# Understanding Factor Effects

## How to Develop the Formulas for the Main and the Interaction Effects of the Basic 22 Factorial Design?

The calculation of the effects from a 22 factorial design of experiment is described with examples in Video 2 (both Video 1 and are the same Video 2).

### Method 1. The Basic

The calculation of the main effect A was described in the Module 5 Factorial Design of Experiments. The main effect of A is calculated by subtracting the average responses at the low-level from the average responses at the high-level.

Therefore, the main effect of the temperature factor can be calculated as in Equation 2. Equation 2

Using the treatment combination notations, the Equation 2 can be written as in Equation 3, and therefore, the Equation 4. Equation 3 Equation 4

Similarly, the equations for the main effect of B and the interaction effect of AB can be developed for the 22 design of experiments. Equation 5 Equation 6

### Method 2. The -1/+1 Coding System

Using the -1/+1 coding system, the total effect can also be calculated by simply multiplying the factor A column with the treatment combination column (Table 1), which results in the Equation 7. Total effect of Equation 7

As there are two replications for A, the average effect is calculated as in Equation 8. Equation 8

Similarly, the equations for the main effect of B and the interaction effect of AB can be developed as in Equation 5 and Equation 6.

### Method 3. The Commonsense System

Another way of developing the equation for the main effect of factor A is subtracting all treatment combinations without “a” in them from all the treatment combinations that contain “a” in them. Think about all the experimental units that received a fertilizer as compared to those did not receive the fertilizer. Therefore, the main effect of Similarly, the equations for the main effect of B and the interaction effect of AB can be developed as in Equation 5 and Equation 6.