The Annals of Applied Statistics

Optimal factorial designs for cDNA microarray experiments

Tathagata Banerjee and Rahul Mukerjee

Full-text: Open access


We consider cDNA microarray experiments when the cell populations have a factorial structure, and investigate the problem of their optimal designing under a baseline parametrization where the objects of interest differ from those under the more common orthogonal parametrization. First, analytical results are given for the 2×2 factorial. Since practical applications often involve a more complex factorial structure, we next explore general factorials and obtain a collection of optimal designs in the saturated, that is, most economic, case. This, in turn, is seen to yield an approach for finding optimal or efficient designs in the practically more important nearly saturated cases. Thereafter, the findings are extended to the more intricate situation where the underlying model incorporates dye-coloring effects, and the role of dye-swapping is critically examined.

Article information

Ann. Appl. Stat. Volume 2, Number 1 (2008), 366-385.

First available in Project Euclid: 24 March 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Admissibility augmented design baseline parametrization dye-swapping interaction main effect orthogonal parametrization saturated design weighted optimality


Banerjee, Tathagata; Mukerjee, Rahul. Optimal factorial designs for cDNA microarray experiments. Ann. Appl. Stat. 2 (2008), no. 1, 366--385. doi:10.1214/07-AOAS144.

Export citation


  • Altman, N. S. and Hua, J. (2006). Extending the loop design for two-channel microarray experiments., Genet. Res. 88 153–163.
  • Amaratunga, D. and Cabrera, J. (2004)., Exploration and Analysis of DNA Microarray and Protein Array Data. Wiley, New York.
  • Banerjee, T. and Mukerjee, R. (2008). Supplement to “Optimal factorial designs for CDNA microarray experiments.” DOI:, 10.1214/07-AOAS144SUPP.
  • Bueno Filho, J. S. S., Gilmour, S. G. and Rosa, G. J. M. (2006). Design of microarray experiments for genetical genomics studies., Genetics 174 945–957.
  • Churchill, G. A. (2002). Fundamentals of experimental design for cDNA microarrays., Nature Genetics (Suppl.) 3 490–495.
  • Dey, A. and Mukerjee, R. (1999)., Fractional Factorial Plans. Wiley, New York.
  • Dobbin, K. and Simon, R. (2002). Comparison of microarray designs for class comparison and class discovery., Bioinformatics 18 1438–1445.
  • Glonek, G. F. V. and Solomon, P. J. (2004). Factorial and time course designs for cDNA microarray experiments., Biostatistics 5 89–111.
  • Grossman, H. and Schwabe, R. (2008). The relationship between optimal designs for microarray and paired comparison experiments., Preprint.
  • Gupta, S. (2006). Balanced factorial designs for cDNA microarray experiments., Comm. Statist. Theory Methods 35 1469–1476.
  • Gupta, S. and Mukerjee, R. (1989)., A Calculus for Factorial Arrangements. Springer, Berlin.
  • Kendziorski, C., Irizarry, R. A., Chen, K. S., Haag, J. D. and Gould, M. N. (2005). On the utility of pooling biological samples in microarray experiments., Proc. Natl. Acad. Sci. USA 102 4252–4257.
  • Kerr, K. F. (2006). Efficient, 2k factorial designs for blocks of size 2 with microarray applications. J. Qual. Technol. 38 309–318.
  • Kerr, M. K. (2003). Design considerations for efficient and effective microarray studies., Biometrics 59 822–828.
  • Kerr, M. K. and Churchill, G. A. (2001a). Experimental design for gene expression microarrays., Biostatistics 2 183–201.
  • Kerr, M. K. and Churchill, G. A. (2001b). Statistical design and the analysis of gene expression microarray data., Genet. Res. 77 123–128.
  • Kiefer, J. C. (1975). Construction and optimality of generalized Youden designs. In, A Survey of Statistical Design and Linear Models (J. N. Srivastava, ed.) 333–353. North-Holland, Amsterdam.
  • Landgrebe, J., Bretz, F. and Brunner, E. (2006). Efficient design and analysis of two colour factorial microarray experiments., Comput. Statist. Data Anal. 50 499–517.
  • Majumdar, D. (1996). Optimal and efficient treatment-control designs. In, Handbook of Statistics 13 (S. Ghosh and C. R. Rao, eds.) 1007–1053. North-Holland, Amsterdam.
  • Nguyen, D., Arpat, A. Wang, N. and Carroll, R. J. (2002). DNA microarray experiments: Biological and technical aspects., Biometrics 58 701–717.
  • Rosa, G. J. M, Steibel, J. P. and Tempelman, R. J. (2005). Reassessing design and analysis of two-color microarray experiments using mixed effects models., Comp. Funct. Genomics 6 123–131.
  • Silvey, S. D. (1980)., Optimal Design. Chapman and Hall, London.
  • Wang, P. C. (2004). Designing two-level fractional factorial experiments in blocks of size two., Sankhyā Ser. A 66 327–342.
  • Wit, E., Nobile, A. and Khanin, R. (2005). Near-optimal designs for dual-channel microarray studies., Appl. Statist. 54 817–830.
  • Wu, C. F. J. and Hamada, M. (2000)., Experiments: Planning, Analysis and Parameter Design Optimization. Wiley, New York.
  • Yang, Y. J., and Draper, N. R. (2003). Two-level factorial and fractional factorial designs in blocks of size two., J. Qual. Technol. 35 294–305.
  • Yang, Y. H. and Speed, T. (2002). Design issues for cDNA microarray experiments., Nature Genetics (Suppl.) 3 579–588.

Supplemental materials