Lesson 6: The \(2^k\) Factorial Design

Introduction Section

The \(2^k\) designs are a major set of building blocks for many experimental designs. These designs are usually referred to as screening designs. The \(2^k\) refers to designs with k factors where each factor has just two levels. These designs are created to explore a large number of factors, with each factor having the minimal number of levels, just two. By screening we are referring to the process of screening a large number of factors that might be important in your experiment, with the goal of selecting those important for the response that you're measuring. We will see that k can get quite large. So far we have been looking at experiments that have one, two or three factors, maybe a blocking factor and one or two treatment factors, but when using screening designs k can be as large as 8, 10 or 12. For those of you familiar with chemical or laboratory processes, it would not be hard to come up with a long list of factors that would affect your experiment. In this context we need to decide which factors are important.

In these designs we will refer to the levels as high and low, +1 and -1, to denote the high and the low level of each factor. In most cases the levels are quantitative, although they don't have to be. Sometimes they are qualitative, such as gender, or two types of variety, brand or process. In these cases the +1 and -1 are simply used as labels.


Upon completion of this lesson, you should be able to do the following:

  • The idea of 2-level Factorial Designs as one of the most important screening designs
  • Defining a “contrast” which is an important concept and how to derive Effects and Sum of Squares using the Contrasts
  • Process of analyzing Unreplicated or Single replicated factorial designs, and 
  • How to use Transformation as a tool in dealing with inadequacy of either variance homogeneity or normality of the data as major hypothetical assumptions.