MIXED
MODELS SITE
ENUMERATION
OF FIXED EFFECTS MODELS
It is well known that there is a onetoone correspondence between fixed effects ANOVA models involving both crossed and nested factors, and combinatorial objects called "posets". The enumeration of nonisomorphic posets is an interesting and nontrivial combinatorial problem for which answers are available for posets of order 14 or less, i.e. for fixed effects ANOVA models with 14 or fewer factors.
Factors 
Nonisomorphic Fixed Effects Models 
Name 
1 
1 
 
2 
2 
 
3 
5 
 
4 
16 
 
5 
63 
 
6 
318 
 
7 
2,045 
J. Wright 1972 
8 
16,999 
S.K. Das 1977 
9 
183,231 
R.H. Mohring 1984 
10 
2,567,284 
J.C. Culberson and G.J.E. Rawlins 1990 
11 
46,749,427 
J.C. Culberson and G.J.E. Rawlins 1990 
12 
1,104,891,746 
C. Chaunier and N. Lygeros 1991 
13 
33,823,827,452 
C. Chaunier and N. Lygeros 1992 
14 
1,338,193,159,771 
N. Lygeros and P. Zimmerman 2000 
ENUMERATION
OF MIXED MODELS
The enumeration of mixed models does not appear to have been considered. The table below gives results for models with up to 5 factors. Note that models where a fixed effect is nested within a random effect are not allowed. For more information, including the enumeration of mixed models WITHOUT the constraint (that a fixed effect cannont be nested within a random effect) see the website of Nik Lygeros.
Factors 
Nonisomorphic
Mixed Modesls 
Name 
1 
2 
 
2 
6 
 
3 
22 
 
4 
101 
 
5 
576 
A. Hess 1999 
6 
4,162 
R. Bayon, N. Lygeros and J.S. Sereni 2002 
7 
38,280 
R. Bayon, N. Lygeros and J.S. Sereni 2002 
8 
451,411 
R. Bayon, N. Lygeros and J.S. Sereni 2002 
9 
6,847,662 
R. Bayon, N. Lygeros and J.S. Sereni 2002 
CONFIDENCE
INTERVALS FOR VARIANCE COMPONENTS
Confidence intervals for variance components in any of these models, with superior coverage properties than those afforded by the application of the Satterthwaite method, may be computed using the methods discussed in the book "Confidence Intervals for Variance Components" by Burdick and Graybill (1992). To facilitate these calculations, we have written a SAS macro that will compute these confidence intervals for any mixed effects saturated ANOVA model involving five or fewer factors. User input to the SAS MACRO is the actual data set along with a matrix indicating whether each factor is fixed or random and the nesting/crossing configuration among the factors. The operation of the MACRO is illustrated using several examples. Such a MACRO is expected to be extremely useful to practitioners in view of the fact that SAS or any other commonly available statistical software package does not have built in commands for obtaining confidence intervals for variance components discussed in Burdick and Graybill (1992).

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