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SAS Textbook Examples
Multilevel Analysis Techniques and Applications by Joop Hox
Chapter 8: The Multilevel Approach to Meta-Analysis


The code for this chapter was provided by Professor Hoffman from the Department of Psychology of the University of Nebraska-Lincoln. We thank Professor Hoffman for her contribution to this chapter.


Page 148, table 8.3 using the data set meta20.sas7bdat. SAS proc mixed is used in all the analyses. The use of the statement parms with the "hold =" option allows us to perform variance-known analysis.

Model 2: multilevel intercept-only

data practice; 
   set ats.meta20; 
   if study=16 and G=.955 THEN study=17; 
run;
* creating dataset of residual variances to hold constant;
* covariance parameters start values need to start at 2 instead of 1;
data resvar; set practice;
	covp = study+1; 
	keep covp varofd; run;
* transposing to multivariate;
proc transpose data=resvar out=resvar;
  id covp; 
  idlabel covp; 
  var varofd; 
run;
* renaming transposed variables to use in PARMS statement;
* adding in start value for intercept variance as covp1;
data resvar; retain covp1; 
	set resvar; drop _name_;
	array old(20) _2-_21;
	array new(20) covp2-covp21;
	do i=1 to 20; new[i]=old[i]; end; 
	drop _2--_21 i; 
	covp1=.14; 
run;

proc mixed data=practice  method=reml;
	class study;			        * study is categorical variable;
	model d =  / solution ddfm=bw;		* empty model for effect size;
	random intercept / subject=study;	* allowing heterogeneity in d;		
	repeated / group=study type=vc;		* separate residual variance per study;
	parms / parmsdata=resvar hold=2 to 21;  * hold residual variances at known values;
run;
       Covariance Parameter Estimates

Cov Parm      Subject    Group       Estimate

Intercept     STUDY                    0.1446
Residual                 STUDY 1      0.08600
Residual                 STUDY 2       0.1060
   ...                     ...
Residual                 STUDY 20      0.1410

           Fit Statistics

-2 Res Log Likelihood            30.4
AIC (smaller is better)          32.4
AICC (smaller is better)         32.7
BIC (smaller is better)          33.4

  PARMS Model Likelihood Ratio Test

    DF    Chi-Square      Pr > ChiSq

     1          0.00          0.9533

                   Solution for Fixed Effects

                         Standard
Effect       Estimate       Error      DF    t Value    Pr > |t|

Intercept      0.5801      0.1080      19       5.37      <.0001

Model 2: multilevel regression

proc mixed data=practice  method=reml;
	class study;					* study is categorical variable;
	model d = weeks / solution cl ddfm=bw;		
	random intercept / subject=study;		* allowing heterogeneity in d;		
	repeated / group=study;				* separate residual variance per study;
	parms / parmsdata=resvar hold=2 to 21;  	* hold residual variances at known values;
run;	
       Covariance Parameter Estimates

Cov Parm      Subject    Group       Estimate

Intercept     STUDY                   0.03658
Residual                 STUDY 1      0.08600
Residual                 STUDY 2       0.1060
   ...                     ...
Residual                 STUDY 20      0.1410

           Fit Statistics

-2 Res Log Likelihood            22.2
AIC (smaller is better)          24.2
AICC (smaller is better)         24.5
BIC (smaller is better)          25.2

  PARMS Model Likelihood Ratio Test

    DF    Chi-Square      Pr > ChiSq

     1          2.60          0.1071

                                    Solution for Fixed Effects

                         Standard
Effect       Estimate       Error      DF    t Value    Pr > |t|     Alpha       Lower       Upper

Intercept     -0.2169      0.2043      18      -1.06      0.3023      0.05     -0.6462      0.2123
WEEKS          0.1399     0.03378      18       4.14      0.0006      0.05     0.06895      0.2109

Page 151, table 8.4: Random-effects model and multilevel meta-analyses on example data

Model 1: Intercept-only (see previous example)

Model 2: Intercept + Ntot

proc mixed data=practice method=reml;
  class study;
  model d = ntot / solution  ddfm=bw;	
  random intercept / subject=study;
  repeated / group=study;
  parms / parmsdata=resvar hold=2 to 21; 
run;
Cov Parm      Subject    Group       Estimate

Intercept     STUDY                    0.1592
Residual                 STUDY 1      0.08600
Residual                 STUDY 2       0.1060
   ...                       ...
Residual                 STUDY 20      0.1410

           Fit Statistics

-2 Res Log Likelihood            38.0
AIC (smaller is better)          40.0
AICC (smaller is better)         40.2
BIC (smaller is better)          41.0

  PARMS Model Likelihood Ratio Test

    DF    Chi-Square      Pr > ChiSq

     1          0.05          0.8157

                   Solution for Fixed Effects

                         Standard
Effect       Estimate       Error      DF    t Value    Pr > |t|

Intercept      0.4430      0.5125      18       0.86      0.3988
NTOT         0.002489    0.009005      18       0.28      0.7854

Model 3:  intercept + reliability

proc mixed data=practice method=reml;
  class study;
  model d = rii / solution ddfm=bw;  
  random intercept / subject=study;
  repeated / group=study;
  parms / parmsdata=resvar hold=2 to 21; 
run;
       Covariance Parameter Estimates

Cov Parm      Subject    Group       Estimate

Intercept     STUDY                    0.1565
Residual                 STUDY 1      0.08600
Residual                 STUDY 2       0.1060
  ...                      ...
Residual                 STUDY 20      0.1410

           Fit Statistics

-2 Res Log Likelihood            27.7
AIC (smaller is better)          29.7
AICC (smaller is better)         30.0
BIC (smaller is better)          30.7

  PARMS Model Likelihood Ratio Test

    DF    Chi-Square      Pr > ChiSq

     1          0.04          0.8419

                                    Solution for Fixed Effects

                         Standard
Effect       Estimate       Error      DF    t Value    Pr > |t|     Alpha       Lower       Upper

Intercept      0.1602      1.2273      18       0.13      0.8976      0.05     -2.4182      2.7386
RII            0.5087      1.4773      18       0.34      0.7346      0.05     -2.5950      3.6124

Model 4:  intercept + duration

proc mixed data=practice method=reml;
  class study;
  model d = weeks / solution ddfm=bw; 
  random intercept / subject=study;
  repeated / group=study;
  parms / parmsdata=resvar hold=2 to 21;
run;
       Covariance Parameter Estimates

Cov Parm      Subject    Group       Estimate

Intercept     STUDY                   0.03658
Residual                 STUDY 1      0.08600
Residual                 STUDY 2       0.1060
   ...                     ...
Residual                 STUDY 20      0.1410

           Fit Statistics

-2 Res Log Likelihood            22.2
AIC (smaller is better)          24.2
AICC (smaller is better)         24.5
BIC (smaller is better)          25.2

  PARMS Model Likelihood Ratio Test

    DF    Chi-Square      Pr > ChiSq

     1          2.60          0.1071

                   Solution for Fixed Effects

                         Standard
Effect       Estimate       Error      DF    t Value    Pr > |t|

Intercept     -0.2169      0.2043      18      -1.06      0.3023
WEEKS          0.1399     0.03378      18       4.14      0.0006

Model 5:  intercept + all

proc mixed data=practice method=reml;
  class study;
  model d = ntot rii weeks / solution ddfm=bw; 
  random intercept / subject=study;
  repeated / group=study;
  parms / parmsdata=resvar hold=2 to 21; 
run;
       Covariance Parameter Estimates

Cov Parm      Subject    Group       Estimate

Intercept     STUDY                   0.04934
Residual                 STUDY 1      0.08600
Residual                 STUDY 2       0.1060
  ...                      ...
Residual                 STUDY 20      0.1410

           Fit Statistics

-2 Res Log Likelihood            27.6
AIC (smaller is better)          29.6
AICC (smaller is better)         29.9
BIC (smaller is better)          30.6

  PARMS Model Likelihood Ratio Test

    DF    Chi-Square      Pr > ChiSq

     1          1.67          0.1965

                   Solution for Fixed Effects

                         Standard
Effect       Estimate       Error      DF    t Value    Pr > |t|

Intercept      0.3843      0.9234      16       0.42      0.6828
NTOT         -0.00357    0.007035      16      -0.51      0.6187
RII           -0.5510      1.2010      16      -0.46      0.6526
WEEKS          0.1506     0.03755      16       4.01      0.0010

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