# Layout/Data Structure and the Level Coding System of the Basic 22 Factorial Design

The low level of a variable (or factor) is generally the current level, exiting level, or the control level, while the high level is used for the treatment level. For example, testing whether a medicine works or not, the medicine is generally the high level while the low level is the control level, which is the placebo in general. Testing a new fertilizer against nothing or the current fertilizer, hoping that the new fertilizer is better or higher in value with respect to the yield (response or the dependent variable) is another example. The word ‘treatment combination” is inherited from the agriculture and medical fields where plants and animals are treated with multiple fertilizers, medicines, and so on. For example, when the low levels of both factors are used, meaning that no treatment is applied, the treatment combination is defined as the “control,” which is denoted by “(1).” Simply “1” without the parenthesis is also used to represent “control” in some equations. Similarly, when only the high level of factor A is applied, the treatment combination is denoted as “a.” When only the high level of factor B is applied, the treatment combination is denoted as “b.” Notation “ab” is used for the treatment combinations when both the high level of both factors are applied. The treatment combination notations, (1), a, b, and ab are used for the total responses at that particular treatment combination.

Table 1. Layout, Data Structure, & Level Coding System of a Factorial Design

# Graphical Representation of the Basic 22 Factorial Design of Experiment

An example graphical representation of the 22 factorial design of experiment is provided in Figure 1.

Figure 1. 22 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.