Chemical Engineering November 2014 - 55

Feature Report
(0 0 1)
C
A
(1 0 0)
B
(0 1 0)
A
B
FIGURE 8. These graphs show the principle of a mixture design. A) Since the sum
of all components must be 1, the possible values are restricted to a triangular part of
a plane. B) Example for a mixture design in the plane of Figure 8A
equipment is in good working
condition; all personnel are well
trained, and so on
* Write down the experimental
and measuring procedure in detail
and practice it beforehand
* Test whether possibly problematic
extreme combinations in
your design still work
* Calibrate your measuring equipment
and do some repeat-measurements
to know the random
variation and pitfalls of your
equipment before, and take appropriate
precautions
* Make sure all runs are done in
the planned order (and not reordered
for convenience) - the
" messed up " (randomized) order
is intentional
* Stay with the experiments, particularly
if they are run in a production
environment or if it is the
first designed experiment. Many
surprising discoveries have been
made in that way
* Document any departures from
the procedures laid down, even if
they seem unimportant at first.
Document environmental and
other relevant conditions runby-run
*
If there are any departures from
the planned design, always use the
actual factor values in modeling
(rather than the design values)
* Make sure that no settings or results
are confused - this could
have a dramatic effect on the
modeling. Check all results for
plausibility
Fit and evaluate models. This
is done using suitable software.
Usually simple polynomial models
are fitted using standard linear or
quasi-linear regression, a common
statistical procedure. Model terms
that are significant (that is, those
that are larger than random variation)
are retained in the model. The
usual criterion is the p-value, which
is the probability with which the observed
term, or an even bigger one,
occurs by chance, given the observed
random variation. If this p-value is
small (often <5%, but this is only a
guideline), the corresponding term
is called significant.
There are many diagnostic plots
of the residuals (that is, deviations
from the fitted model) to help in
checking the results. The most important
ones are as follows:
* Normal plot of residuals: Deviations
from a normal distribution
can be recognized, in particular
outliers - these are results that
are unusual. The reason may be a
mistake in the experiment, but it
may also indicate that something
new happens at the factor-level
combination (setting) involved
* Residuals in the run order:
Trends or sudden changes may
be recognized
* Residuals versus factors: Systematic
deviations from the model
can be detected
* Box-Cox-plot: It shows whether
a transformation of the response
(for instance, taking the logarithm)
leads to a model that fits
the data better
A wide variety of further diagnostic
information is given, such as the
following:
* Lack-of-fit: It checks whether the
systematic deviations from the
model are comparable to the random
variation found from replications
at identical settings of the
factors
* Predicted r2 or cross-validation:
It checks the capability of the
model to predict responses at
settings not used for fitting the
model, to avoid overfitting
Deduce improvements. If the
model is good, it can be plotted for
a response-surface design as shown
in Figures 2, 3 and 4. As shown in
the introductory example, these
plots can be used for a better understanding
of the cause-and-effect
relationships in the process or product
studied and for finding optimal
compromises between conflicting
aims. A confirmation experiment at
this setting rounds off the program.
For screening designs, the results
can be used to distinguish between
important and not so important factors
to then investigate the important
factors in a response-surface
design. Or the results can be used
to identify the direction of steepest
improvement, if one is far from the
optimum - and then proceed in
that direction.
In other words, either the improvement
is complete after the
current cycle or a new cycle starts.
In any case, important new information
has been gained.
Typical experimental designs
There are many different experimental
designs. They all have in
common that they are efficient for
a particular purpose. What follows
are some typical designs.
Two-level factorial designs. Figure
6 shows a two-level factorial
design for three factors. The full
factorial design consists of all 23 =
8 corners of the cube (the red and
the blue points in Figure 6). With
it, only linear effects of the factors
and all their interactions can be determined.
For k factors, the design
consists of the 2k corners of a k-dimensional
cube.
If all factors are numerical (as in
the introductory example, where
any value between, for example,
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