#### Abstract

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. Such an enumeration problem does not appear to have been considered for mixed effects models. In this paper we study this problem and obtain a list of nonisomorphic mixed effects models involving five or fewer factors. For instance, it is shown that there are 576 nonisomorphic mixed effects ANOVA models involving five factors. 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 [2]. 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 an example involving real data. 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 [2]. A copy of this MACRO is available at www.stat.colostate.edurhess/MixedModels.htm.

#### Creative Commons License

This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.

ENUMERATION OF MIXED LINEAR MODELS AND A SAS MACRO FOR COMPUTATION OF CONFIDENCE INTERVALS FOR VARIANCE COMPONENTS

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. Such an enumeration problem does not appear to have been considered for mixed effects models. In this paper we study this problem and obtain a list of nonisomorphic mixed effects models involving five or fewer factors. For instance, it is shown that there are 576 nonisomorphic mixed effects ANOVA models involving five factors. 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 [2]. 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 an example involving real data. 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 [2]. A copy of this MACRO is available at www.stat.colostate.edurhess/MixedModels.htm.