Factorial
Experiments
Factorial
Experiments
are experiments that investigate the effects of two or more factors or
input parameters on the output response of a process. Factorial
experiment design, or simply factorial design, is a systematic method
for formulating
the steps needed to successfully implement a factorial experiment.
Estimating the effects of various factors on the output of a process
with a minimal number of observations is
crucial to being able to optimize the output of the process.
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In a
factorial design,
the effects of varying the levels of the various factors affecting the
process output is investigated. Each complete trial or replication of
the experiment takes into account all the possible combinations of the
varying levels of these factors. Effective factorial design
ensures that the least number of experiment runs are conducted to
generate the maximum amount of information about how input variables
affect the output of a process.
For instance,
if the effects of two factors A and B on the output of a process are
investigated, and A has 3 levels of intensity (e.g., weak, moderate, and
strong presence) while B has 2 levels (weak and strong), then one would
need to run 6 treatment combinations to complete the experiment,
observing the process output for each of the combinations: weakAweakB,
weakAstrongB, moderateAweakB, moderateAstrongB, strongAweakB,
strongA, strongB.
The amount of
change produced in the process output for a change in the 'level' of a
given factor is referred to as the 'main effect' of that factor. Table 1
shows an example of a simple factorial experiment involving two factors
with two levels each. The two levels of each factor may be denoted as
'low' and 'high', which are usually symbolized by '' and '+' in
factorial designs, respectively.
Table 1.
A Simple 2Factorial Experiment

A () 
A(+) 
B() 
20 
40 
B(+) 
30 
52 
The
main effect
of a factor is basically the 'average' change in the output response as
that factor goes from '' to '+'. Mathematically, this is the
average of two numbers: 1) the change in output when the factor goes
from low to high level as the other factor stays low, and 2) the change in
output when the factor goes from low to high level as the other factor
stays high.
In the
example in Table 1, the output of the process is just 20 (lowest output)
when both A and B are at their '' level, while the output is maximum at
52 when both A and B are at their '+' level. The main effect of A is the
average of the change in output response when B stays '' as A goes from
'' to '+', or (4020) = 20, and the change in
output response when B stays '+' as A goes from '' to '+', or (5230) =
22. The main effect of A, therefore, is equal to 21.
Similarly,
the main effect of B is the average change in output as it goes from ''
to '+' , i.e., the average of 10 and 12, or 11. Thus, the main effect of
B in this process is 11. Here, one can see that the factor A exerts a
greater influence on the output of process, having a main effect of 21
versus factor B's main effect of only 11.
It must be
noted that aside from 'main effects', factors can likewise result in
'interaction effects.'
Interaction
effects are changes in the process output caused by two or more factors
that are interacting with each other. Large interactive effects
can make the main effects insignificant, such that it becomes more
important for the engineer to pay attention to the interaction of the
involved factors than to investigate them individually.
The running
of factorial combinations and the mathematical interpretation of the
output responses of the process to such combinations is the essence of
factorial experiments. It allows an engineer to understand which
factors affect his or her process most so that improvements (or
corrective actions) may be geared towards these.
See
also:
2Level Factorial Design;
Factorial Design Tables;
Example of a 2Level Factorial
Experiment
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