A note on robustness of D-optimal block designs for two-colour microarray experiments

Two-colour microarray experiments form an important tool in gene expression analysis. Due to the high risk of missing observations in microarray experiments, it is fundamental to concentrate not only on optimal designs but also on designs which are robust against missing observations. As an extension of Latif et al. (2009), we define the optimal breakdown number for a collection of designs to describe the robustness, and we calculate the breakdown number for various D-optimal block designs. We show that, for certain values of the numbers of treatments and arrays, the designs which are D-optimal have the highest breakdown number. Our calculations use methods from graph theory.

[1]  J. Kiefer.  Construction and optimality of generalized Youden designs II , 1975 .

[2]  Improved conditions for the robustness of binary block designs against the loss of whole blocks , 2011 .

[3]  D. V. Gokhale,et al.  A Survey of Statistical Design and Linear Models. , 1976 .

[4]  R. A. Fisher,et al.  Design of Experiments , 1936 .

[5]  Gary A. Churchill,et al.  Analysis of Variance for Gene Expression Microarray Data , 2000, J. Comput. Biol..

[6]  Y. S. Sathe,et al.  Some more robust block designs against the unavailability of data , 1992 .

[7]  D. Whittinghill,et al.  Optimality and Robustness to the Unavailability of Blocks in Block Designs , 1991 .

[8]  Béla Bollobás,et al.  Graph Theory: An Introductory Course , 1980, The Mathematical Gazette.

[9]  Peter J. Cameron,et al.  Combinatorics of optimal designs , 2008 .

[10]  J. A. Bondy,et al.  Graph Theory , 2008, Graduate Texts in Mathematics.

[11]  John P. Morgan,et al.  Optimal Incomplete Block Designs , 2007 .

[12]  Edgar Brunner,et al.  Robustness considerations in selecting efficient two-color microarray designs , 2009, Bioinform..

[13]  Ching-Shui Cheng,et al.  Do nearly balanced multigraphs have more spanning trees? , 1985, J. Graph Theory.

[14]  K. R. Shah,et al.  Theory of Optimal Designs , 1989 .

[15]  R. A. Bailey.  Designs for two‐colour microarray experiments , 2007 .

[16]  J. K. Baksalary,et al.  Conditions for the robustness of block designs against the unavailability of data , 1987 .

[17]  Raya Khanin,et al.  Near‐optimal designs for dual channel microarray studies , 2005 .

[18]  D. Cox,et al.  Applied Statistics: Principles and Examples , 1982 .

[19]  G. Churchill,et al.  Experimental design for gene expression microarrays. , 2001, Biostatistics.

[20]  Peter J. Cameron,et al.  Using graphs to find the best block designs , 2011, 1111.3768.

[21]  S. Ghosh.  Robustness of BIBD against the unavailability of data , 1982 .

[22]  Russ B. Altman,et al.  Missing value estimation methods for DNA microarrays , 2001, Bioinform..

[23]  Ching-Shui Cheng,et al.  Maximizing the total number of spanning trees in a graph: Two related problems in graph theory and optimum design theory , 1981, J. Comb. Theory B.

[24]  N. Gaffke,et al.  D-optimal block designs with at most six varieties , 1982 .