A 2 means that the independent variable has two levels, a 3 means that the independent variable has three levels, a 4 means it has four levels, etc. To illustrate a 3 x 3 design has two independent variables, each with three levels, while a 2 x 2 x 2 design has three independent variables, each with two levels.

A factorial design is one involving two or more factors in a single experiment. Such designs are classified by the number of levels of each factor and the number of factors. So a 2×2 factorial will have two levels or two factors and a 2×3 factorial will have three factors each at two levels.

In its simplest form, an experiment aims at predicting the outcome by introducing a change of the preconditions, which is represented by one or more independent variables, also referred to as “input variables” or “predictor variables.” The change in one or more independent variables is generally hypothesized to result …

Let’s take the case of 2×2 designs. There will always be the possibility of two main effects and one interaction. You will always be able to compare the means for each main effect and interaction.

The three-level design is written as a 3k factorial design. It means that k factors are considered, each at 3 levels. These are (usually) referred to as low, intermediate and high levels. Unfortunately, the three-level design is prohibitive in terms of the number of runs, and thus in terms of cost and effort.

The terms “three-way”, “two-way” or “one-way” in ANOVA refer to how many factors are in your test. A three-way ANOVA (also called a three-factor ANOVA) has three factors (independent variables) and one dependent variable.

Similarly, a 25 design has five factors, each with two levels, and 25=32 experimental conditions; and a 32 design has two factors, each with three levels, and 32=9 experimental conditions. Factorial experiments can involve factors with different numbers of levels.