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Optimal Design of Blocked Experiments in the Presence of Supplementary Information About the Blocks∗

In some designed experiments, measurements of characteristics of the experimental units may be available prior to performing the runs. If the investigators believe that these measured characteristics may have some effect on the response of interest, then it seems natural to include these characteristics as factors in the experiment even though they are not under direct control. It may also be possible to apply multiple runs to a given experimental unit by subdividing it into multiple pieces, each having the same characteristics. A similar scenario involves using a person or an animal as an experimental unit multiple times but with different treatments. Here, the measured information about the subjects may not change over the experiment. In either of these cases, the fact that several runs employ the same experimental unit means that the responses for those runs are correlated. This correlation, in addition to the natural variability of the measured characteristics over the sample of available experimental units, requires new methodology for creating optimal designs. Specifically, the methodology must choose a subset of the experimental units and determine the number of treatments applied to each experimental unit in addition to choosing the level combinations of the controllable factors for each run. In this article, we provide two methods for generating optimal designs in the presence of additional information about the experimental units. The first method fixes the number of runs performed on each experimental unit. The second method allows for varying numbers of runs applied to each experimental unit, subject to a constraint on the total number of runs. We discuss several illustrative examples using each method as well as a real experiment using previously fabricated batches of polypropylene as experimental units in a study on the effects of a subsequent plasma treatment.