Designs with blocks of size two and applications to microarray experiments

Abstract

Designs with blocks of size two have numerous applications. In experimental situations where observation loss is common, it is important for a design to be robust against breakdown. For designs with one treatment factor and a single blocking factor, with blocks of size two, conditions for connectivity and robustness are obtained using combinatorial arguments and results from graph theory. Lower bounds are given for the breakdown number in terms of design parameters. For designs with equal or near equal treatment replication, the concepts of treatment and block partitions, and of linking blocks, are used to obtain information on the number of blocks required to guarantee various levels of robustness. The results provide guidance for construction of designs with good robustness properties. Robustness conditions are also established for row column designs in which one of the blocking factors involves blocks of size two. Such designs are particularly relevant for microarray experiments, where the high risk of observation loss makes robustness important. Disconnectivity in row column designs can be classified as three types. Techniques are given to assess design robustness according to each type, leading to lower bounds for the breakdown number. Guidance is given for robust design construction. Cyclic designs and interwoven loop designs are shown to have good robustness properties.