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

• Introduces the common variance fractional factorial designs to identify a class of models explaining the dependence of response on factors.
• Concept of the optimal common variance design having the smallest common variance with a class of designs is also introduced.
• Presents many general series of optimal common variance designs for two level factorial experiments.

Fractional factorial designs with n   treatments for 2m2m factorial experiments are considered to identify a class of (m2) 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 Pg(v1,…,vg)Pg(v1,…,vg) 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≤(m2), i=1,…,gi=1,…,g, v1+⋯+vg=(m2). The property P1(v1)P1(v1) is desirable to have for the fractional factorial designs to identify the (m2) models. The concept of CV designs having the property P1(v1)P1(v1) 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  , Dn,mDn,m represents the class of all fractional factorial CV designs having the property P1(v1)P1(v1). CV designs in Dn,mDn,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 Dn,mDn,m is also introduced. This paper presents some OPTCV designs.