Row–Column Designs for 2 n factorial 2-Colour Microarray Experiments for Estimation of Main Effects and Two-Factor Interactions with Orthogonal Parameterization

Abstract

A method of construction of row-column designs for estimation of main effects and two factor interaction effects in 2 n factorial microarray experiments based on orthogonal parameterization has been developed in minimum number of replications. A catalogue of designs for 2 B n B 9 has been prepared. The catalogue also gives the main effects and two-factor interactions confounded in different replications and the factorial effects that are not confounded in a replication. The efficiency factor of estimable main effects and two-factor interactions has been given. For each 2 n factorial, two designs have been given, one in which main effects are estimated with more efficiency and another in which two-factor interactions are estimated with more efficiency. A procedure of construction of row-column designs for estimation of all factorial effects with odd number of factors has been given. Row-column designs with unequal replication of different treatment combinations have also been obtained for estimation of all main effects and two-factor interactions.

Construction of Row–Column Factorial Designs

“…Dash et al . () provided designs in \mathfrak m replicates of a 2 q +1 ‐factorial arranged in 2×2 q arrays for 2⩽ n = q +1⩽9 which enable estimation of all effects of interest. Theorem 4 gives designs with equivalent estimability properties to those of Table 3 of Dash et al .…”

“…Focusing on those effects estimated with smallest relative information, from the design summaries in expressions (4.6) and (4.7) it is observed that in D81 one main effect and five interactions are estimable from only one replicate but that in D82 two main effects and four interactions have this property. Dash et al (2013) provided designs in ᒊ replicates of a 2 q+1 -factorial arranged in 2 × 2 q arrays for 2 n = q + 1 9 which enable estimation of all effects of interest. Theorem 4 gives designs with equivalent estimability properties to those of Table 3 of Dash et al (2013), in the sense that the numbers and distributions of independent estimates of main effects and two-factor interactions are the same.…”

“…Recent applications are found in Dash et al . (), who described microarray experiments with dye and microarray as blocking factors, and Datta et al . (), who outlined doubly confounded factorial designs for experiments involving livestock.…”

“…Such an experiment with batches of raw material and machines as blocking factors is described in chapter 9 of Hinkelmann and Kempthorne (2005). Recent applications are found in Dash et al (2013), who described microarray experiments with dye and microarray as blocking factors, and Datta et al (2017), who outlined doubly confounded factorial designs for experiments involving livestock.…”

“…Choi and Gupta (2008) produced several designs by selecting subsets of factorial effects to be confounded with blocks. For n 9, Dash et al (2013) gave designs for s replicates of a 2 n -factorial in a 2 × 2 sn−1 array that provide estimates of all main effects and all two-factor interactions. Cheng and Tsai (2013) built on work of Patterson and Bailey (1978) to give the structure for a design key for a 2 n -factorial arranged in a rectangular array.…”