# Develop Generic Formulas for 2K Design

To develop the generic formulas, let’s start the design with three variables can be found in Table 6. The use of the -1/+1 coding system in MS Excel provides an advantage to multiply columns easily to find the interaction effects. For an example, the AB interaction column is simply the multiplication of the main effect columns A and B.

Table 6. 23 factorial design

Now, the total effect of A (known as the contrast of A) can be calculated from the Table 6 by simply multiply the column A and the treatment combination column, TRT. Another way of find the total effect of A is subtracting all treatment combinations without “a” from all the treatment combinations with “a” in them. Think about all the experimental units that received a fertilizer (e.g. factor A) and all the experimental units that did not receive the fertilizer. Therefore, the total effect of A (contrast of A) can be written as in Equation 14.

Note. In the (a-1), the 1 (one) represents the total response by the control group, which is generally represented using (1) notation rather than the simply “1.” The parenthesis is taken out for the visual cleanness of the formula. Equation 14

Table 6 also shows that there are four replications for each level of the factor A. Therefore, the average effect can be calculated as.

Equation 15

As the replication for the factor increases by the power of 2 for each additional factor in a 2K design (Equation 1), to generalize for the higher number of factors, the Equation 15 can be modified as Equation 16.

Equation 16

Therefore, the main effect for A and the other effects can be generalized for k number of variables (factors) with n replications as in Equation 17.

Equation 17

Similarly, the Equation 18 represents the estimates (one-half of the effect) for the 2K design, which can be developed following the basic 22 design in Equation 10.

Equation 18

Similarly, the Equation 19 represents the sum of square for the 2K design, which can be developed following the basic 22 design described in Equation 11.

Equation 19

## ANOVA Table for a 2k Factorial Design of Experiment

The ANOVA table for a 2k factorial design of experiment can be developed as Table 2. One exception is the levels of A and B are denoted by small a and b respectively.

Table 2. ANOVA table for a 2k factorial design of experiment