Primary Basics Fractional Design

The One-Half Fraction Design and Its Associated Terminology

An introduction to the one-half fraction and its justification is provided in Video 2.

Video 3 demonstrates the details on the primary basics, including the definitions and explanations for associated terms and the systematic process of developing the fractional factorial design.

Table 1 shows the one full replication of the 23 design. Assume that we just want to screen the factors or to see the importance of the three variables first before we invest more time into them. Three degrees of freedom are required to get information on three variables/factors. Three degrees of freedom can be obtained from four experiments. Therefore, half of the experiments, four out of the eight can be systematically conducted to determine the effects of the three factors, A, B, and C.

Table 1. The 23 Full Factorial Design of Experiments

As the higher-order interactions are considered the least important, the ABC effect can be sacrificed. To sacrifice ABC, either all the positive (or highs) treatment combinations (a, b, c, and abc) or all the negative (or lows) treatment combinations ((1), ab, ac, bc) of the ABC effect can only be run (Table 2). By dividing the experiments into two, the one fraction is known as the One-Half Fraction of the 2k design (23 design in this example).

Table 2. The 23 Factorial Design of Experiments with two blocks

Video 4 shows the details on the two one-half fractions of the 23 design.

Assume that the positive fraction is used for the fractional design shown in Table 3. This one-half fraction with all positive treatment combinations is known as the principal fraction (Table 3), while the other half with the negative treatment combinations is known as the alternative or complementary fraction. The ABC interaction for the 23 design is known as the design generator, simply because the word ABC is generating this design. The Generator is also known as word, probably because of it is used very often in fractional designs and easier to say. Moreover, the higher order interaction terms also look like a word.

Table 3. The Principal One-Half Fraction of the 23 Factorial Design of Experiments

Table 4. The Effects of the One-Half Fraction of the 23 Factorial Design

Table 4 shows the signs for the treatment combination of a factor stays the exact same for the multiplications between the generator ABC and the factor A, B, and C. Therefore, the word ABC is called the identity. Therefore, a relationship for the word ABC or the generator of the design can be developed as in Equation 1, which is known as the defining relation.

Equation 1

The rest of the relationships can be developed using the defining relation of the design such as in Equation 2. Any relationship can be developed by multiplying both sides of the defining relation. For an example, if the relationship for the factor A is desired, A is multiplied with both sides of the defining relation, results in A = BC (Equation 2).

Equation 2

Similarly, the relationships for all other effects (e.g. B =AC, and C=AB in this 23 design) can be developed. As A and BC are related and indistinguishable from each other for the one-half of the 23 design, A and BC is aliased with each other. Similarly, B is aliased with AC, and C is aliased with AB. The word “alias” is used to call the relationship between effects or when they are indistinguishable from each other.

Table 4 shows the main, the interaction effects and their treatment combinations. An effect can be easily estimated by simply multiplying the factor/effect column with the treatment combination column. The effect is estimated by the half of the contrast or the linear combinations (Equation 3).

Equation 3

Similarly, the interaction effects can also be estimated by multiplying the interaction column with the treatment combination column, results in Equation 4.

Equation 4

Again, the Equation 3 & Equation 4 show that [A] =[BC], [B] = [AC], and [C]=[AB]. Therefore, the A effect is indistinguishable from the BC effect and so on.

The other one-fraction (the alternative or complementary fraction) of the design is shown in Table 5. While, the defining relation I = ABC used for the principle fraction, I = -ABC defining relation for the complementary fraction produces the exact same results (Table 6).

Table 5. The Complementary One-Half Fraction of the 23 Factorial Design

Table 6. The Effects of the Complementary One-Half Fraction of the 23 Factorial Design

And the relationships for each effect can be developed from this defining relation as in Equation 5.

Equation 5

Similarly, the relationships for all other effects (e.g. B =-AC, and C=-AB in this 23 design) can be developed.

Table 6 shows the main and the interaction effects and their treatment combinations for the complementary fraction of the one-half fraction. An effect can be easily estimated by simply multiplying the effect column with the treatment combination column. The effect is estimated by the half of the contrast or the linear combinations (Equation 6).

Equation 6

Similarly, the interaction effects can also be estimated by multiplying the interaction column with the treatment combination column, results in Equation 7.

Equation 7

As the principal and complementary one-half fractions complete the full replication, the full estimate of the main effect can be calculated using the Equation 3 & Equation 6. For an example,

Equation 8

Which can be verified from the 23 full factorial design in Table 1.

Similarly, the interaction effects can be estimated from Equation 4 & Equation 7.

Equation 9

Which can be verified from the 23 full factorial design in Table 1.

2.1 Graphical Representation of the One-Half Fraction

Figure 1 shows the graphical representation of the two one-half fractions of the 23 design.

Figure 1. The Two One-Half Fractions of the 23 Design

Video 5 demonstrates a simple process of designing a one-half fraction of the 23 design.