Defense Date


Document Type


Degree Name

Master of Science



First Advisor

Walter B. Cummings


The analysis of variance for experiments where the fixed effects or random effects model is appropriate is generally agreed upon with regard to testing procedures and covariance structure. It is only in experiments involving both random and fixed factors, i.e. mixed effects models, that controversy occurs as to the proper analysis. The mixed effect model has been considered by many statisticians, and several techniques have been developed for explaining its structure and performing its analysis for balanced data sets. The relationship of these techniques have been discussed in several papers as well.

The simplest case of the difficulties presented by the mixed effects models occurs in the two-way cross classification model with interaction. The various models for the two-way mixed situation were examined and compared. It was found that Scheffe's model defined the effects in a meaninful way, is completely general, and provides exact tests. In situations where Scheffe's model cannot be applied, it was found that Kempthorne's model or Graybill's model should be used since they define effects in a meaningful way and, under certain assumptions, gives exact tests. Searle's model does not define the effects in the same manner as the former three models. Searle's effects are defined more for mathematical appeal and his model is designed for easy application to unbalanced cases. Consequently, his model was not found to be desirable in balanced two-way mixed effect designs.

In higher order models, Scheffe's modeling techniques were found not to be practical since his test for fixed effect differences in models with more than two random effects cannot be computed. Kempthorne's models and Graybill's models both, under certain assumptions, provide straightforward tests for all effects. For this reason, their modeling techniques are recommended for higher order mixed models involving balanced data sets. Searle's modeling technique was again found unapplicable for balanced data sets in higher order mixed models for the same reasons as those in the two-way case.

The results of the investigation recommends Scheffe's model for two-way situations, but Kempthorne's modeling technique and Graybill's modeling technique seem the most versatile. Although the task would be very cumbersome, further investigation is suggested in comparing Kempthorne's procedure and Graybill's procedure to Scheffe's procedure for testing fixed effect differences.


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