5Completely Randomized Design with Multiple Treatment Factors

Introduction

Processes and natural phenomena generally involve multiple variables or “factors,” for example, cookies. Producing cookies from a box of mix requires adding certain amounts of other ingredients to the mix, mixing the dough by hand or mixer for a specified time, preheating an oven, and then baking cookies for a specified time at a specified temperature. A cake-mix manufacturer like Betty Crocker or Duncan Hines needs to know how the various factors (steps and ingredients in the recipe) affect cookie quality, perhaps as measured by a panel of tasters. The manufacturer can then use this information to determine the preparation and baking instructions on the box. In addition to maximizing quality, the manufacturer would like the process settings (recipe) to be “robust,” meaning that if you and I don’t get the ingredients or settings quite right or if our ovens don’t quite achieve the desired temperature, we will still get good cookies. Similar “multifactor” problems need to be solved in many other contexts. It is not a coincidence that the silicon chips that so much of our technology relies on are called “cookies” in their production.

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In Chapter 2, we described how treatments (or blocks) may be defined by two or more factors and those factors could be either crossed or nested. They can also be either qualitative ...

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