On a Special Subset Giving an Irregular Fractional Replicate of a 2N Factorial Experiment

Abstract

Summary A special class of fractional replicates of 2n factorial experiments has been studied in this paper. Such fractional replicates are expected to be useful in practice where, for very large values of n, we may be interested only in the main effects. Analysis of such designs is simple in that the entire computational procedure is reducible to a simple routine.

Results on Factorial Designs of Resolution IV for the 2nand 2n3mSeries

“…Non-orthogonal 2" designs of resolution IV have been constructed by John ([7] and [8]) and by Banerjee and Federer 432 B. Ii. MARGOLIN [2]. Now a new paper by Webb [15] has presented some further results for 2" non-orthogonal designs of resolution IV and has stimulated the observations and results which follow.…”

“…x = (X-J = I I xl * -X1 Theorem 1 of Banerjee and Federer [2] is applicable here. If Y, denotes the response vector for the first n runs in the 2"//2n (from the weighing design) and Y, denotes the response vector for the second n runs in the 2"//2n (from the fold-over), then 6, the estimate of the n main effects, is: (6) 6 = %PvGl-'x:~l -Yz).…”

“…Then from (6) and (8) If Et-1 Xik = Si , j = 1, * * * , n, then for k = 1, * * * , n, @k = (2n -2)-l 2 xik(Yli -Yzi) -((an -U@n -2) ,$ Si(Yli -YZi))' Theorem 1 of Banerjee and Federer [2] states, in effect, that for purposes of estimation, the difference between the response to a run and the response to the fold-over of that run is the only pertinent datum as far as the two runs are concerned. This suggests that all minimal 2"//2n resolution IV designs can be run in n blocks of size 2, each block consisting of a run and its fold-over run.…”

“…The Box-Wilson technique has been used primarily to produce orthogonal resolution IV designs from orthogonal resolution III designs, with the noted exception of the papers by John [7, S] and the paper by Banerjee and Federer [2]. To discuss the efficiency of a design we need to decide on one measure of efficiency.…”

Near Minimal Resolution IV Designs for the s n Factorial Experiment

“…Other work related to foldover designs include the nonorthogonal constructions of Webb [13], John [7] and Banerjee and Federer [3], the optimal balanced resolution IV designs of Srivastava and Anderson [12], and some more efficient imbalanced designs given by Mitchell Other work related to foldover designs include the nonorthogonal constructions of Webb [13], John [7] and Banerjee and Federer [3], the optimal balanced resolution IV designs of Srivastava and Anderson [12], and some more efficient imbalanced designs given by Mitchell…”