Characterization of Special Variance Structures for Designs in Model Identification and Discrimination
Models containing the general mean, main effects, and all possible k two-factor interaction effects are considered for factorial experiments with m factors observed at two levels each. Specifically, fractional factorial designs consisting of n runs which permit the identification and discrimination of the models of interest are evaluated and classified. The classifications are dependent on a new property introduced in this dissertation, denoted PV, g≥1, which relies on the structure of the variance-covariance matrix for the estimates of the model parameters. Designs with the property, PV, g≥1, permit to divide the models in the class considered into g groups so that all models in a group have equal variances for the least squares estimates of the k two-factor interaction effects. Ghosh-Tian optimum designs [Ghosh and Tian (2006)] for m=4, n=6,...,11, k=1, and m=5, n=7,...,16, k=1 are classified with respect to g values, g≥1, in the property PV and presented through illustrative examples.
Several characterizations of designs with PV, g≥1, are provided for the case when k=1. Designs such as balanced designs, isomorphic designs and complementary designs with the property PV are proven to have P<1V for k=1. Tables identifying all such balanced designs are provided. It is noted that although D9.2 and D14 are not balanced, these designs in fact take a special form resulting in P<1V for k=1. These special forms are investigated in depth.
In addition, the construction of all designs giving P<1V for m=3, n=5, 6, 7, 8, k=1 and m=4, n=6,...,16, k=1 are described. Further, occurrences of P<1V are presented for fractional factorial designs when m=5, n=7, 8, 9, k=1.
Finally, additional characterizations of PV, g≥1, when P>1 are given and illustrated through various examples. Special designs are presented with the property P<1V for maxk which is the largest value of k in the models of interest that the design has the ability to identify and discriminate. It is shown that balanced designs will have P<1V for k=mC2 . Tables identifying all such balanced designs are provided.