Multiple Membership with Correlated Random Effects

[jmoyer]

I'd like to fit a multiple membership model in which clusters are crossed on some dichotomous variable X. There will be a random effect for cluster when X = 0 as well as a random effect for cluster when X = 1. These two random effects will covary. My specific question focuses on what structure should be specified using the TYPE option of the RANDOM statement to model this covariance in the G matrix.

The variables in my dataset are as follows:

y: continuous outcome

x: dichotomous variable

participant: participant ID

cluster1-cluster2: clusters participant interacts with (there are six clusters)

weight1-weight2: weights for clusters (both 0.50)

Code to generate an example data set is placed behind the spoiler block below. 

Spoiler

data WORK.D;
infile datalines dsd truncover;
input y:BEST12. x:BEST12. participant:$5. cluster1:BEST12. cluster2:BEST12. weight1:BEST12. weight2:BEST12.;
format y BEST12. x BEST12. cluster1 BEST12. cluster2 BEST12. weight1 BEST12. weight2 BEST12.;
datalines4;
1.7116292881,0,01,1,3,0.5,0.5
-5.933128683,0,02,1,6,0.5,0.5
-13.5337454,0,03,1,4,0.5,0.5
7.8635381751,0,04,1,2,0.5,0.5
4.375759951,0,05,1,3,0.5,0.5
-2.805837926,0,06,1,5,0.5,0.5
-4.125339503,0,07,1,5,0.5,0.5
-4.502606231,0,08,1,2,0.5,0.5
-3.07787325,0,09,1,2,0.5,0.5
-8.712734561,0,010,1,4,0.5,0.5
3.5771897824,0,011,2,6,0.5,0.5
-5.569639235,0,012,2,3,0.5,0.5
5.0164446473,0,013,2,5,0.5,0.5
0.9662273736,0,014,2,6,0.5,0.5
-3.053160929,0,015,2,6,0.5,0.5
5.7631686365,0,016,2,6,0.5,0.5
-4.061618169,0,017,1,2,0.5,0.5
-5.675804875,0,018,2,6,0.5,0.5
-1.998492309,0,019,1,2,0.5,0.5
-4.877700179,0,020,2,4,0.5,0.5
0.5826871124,0,021,3,5,0.5,0.5
1.5678925913,0,022,2,3,0.5,0.5
-1.662167585,0,023,3,5,0.5,0.5
-1.852380966,0,024,1,3,0.5,0.5
-1.306121158,0,025,3,6,0.5,0.5
0.1193967378,0,026,2,3,0.5,0.5
-4.690690418,0,027,3,4,0.5,0.5
-7.514667431,0,028,3,5,0.5,0.5
1.2045188081,0,029,3,4,0.5,0.5
1.1362348613,0,030,3,4,0.5,0.5
-3.448389572,0,031,4,5,0.5,0.5
-4.637042638,0,032,4,5,0.5,0.5
-5.800713663,0,033,4,5,0.5,0.5
-10.46251835,0,034,4,5,0.5,0.5
-3.226708786,0,035,3,4,0.5,0.5
-0.108735954,0,036,1,4,0.5,0.5
1.4564796198,0,037,4,5,0.5,0.5
5.3981391556,0,038,3,4,0.5,0.5
5.044206624,0,039,2,4,0.5,0.5
-4.990751737,0,040,1,4,0.5,0.5
4.4013568068,0,041,2,5,0.5,0.5
2.8935250611,0,042,3,5,0.5,0.5
4.3287629943,0,043,1,5,0.5,0.5
-7.972389136,0,044,1,5,0.5,0.5
5.6880095481,0,045,4,5,0.5,0.5
5.2309207532,0,046,2,5,0.5,0.5
-5.75182201,0,047,2,5,0.5,0.5
-0.391128977,0,048,1,5,0.5,0.5
0.8261384611,0,049,3,5,0.5,0.5
0.5012871882,0,050,5,6,0.5,0.5
-2.502664486,0,051,2,6,0.5,0.5
-2.283545937,0,052,4,6,0.5,0.5
5.667798606,0,053,2,6,0.5,0.5
2.6738632685,0,054,4,6,0.5,0.5
-0.348288503,0,055,5,6,0.5,0.5
-0.721466022,0,056,1,6,0.5,0.5
-3.347618332,0,057,1,6,0.5,0.5
-0.543900391,0,058,3,6,0.5,0.5
1.7437789482,0,059,4,6,0.5,0.5
2.2261827522,0,060,4,6,0.5,0.5
-4.027513544,1,11,1,3,0.5,0.5
2.4456106753,1,12,1,2,0.5,0.5
1.7491766973,1,13,1,6,0.5,0.5
3.8937891424,1,14,1,2,0.5,0.5
-1.687468755,1,15,1,5,0.5,0.5
-3.464784494,1,16,1,3,0.5,0.5
-0.094955504,1,17,1,6,0.5,0.5
-2.894264012,1,18,1,4,0.5,0.5
-0.01356603,1,19,1,6,0.5,0.5
-4.708004753,1,110,1,4,0.5,0.5
-3.392529479,1,111,2,3,0.5,0.5
-3.817671171,1,112,1,2,0.5,0.5
3.0014733968,1,113,2,4,0.5,0.5
-1.56760315,1,114,2,5,0.5,0.5
-5.872920416,1,115,2,4,0.5,0.5
3.7860683279,1,116,1,2,0.5,0.5
2.7506424395,1,117,2,6,0.5,0.5
-9.507855019,1,118,2,5,0.5,0.5
-5.690620937,1,119,2,5,0.5,0.5
-1.502637305,1,120,2,5,0.5,0.5
0.6257569539,1,121,1,3,0.5,0.5
4.1823693449,1,122,3,6,0.5,0.5
-0.988326345,1,123,3,4,0.5,0.5
-0.519023566,1,124,3,4,0.5,0.5
-5.893752026,1,125,2,3,0.5,0.5
0.8888285888,1,126,1,3,0.5,0.5
-4.85976943,1,127,1,3,0.5,0.5
3.8821360339,1,128,3,6,0.5,0.5
1.740925357,1,129,2,3,0.5,0.5
-2.360578263,1,130,3,4,0.5,0.5
0.48730412,1,131,2,4,0.5,0.5
-4.090089012,1,132,4,5,0.5,0.5
-4.981695651,1,133,4,6,0.5,0.5
6.9505380042,1,134,4,6,0.5,0.5
5.3731396082,1,135,4,6,0.5,0.5
2.161390774,1,136,4,6,0.5,0.5
-9.882399516,1,137,4,5,0.5,0.5
-1.364193916,1,138,3,4,0.5,0.5
-0.601594245,1,139,2,4,0.5,0.5
-3.166185748,1,140,4,6,0.5,0.5
-2.247222929,1,141,1,5,0.5,0.5
4.3654810524,1,142,5,6,0.5,0.5
2.5255637516,1,143,5,6,0.5,0.5
0.3662445094,1,144,5,6,0.5,0.5
-2.467531122,1,145,4,5,0.5,0.5
-7.228769228,1,146,1,5,0.5,0.5
2.5481005061,1,147,2,5,0.5,0.5
-2.546195615,1,148,2,5,0.5,0.5
-2.222218085,1,149,2,5,0.5,0.5
-2.864998875,1,150,3,5,0.5,0.5
1.5270292508,1,151,1,6,0.5,0.5
1.137488738,1,152,1,6,0.5,0.5
1.6653747585,1,153,5,6,0.5,0.5
4.9831078878,1,154,5,6,0.5,0.5
10.437312685,1,155,4,6,0.5,0.5
4.5063732354,1,156,3,6,0.5,0.5
5.1041559153,1,157,2,6,0.5,0.5
-2.311161319,1,158,4,6,0.5,0.5
5.0289995693,1,159,5,6,0.5,0.5
-4.905688138,1,160,5,6,0.5,0.5
;;;;

Here is the PROC GLIMMIX code that has gotten me closest to where I'd like to be:

proc glimmix data=D;
class cluster1-cluster2;
effect clusterg1=mm(cluster1-cluster2/weight=(weight1-weight2));
model Y = X /dist=normal link=identity;
random clusterg1 clusterg1*X / G;
run;

The G matrix from this model is as follows:

What's missing are the covariances between random effects, which I've indicated with light gray cells. I have not been able to find a covariance structure in the list of TYPE options that matches this scenario. Can PROC GLIMMIX work in this situation?