MIXED MODELS SITE

ENUMERATION OF FIXED EFFECTS MODELS

It is well known that there is a one-to-one 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).