In experimental design when two interactions,
or a main effect and an interaction, share
the same column, and so cannot be individually
analyzed then their effects are aliased.
In the 23-1 design the factor
C is aliased with the interaction AB:
| Run |
A |
B |
AB +
C |
| 1 |
-1 |
-1 |
+1 |
| 2 |
+1 |
-1 |
-1 |
| 3 |
-1 |
+1 |
-1 |
| 4 |
+1 |
+1 |
+1 |
Also Balanced Experiment.
A factorial design in which each factor
is run the same number of times at the high
and low levels. Most factorial designs are
balanced, unbalanced designs are only used
in exceptional circumstances.
Blocking is a technique used
to eliminate identified possible sources
of variation that cannot be randomized.
The experiment is organized into blocks,
where the possible source of variation is
held level in each block.
Suppose that there was insufficient material
in one batch to carry out all the runs in
an experiment. Half the runs could be done
on one batch and half on another. The experiment
would have to be carried out in such a way
that the aggregate result was balanced between
the two batches. At its simplest, if the
experiment involved an even number of replications
of each run then this would simply involve
carrying out half the replications on one
batch, and half on the other. Things are
not usually so simple, and there are many
ways of approaching more challenging situations.
In experimental design it is usual to substitute
'coded factors' for the actual physical
values. In a factorial design the coded
factors are given the levels '-1' and '+1'.
This assists the analysis in various ways.
| Completely
Randomized Design |
|
A design where the experimental units are
to the treatments in a completely random
fashion. Suppose we wanted to compare four
brands of washing powder. We decide to carry
out four replications for each brand.
We would prepare 16 apparently identical
pieces of soiled cloth and allocate them
to the powders at random. To do this we
could mix them up in a tub and pull them
out one at a time. To be sure there is no
bias, we might take 16 slips of paper and
mark them Brand 1#1 up to Brand 4#4. We
would then mix the pieces of paper up in
a hat and match a slip of paper drawn at
random with a piece of cloth drawn at random.
Factors or interactions are confounded
when the design array is configured so that
the effect of one factor is combined with
the other. The effect of the individual
factors/interactions cannot be isolated
by the analysis.
Confounding is very similar to aliasing,
although aliasing is used to describe factors/interactions
that are fully confounded, rather than partially
confounded.
The relationship that determines the aliasing
structure in a design.
| Run |
A |
B |
AB
+ C |
| 1 |
-1 |
-1 |
+1 |
| 2 |
+1 |
-1 |
-1 |
| 3 |
-1 |
+1 |
-1 |
| 4 |
+1 |
+1 |
+1 |
The Defining Relationship is I=ABC
When an experiment is conducted the variables
manipulated by the experimenter are called
"independent variables" or factors
and the response or output variables are
the "dependent variables".
The relationship used to create a fractional
factorial design, and which creates the
defining relation:
| Run |
A |
B |
AB
+ C |
| 1 |
-1 |
-1 |
+1 |
| 2 |
+1 |
-1 |
-1 |
| 3 |
-1 |
+1 |
-1 |
| 4 |
+1 |
+1 |
+1 |
The Design Generator is A=BC
Experimental design involves conducting
a systematic series of tests to discover
the relationship between the factors effecting
a process (X) and the critical outputs (responses).
The inputs are varied in a systematic fashion
and the effects on the response(s) are observed.
The Design of Experiments (Experimental
Design) is used when a process is affected
by many separate factors. It is far more
efficient than the 'One Factor at a Time'
(OFAT) method in which each factor is varied
in turn whilst the others remain constant.
The Design of Experiments will give more
information for less testing. It will also
allow 'interactions' between factors to
be evaluated, and the effects of interactions
are usually important.
The two main approaches to experimental
design are the 'classical' and 'Taguchi'
methods. Experts are divided on the merits
of the Taguchi method, but the emphasis
on variation and the methods it uses to
address variation are important. The types
of design commonly used included Full Factorial
Designs, Fractional Factorial Designs and
Plackett-Burman Designs.
A more powerful approach is to use Response
Surface Methods; this group includes the
Box Behnken and Central Composite Designs
(CCD).
Designs that involve mixtures require a
different method of analysis, see the topic
on Mixture Designs.
Used in experimental design to help select
a fractional factorial design:
| Resolution
III |
RIII |
two factor interactions
are aliased with main effects |
| Resolution
IV |
RIV |
two factor interactions
are aliased with other two factor interactions,
but not with main effects. Main effects
are aliased with three factor interactions. |
| Resolution
5 |
RV |
two factor interactions
are not aliased with each other, but
are aliased with three factor interaction |
An RIII design would usually be used for
screening experiments. There would rarely
be any point in going beyond an RV
design.
In general the resolution is the length
of the shortest word in the defining relationship.
The degree to which data tend to spread
about the mean. It is measured by the standard
deviation, or variance.
Evolutionary Operation. An experimental
design technique that involves making small
changes to a process progressively, and
during normal operation, to find the optimum
operating conditions.
See Design of Experiments
A factor is a parameter whose potential
effect on the response is being evaluated
in the experiment.
The experiment involves varying the levels
of 'factors' to determine their effect on
the 'response'.
An experimental design where the factor
levels are specifically selected by the
experimenter (cf. Random Effects Model)
See Dependent Variable for an explanation
In many processes the factors interact,
the combined effect is not the sum of the
individual effects. The figure below uses
the well known danger of combining alcohol
with some medications to illustrate the
idea:
Interactions are an important
consideration in experimental design.
During the experiment the
factors are set to different values or 'levels'.
In a factorial experiment two levels, or
sometimes three, are selected for each factor.
The effect of a factor, as opposed to an
interaction effect.
A design is orthogonal if each factor can
be evaluated independently of all other
factors. In a two level factorial design,
this is achieved by matching each level
of each factor with an equal number of each
level of the other factors.
For example, in the array the '+1' level
of Factor A (runs 2 and 4) is matched with
one instance of Factor B at '-1' and one
at '+1'. If any two columns are compared,
the same thing will be found for both factor
levels.
| Run |
A |
B |
AB
+ C |
| 1 |
-1 |
-1 |
+1 |
| 2 |
+1 |
-1 |
-1 |
| 3 |
-1 |
+1 |
-1 |
| 4 |
+1 |
+1 |
+1 |
The term 'orthogonal array' is often used
in the context of Taguchi
designs; 'Taguchi Orthogonal
Arrays'.
An experimental design where the factor
levels are selected at random from a large
number of possible levels. The analysis
is based on estimating the variance.
This compares to a fixed effects model
where a restricted number of levels are
selected and set by the experimenter.
Experiments should be run in random order
to randomize the effects of any variables
that are varying during the experiment,
and might impose a pattern on the results.
Randomizing destroys any systematic variation,
which could not be detected by the analysis,
and converts it to either 'common cause'
variation, or variation that will be detected
in the residual analysis.
Suppose we wanted to compare four brands
of washing powder. We decide to carry out
four replications for each brand. To make
the test realistic we give four volunteers
four clean handkerchiefs each; they are
to use each for one day.
In a Completely Randomized design we would
allocate the handkerchiefs to powder at
random, so that one powder might get one,
two, three or even all four handkerchiefs
from one person.
In a Randomized Block design we would block
on the handkerchiefs. We would allocate
one, and only one, handkerchief from each
volunteer to each powder. The four handkerchiefs
from each volunteer would be allocated at
random to the powders, using a similar scheme
to the one described in the Completely Randomized
design.
See Replication
Carrying out several tests at each treatment.
If each treatment is tested four times then
there are four replications of the design.
It is important that the order of the runs
is randomized to remove systematic error.
If this is not done, but each treatment
is repeated several times before moving
on to the next, then that is 'repetition'.
The ANOVA analysis should not be carried
out using the repeats as individual readings.
Instead the mean of the repeats should be
treated as a single reading
An experimental design explores the effect
of input factors on the process response.
For example, an experiment might investigate
the effects of altering the amounts of flour,
water, yeast, oven temperature etc. on the
consistency of bread. The consistency quality
of the bread would be the response variable.
A desirable property of Response Surface
Designs. It refers to designs where the
variance is the same at all points that
are the same distance from the design center.
A type of experimental design used when
a large number of factors that may affect
the process have been identified. The screening
design identifies the factors that do not
affect the process so those remaining can
be studied more closely. Screening designs
are Resolution III designs.
Applying a mathematical operation to response
data to make it conform to a normal distribution.
Often necessary with experiments that involve
rates because the results typically do not
conform to a normal distribution.
Common transforms include natural log,
reciprocal and squares.
A certain combination of factor levels
whose effect on the response variable is
of interest. Often replaced by the term
'run'.
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