Carotenoids are fat-soluble pigments that occur naturally in fruits in vegetables that are recommended for healthy diets. A well-known carotenoid is beta-carotene. Astaxanthin is another carotenoid that is a strong antioxidant and commercially produced. An exercise later in this chapter describes an experiment in Biotechnology Progress to promote astaxanthin production. Seven variables were considered important to production: photon flux density, and concentrations of nitrogen, phosphorous, magnesium, acetate, ferrous, NaCl. It was important to study the effects of these factors, but also the effects of combinations on the production. Even with only a high and low setting for each variable, an experiment that uses all possible combinations requires 27 = 128 tests. There are a number of disadvantages of such a large experiment and a question is whether a fraction of the full set of tests can be selected to provide the most important information about the effects of these variables in many fewer runs. The example used a surprisingly small set of 16 runs (16/128 = 1/8 fraction). The design and analysis of experiments of this type is the focus of this chapter. Such experiments are widely-used throughout modern engineering development and scientific studies.

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