Central composite design

In experimental design strategy, a central composite design is

an experimental design, useful in creating relatively sophisticated
models, usually, a second order (quadratic) model for the output
function without needing to use other designs, for instance a
complete three-level factorial design (requiring more
experimental work).
After the designed experiment is performed, the calculation
algorithm follows the regression strategy, sometimes iteratively,
to obtain results. Coded variables are often used when
constructing this design.
Graphical presentation
 Legend to the figure
• The figure presented in the previous slide demonstrates the
design for 3 input factors as central composite design;
• The total number of required experiments is 15 as follows:
• 8 experiments according to the scheme of FFD 2^3 (totally
8);
• 6 experiments at so called “star” or “axial” points;
• 1 experiment in the center of the design (center point).
The choice of various “points” for the design
 Corner points
• Previous slide shows the simplest central composite design for two
input factors – it needs totally 9 experimental points;
• Corner points: these are the experiments according to the design of
the type full factorial design at two levels of variation of the factors;
• The choice of the intervals of variation for the factors is an important
task for the scientist and, as we already know, depends on
preliminary experience and information; the factors are respective
coded as “+1” and “-1”;
• These are the experiments at the corners of the quadrat (4 points).
 Axial or star points
• These points are usually marked by “α”;
• They are located symmetrically around the design center and
the value of α depends on the number of input factors
involved (this value is determined a prioi and could be found
in statistical tables (software products); the experiments at
the axial points are carried out with coordinates (0, α) or (0, -
α) and for 2 factors the total number of axial experiments is
2x2 = 4; this part of the central composite design is the
“superstructure” to the full factorial design.
The central point of the design marked by 0
 Some clarifications
• The central experiment is performed when the input factors have as
coordinates the middle of the interval of variation (0, 0);
• In the scale of the coded input variables the corner experiments have
as coordinates combinations of +1 and -1; the axial experiments –
combinations of 0 and α (the α value for two factors is 1.4, so the real
value for the experiment could be easily calculated); the center
experiment was considered above.
• All necessary steps for carrying out the real experiments are as in case
of full factorial experiment – assessment of experimental error,
randomization etc.
 The statistical model
• It fits a complete quadratic model;
• The model is checked for error homogeneity, significance of
the regression coefficients and validity (comparison between
calculated by the model and experimentally found value)
 Optimization experiments
• Very often the final goal of a research (experimental) study is
to reach an extreme (minimal or maximal) value of the
response (output) function, e.g. the response has to be
optimized;
• Possible actions are:
• Mapping experiments (experiments around the maximum);
• Simplex optimization;
• Box – Wilson gradient methods (steepest slope approach).
 “Mapping” experiments approach
• After carrying out some kind of experimental design the
research will reach a maximal value of the response for
certain combination of input factor levels; Is this the real
optimum?
• In order to check this assumption single-at-a-time
experiments around the condition accepted as “optimal” for
the response are organized and performed; it is easy to find
out if higher (or lower) values as compared to that by the
design are obtained as a result of the mapping.
Graphical example
 Simplex optimization procedure
• What is a simplex? Geometric figure with number of apexes
equal to the number of input factors + 1; it means that the
simplex for 2 factors has three vertexes, i.e. it is triangle, for
3 factors is a pyramid and for more factors is a topological
figure which could not be drawn on the plane of the sheet;
• In order to carry out experiments following the algorithm for
Simplex optimization coordinates of the initial Simplex are
determined (by the researcher according his/her goals and
information).
 The movement of the Simplex towards
optimum
• Let’s consider “triangle” case (2 factors) – three initial experiment at
the apexes; three responses – the worst one is eliminated and by
reflection a new Simplex is formed (the coordinates of the newly
introduced point are easily calculated.
 Next steps for Simplex
• The movement continues using the same algorithm for the next
simplexes:
 Basic rules
• The movement of the simplex needs some basic rules:
1. Rotation – the simplex could stop at certain point and starts rotation
around its centroid; the direction of movement has to be changes by
elimination of the second worst response;
2. At a certain point the Simplex could jump over the coordinate system
(movement in not allowed space, e.g. negative concentration); then the
direction should be changed by using the second worst response;
3. Appearance of one and the same vertex in three consecutive
simplexes (probably the region of optimum is achieved)
4. There are options for acceleration of the movement (expansion) or
delay (contraction).
Graphical presentation of the rules
 Optimal vertex or area of optimum
• It has to be kept in mind that the Simplex could miss the optimal
vertex due to many reasons; that is why a satisfactory outcome is the
location of area of optimum, plateau.
 Is this the real optimum?
• One good option to check if the real optimum is achieved is to start a
simplex with different initial coordinates. If the same peak or region
of optimum is reached, then the final goal is validated.
 Box – Wilson gradient method for
optimization
• In statistics, response surface methodology (RSM) explores the relationships
between several input factors and one or more output functions. The main idea
of RSM is to use a sequence of designed experiments to obtain an optimal
response. The authors acknowledge that this model is only an approximation, but
they use it because such a model is easy to estimate and apply, even when little is
known about the process.
Steepest descent method
Steepest ascent method
 General Steps of the method
• 1. Full factorial design
• 2. Determination of the steepest ascent – the factor with the most
significant weight
• 3. Determination of the new intervals of variation with respect to
steepest ascent and carrying out next FFD
• 4. Mental or real experiments toward the trend of optimum
• 5. Change of direction (sign of the regression coefficient), if necessary
• 6. Location of optimal region
Simple software package CHEMOFACE
 Some interesting outputs from Chemoface
• Pareto charts to indicate the significance of the input factors
 More outputs…
• Response surface
Before starting with designs
The road to multivariate statistics