Multi-Factor 2-Level Factorial Experiments

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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.

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