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 2^{2}
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 2^{2}
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 2^{3}
Experiment

Treatment
Comb |
Factorial Effects |

I |
A |
B |
AB |
C |
AC |
BC |
ABC |

(1) |
+ |
- |
- |
+ |
- |
+ |
+ |
- |

a |
+ |
+ |
- |
- |
- |
- |
+ |
+ |

b |
+ |
- |
+ |
- |
- |
+ |
- |
+ |

ab |
+ |
+ |
+ |
+ |
- |
- |
- |
- |

c |
+ |
- |
- |
+ |
+ |
- |
- |
+ |

ac |
+ |
+ |
- |
- |
+ |
+ |
- |
- |

bc |
+ |
- |
+ |
- |
+ |
- |
+ |
- |

abc |
+ |
+ |
+ |
+ |
+ |
+ |
+ |
+ |

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

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