Optimum two level fractional factorial plans for model identification and discrimination

Abstract

Model identification and discrimination are two major statistical challenges. In this paper we consider a set of models M k for factorial experiments with the parameters representing the general mean, main effects, and only k out of all two-factor interactions. We consider the class D of all fractional factorial plans with the same number of runs having the ability to identify all the models in M k , i.e., the full estimation capacity.The fractional factorial plans in D with the full estimation capacity for k 2 are able to discriminate between models in M u for u k * , where k * = (k/2) when k is even, k * = ((k − 1)/2) when k is odd. We obtain fractional factorial plans in D satisfying the six optimality criterion functions AD, AT, AMCR, GD, GT, and GMCR for 2 m factorial experiments when m = 4 and 5. Both single stage and multi-stage (hierarchical) designs are given. Some results on estimation capacity of a fractional factorial plan for identifying models in M k are also given. Our designs D4.1 and D10 stand out in their performances relative to the designs given in Li and Nachtsheim [Model-robust factorial designs, Technometrics 42(4) (2000) 345-352.] for m = 4 and 5 with respect to the criterion functions AD, AT, AMCR, GD, GT, and GMCR. Our design D4.2 stands out in its performance relative the Li-Nachtsheim design for m = 4 with respect to the four criterion functions AT, AMCR, GT, and GMCR. However, the Li-Nachtsheim design for m = 4 stands out in its performance relative to our design D4.2 with respect to the criterion functions AD and GD. Our design D14 does have the full estimation capacity for k = 5 but the twelve run Li-Nachtsheim design does not have the full estimation capacity for k = 5.

J.N. Srivastava and experimental design

“…The AD criterion was used by Shirakura and Ohnishi (1985) in designs for which β 1 contains up to -factor interactions and β 2 contains ( + 1)-factor interactions only. Ghosh and Tian (2006) presented tables of optimal search designs with respect to all six criteria for Scenario 4, for the maximum value of k in 2 4 and 2 5 factorial experiments.…”

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

“…The criterion functions GD, GT, and GMCR are the geometric means of the same. Shirakura and Ohnishi (1985) considered AD optimal designs for a 2 m factorial experiment, Ghosh and Tian (2006) presented optimum designs with respect to the criterion functions AD, AT, AMCR, GD, GT, and GMCR. The new criterion function OPTCV in this paper is based on Varð b β 2i Þ which is only the last diagonal element of Varð b β ðiÞ Þ.…”

“…The design d 1 which is the design D 4:1 for m¼4 and n ¼8 in Ghosh and Tian (2006), has the property P 2 ð3; 3Þ with the common variance in the two groups 0.136 s 2 and 0.188 s 2 , respectively.…”

“…For m ¼4 and n ¼9, we consider the design d 5 which is the design D5 in Ghosh and Tian (2006). An orthogonal array of strength 3 is a special balanced array of strength 3 and therefore has the property P 1 ðv 1 Þ and is a CV design.…”

Design and Inference for Discriminating between Two Competing Linear Regression Models

“…Many researchers have done work on designs for discriminating between two or more competing models; Atkinson and Cox (1974); Atkinson and Fedorov (1975a and b); Srivastava (1975Srivastava ( , 1977; Srivastava and Mallenby (1985); Pukelsheim and Rosenberger (1993); Dette (1994); Shirakura et al (1996); Biswas and Chaudhuri (2002); Ghosh and Teschmacher (2002); Dette and Kwiecien (2004); Ghosh and Tian (2006), and numerous others. In this paper we present new criterion functions for discriminating between two competing models.…”