Isomorphism Check for Two-level Multi-Stage Factorial Designs with Randomization Restrictions via an R Package: IsoCheck

  • Pratishtha Batra
  • Neil A. Spencer
  • Pritam Ranjan Indian Institute of Management Indore
Keywords: Isomorphism; Spreads; Factorial designs; Design ranking


Factorial designs are often used in various industrial and sociological experiments to identify significant factors and factor combinations that may affect the process re- sponse. In the statistics literature, several studies have investigated the analysis, con- struction, and isomorphism of factorial and fractional factorial designs. When there are multiple choices for a design, it is helpful to have an easy-to-use tool for identifying which are distinct, and which of those can be efficiently analyzed/has good theoretical properties. For this task, we present an R library called IsoCheck that checks the isomorphism of multi-stage 26n factorial experiments with randomization restrictions. Through representing the factors and their combinations as a finite projective geometry, IsoCheck recasts the problem of searching over all possible relabelings as a search over collineations, then exploits projective geometric properties of the space to make the search much more efficient. Furthermore, a bitstring representation of the factorial effects is used to characterize all possible rearrangements of designs, thus facilitating quick comparisons after relabeling. This paper presents several detailed examples with R codes that illustrate the usage of the main functions in IsoCheck. Besides checking equivalence and isomorphism of 2^n multi-stage factorial designs, we demonstrate how the functions of the package can be used to create a catalog of all non-isomorphic designs, and good designs as per a suitably defined ranking criterion.


Addelman, S. (1964). Some two-level fractional plans with split-plot confounding. Technometrics, 30:253–258.

Andr ́e, J. (1954). Uber nicht-desarguessche ebenen mit transitiver translationsgruppe. Mathematische Zeitschrift, 60(1):156–186.

Bailey, R. A. (2004). Association schemes: Designed experiments, algebra and combinatorics, volume 84. Cambridge University Press.

Batten, L. M. (1997). Combinatorics of finite geometries. Cambridge University Press.

Bingham, D. and Sitter, R. (1999a). Minimum-aberration two-level fractional factorial split- plot designs. Technometrics, 41:62–70.

Bingham, D., Sitter, R., Kelly, E., Moore, L., and Olivas, J. D. (2008). Factorial designs with multiple levels of randomization. Statistica Sinica, pages 493–513.

Bingham, D. and Sitter, R. R. (1999b). Minimum-aberration two-level fractional factorial split-plot designs. Technometrics, 41(1):62–70.

Bisgaard, S. (1994). Blocking generators for small 2k−p designs. Journal of Quality Technology, 26:288–296.

Bose, R. C. (1947). Mathematical theory of the symmetrical factorial design. Sankhy ̄a: The Indian Journal of Statistics, pages 107–166.

Box, G. E., Hunter, W. G., Hunter, J. S., et al. (1978). Statistics for experimenters. Butler, N. A. (2003). Some theory for constructing minimum aberration fractional factorial designs. Biometrika, 90(1):233–238.

Cheng, C.-S. and Tsai, P.-W. (2011). Multistratum fractional factorial designs. Statistica Sinica, 21:1001–1021.

Fisher, R. A. (1942). The theory of confounding in factorial experiments in relation to the theory of groups. Annals of Eugenics, 11:290–299.

Franklin, M. and Bailey, R. (1977). Selection of defining contrasts and confounded effects in two-level experiments. Applied Statistics, 26:321–326.

Hirschfeld, J. (1998). Projective Geometries Over Finite Fields. Oxford Mathematical Monographs. Oxford University Press New York.

Lin, C. and Sitter, R. (2008). n isomorphism check for two-level fractional factorial designs. Journal of Statistical Planning and Inference, 134:1085–1101.

Mee, R. and Bates, R. (1998). Split-lot designs: Experiments for multistage batch processes. Technometrics, 40:127–140.

Miller, A. (1997). Strip-plot configurations of fractional factorials. Technometrics, 39:153–161.

Nelder, J. (1965a). The analysis of randomized experiments with orthogonal block structure. i. block structure and the null analysis of variance. Proc. R. Soc. Lond. A, 283:147–162.

Nelder, J. (1965b). The analysis of randomized experiments with orthogonal block structure. ii. treatment structure and the general analysis of variance. Proc. R. Soc. Lond. A, 283:163– 178.

R Core Team (2020). R: A Language and Environment for Statistical Computing. R Foun- dation for Statistical Computing, Vienna, Austria.

Ranjan, P., Bingham, D., and Mukerjee, R. (2010). Stars and regular fractional factorial designs with randomization restrictions. Statistica Sinica, pages 1637–1653.

Ranjan, P., Bingham, D. R., and Dean, A. M. (2009). Existence and construction of random- ization defining contrast subspaces for regular factorial designs. The Annals of Statistics, pages 3580–3599.

Soicher, L. (2000). Computation of partial spreads, web preprint.

Speed, T. P. and Bailey, R. A. (1982). On a class of association schemes derived from lattices of equivalence relations. Algebraic Structures and Applications, Marcel Dekker, New York, pages 55–74.

Spencer, N., Ranjan, P., and Mendivil, F. (2019). Isomorphism check for 2n factorial designs with randomization restriction. Journal of Statistical Theory and Practice, 13(60):1–24.

Tjur, T. (1984). Analysis of variance models in orthogonal designs. Int. Statist. Rev., 52:33– 81.

Xu, H., Wu, C. J., et al. (2001). Generalized minimum aberration for asymmetrical fractional factorial designs. The Annals of Statistics, 29(4):1066–1077.

Yates, F. (1937). Design and Analysis of Factorial Experiments. Technical Communication 35, Harpenden, United Kingdom: Imperial Bureau of Soil Science.

How to Cite
Batra, P., Spencer, N., & Ranjan, P. (2023). Isomorphism Check for Two-level Multi-Stage Factorial Designs with Randomization Restrictions via an R Package: IsoCheck. Statistics, Optimization & Information Computing, 11(4), 892-910.
Research Articles