Summary
For a given fractional 2m factorial (2m-FF) designT, the constitution of a block plan to divideT intok (2r−1<k≦2r) blocks withr block factors each at two levels is proposed and investigated. The well-known three norms of the confounding matrix are used as measures for determining a “good” block plan. Some theorems concerning the constitution of a block plan are derived for a 2m-FF design of odd or even resolution. Two norms which may be preferred over the other norm are slightly modified. For each value ofN assemblies with 11≦N≦26, optimum block plans fork=2 blocks with block sizes [N/2] andN−[N/2] minimizing the two norms are presented forA-optimal balanced 24-FF designs of resolutionV given by Srivastava and Chopra (Technometrics,13, 257–269).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Box, G. E. P. and Hunter, J. S. (1961). The 2k−p fractional designs, Part I, II,Technometrics,3, 311–351, 449–458.
Chopra, D. V. (1975). Balanced optimal 28 fractional fractorial designs of resolution V, 52≦N≦59, A survey of statistical designs and linear models (ed. J. N. Srivastava), 91–100, Amsterdam, North-Holland.
Chopra, D. V. and Srivastava, J. N. (1973). Optimal balanced 27 fractional factorial designs of resolution V, withN≦42,Ann. Inst. Statist. Math.,25, 587–604.
Hedayat, A., Raktoe, B. L. and Federer, W. T. (1974). On a measure of aliasing due to fitting an incomplete model,Ann. Statist. 2, 650–660.
Kiefer, J. (1959). Optimum experimental designs,J. Roy. Statist. Soc., Ser. B,21, 272–319.
Shirakura, T. (1976). Optimal balanced fractional 2m factorial designs of resolution VII, 6≦m≦8,Ann. Statist.,4, 515–531.
Shirakura, T. (1977). Contributions to balanced fractional 2m factorial designs derived from balanced arrays of strength 2l, Hiroshima Math. J.,7, 217–285.
Shirakura, T. and Kuwada, M. (1976). Covariance matrices of the estimates for balanced fractional 2m factorial designs of resolution 2l+1,J. Japan Statist. Soc.,6, 27–31.
Srivastava, J. N. (1970). Optimal balanced 2m fractional factorial designs,S. N. Roy Memorial Volume, Univ. of North Carolina and Indian Statistical Institute, 689–706.
Srivastava, J. N. and Chopra, D. V. (1971). Balanced optimal 2m fractional factorial designs of resolution V,m≦6,Technometrics,13, 257–269.
Srivastava, J. N. and Chopra, D. V. (1973). Balanced optimal 27 fractional factorial designs of resolution V, with 56 to 68 runs,Utilitas Mathematica,5, 263–279.
Srivastava, J. N., Raktoe, B. L. and Pesotan, H. (1976). On invariance and randomization in fractional replication,Ann. Statist.,4, 423–430.
Yamamoto, S., Shirakura, T. and Kuwada, M. (1975). Balanced arrays of strength 2l and balanced fractional 2m factorial designs,Ann. Inst. Statist. Math.,27, 143–157.
Yamamoto, S., Shirakura, T. and Kuwada, M. (1976). Characteristic polynomials of the information matrices of balanced fractional 2m factorial designs of higher (2l+1) resolution,Essays in Prob. and Statist., Birthday Volume in Honor of Prof. J. Ogawa (ed. S. Ikeda, et al.), 73–94, Tokyo: Shinko-Tsusho.
Author information
Authors and Affiliations
About this article
Cite this article
Shirakura, T. Block plan for a fractional 2m factorial design derived from a 2r factorial design. Ann Inst Stat Math 38, 145–159 (1986). https://doi.org/10.1007/BF02482507
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02482507