2^k Factorial Experiments

A frequently used factorial experiment design in the semiconductor industry is known as the 2^k factorial design, which is basically an experiment involving k factors, each of which has two levels ('low' and 'high').  In such a multi-factor two-level experiment, the number of treatment combinations needed to get complete results is equal to 2^k. Thus, a 2^k factorial experiment that deals with 3 factors would require 8 treatment combination, while one that deals with 4 factors would require 16 of them.  One can easily see that the number of runs needed to complete a factorial experiment, even if only two levels are explored for each factor, can become very large.

The first objective of a factorial experiment is to be able to determine, or at least estimate, the factor effects, which indicate how each factor affects the process output.  Factor effects need to be understood so that the factors can be adjusted to optimize the process output. 

The effect of each factor on the output can be due to it alone (a main effect of the factor), or a result of the interaction between the factor and one or more of the other factors (interactive effects).  When assessing factor effects (whether main or interactive effects), one needs to consider not only the magnitudes of the effects, but their directions as well.  The direction of an effect determines the direction in which the factors need to be adjusted in a process in order to optimize the process output.

In factorial designs, the main effects are referred to using single uppercase letters, e.g., the main effects of factors A and B are referred to simply as 'A' and 'B', respectively.  An interactive effect, on the other hand, is referred to by a group of letters denoting which factors are interacting to produce the effect, e.g., the interactive effect produced by factors A and B is referred to as 'AB'. 

Each treatment combination in the experiment is denoted by the lower case letter(s) of the factor(s) that are at 'high' level (or '+' level). Thus, in a 2-factorial experiment, the treatment combinations are: 1) 'a' for the combination wherein factor A = 'high' and factor B = 'low'; 2) 'b' for factor A = 'low' and factor B = 'high'; 3) 'ab' for the combination wherein both A and B = 'high'; and 4) '(1)', which denotes the treatment combination wherein both factors A and B are 'low'.

Based on discussions in this link: Factorial Experiments, the main effect of a factor A in a two-level two-factor design is the change in the level of the output produced by a change in the level of A (from 'low' to 'high'), averaged over the two levels of the other factor B. On the other hand, the interaction effect of A and B is the average difference between  the effect of A when B is 'high' and the effect of A when B is 'low.'  This is also the average difference between  the effect of B when A is 'high' and the effect of B when A is 'low.'

The magnitude and polarity (or direction) of the numerical values of main and interaction effects indicate how these effects influence the process output. A higher absolute value for an effect means that the factor responsible for it affects the output significantly. A negative value means that increasing the level(s) of the factor(s) responsible for that effect will decrease the output of the process.

In a 2² factorial experiment wherein n replicates were run for each combination treatment, the main and interactive effects of A and B on the output may be mathematically expressed as follows:

A = [ab + a - b - (1)] / 2n;   (main effect of factor A)

B = [ab + b - a - (1)] / 2n;   (main effect of factor B)

AB = [ab + (1) - a - b] / 2n;   (interactive effect of factors A and B)

where n is the number of replicates per treatment combination; a is the total of the outputs of each of the n replicates of the treatment combination a (A is 'high and B is 'low); b is the total output for the n replicates of the treatment combination b (B is 'high' and A is 'low); ab is the total output for the n replicates of the treatment combination ab (both A and B are 'high'); and (1) is the total output for the n replicates of the treatment combination (1) (both A and B are 'low.

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

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

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