Common variance fractional factorial designs and their optimality to identify a class of models

Fractional factorial designs with n treatments for 2 m factorial experiments are considered to identify a class of ( m 2 ) models with the common parameters representing the general mean and the main effects while the uncommon parameter in each model represents a two factor interaction. A new property P g ( v 1 , … , v g ) of designs is introduced in this context to least squares estimate the uncommon parameters in g groups of models so that the estimates of vi such parameters in the ith group have a common variance (CV), where g is an integer satisfying 1 ≤ g ≤ ( m 2 ) , i = 1 , … , g , v 1 + ⋯ + v g = ( m 2 ) . The property P 1 ( v 1 ) is desirable to have for the fractional factorial designs to identify the ( m 2 ) models. The concept of CV designs having the property P 1 ( v 1 ) is introduced for the model identification. Several series of CV designs for general m and n are presented. For fixed values of n and m, D n , m represents the class of all fractional factorial CV designs having the property P 1 ( v 1 ) . CV designs in D n , m have possible unequal values for the common variance. The smaller the common variance, the better the CV designs for the model identification. The concept of optimum common variance (OPTCV) design having the smallest common variance in D n , m is also introduced. This paper presents some OPTCV designs.