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Introduction to Design of Experiments
(DOE) -
DOE Types
This article continues the discussion of
Design of Experiments (DOE) that started in
last month's issue of the
Reliability HotWire. This article gives a summary of the various
types of DOE. Future articles will cover more DOE fundamentals in addition
to applications and discussion of DOE analyses accomplished with the
soon-to-be-introduced
DOE++ software!
The design and analysis of experiments
revolves around the understanding of the effects of different variables on
other variable(s). In mathematical jargon, the objective is to establish a
cause-and-effect relationship between a number of independent
variables and a dependent variable of interest. The dependent
variable, in the context of DOE, is called the response, and the
independent variables are called factors. Experiments are run at
different factor values, called levels. Each run of an experiment
involves a combination of the levels of the investigated factors. Each of
the combinations is referred to as a treatment. In a single factor
experiment, each level of the factor is referred to as a treatment. In
experiments with many factors, each combination of the levels of the factors
is referred to as a treatment. When the same number of response observations
are taken for each of the treatments of an experiment, the design of the
experiment is said to be balanced. Repeated observations at a given
treatment are called
replicates. The number of treatments of an experiment is determined
on the basis of the number of factor levels being investigated in the
experiment. For example, if an experiment involving two factors is to be
performed, with the first factor having x levels and the second
factor having z levels, then
x z treatment combinations can possibly be run, and the experiment is an
x z factorial design. If all
x z combinations are run, then the experiment is a full factorial. If
only some of the
x z treatment combinations are run, then the experiment is a fractional
factorial. In full factorial experiments, all of the factors and their
interactions are investigated, whereas in fractional factorial experiments,
all interactions are not considered because not all treatment combinations
are run.
It can be seen that the size of an
experiment escalates rapidly as the number of factors, or the number of the
levels of the factors, increases. For example, if two factors at three
levels each are to be used, nine different treatments are required for a
full factorial experiment (3 3 = 9). If a third factor with three levels is
added, 27 treatments are required (333 = 27) and 81 treatments are required
if a fourth factor with three levels is added (3333 = 81). If only two
levels are used for each factor, then in the four factor case, 16 treatments
are required (2 2 2 2 = 16). For this reason, many experiments are
restricted to two levels. Fractional factorial experiments further reduce
the number of treatments to be executed in an experiment.
DOE Types
The
following is a summary of some of the most common DOE types.
1 One Factor Designs
These are the designs where only one factor is under investigation, and the
objective is to determine whether the response is significantly different at
different factor levels. The factor can be qualitative or
quantitative. In the case of qualitative factors (e.g. different
suppliers, different materials, etc.), no extrapolations (i.e.
predictions) can be performed outside the tested levels, and only the effect
of the factor on the response can be determined. On the other hand, data
from tests where the factor is quantitative (e.g. temperature,
voltage, load, etc.) can be used for both effect investigation and
prediction, provided that sufficient data are available.
2 Factorial Designs
In factorial designs, multiple factors are investigated simultaneously
during the test. As in one factor designs, qualitative and/or quantitative
factors can be considered. The objective of these designs is to identify the
factors that have a significant effect on the response, as well as
investigate the effect of interactions (depending on the experiment design
used). Predictions can also be performed when quantitative factors are
present, but care must be taken since certain designs are very limited in
the choice of the predictive model. For example, in two level designs only a
linear relationship between the response and the factors can
be used, which may not be realistic.
General Full Factorial Designs
In general full factorial designs, each factor can have a different
number of levels, and the factors can be quantitative, qualitative or
both.
Two Level Full Factorial Designs
These are factorial designs where the number of levels for each factor
is restricted to two. Restricting the levels to two and running a full
factorial experiment reduces the number of treatments (compared to a
general full factorial experiment) and allows for the investigation of
all the factors and all their interactions. If all factors are
quantitative, then the data from such experiments can be used for
predictive purposes, provided a linear model is appropriate for modeling
the response (since only two levels are used, curvature cannot be
modeled).
Two Level Fractional Factorial
Designs
This is a special category of two level designs where not all factor
level combinations are considered and the experimenter can choose which
combinations are to be excluded. Based on the excluded combinations,
certain interactions cannot be determined.
Plackett-Burman Designs
This is a special category of two level fractional factorial designs,
proposed by R. L. Plackett and J. P. Burman, where only a few
specifically chosen runs are performed to investigate just the main
effects (i.e. no interactions).
Taguchis Orthogonal Arrays
Taguchis orthogonal arrays are highly fractional designs, used to
estimate main effects using only a few experimental runs. These designs
are not only applicable to two level factorial experiments, but also can
investigate main effects when factors have more than two levels. Designs
are also available to investigate main effects for certain mixed level
experiments where the factors included do not have the same number of
levels.
3 Response Surface Method Designs
These are special designs that are used to determine the settings of the
factors to achieve an optimum value of the response.
4 Reliability DOE
This is a special category of DOE where traditional designs, such as the two
level designs, are combined with reliability methods to investigate effects
of different factors on the life of a unit. In Reliability DOE, the response
is a life metric (e.g. age, miles, cycles, etc.), and the data may
contain censored observations (suspensions, interval data). One factor
designs and two level factorial designs (full, fractional, and
Plackett-Burman) are available in DOE++ to conduct a Reliability DOE
analysis.
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