Abstract
The problem of ascertaining conditions that ensure that an m-way design is connected has occupied the attention of research workers for very many years. One of the significant advances, as well as one of the earliest contributions, was provided by the classic work of J. N. Srivastava and D. A. Anderson in 1970, which gives a necessary and sufficient rank condition for an m-way design to be completely connected. In this article it is shown that the class of estimable parametric functions for an individual factor is derived directly from a simple extension of the Srivastava-Anderson result. This takes the form of a necessary and sufficient rank condition that is expressed in terms of the dimension of a segregated component of the kernel of the design matrix. The result has the interesting property that the connectivity status for all of the individual factors can be found simultaneously. Furthermore, it enables the formulation of several general results, which include the specification of conditions on designs exhibiting adjusted orthogonality. A number of examples are given to illustrate these results.
Similar content being viewed by others
References
Bérubé, J., and G. P. H. Styan. 1993. Decomposable three-way layouts. J. Stat. Plan. Inference, 36, 311–322.
Birkes, D., Y. Dodge, and J. Seely. 1976. Spanning sets for estimable contrasts in classification models. Ann. Stat., 4, 86–107.
Bose, R.C. 1944. The fundamental theorem in linear estimation. Proc. 31st Ind. Sci. Congress, 2–3 (abstract).
Butz, L. 1982. Connectivity in multi-factor designs. Berlin, Germany: Heldermann Verlag.
Ceranka, B., and M. Kozlowska. 1991. Connectedness of row and column designs. Zast. Mat. Appl. Math. 21, 27–31.
Chakrabarti, M. C. 1963. On the C-matrix in design of experiments. J. Indian Stat. Assoc., 1, 23.
Christensen, R. 2002. Plane answers to complex questions, 3rd ed. New York, NY, Springer.
Dodge, Y. 1985. Analysis of experiments with missing data. New York, NY, Wiley.
Eccleston, J., and A. S. Hedayat. 1974. On the theory of connected designs: Characterisation and optimality. Ann. Stat., 2, 1238–1255.
Eccleston, J., and K. Russell. 1975. Connectedness and orthogonality in multi-factor designs. Biometrika, 62, 341–345.
Eccleston, J., and K. Russell. 1977. Adjusted orthogonality in non-orthogonal designs. Biometrika, 64, 339–345.
Ghosh, S. 1986. On a new graphical method of determining the connectedness in three dimensional designs. Sankya, B 48, 207–215.
Godolphin, J. D. 2004. Simple pilot procedures for the avoidance of disconnected experimental designs. Appl. Stat., 53, 133–147.
Godolphin, J. D., and E. J. Godolphin. 2001. On the connectivity of row-column designs. Util. Math., 60, 51–65.
Harville, D. A. 1997. Matrix algebra from a statistician’s perspective. New York, NY, Springer.
John, J.A., and E. R. Williams, 1995. Cyclic and computer generated designs, 2nd ed. London, UK, Chapman and Hall.
Katyal, V., and S. Pal. 1991. Analysis and connectedness of four-dimensional designs. J. Indian Soci. Agri. Stat., 43, 296–309.
Park, D. K., and A. M. Dean. 1990. Average efficiency factors and adjusted orthogonality in multidimensional designs. J. R. Stat. Soc., Ser. B, 52, 361–368.
Park, D.K., and K. R. Shah. 1995. On the connectedness of row-column designs. Commun. Stat., 24, 87–96.
Preece, D.A. 1996. Multifactor balanced designs with complete adjusted orthogonality for all pairs of treatment factors. Aust. J. Stat., 38, 223–230.
Raghavarao, D., and W. T. Federer. 1975. On connectedness in two-way elimination of heterogeneity designs. Ann. Stat., 2, 730–735.
Searle, S. R. 1987. Linear models for unbalanced data. New York, NY, Wiley.
Sengupta, D., and S. R. Jammalamadaka. 2003. Linear models: an integrated approach. London, UK, World Scientific.
Shah, K. R., and Y. Dodge. 1977. On the connectedness of designs. Sankya, B 39, 284–287.
Shah, K. R., and C. G. Khatri. 1973. Connectedness in row-column designs. Commun. Stat., 2, 571–573.
Srivastava, J. N., and D. A. Anderson. 1970. Some basic properties of multidimensional partially balanced designs. Ann. Math. Stat., 41, 1438–1445.
Wynn, H. P. 2008. Algebraic solutions to the connectivity problem for m-way layouts: Interaction-contrast aliasing. J. Stat. Plan. Inference, 138, 259–271.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Godolphin, J.D. On the Connectivity Problem for m-Way Designs. J Stat Theory Pract 7, 732–744 (2013). https://doi.org/10.1080/15598608.2013.782193
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1080/15598608.2013.782193