confounding of interaction term with residual variance, MIXED or HPMIXED

[brgray]

can someone point me to where SAS or a SAS user discusses how procedures with an obligate residual variance term (eg MIXED, HPMIXED) address situations where, due to an absence of replication, an interaction variance is confounded with what these procs call residual variance? in such a situation and when the levels of the factors are treated as random, i obtain estimates for both the interaction term and the residual variance that are roughly half that supplied for the residual variance when the confounded/aliased term is omitted.

here's example code:

ods select off;
proc hpmixed data=A;
class reader structure;
model agenum = structure / s;
random reader / s;
random structure*reader / s;
ods output covparms=covparms;
run;
ods select all;

proc print data=covparms; run;

* as above except omitting structure*reader term;
ods select off;
proc hpmixed data=A;
class reader structure;
model agenum = structure / s;
random reader / s;
*random structure*reader / s;
ods output covparms=covparms2;
run;
ods select all;

proc print data=covparms2; run;

[brgray]

PS i neglected to highlight that the number of readers (11), structures (6) and observations (66=11x6) precludes replication. hence, the confounding if not aliasing of the interaction and residual terms. i used REML.

[SteveDenham · Accepted solution]

Actually, I am not surprised at this "result".  Think about how the mixed model equations are parameterized and then imagine you could split the residual into separate variance components.  If there is equal weighting of observations, half of the variability goes to each of the separate variance components.  Now I don't think specifying the interaction as a separate source of variability is quite kosher given the design, but as you have demonstrated, it can be done. Whether the value obtained is meaningful or not is a matter for discussion.  Off the top of my head, I vote for "not meaningful".

SteveDenham