 20.91 <.0001 Residual 1256.72 248.25 5.06 <.0001

           Fit Statistics
-2 Res Log Likelihood          1265.9
AIC (smaller is better)        1269.9
AICC (smaller is better)       1270.0
BIC (smaller is better)        1273.0

  Null Model Likelihood Ratio Test
    DF    Chi-Square      Pr > ChiSq
     1        115.13          <.0001

Solution for Fixed Effects

Standard Effect Estimate Error DF t Value Pr > |t| Intercept 164.34 5.9797 33 27.48 <.0001 TIME 27.1979 1.8689 103 14.55 <.0001 CCOG -0.03324 0.4856 33 -0.07 0.9458 TIME*CCOG 0.4197 0.1518 103 2.77 0.0067

Table 7.3: Conventional multilevel model for change: Opposite naming data
Comparing alternative error structures at level-1, with none at level-2 (REML)
ar(1)heterogeneous

The Mixed Procedure

Estimated R Matrix for ID 1

Row Col1 Col2 Col3 Col4 1 1340.67 1000.67 857.30 708.87 2 1000.67 1111.10 951.90 787.10 3 857.30 951.90 1213.18 1003.14 4 708.87 787.10 1003.14 1233.93

                 Covariance Parameter Estimates
Cov                               Standard         Z
Parm       Subject    Estimate       Error     Value        Pr Z
Var(1)     ID          1340.67      316.29      4.24      <.0001
Var(2)     ID          1111.10      265.70      4.18      <.0001
Var(3)     ID          1213.18      290.11      4.18      <.0001
Var(4)     ID          1233.93      293.63      4.20      <.0001
ARH(1)     ID           0.8199     0.04105     19.97      <.0001

           Fit Statistics
-2 Res Log Likelihood          1264.8
AIC (smaller is better)        1274.8
AICC (smaller is better)       1275.3
BIC (smaller is better)        1282.6

  Null Model Likelihood Ratio Test
    DF    Chi-Square      Pr > ChiSq
     4        116.17          <.0001

Solution for Fixed Effects

Standard Effect Estimate Error DF t Value Pr > |t| Intercept 164.64 6.0588 33 27.17 <.0001 TIME 27.1638 1.9131 103 14.20 <.0001 CCOG -0.1111 0.4920 33 -0.23 0.8228 TIME*CCOG 0.4281 0.1554 103 2.76 0.0069

Table 7.3: Conventional multilevel model for change: Opposite naming data
Comparing alternative error structures at level-1, with none at level-2 (REML)
Toeplitz

The Mixed Procedure

Estimated R Matrix for ID 1

Row Col1 Col2 Col3 Col4 1 1246.90 1029.33 896.59 624.06 2 1029.33 1246.90 1029.33 896.59 3 896.59 1029.33 1246.90 1029.33 4 624.06 896.59 1029.33 1246.90

Covariance Parameter Estimates

Standard Z Cov Parm Subject Estimate Error Value Pr Z TOEP(2) ID 1029.33 239.57 4.30 <.0001 TOEP(3) ID 896.59 232.76 3.85 0.0001 TOEP(4) ID 624.06 234.89 2.66 0.0079 Residual 1246.90 242.67 5.14 <.0001

           Fit Statistics
-2 Res Log Likelihood          1258.1
AIC (smaller is better)        1266.1
AICC (smaller is better)       1266.4
BIC (smaller is better)        1272.3

Null Model Likelihood Ratio Test DF Chi-Square Pr > ChiSq 3 122.93 <.0001

Solution for Fixed Effects

Standard Effect Estimate Error DF t Value Pr > |t| Intercept 165.10 5.9225 33 27.88 <.0001 TIME 26.8954 1.9429 103 13.84 <.0001 CCOG -0.00070 0.4810 33 -0.00 0.9988 TIME*CCOG 0.4364 0.1578 103 2.77 0.0067

Table 7.4 on page 265.

title1 'Table 7.4: Comparing fixed effects in models with alternative error structures';
proc mixed data='c:\alda\opposites_pp' noclprint noinfo covtest noitprint method=reml;
  title2 'Standard error covariance structure';
  class id ;
  model opp=time ccog time*ccog / solution ddfm=bw notest;
  random intercept time / subject=id type=un;
run;
 
proc mixed data='c:\alda\opposites_pp' noclprint noinfo covtest noitprint method=reml;
  title3 'Toeplitz error covariance matrix';
  class id wave;
  model opp=time ccog time*ccog / solution ddfm=bw notest;
  repeated wave / subject=id type=toep r;
run;
 
proc mixed data='c:\alda\opposites_pp' noclprint noinfo covtest noitprint method=reml;
  title2 'Unstructured error covariance matrix';
  class id wave;
  model opp=time ccog time*ccog / solution ddfm=bw notest;
  repeated wave / subject=id type=un r;
run;

Table 7.4: Comparing fixed effects in models with alternative error structures
Standard error covariance structure

The Mixed Procedure

Covariance Parameter Estimates

           Fit Statistics
-2 Res Log Likelihood          1260.3
AIC (smaller is better)        1268.3
AICC (smaller is better)       1268.6
BIC (smaller is better)        1274.5

  Null Model Likelihood Ratio Test
    DF    Chi-Square      Pr > ChiSq
     3        120.72          <.0001

Solution for Fixed Effects

Table 7.4: Comparing fixed effects in models with alternative error structures
Standard error covariance structure
Toeplitz error covariance matrix

The Mixed Procedure

Estimated R Matrix for ID 1

Row Col1 Col2 Col3 Col4 1 1246.90 1029.33 896.59 624.06 2 1029.33 1246.90 1029.33 896.59 3 896.59 1029.33 1246.90 1029.33 4 624.06 896.59 1029.33 1246.90

Covariance Parameter Estimates

           Fit Statistics
-2 Res Log Likelihood          1258.1
AIC (smaller is better)        1266.1
AICC (smaller is better)       1266.4
BIC (smaller is better)        1272.3

  Null Model Likelihood Ratio Test
    DF    Chi-Square      Pr > ChiSq
     3        122.93          <.0001

Solution for Fixed Effects

Table 7.4: Comparing fixed effects in models with alternative error structures
Unstructured error covariance matrix

The Mixed Procedure

Estimated R Matrix for ID 1

Row Col1 Col2 Col3 Col4 1 1344.84 1005.57 946.05 583.12 2 1005.57 1150.30 1028.43 846.49 3 946.05 1028.43 1235.65 969.24 4 583.12 846.49 969.24 1205.89

                 Covariance Parameter Estimates
                                   Standard         Z
Cov Parm    Subject    Estimate       Error     Value        Pr Z
UN(1,1)     ID          1344.84      330.16      4.07      <.0001
UN(2,1)     ID          1005.57      277.88      3.62      0.0003
UN(2,2)     ID          1150.30      282.13      4.08      <.0001
UN(3,1)     ID           946.05      276.91      3.42      0.0006
UN(3,2)     ID          1028.43      272.75      3.77      0.0002
UN(3,3)     ID          1235.65      301.84      4.09      <.0001
UN(4,1)     ID           583.12      243.58      2.39      0.0167
UN(4,2)     ID           846.49      252.08      3.36      0.0008
UN(4,3)     ID           969.24      270.78      3.58      0.0003
UN(4,4)     ID          1205.89      296.70      4.06      <.0001

           Fit Statistics
-2 Res Log Likelihood          1255.8
AIC (smaller is better)        1275.8
AICC (smaller is better)       1277.5
BIC (smaller is better)        1291.3

  Null Model Likelihood Ratio Test
    DF    Chi-Square      Pr > ChiSq
     9        125.22          <.0001

Solution for Fixed Effects