======================================================================================
======================================================================================
1. - Intraclass Correlation for Continuous Outcome Variable using PROC NLMIXED:
======================================================================================


PROC NLMIXED DATA=<SAS Dataset> METHOD=FIRO QPOINTS=##;
  
  /*** List of parameter coefficient names and starting values ***/
  PARMS l2_int=## var1=##
      l2_c1=## l2_c2=## ... l2_cn=## 
      l1_c1=## l1_c2=## ... l1_cn=##; 

  /*** Cluster or Class level variables (Level 2) ***/
  level2 = l2_int + l2_c1*l2_v1 + l2_c1*l2_v2 + ... + l2_cn*l2_vn + r1;
  
  /*** Prediction function ***/
  pred = level2 + l1_c1*l1_v1 + l1_c2*l1_v2 + ... + l1_cn*l1_vn;
    
  MODEL <Continuous Outcome Variable> ~ NORMAL(pred,var2); 
  RANDOM r1 ~ NORMAL(0,var1) SUBJECT=<Class or cluster identifier>; 
  ESTIMATE 'ICC' var1/(var1+var2); 

RUN;

NOTE: SAS keywords that are in all caps are required code.

=============
Example 1:
=============

proc nlmixed data=<Input Data> method=firo qpoints=64;

  parms l2_int=0.25 var1=0.5 fs=0.25 mms=0.25 teen=0.25 sex=0.25 bo=0.25;
  level2 = l2_int + fs*famsize + mms*mar_1stbirth + teen*mom_1stbirth_under20 + r1;
  pred = level2 + sex*male + bo*order;
    
  model la_index ~ normal(pred,var2); 
  random r1 ~ normal(0,var1) subject=famnum; 
  estimate 'ICC' var1/(var1+var2); 

run;



======================================================================================
======================================================================================
2. - Intraclass Correlation for Binary Outcome Variable using PROC NLMIXED:
======================================================================================


PROC NLMIXED DATA=<SAS Dataset>;
  
  /*** List of parameter coefficient names and starting values ***/
  PARMS 	l2_int=## var1=##
      l2_c1=## l2_c2=## ... l2_cn=## 
      l1_c1=## l1_c2=## ... l1_cn=##; 

  /*** Cluster or Class level variables (Level 2) ***/
  level2 = l2_int + l2_c1*l2_v1 + l2_c1*l2_v1 + ... + l2_cn*l2_vn + r1;
  
  /*** Link function ***/
  exprhs = exp(level2 + l1_c1*l1_v1 + l1_c2*l1_v2 + ... + l1_cn*l1_vn);

  /*** Prediction function ***/
  pred = exprhs/(1+exprhs);
    
  MODEL <Binary Outcome Variable> ~ BINARY(pred);  
  RANDOM r1 ~ NORMAL(0,var1) SUBJECT=<Class or cluster identifier>; 
  ESTIMATE 'ICC' var1/(var1 + 3.289868134); 
/* 3.289868134 is an approximate value for the variance of a standard logistic function */
/* The real value is pi-squared divided by 3 */

RUN;

=============
Example 2:
=============

proc nlmixed data=<Input Data>;

  parms l2_int=0.25 var1=0.5 fs=0.25 mms=0.25 teen=0.25 sex=0.25 bo=0.25;
  level2 = l2_int + fs*famsize + mms*mar_1stbirth + teen*mom_1stbirth_under20 + r1;
  exprhs = exp(level2 + sex*male + bo*order);
  pred = exprhs/(1+exprhs);
  
  model g12by17 ~ binary(pred); 
  random r1 ~ normal(0,var1) subject=famnum; 
  estimate 'ICC' var/(var1 + 3.289868134); 

run;


======================================================================================
======================================================================================
3. - Intraclass Correlation for Continuous Outcome Variable using PROC MIXED:
======================================================================================


ODS OUTPUT MIXED.COVPARMS=<Output Dataset for Covariance Parameters>; 
PROC MIXED DATA=<SAS Dataset>;
  CLASS <Cluster or Level 2 Identifier> <Other categorical variables>;
  MODEL <Continuous Outcome> = <Other categorical variables>;
  RANDOM <Cluster or Level 2 Identifier>;
RUN;

Output:

Covariance Parameter Estimates 

Cov Parm		Estimate   

<Cluster>		Var1 

Residual		Var2

=============
Example 3:
=============

proc mixed data=<Input Data>;
  class famnum famsize mar_1stbirth mom_1stbirth_under20 male order;
  model la_index = famsize mar_1stbirth mom_1stbirth_under20 male order;
  random famnum;
run;

NOTE 1: Intraclass correlation cannot be computed for binary outcome variables using PROC MIXED.

NOTE 2: Intraclass correlation has to be computed manually using PROC MIXED.

      Formula:	ICC = Var1 /(Var1 + Var2)

NOTE 3: The ICC denotes the variability accounted for by the “between-cluster” factor with respect to 
the overall variability, or in other words, it denotes the degree of homogeneity within clusters.

NOTE 4: When using PROC MIXED and there are many clusters (levels or groups in the RANDOM variable), this can be problematic for SAS can run out of memory.

======================================================================================
======================================================================================