Optimal designs for two-color microarray experiments in multi-factorial models

Two-color microarray experiments form an important tool in gene expression analysis. They are often used to identify candidate genes that can be made accountable for the genesis of a certain disease. Due to the high costs of microarray experiments it is fundamental to design these experiments carefully and specifically give instructions, which samples should be allocated on the same microarray. Thereby, two samples are hybridized together on one array and the assignment of samples to arrays influences the precision of the results. Therefore, design issues for microarray experiments have been investigated intensively in the last years. However, only few authors, e.g., Stanzel [37], focused on more than one factor of interest. We extend Stanzel’s work and derive approximate optimal designs for estimating interactions in multi-factorial settings. Thereby, optimality of candidate designs is shown using equivalence theorems (Pukelsheim [33]). Another practical important but less studied topic is the derivation of exact optimal designs. Most research considers approximate designs or exact designs for special contrast sets and selected numbers of arrays. Therefore, we focus on exact designs and present a method to construct A-optimal microarray designs for arbitrary numbers of arrays and arbitrary contrast sets. This method is applied to derive optimal designs for estimating treatment-control comparisons, all-tonext contrasts, Helmert contrasts and all pairwise comparisons. Furthermore, we derive robust designs, which achieve efficient results even if observations are missing. Missing values are a crucial topic in the context of microarray experiments, since they often occur due to scratches on the slide or other damaging. In applications recommendations for the choice of efficient experimental layouts can be derived from our constructed designs.