5.
Process Improvement
5.5.
Advanced topics
5.5.9.
An EDA approach to experimental design
5.5.9.4.

Interaction effects matrix plot


Purpose

The interaction effects matrix plot is an extension of the
DOE mean plot to include both main effects
and 2factor interactions (the DOE mean plot focuses on main effects
only). The interaction effects matrix plot answers the following two
questions:
 What is the ranked list of factors (including 2factor
interactions), ranked from most important to least important;
and
 What is the best setting for each of the k factors?
For a kfactor experiment, the effect on the response
could be due to main effects and various interactions all the way up
to kterm interactions. As the number of factors, k,
increases, the total number of interactions increases
exponentially. The total number of possible interactions of all
orders = 2^{k}  1  k. Thus for k = 3,
the total number of possible interactions = 4, but for k = 7
the total number of possible interactions = 120.
In practice, the most important interactions are likely to be 2factor
interactions. The total number of possible 2factor interactions is
\[ \left( \begin{array}{c}
k \\ 2
\end{array}
\right)
= \frac{k!} {2!(k2)!} = \frac{k(k1)}{2}
\]
Thus for k = 3, the number of 2factor interactions = 3, while
for k = 7, the number of 2factor interactions = 21.
It is important to distinguish between the number of interactions
that are active in a given experiment versus the number of
interactions that the analyst is capable of making definitive
conclusions about. The former depends only on the physics and
engineering of the problem. The latter depends on the number of
factors, k, the choice of the k factors, the constraints
on the number of runs, n, and ultimately on the experimental
design that the analyst chooses to use. In short, the number of
possible interactions is not necessarily identical to the
number of interactions that we can detect.
Note that
 with full factorial designs, we can uniquely
estimate interactions of all orders;
 with fractional factorial designs, we can uniquely estimate
only some (or at times no) interactions; the more fractionated
the design, the fewer interactions that we can estimate.

Output

The output for the interaction effects matrix plot is
 Primary: Ranked list of the factors (including 2factor
interactions) with the factors are ranked from important to
unimportant.
 Secondary: Best setting for each of the k factors.

Definition

The interaction effects matrix plot is an upper righttriangular
matrix of mean plots
consisting of k main effects plots on the diagonal and
k*(k1)/2 2factor interaction effects plots
on the offdiagonal.
In general, interactions are not the same as the usual
(multiplicative) crossproducts. However, for the special case of
2level designs coded as (, +) = (1, +1), the interactions
are identical to crossproducts. By way of contrast,
if the 2level designs are coded otherwise (e.g., the (1, 2) notation
espoused by Taguchi and others), then this equivalance is not
true. Mathematically,
{1, +1} x {1, +1} => {1, +1}
but
{1, 2} x {1, 2} => {1, 2, 4}
Thus, coding does make a difference. We recommend the use of
the (, +) coding.
It is remarkable that with the  and + coding, the 2factor
interactions are dealt with, interpreted, and compared in the same
way that the k main effects are handled. It is thus natural to
include both 2factor interactions and main effects within the same
matrix plot for ease of comparison.
For the offdiagonal terms, the first construction step is to form
the horizontal axis values, which will be the derived values (also
 and +) of the crossproduct. For example, the settings for the
X_{1}*X_{2} interaction are derived by simple multiplication
from the data as shown below.
X_{1}

X_{2}

X_{1}*X_{2}





+

+







+



+

+

+

Thus X_{1}, X_{2}, and X_{1}*X_{2} all form a closed
(, +) system. The advantage of the closed system is that graphically
interactions can be interpreted in the exact same fashion as the
k main effects.
After the entire X_{1}*X_{2} vector of settings has been
formed in this way, the vertical axis of the X_{1}*X_{2}
interaction plot is formed:
 the plot point above X_{1}*X2 = "" is simply the
mean of all response values for which X_{1}*X_{2} = ""
 the plot point above X_{1}*X_{2} = "+" is simply the
mean of all response values for which X_{1}*X_{2} = "+".
We form the plots for the remaining 2factor interactions in a
similar fashion.
All the mean plots, for both main effects and 2factor interactions,
have a common scale to facilitate comparisons. Each mean plot has
 Vertical Axis: The mean response for a given setting ( or +)
of a given factor or a given 2factor interaction.
 Horizontal Axis: The 2 settings ( and +) within each factor,
or within each 2factor interaction.
 Legend:
 A tag (1, 2, ..., k, 12, 13, etc.), with
1 = X_{1}, 2 = X_{2}, ..., k =
X_{k}, 12 = X_{1}*X_{2},
13 = X_{1}*X_{3}, 35 = X_{3}*X_{5},
123 = X_{1}*X_{2}*X_{3}, etc.) which
identifies the particular mean plot; and
 The least squares estimate of the factor (or 2factor
interaction) effect. These effect estimates are large
in magnitude for important factors and nearzero in
magnitude for unimportant factors.
In a later section, we discuss in detail
the models associated with full and fractional factorial 2level designs.
One such model representation is
\( Y = \mu + \beta_{1} X_{1} + \beta_{2} X_{2} + \beta_{12} X_{1} X_{2} +
\cdots + \epsilon \)
For factor variables coded with + and  settings, the
β_{i} coefficient is one half of the effect estimate
due to factor X_{i}. Thus, if we multiply the leastsquares
coefficients by two, due to orthogonality, we obtain the simple difference
of means at the + setting and the  setting. This is true for the
k main factors. It is also true for all twofactor and
multifactor interactions.
Thus, visually, the difference in the mean values on the plot is
identically the least squares estimate for the effect. Large
differences (steep lines) imply important factors while small
differences (flat lines) imply unimportant factors.

Motivation

As discussed in detail above, the next logical step beyond main effects
is displaying 2factor interactions, and this plot matrix provides a
convenient graphical tool for examining the relative importance of
main effects and 2factor interactions in concert. To do so, we make
use of the striking aspect that in the context of 2level designs,
the 2factor interactions are identical to crossproducts and the
2factor interaction effects can be interpreted and compared the same
way as main effects.

Plot for defective springs data

Constructing the interaction effects matrix plot for the defective
springs data set yields the following plot.

How to interpret

From the interaction effects matrix, we can draw three
important conclusions:
 Important Factors (including 2factor interactions);
 Best Settings;
 Confounding Structure (for fractional factorial designs).
We discuss each of these in turn.
 Important factors (including 2factor interactions):
Jointly compare the k main factors and the
k*(k1)/2
2factor interactions. For each of these subplots, as we go
from the "" setting to the "+" setting within a subplot, is
there a shift in location of the average data (yes/no)?
Since all subplots have a common (1, +1) horizontal axis,
questions involving shifts in location translate into
questions involving steepness of the mean lines (large
shifts imply steep mean lines while no shifts
imply flat mean lines).
 Identify the factor or 2factor interaction that has the
largest shift (based on averages). This defines
the "most important factor". The largest shift is
determined by the steepest line.
 Identify the factor or 2factor interaction that has the
next largest shift (based on averages). This
defines the "second most important factor". This shift
is determined by the next steepest line.
 Continue for the remaining factors.
This ranking of factors and 2factor interactions
based on local means is a major step in
building the definitive list of ranked factors
as required for screening experiments.
 Best settings:
For each factor (of the k main factors along
the diagonal), which setting ( or +) yields
the "best" (highest/lowest) average response?
Note that the experimenter has the ability to change settings
for only the k main factors, not for any 2factor
interactions. Although a setting of some 2factor interaction
may yield a better average response than the alternative
setting for that same 2factor interaction, the experimenter is
unable to set a 2factor interaction setting in practice.
That is to say, there is no "knob" on the machine that
controls 2factor interactions; the "knobs" only control the
settings of the k main factors.
How then does this matrix of subplots serve as
an improvement over the k best settings that
one would obtain from the DOE mean
plot? There are two common possibilities:
 Steep Line:
For those main factors along the diagonal that have
steep lines (that is, are important), choose the best
setting directly from the subplot. This will be the same
as the best setting derived from the DOE mean plot.
 Flat line:
For those main factors along the diagonal that have flat
lines (that is, are unimportant), the naive conclusion to
use either setting, perhaps giving preference to the
cheaper setting or the easiertoimplement setting, may
be unwittingly incorrect. In such a case, the use of the
offdiagonal 2factor interaction information from the
interaction effects matrix is critical for deducing the
better setting for this nominally "unimportant" factor.
To illustrate this, consider the following example:
 Suppose the factor X_{1} subplot is steep
(important) with the best setting for X_{1}
at "+".
 Suppose the factor X_{2} subplot is flat
(unimportant) with both settings yielding about
the same mean response.
Then what setting should be used for X_{2}? To answer
this, consider the following two cases:
 Case 1. If the X_{1}*X_{2} interaction plot
happens also to be flat (unimportant), then choose
either setting for X_{2} based on cost or ease.
 Case 2. On the other hand, if the
X_{1}*X_{2} interaction plot is steep
(important), then this dictates a prefered setting
for X_{2} not based on cost or ease.
To be specific for case 2, if X_{1}*X_{2} is
important, with X_{1}*X_{2} = "+" being the
better setting, and if X_{1} is important, with
X_{1} = "+" being the better setting, then
this implies that the best setting for X_{2} must be
"+" (to assure that X_{1}*X2 (= +*+) will also
be "+"). The reason for this is that since we are already
locked into X_{1} = "+", and since X_{1}*X_{2}
= "+" is better, then the only way we can obtain
X_{1}*X_{2} = "+" with X_{1} = "+" is for
X_{2} to be "+" (if X_{2} were "", then
X_{1}*X_{2} with X_{1} = "+" would yield
X_{1}*X_{2} = "").
In general, if X_{1} is important, X_{1}*X_{2}
is important, and X_{2} is not important, then
there are four distinct cases for deciding
what the best setting is for X_{2}:
X_{1}

X_{1}*X_{2}

=> X_{2}

+

+

+

+







+







+

By similar reasoning, examining each factor and pair of
factors, we thus arrive at a resulting vector of the
k best settings:
(x1best, x2best, ..., xkbest)
This averagebased kvector should be compared
with best settings kvectors obtained from
previous steps (in particular, from step 1 in which the
best settings were drawn from the best data value).
When the averagebased best settings and the databased
best settings agree, we benefit from the increased
confidence given our conclusions.
When the averagebased best settings and the
databased best settings disagree, then what settings
should the analyst finally choose? Note that in general
the averagebased settings and the databased settings
will invariably be identical for all "important" factors.
Factors that do differ are virtually always
"unimportant". Given such disagreement,
the analyst has three options:
 Use the averagebased settings for minor
factors. This has the advantage of a broader
(average) base of support.
 Use the databased settings for minor factors.
This has the advantage of demonstrated local
optimality.
 Use the cheaper or more convenient settings for
the local factor. This has the advantage of
practicality.
Thus the interaction effects matrix yields important
information not only about the ranked list of factors, but
also about the best settings for each of the k main
factors. This matrix of subplots is one of the most important
tools for the experimenter in the analysis of 2level
screening designs.
 Confounding Structure (for Fractional Factorial Designs)
When the interaction effects matrix is used to analyze
2level fractional (as opposed to full) factorial designs,
important additional information can be extracted from the
matrix regarding confounding structure.
It is wellknown that all fractional factorial designs have
confounding, a property whereby every estimated main effect is
confounded/contaminated/biased by some highorder
interactions. The practical effect of this is that the analyst
is unsure of how much of the estimated main effect is due to the
main factor itself and how much is due
to some confounding interaction. Such contamination is the price
that is paid by examining k factors with a sample size
n that is less than a full factorial
n = 2^{k} runs.
It is a "fundamental theorem" of the discipline of experimental
design that for a given number of factors k and a given
number of runs n, some fractional factorial designs are
better than others. "Better" in this case means that the
intrinsic confounding that must exist in all
fractional factorial designs has been minimized by the choice
of design. This minimization is done by constructing the design
so that the main effect confounding is pushed to as high an
order interaction as possible.
The rationale behind this is that in physical science and
engineering systems it has been found that the
"likelihood" of highorder interactions being significant is
small (compared to the likelihood of main effects and 2factor
interactions being significant). Given this, we would prefer
that such inescapable main effect confounding be with the highest
order interaction possible, and hence the bias to the estimated
main effect be as small as possible.
The worst designs are those in which the main effect confounding
is with 2factor interactions. This may be dangerous
because in physical/engineering systems, it is quite common for
Nature to have some real (and large) 2factor interactions. In
such a case, the 2factor interaction effect will be inseparably
entangled with some estimated main effect, and so the experiment
will be flawed in that
 ambiguous estimated main effects and
 an ambiguous list of ranked factors
will result.
If the number of factors, k, is large and the
number of runs, n, is constrained to be small, then
confounding of main effects with 2factor interactions is
unavoidable. For example, if we have k = 7 factors and
can afford only n = 8 runs, then the corresponding
2level fractional factorial design is a 2^{74}
which necessarily will have main effects confounded with (3)
2factor interactions. This cannot be avoided.
On the other hand, situations arise in which 2factor interaction
confounding with main effects results not from constraints on
k or n, but on poor design construction. For
example, if we have k = 7 factors and can afford n
= 16 runs, a poorly constructed design might have main effects
counfounded with 2factor interactions, but a wellconstructed
design with the
same k = 7, n = 16 would have main effects
confounded with 3factor interactions but no 2factor
interactions. Clearly, this latter design is preferable in
terms of minimizing main effect confounding/contamination/bias.
For those cases in which we do have main effects confounded
with 2factor interactions, an important question arises:
For a particular main effect of interest,
how do we know which 2factor interaction(s)
confound/contaminate that main effect?
The usual answer to this question is by means of generator theory,
confounding tables, or alias charts. An alternate complementary
approach is given by the interaction effects matrix. In
particular, if we are examining a 2level fractional factorial
design and
 if we are not sure that the design has main effects
confounded with 2factor interactions, or
 if we are sure that we have such 2factor
interaction confounding but are not sure
what effects are confounded,
then how can the interaction effects matrix be of assistance?
The answer to this question is that the confounding structure
can be read directly from the interaction effects matrix.
For example, for a 7factor experiment, if, say, the factor
X_{3} is confounded with the 2factor interaction
X_{2}*X_{5}, then
 the appearance of the factor X_{3} subplot
and the appearance of the 2factor
interaction X_{2}*X_{5} subplot will
necessarily be identical, and
 the value of the estimated main effect for X_{3}
(as given in the legend of the main effect subplot) and
the value of the estimated 2factor interaction effect for
X_{2}*X_{5} (as given in the legend of the
2factor interaction subplot) will also necessarily be
identical.
The above conditions are necessary, but not sufficient for the
effects to be confounded.
Hence, in the abscence of tabular descriptions (from your
statistical software program) of the confounding structure, the
interaction effect matrix offers the following graphical
alternative for deducing confounding structure in fractional
factorial designs:
 scan the main factors along the diagonal subplots and
choose the subset of factors that are "important".
 For each of the "important" factors, scan all of the
2factor interactions and compare the main factor subplot
and estimated effect with each 2factor interaction
subplot and estimated effect.
 If there is no match, this implies that the main effect
is not confounded with any 2factor interaction.
 If there is a match, this implies that the main
effect may be confounded with that 2factor
interaction.
 If none of the main effects are confounded with any
2factor interactions, we can have high confidence in the
integrity (noncontamination) of our estimated main
effects.
 In practice, for highlyfractionated designs, each main
effect may be confounded with several 2factor
interactions. For example, for a 2^{74}
fractional factorial design, each main effect will be
confounded with three 2factor interactions. These
1 + 3 = 4 identical subplots will be blatantly obvious
in the interaction effects matrix.
Finally, what happens in the case in which the design
the main effects are not confounded with 2factor
interactions (no diagonal subplot matches any offdiagonal
subplot). In such a case, does the interaction effects matrix
offer any useful further insight and information?
The answer to this question is yes because even though such
designs have main effects unconfounded with 2factor interactions,
it is fairly common for such designs to have 2factor interactions
confounded with one another, and on occasion it may be of
interest to the analyst to understand that confounding. A
specific example of such a design is a 2^{41} design
formed with X_{4} settings = X_{1}*X_{2}*X_{3}.
In this case, the 2factorinteraction confounding structure may
be deduced by comparing all of the 2factor interaction subplots
(and effect estimates) with one another. Identical subplots and
effect estimates hint strongly that the two 2factor interactions
are confounded. As before, such comparisons provide necessary
(but not sufficient) conditions for confounding. Most statistical
software for analyzing fractional factorial experiments will
explicitly list the confounding structure.

Conclusions for the defective springs data

The application of the interaction effects matrix plot to the
defective springs data set results in the following conclusions:
 Ranked list of factors (including 2factor interactions):
 X_{1} (estimated effect = 23.0)
 X_{1}*X_{3} (estimated effect = 10.0)
 X_{2} (estimated effect = 5.0)
 X_{3} (estimated effect = 1.5)
 X_{1}*X_{2} (estimated effect = 1.5)
 X_{2}*X_{3} (estimated effect = 0.0)
Factor 1 definitely looks important. The X_{1}*X_{3}
interaction looks important. Factor 2 is of lesser importance.
All other factors and 2factor interactions appear to be
unimportant.
 Best Settings (on the average):
(X_{1}, X_{2}, X_{3}) = (+, , +) = (+1, 1, +1)
