| Definitions
for key DOE terms |
This page gives definitions and information for many of
the basic terms used in DOE.
Alias: When the estimate of an effect
also
includes the influence of one or more other effects (usually high order
interactions)
the effects are said to be aliased
(see confounding).
For example, if the estimate of effect D in a four factor experiment actually
estimates (D + ABC), then the main effect D is aliased with the 3-way interaction
ABC. Note: This causes no difficulty when the higher order interaction
is either non-existent or insignificant.
Analysis of Variance (ANOVA):
A mathematical process for separating the variability of a group of observations
into assignable causes and setting up various significance tests.
Balanced Design: An experimental design
where all cells (i.e. treatment combinations) have the same number of observations.
Blocking: A schedule for conducting
treatment
combinations in an experimental study such that any effects on the
experimental results due to a known change in raw materials, operators,
machines, etc., become concentrated in the levels of the blocking variable.
Note:
the reason for blocking is to isolate a systematic effect and prevent it
from obscuring the main effects. Blocking is achieved by restricting randomization.
Center Points: Points at the center
value of all factor ranges.
Coding Factor Levels: Transforming
the scale of measurement for a factor so that the high value becomes +1
and the low value becomes -1 (see scaling). After coding all factors
in a 2-level full factorial experiment, the design matrix has all orthogonal
columns.
Coding is a simple linear transformation of the original measurement
scale. If the "high" value is Xh and the "low" value is XL
(in
the original scale), then the scaling transformation takes any original
X value and converts it to (X - a)/b, where
a = (Xh + XL)/2 and b = ( Xh -X L)/2.
To go back to the original measurement scale, just take the coded value
and multiply it by "b" and add "a" or, X = b(coded value) + a.
As an example, if the factor is temperature and the high setting is
65oC and the low setting is 55oC, then a = (65 +
55)/2 = 60 and b = (65 - 55)/2 = 5. The center point (where the coded value
is 0) has a temperature of 5(0) + 60 = 60oC.
Comparative Designs: A design aimed
at making conclusions about one a priori important factor, possibly in
the presence of one or more other "nuisance" factors.
Confounding: A confounding
design is one where some treatment effects (main
or interactions) are estimated by the same linear combination of the experimental
observations as some blocking effects. In this
case, the treatment effect and the blocking effect are said to be confounded.
Confounding is also used as a general term to indicate that the value of
a main effect estimate comes from both the main effect
itself and also contamination or bias from higher order interactions.
Note: Confounding designs naturally arise when full
factorial designs have to be run in blocks and the block size is smaller
than the number of different treatment combinations. They also occur whenever
a fractional factorial design is chosen
instead of a full factorial design.
Crossed Factors: See factors
below.
Design: A set of experimental runs
which allows you to fit a particular model and estimate your desired effects.
Design Matrix: A matrix
description of an experiment that is useful for constructing and analyzing
experiments.
Effect: How changing the settings of
a factor changes the response. The effect of a single factor is also called
a main effect. Note: For a factor
A with two levels, scaled so that low = -1
and high = +1, the effect of A is estimated by subtracting the average
response when A is -1 from the average response when A = +1 and dividing
the result by 2 (division by 2 is needed because the -1 level is 2 scaled
units away from the +1 level).
Error: Unexplained variation in a collection
of observations. Note: DOE's typically require understanding of
both random error and lack of fit error.
Experimental Unit: The entity
to which a specific treatment combination is applied. Note: an experimental
unit can be a
-
PC board
-
silicon wafer
-
tray of components simultaneously treated
-
individual agricultural plants
-
plot of land
-
automotive transmissions
-
etc.
Factors: Process inputs an
investigator manipulates to cause a change in the output. Some factors
cannot be controlled by the experimenter but may effect the responses.
If their effect is significant, these uncontrolled factors should
be measured and used in the data analysis. Note: The inputs can
be discrete or continuous.
-
Crossed Factors: Two factors are crossed
if
every level of one occurs with every level of the other in the experiment.
-
Nested Factors: A factor "A" is nested
within another factor "B" if the levels or values of "A" are different
for every level or value of "B". Note: Nested factors or effects
have a hierarchical relationship.
Fixed Effect: An effect associated with
an input variable that has a limited number of levels or in which only
a limited number of levels are of interest to the experimenter.
Interactions: Occurs when
the effect of one factor on a response depends on the level of another
factor(s).
Lack of Fit Error: Error
that occurs when the analysis omits one or more important terms or factors
from the process model. Note: Including replication in a DOE allows
separation of experimental error into its components: lack of fit and random
(pure) error.
Model: Mathematical relationship
which relates changes in a given response to changes in one or more factors.
Nested Factors: See factors
above.
Orthogonality: Two vectors of
the same length are orthogonal if the sum of the products of their corresponding
elements is 0. Note: An experimental design is orthogonal if the
effects of any factor balance out (sum to zero) across the effects of the
other factors.
Random Effect: An effect associated
with input variables chosen at random from a population having a large
or infinite number of possible values.
Random error: Error that occurs
due to natural variation in the process. Note: Random error is typically
assumed
to
be normally distributed with zero mean and a constant variance. Note:
Random error is also called experimental error.
Randomization: A schedule
for allocating treatment material and for conducting treatment combinations
in a DOE such that the conditions in one run neither depend on the conditions
of the previous run nor predict the conditions in the subsequent runs.
Note:
The importance of randomization cannot be over stressed. Randomization
is necessary for conclusions drawn from the experiment to be correct, unambiguous
and defensible.
Replication: Performing
the same treatment combination more than once. Note: Including replication
allows an estimate of the random error independent of any lack of fit error.
Resolution: A term which describes
the degree to which estimated main effects are aliased
(or
confounded)
with estimated 2-level interactions, 3-level
interactions, etc. In general, the resolution of a design is one more than
the smallest order interaction that some main effect is confounded (aliased)
with. If some main effects are confounded with some 2-level interactions,
the resolution is 3. Note: Full
factorial designs have no confounding and are said to have resolution
"infinity". For most practical purposes, a resolution 5 design is excellent
and a resolution 4 design may be adequate. Resolution 3 designs are useful
as economical screening designs.
Responses: The output(s) of
a process. Sometimes called dependent variable(s).
Response Surface Designs:
A DOE that fully explores the process window and models the responses.
Note:
These designs are most effective when there are less than 5 factors. Quadratic
models are used for response surface designs and at least three levels
of every factor are needed in the design.
Rotatability: A design is rotatable
if the variance of the predicted response at any point x depends
only on the distance of x from the design center
point. A design with this property can be rotated around its center
point without changing the prediction variance at x. Note:
Rotatability is a desirable property for response surface designs (i.e.
quadratic model designs).
Scaling Factor Levels: Transforming
factor levels so that the high value becomes +1 and the low value becomes
-1.
Screening Designs: A DOE that
identifies which of many factors have a significant effect on the response.
Note:
Typically screening designs have more than 5 factors.
Treatment: A treatment is a specific
combination of factor levels whose effect is to be compared with other
treatments.
Treatment Combination: The
combination of the settings of several factors in a given experimental
trial. Also known as a run.
Variance Components: Partitioning
of the overall variation into assignable components.
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