### SAS Textbook Examples Modeling Longitudinal Data by Robert Weiss Chapter 9: Random Effects Models

Table 9.1, page 314.

data bsi;
set bsitotal;
knot18 = (true_month - 18)*(true_month >= 18);
knot36 = (true_month - 36)*(true_month >= 36);
spring = (season = "spring (3-6)");
summer = (season = "summer (7-10)");
l2bsi_gsi = log(bsi_gsi + 1/53)/log(2);
run;

proc mixed data = bsi method = reml noitprint noclprint covtest;
class pid rounded3_true_month gender drug_status;
model l2bsi_gsi = true_month knot18 knot36 spring summer gender parent_alcohol parent_marijuana drug_status/
notest;
repeated rounded3_true_month / subject = pid type = arma(1,1);
run;

The Mixed Procedure

Model Information
Data Set                     WORK.BSI
Dependent Variable           l2bsi_gsi
Covariance Structure         Autoregressive
Moving Average
Subject Effect               pid
Estimation Method            REML
Residual Variance Method     Profile
Fixed Effects SE Method      Model-Based
Degrees of Freedom Method    Between-Within

Dimensions
Covariance Parameters             3
Columns in X                     13
Columns in Z                      0
Subjects                        329
Max Obs Per Subject              13

Number of Observations
Number of Observations Used            1907
Number of Observations Not Used        2950

Covariance Parameter Estimates
Standard         Z
Cov Parm     Subject    Estimate       Error     Value        Pr Z
Rho          pid          0.9415     0.01068     88.19      <.0001
Gamma        pid          0.5855     0.02435     24.05      <.0001
Residual                  4.0431      0.1954     20.69      <.0001

Fit Statistics
-2 Res Log Likelihood          7367.7
AIC (smaller is better)        7373.7
AICC (smaller is better)       7373.7
BIC (smaller is better)        7385.1

Null Model Likelihood Ratio Test
DF    Chi-Square      Pr > ChiSq
2        761.23          <.0001

proc mixed data = bsi method = reml noitprint noclprint covtest;
class pid rounded3_true_month gender drug_status parent;
model l2bsi_gsi = true_month knot18 knot36 spring summer gender parent_alcohol parent_marijuana drug_status/
notest;
repeated rounded3_true_month / subject = pid(parent) type = arma(1,1);
random intercept / subject = parent type = un;
run;

The Mixed Procedure

Model Information

Data Set                     WORK.BSI
Dependent Variable           l2bsi_gsi
Covariance Structures        Unstructured,
Autoregressive
Moving Average
Subject Effects              parent, pid(parent)
Estimation Method            REML
Residual Variance Method     Profile
Fixed Effects SE Method      Model-Based
Degrees of Freedom Method    Containment

Dimensions
Covariance Parameters             4
Columns in X                     13
Columns in Z Per Subject          1
Subjects                        220
Max Obs Per Subject              44

Number of Observations
Number of Observations Used            1907
Number of Observations Not Used        2950

Covariance Parameter Estimates
Standard         Z
Cov Parm     Subject        Estimate       Error     Value        Pr Z

UN(1,1)      parent           0.7199      0.2348      3.07      0.0011
Rho          pid(parent)      0.9047     0.02247     40.25      <.0001
Gamma        pid(parent)      0.5036     0.03542     14.22      <.0001
Residual                      3.3345      0.2248     14.83      <.0001

Fit Statistics
-2 Res Log Likelihood          7357.2
AIC (smaller is better)        7365.2
AICC (smaller is better)       7365.2
BIC (smaller is better)        7378.8

Null Model Likelihood Ratio Test
DF    Chi-Square      Pr > ChiSq
3        771.70          <.0001
Table 9.2, page 315.
data weight2_1;
set weight2;
d1 = day;
d2 = day*day/100;
run;

*Model 1;
proc mixed data = weight2_1 method = reml noitprint noclprint;
class id ;
model weight = d1 d2 / solution notest;
random intercept d1 / subject = id type = un;
run;

The Mixed Procedure

Model Information
Data Set                     WORK.WEIGHT2
Dependent Variable           weight
Covariance Structure         Unstructured
Subject Effect               id
Estimation Method            REML
Residual Variance Method     Profile
Fixed Effects SE Method      Model-Based
Degrees of Freedom Method    Containment

Dimensions
Covariance Parameters             4
Columns in X                      3
Columns in Z Per Subject          2
Subjects                         38
Max Obs Per Subject               8

Number of Observations
Number of Observations Used             265
Number of Observations Not Used          39

Covariance Parameter Estimates
Cov Parm     Subject    Estimate
UN(1,1)      id           666.30
UN(2,1)      id          -0.6606
UN(2,2)      id          0.01148
Residual                  4.5231

Fit Statistics
-2 Res Log Likelihood          1474.1
AIC (smaller is better)        1482.1
AICC (smaller is better)       1482.3
BIC (smaller is better)        1488.7

Null Model Likelihood Ratio Test
DF    Chi-Square      Pr > ChiSq
3        986.16          <.0001

Solution for Fixed Effects
Standard
Effect       Estimate       Error      DF    t Value    Pr > |t|
Intercept      193.72      4.1976      37      46.15      <.0001
d1            -0.2653     0.03428      37      -7.74      <.0001
d2             0.2079     0.06208     188       3.35      0.0010

*Model 2;
proc mixed data = weight2_1 method = reml noitprint noclprint;
class id;
model weight = d1 d2 visit/ solution notest;
random intercept d1 / subject = id type = un;
run;

The Mixed Procedure

Model Information
Data Set                     WORK.WEIGHT2
Dependent Variable           weight
Covariance Structure         Unstructured
Subject Effect               id
Estimation Method            REML
Residual Variance Method     Profile
Fixed Effects SE Method      Model-Based
Degrees of Freedom Method    Containment

Dimensions
Covariance Parameters             4
Columns in X                      4
Columns in Z Per Subject          2
Subjects                         38
Max Obs Per Subject               8

Number of Observations
Number of Observations Used             265
Number of Observations Not Used          39

Covariance Parameter Estimates
Cov Parm     Subject    Estimate
UN(1,1)      id           666.06
UN(2,1)      id          -0.6833
UN(2,2)      id          0.01179
Residual                  3.4397

Fit Statistics
-2 Res Log Likelihood          1420.7
AIC (smaller is better)        1428.7
AICC (smaller is better)       1428.9
BIC (smaller is better)        1435.3

Null Model Likelihood Ratio Test
DF    Chi-Square      Pr > ChiSq
3       1034.39          <.0001

Solution for Fixed Effects
Standard
Effect       Estimate       Error      DF    t Value    Pr > |t|
Intercept      193.46      4.1945      37      46.12      <.0001
d1            -0.1732     0.03337      37      -5.19      <.0001
d2             0.1104     0.05577     187       1.98      0.0492
visit         -2.3782      0.3025     187      -7.86      <.0001

*Model 3;
proc mixed data = weight2_1 method = reml noitprint noclprint;
class id;
model weight = d1 d2 visit/ solution notest;
random intercept d1 / subject = id type = un;
random visit /subject = id type = un;
run;

The Mixed Procedure

Model Information
Data Set                     WORK.WEIGHT2
Dependent Variable           weight
Covariance Structure         Unstructured
Subject Effects              id, id
Estimation Method            REML
Residual Variance Method     Profile
Fixed Effects SE Method      Model-Based
Degrees of Freedom Method    Containment

Dimensions
Covariance Parameters             5
Columns in X                      4
Columns in Z Per Subject          3
Subjects                         38
Max Obs Per Subject               8

Number of Observations
Number of Observations Used             265
Number of Observations Not Used          39

Covariance Parameter Estimates
Cov Parm     Subject    Estimate
UN(1,1)      id           667.10
UN(2,1)      id          -0.5547
UN(2,2)      id          0.01195
UN(1,1)      id           4.0915
Residual                  2.5165

Fit Statistics
-2 Res Log Likelihood          1399.4
AIC (smaller is better)        1409.4
AICC (smaller is better)       1409.6
BIC (smaller is better)        1417.6

Null Model Likelihood Ratio Test
DF    Chi-Square      Pr > ChiSq
4       1055.75          <.0001

Solution for Fixed Effects
Standard
Effect       Estimate       Error      DF    t Value    Pr > |t|
Intercept      193.45      4.1957      37      46.11      <.0001
d1            -0.1715     0.03021      37      -5.68      <.0001
d2             0.1081     0.04824     150       2.24      0.0266
visit         -2.3786      0.4219      37      -5.64      <.0001

*Model 4;
proc mixed data = weight2_1 method = reml noitprint noclprint;
class id ;
model weight = d1 d2 visit/ solution notest;
random intercept d1 visit / subject = id type = un;
run;

The Mixed Procedure

Model Information
Data Set                     WORK.WEIGHT2
Dependent Variable           weight
Covariance Structure         Unstructured
Subject Effect               id
Estimation Method            REML
Residual Variance Method     Profile
Fixed Effects SE Method      Model-Based
Degrees of Freedom Method    Containment

Dimensions
Covariance Parameters             7
Columns in X                      4
Columns in Z Per Subject          3
Subjects                         38
Max Obs Per Subject               8

Number of Observations
Number of Observations Used             265
Number of Observations Not Used          39

Covariance Parameter Estimates
Cov Parm     Subject    Estimate
UN(1,1)      id           667.00
UN(2,1)      id          -0.4385
UN(2,2)      id          0.01223
UN(3,1)      id         -13.5255
UN(3,2)      id         -0.02733
UN(3,3)      id           4.3062
Residual                  2.5039

Fit Statistics
-2 Res Log Likelihood          1397.3
AIC (smaller is better)        1411.3
AICC (smaller is better)       1411.7
BIC (smaller is better)        1422.7

Null Model Likelihood Ratio Test
DF    Chi-Square      Pr > ChiSq
6       1057.86          <.0001

Solution for Fixed Effects
Standard
Effect       Estimate       Error      DF    t Value    Pr > |t|
Intercept      193.45      4.1954      37      46.11      <.0001
d1            -0.1721     0.03028      37      -5.69      <.0001
d2             0.1091     0.04810     150       2.27      0.0248
visit         -2.3724      0.4285      37      -5.54      <.0001

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