 Open Access Publications from the University of California

## Common Variance Fractional Factorial Designs for Model Comparisons

• Author(s): CHOWDHURY, SHRABANTI
• Advisor(s): Ghosh, Subir
• et al.
Abstract

In designing a fractional factorial experiment, a class of models with some common parameters is considered for describing the data to be obtained from the experiment. The uncommon parameters of these models are to be estimated with the same variance as best as possible. Fractional factorial designs are obtained with the various variance structures in terms of their equalities. A special variance structure having the equal variances of the estimators of all uncommon parameters is the main theme of this thesis. In particular the 2-factor interaction effect is considered as the uncommon parameter in each model. Such plans with the ability of estimating the uncommon parameter with equal precision are called Common Variance (CV) designs. From the class of all CV designs for particular values of the number of factors m and the number of runs n designs giving smallest value of CV are obtained. Such designs are called Optimum CV designs. Both symmetric and asymmetric factorial experiments are considered with factors at two and three levels.

Two series of CV designs are obtained for general 3^m factorial experiment with different number of runs. The common variance property is characterized for general fractional factorial designs. Several sufficient conditions are obtained using projection matrix and runs of the designs. The projection matrices of the series of CV designs for general m are investigated and a special structure of the projection matrix is presented for the CV designs including the optimum CV designs. Optimum CV designs are also presented for these two series for different m. CV designs are obtained with replicated runs. It is shown that a 3^2 CV design which is optimum in the class of all CV designs for n=6 remains CV after replicating any of its six runs any number of times. Several other 3^2 CV designs for n=6 are presented which satisfy this general replication property. Condition is derived for obtaining hierarchical CV designs for a general fractional factorial experiment. The determination of CV designs was also extended to a mixed level factorial experiment with factors at two and three levels. For a 2x3 factorial experiment CV designs exist only under a constraint of replications, for 2^mx3 and 2^mx3^3 factorial experiments designs are presented which give common variance within groups of similar structured interactions.