2^k Factorial Experiments (Page 2 of 2)

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The analysis of factor effects in the conduct of 2^k factorial experiments requires a lot of number crunching (even if only two levels per factor are considered in such experiments), especially if the number of factors being investigated is high.  Fortunately, there's a systematic method for doing the required math in the analysis of factor effects.

In the previous page,  the main and interactive effects of A and B in a 2² factorial experiment involving n replicates were given as follows: A = [ab + a - b - (1)] / 2n; B = [ab + b - a - (1)] / 2n; and AB = [ab + (1) - a - b] / 2n.

Note that each of these formulas involves a 'contrast', or a special linear combination of parameters whose coefficients equal zero.  For instance, the contrast for A is ab+a-b-1 while the contrast for B is ab+b-a-1.  The coefficients of A's contrast are -1, +1, -1,and +1 if the contrast were written in what is known as Yates' Order, i.e., (1), a, b, ab. Furthermore, in Yates order, the coefficients of B's contrast are -1, -1, +1, +1 while those of AB's contrast are +1, -1, -1, +1.

The significance of Yates' Order (also known as the 'Standard Order') is that it facilitates the determination of the algebraic signs of the coefficients needed for calculating the main and interaction effects of each factor in a factorial experiment.  As discussed earlier, one can easily compute for the numerical values of factor effects if one knows the formulas to use and the output of each replicate for each combination treatment of the factorial experiment. Unfortunately, the formulas look daunting to people not accustomed to them.  There is, however, an easy way to reconstruct these formulas using Yates Order.

The secret is in knowing how to list the combination treatments in Yates' Order and how to assign the '+' and '-' signs to them.  The Yates order for the combination treatments of 2-factor, 3-factor, and 4-factor experiments are:

2 factors: (1), a, b, ab;

3 factors: (1), a, b, ab, c, ac, bc, abc

4 factors: (1), a, b, ab, c, ac, bc, abc, d, ad, bd, abd, cd, acd, bcd, abcd

To facilitate the determination of the algebraic signs of the coefficients needed for calculating the factor effects, one needs to construct a matrix of '+' and '-' signs that map to the factor effects and their combination treatments.  The matrix of '+' and '-' signs is usually constructed with the factorial effects forming the column headers and the combination treatments in Yates' Order forming the row headers.

The first rule for filling the matrix with '+' and '-' signs is this: treatment combinations wherein the factor in the column being filled is 'high' will get a '+' sign. On the other hand, treatment combinations wherein that factor is 'low' will get a '-' sign.  The second rule is: for interaction factors, the signs of their individual factors  simply needs to be multiplied for each treatment. Lastly, the 'identity' column (wherein all factors are 'low') gets a '+' sign for all combination treatments.  

Once the matrix is finished, it can be used to look up the algebraic signs of the coefficients of each term in the contrast of each factor, allowing reconstruction of its 'effect' formula.  Table 1 shows such a matrix for a 3-factor experiment.  Here's an example of how to use this table: to derive the formula for the full effect of factor A, look at the signs under 'A' and assign them to the treatment combinations on their left.  Thus, A = [-(1) + a - b + ab - c + ac - bc + abc]/4n. 

Table 1. Algebraic Signs for Calculating the Factor Effects in a 2³ Experiment

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See also:   Factorial Experiments; Factorial Design Tables; Example of a 2-Level Factorial Experiment

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