# What is a Factorial Design of an Experiment?

The factorial design of experiment is described with examples in Video 1.

Video 1. Introduction to Factorial Design of Experiment DOE and the Main Effect Calculation Explained Example.

In a Factorial Design of Experiment, all possible combinations of the levels of a factor can be studied against all possible levels of other factors. Therefore, the factorial design of experiments is also called the *crossed factor design of experiments*. Due to the crossed nature of the levels, the factorial design of experiments can also be called the *completely randomized design (CRD) of experiments*. Therefore, the proper name for the factorial design of experiments would be *completely randomized factorial design of experiments*.

In an easy to understand study of human comfort, two levels of the temperature factor (or independent variable), including 0^{O}F and 75^{O}F; and two levels of the humidity factor, including 0% and 35% were studied with all possible combinations (Figure 1). Therefore, the four (2X2) possible treatment combinations, and their associated responses from human subjects (experimental units) are provided in Table 1.

*Table 1. Data Structure/Layout of a Factorial Design of Experiment*

## Coding Systems for the Factor Levels in the Factorial Design of Experiment

As the factorial design is primarily used for screening variables, only two levels are enough. Often, coding the levels as (1) low/high, (2) -/+, (3) -1/+1, or (4) 0/1 is more convenient and meaningful than the actual level of the factors, especially for the designs and analyses of the factorial experiments. These coding systems are particularly useful in developing the methods in factorial and fractional factorial design of experiments. Moreover, general formula and methods can only be developed utilizing the coding system. Coding systems are also useful in response surface methodology. Often, coded levels produce smooth, meaningful and easy to understand contour plots and response surfaces. Moreover, especially in complex designs, the coded levels such as the low- and high-level of a factor are easier to understand.

## How to graphically represent the design?

An example graphical representation of a factorial design of experiment is provided in Figure 1 .

*Figure 1. Factorial Design of Experiments with two levels for each factor (independent variable, x). The response (dependent variable, y) is shown using the solid black circle with the associated response values. *