ISO/TR 13195:2015
(Main)Selected illustrations of response surface method - Central composite design
Selected illustrations of response surface method - Central composite design
ISO/TR 13195:2015 describes the steps necessary to understand the scope of Response Surface Methodology (RSM) and the method to analyse data collected using Central Composite Designs (CCD) through illustration with four distinct applications of this methodology. Response surface methodology (RSM) is used in order to investigate a relation between the response and the set of quantitative predictor variables or factors. Especially after specifying the vital few controllable factors, RSM is used in order to find the factor setting which optimizes the response.
Illustrations choisies de méthodologie à surface de réponse — Plans composites centrés
General Information
Overview
ISO/TR 13195:2015 - "Selected illustrations of response surface method - Central composite design" - is a Technical Report that explains how to plan, conduct and analyse Response Surface Methodology (RSM) experiments using Central Composite Designs (CCD). The report provides a generic description of CCDs and illustrates the methodology with four detailed, real-world examples across agriculture, consumer products, semiconductors and chemical synthesis. It is intended to help practitioners identify, model and optimize quantitative factor settings to improve a chosen response.
Key topics and technical requirements
- Scope and purpose: Steps to understand RSM and to analyse data collected with CCDs; finding factor settings that optimize the response.
- Definitions and symbols: Standardized terms (experiment, response variable, predictor variable, model, factor, run, residual error) consistent with ISO practice.
- Structure of a CCD:
- Factorial runs (coded ±1)
- Star (axial) runs (coded ±α) to estimate quadratic terms
- Centre runs (coded 0) for pure error estimation and model checking
- Coding of factor levels (natural → coded) and software coding alternatives; note on face‑centred CCD (levels −1, 0, +1) and rotatability considerations
- Design planning: identification of response(s), selection of controllable factors and levels, measurement-system considerations, randomization and layout planning.
- Analysis and presentation: numerical summaries, graphical displays, model-fitting diagnostics and confirmation runs to validate optimal settings.
- Supporting material: four informative annexes with worked examples and an annex with background on response surface designs.
Practical applications and target users
ISO/TR 13195:2015 is useful for:
- Statisticians and data scientists working on experimental design and modelling
- Process and quality engineers in manufacturing, semiconductor fabrication and chemical processes
- R&D scientists in agriculture, consumer products and catalysts development
- Six Sigma and continuous improvement practitioners using DOE/RSM to optimize product and process performance
Typical applications include process optimization, product formulation, yield improvement and sensitivity analysis where a small set of quantitative factors governs a measurable outcome.
Related standards (if applicable)
- ISO 3534-3:2013 - statistical vocabulary and related definitions referenced throughout the report
- Work produced under ISO/TC 69 (Applications of statistical methods), Subcommittee SC 7 (Six Sigma and statistical techniques)
Keywords: ISO/TR 13195:2015, Response Surface Methodology, RSM, Central Composite Design, CCD, design of experiments, DOE, factorial runs, star points, centre runs, experiment optimization.
Standards Content (Sample)
TECHNICAL ISO/TR
REPORT 13195
First edition
Selected illustrations of response
surface method — Central composite
design
Illustrations choisies de méthodologie à surface de réponse — Plans
composites centrés
PROOF/ÉPREUVE
Reference number
©
ISO 2015
© ISO 2015, Published in Switzerland
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ii © ISO 2015 – All rights reserved
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Terms and definitions . 1
3 Symbols and abbreviated terms . 6
3.1 Symbols . 6
3.2 Abbreviated terms . 6
4 Generic descriptions of central composite designs . 7
4.1 Overview of the structure of the examples in Annexes A to D . 7
4.2 Overall objective(s) of a response surface experiment . 7
4.3 Description of the response variable(s) . 8
4.4 Identification of measurement systems . 8
4.5 Identification of factors affecting the response(s) . 8
4.6 Selection of levels for each factor . 8
4.6.1 Factorial runs . 9
4.6.2 Star runs . 9
4.6.3 Centre run . 9
4.7 Layout plan of the CCD with randomization principle .10
4.8 Analyse the results — Numerical summaries and graphical displays .10
4.9 Present the results .11
4.10 Perform confirmation run .12
5 Description of Annexes A through D .12
5.1 Comparing and contrasting the examples .12
5.2 Experiment summaries .13
Annex A (informative) Effects of fertilizer ingredients on the yield of a crop .14
Annex B (informative) Optimization of the button tactility using central composite design .29
Annex C (informative) Semiconductor die deposition process optimization .42
Annex D (informative) Process yield-optimization of a palladium-copper catalysed C-C-
bond formation .53
Annex E (informative) Background on response surface designs .71
Bibliography .82
Foreword
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bodies (ISO member bodies). The work of preparing International Standards is normally carried out
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described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www.iso.org/directives).
Attention is drawn to the possibility that some of the elements of this document may be the subject of
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Barriers to Trade (TBT) see the following URL: Foreword - Supplementary information
The committee responsible for this document is ISO/TC 69, Applications of statistical methods,
Subcommittee SC 7, Applications of statistical and related techniques for the implementation of Six Sigma.
iv PROOF/ÉPREUVE © ISO 2015 – All rights reserved
Introduction
The present Technical Report takes one specific statistical tool (Central Composite Designs in Response
Surface Methodology) and develops the topic somewhat generically (in the spirit of International
Standards) but then illustrates it through the use of four detailed and distinct applications. The generic
description focuses on the Central Composite Designs.
The annexes containing the four illustrations follow the basic framework but also identify the nuances
and peculiarities in the specific applications. Each example offers at least one “wrinkle” to the problem,
which is generally the case for real applications. It is hoped that practitioners can identify with at least
one of the four examples, if only to remind them of the basic material on response surface method that
was encountered during their training.
Each of the four examples is developed and analysed using statistical software of current vintage. The
explanations throughout are devoid of mathematical detail—such material can be readily obtained from
the many design and analysis of experiments textbooks (such as those given in References [1] to [7]).
TECHNICAL REPORT ISO/TR 13195:2015(E)
Selected illustrations of response surface method —
Central composite design
1 Scope
This Technical Report describes the steps necessary to understand the scope of Response Surface
Methodology (RSM) and the method to analyse data collected using Central Composite Designs (CCD)
through illustration with four distinct applications of this methodology.
Response surface methodology (RSM) is used in order to investigate a relation between the response
and the set of quantitative predictor variables or factors. Especially after specifying the vital few
controllable factors, RSM is used in order to find the factor setting which optimizes the response.
2 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
2.1
experiment
purposive investigation of a system through selective adjustment of controllable conditions and
allocation of resources
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.1. (The notes are not reproduced here.)
2.2
response variable
variable representing the outcome of an experiment (2.1)
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.3. (Except for NOTE 3 the notes are not
reproduced here.)
Note 2 to entry: A common synonym is “output variable”.
Note 3 to entry: The response variable is likely to be influenced by one or more predictor variables (2.3), the
nature of which can be useful in controlling or optimizing the response variable.
2.3
predictor variable
variable that can contribute to the explanation of the outcome of an experiment (2.1)
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.4. (The notes are not reproduced here.)
Note 2 to entry: Natural predictor variables are expressed in natural units of measurement such as degrees
Celcius (°C) or grams per liter, for example. In RSM work, it is convenient to transform the natural variables to
coded variables which are dimensionless variables, symmetric around zero and all with the same spread.
2.4
model
formalized representation of outcomes of an experiment (2.1)
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.2. (The notes and examples are not reproduced
here except for NOTE 2 which is NOTE 1 in ISO 3534-3.)
Note 2 to entry: The model consists of three parts. The first part is the response variable (2.2) that is being
modelled. The second part is the deterministic or the systematic part of the model that includes predictor
variable(s) (2.3). Finally, the third part is the residual error (2.12) that can involve pure random error (2.13)
and misspecification error (2.14). The model applies for the experiment as a whole and for separate outcomes
denoted with subscripts. The model is a mathematical description that relates the response variable to predictor
variables and includes associated assumptions. Outcomes refer to recorded or measured observations of the
response variable.
Note 3 to entry: In some areas the term transfer function is used for the systematic part of the model.
EXAMPLE In the models considered in response surface methodology the deterministic or systematic part
are polynomials in the predictor variables. A second order model with two predictor variables is written as
2 2
yx =+ββ ++ββxx xx++ββ x +ε
0111 22 12 12 11 22 2
where ε is the random error. The associated assumptions on the random error could be either that individual
random errors are uncorrelated with constant variance or independent and normally distributed. The
deterministic part of the model is the second degree polynomial in the predictor variables x and x
1 2
2 2
E yx=+ ββ ++ββxx xx++ββ x
0111 22 12 12 11 22 2
which explains the mean (Ey) of the response variable as a function of the predictor variables.
2.5
factor
feature under examination as a potential cause of variation
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.5. (The notes are not reproduced here.)
Note 2 to entry: Generally the symbol k is used to indicate the number of factors in the experiment.
2.6
factor level
setting, value or assignment of a factor (2.5)
Note 1 to entry: Adapted from ISO 3534-3:2013, definition (3.1.12). (The notes are not reproduced here.)
2.7
coding of factor levels
one-to-one relabelling of factor levels
Note 1 to entry: The coding of factor levels facilitates the identification of the design and the properties of the design.
Note 2 to entry: In response surface experiments the actual (or natural or operational) levels are relabelled such
that the coded levels are numeric and symmetric around 0.
Note 3 to entry: A two-level factor is usually coded to have coded levels −1 and +1. A factorial design where all
factors are two-level factors can be coded such that all runs are represented as factorial runs (2.9).
Note 4 to entry: In central composite designs numeric (or continuous) factors with five levels are considered,
except for the face-centred central composite deigns, where only three levels are needed, see note 6 to 2.7. If the
actual (or natural or operational) levels are l < l < l < l < l then the middle level l shall be the average of the
1 2 3 4 5 3
lowest level l and the highest level l , and, furthermore, l shall be the average of the intermediate levels l and l .
1 5 3 2 4
The form of the coding operation can be expressed as
actual value− l
coded value=
C
where C is half the distance from l to l . With this coding of the factors each run (2.8) of a central composite
2 4
design can be identified as either a factorial point (2.9), a centre point (2.10), or an star point (2.11). This is the
coding used in textbooks for discussing central composite designs.
2 PROOF/ÉPREUVE © ISO 2015 – All rights reserved
Note 5 to entry: An alternative coding is sometimes applied in the computations in software programs. The form
of the coding operation can be expressed as
actual value−l
coded value=
M
where M is half the distance from the lowest level l to the highest level l . This coding will be referred to as
1 5
software coding in this Technical Report.
Note 6 to entry: In the face-centred CCD, only three levels of each factor are needed, so l = l < l < l = l , and l
1 2 3 4 5 3
shall be the average of the lowest level l and the highest level l . This design could be of interest if it is difficult to
1 5
select five levels of the factors. For the face-centred CCD, the possible coded values of a factor are only −1, 0, 1.The
face-centred CCD is not rotatable, see 2.18.
Note 7 to entry: A class of designs that can be used to fit second order models and only require three equidistant
levels of each factor are Box-Behnken designs. Box-Behnken designs are not central composite designs and are
therefore not treated in this Technical Report. But they may be a useful alternative, if only three equidistant
levels of each factor can be used, see References [5], [2] and [7].
2.8
run
experimental treatment
specific settings of every factor (2.5) used on a particular experimental unit
(2.15)
Note 1 to entry: Ultimately, the impact of the factors will be captured through their representation in the
predictor variables (2.3) and the extent to which the model matches the outcome of the experiment (2.1).
EXAMPLE Consider a chemical process experiment (2.1) in which a high yield is the objective and the
predictor variables are temperature, duration, and concentration of a catalyst. A run could be a setting of
temperature of 350 °C, 30 min duration and 10 % concentration of the catalyst, assuming that all of these settings
are possible and permissible.
Note 2 to entry: Adapted from ISO 3534-3:2013, definition 3.1.13.
2.9
factorial point
factorial run
cube point
cube run
vector of factor level settings of the form (a , a , ., a ), where each a equals −1 or +1 as a notation for
1 2 k i
the coded levels of the factors
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.37. (The notes are not reproduced here.)
2.10
centre point
centre run
vector of factor level settings of the form (a , a , ., a ), where all a equal 0, as notation for the coded
1 2 k i
levels of the factors
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.38. (The notes are not reproduced here.)
2.11
star point
axial point
star run
axial run
vector of factor level (2.7) settings of the form (a , a , …, a ), where one a equals α or −α and the other
1 2 k i
a ’s equal +α, as notation for the coded levels of the factors (2.6)
i
Note 1 to entry: For a k factor experiment, this process yields 2 -star points of the form: (±α, 0, …, 0), (0, ±α, 0, …,
k
0), …, (0, 0, …, ±α).
Note 2 to entry: Star points are added to the design in order to estimate a quadratic response surface.
Note 3 to entry: Special values of α give a nice geometric structure. For a k factor experiment, if α = k then the
factorial points and the star points are all on the sphere with radius k . This design is therefore called a
spherical CCD. If α = 1, the star points are on the faces of the unit cube and the design is a face-centred CCD.
2.12
residual error
error term
random variable representing the difference between the response variable (2.3) and its prediction
based on an assumed model (2.4)
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.6. (The notes are not reproduced here.)
2.13
pure random error
pure error
part of the residual error (2.12) associated with replicated observations
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.9. (The notes are not reproduced here.)
2.14
misspecification error
part of the residual error (2.12) not accounted for by pure random error (2.13)
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.9. (The notes are not reproduced here.)
2.15
experimental unit
basic unit of the experimental material
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.24. (The notes are not reproduced here.)
2.16
designed experiment
experiment (2.1) with an explicit objective and structure of implementation
Note 1 to entry: The purpose of a properly designed experiment is to provide the most efficient and economical
method of reaching valid and relevant conclusions from the experiment.
Note 2 to entry: Associated with a designed experiment is an experimental design (2.17) that includes the response
variable (2.2) or variables and the experimental treatments (2.8) with prescribed factor levels (2.6). A class of
models that relates the response variable to the predictor variables could also be envisaged.
Note 3 to entry: Adapted from ISO 3534-3:2013, definition 3.1.27.
2.17
experimental design
assignment of experimental treatments (2.7) to each experimental unit (2.15)
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.28. (The notes are not reproduced here.)
4 PROOF/ÉPREUVE © ISO 2015 – All rights reserved
2.18
rotatability
characteristic of a designed experiment (2.16) for which the response variable (2.2) that is predicted
from a fitted model (2.4) has the same variance at all equal distances from the centre of the design
Note 1 to entry: A design is rotatable if the variance of the predicted response at any point x depends only on
the distance of x from the centre point (2.10). A design with this property can be rotated around its centre point
without changing the prediction variance at x.
Note 2 to entry: Rotatability is a desirable property for response surface designs (2.25).
Note 3 to entry: Rotatability of a central composite design is obtained setting α equal to the fourth root of the
number of factorial points, i.e
14/
α = ()n
F
where n denotes the number of factorial points in a CCD.
F
Note 4 to entry: The definition and notes 1 and 2 are adapted from ISO 3534-3:2013, definition 3.1.40.
2.19
interaction
influence of one factor (2.6) on one or more other factors’ impact on the response variable (2.2)
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.17. (The notes are not reproduced here.)
2.20
factorial experiment
designed experiment (2.16) with one or more factors (2.5) and with at least two levels applied for one
of the factors
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.2.1. (The notes are not reproduced here.)
2.21
full factorial experiment
factorial experiment (2.12) consisting of all possible combinations of the levels of the factors (2.6)
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.2.2. (The notes are not reproduced here.)
2.22
fractional factorial experiment
factorial experiment (2.12) consisting of a subset of the full factorial experiment (2.21)
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.2.3. (The notes are not reproduced here.)
2.23
randomization
process used to assign treatments to experimental units so that each experimental unit has an equal
chance of being assigned a particular treatment
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.26. (The notes are not reproduced here.)
2.24
replication
performance of an experiment more than once for a given set of predictor variables
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.35. (The notes are not reproduced here.)
2.25
response surface design
designed experiment (2.16) that identifies a subset of factors (2.5) to be optimized
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.2.19. (The notes are not reproduced here.)
2.26
analysis of variance
ANOVA
technique which subdivides the total variation of a response variable (2.3) into components associated
with defined sources of variation
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.3.8. (The notes are not reproduced here.)
3 Symbols and abbreviated terms
3.1 Symbols
y Response variable
Predicted response variable
ˆy
Predicted response variable at the stationary point
ˆ
y
S
x Stationary point of fitted response surface
S
D Distance of stationary point to the design centre
S
A, B, C, D Factors
k Number of factors
k
2 Number of runs in a full factorial experiment with k factors all having two levels
k−p −p
2 Number of runs in a fractional factorial experiment with k factors and fraction 2
n Number of factorial points in a CCD
F
n Number of star points in a CCD
S
n Number of centre points in a CCD
a , b , l Levels of factors
i i i
+1, −1 High and low coded factorial levels
−α, α Axial levels of coded factors
σ Standard deviation
3.2 Abbreviated terms
ANOVA analysis of variance
CCD central composite design
DOE design of experiments
6 PROOF/ÉPREUVE © ISO 2015 – All rights reserved
RSM response surface methodology
R&R repeatability and reproducibility
4 Generic descriptions of central composite designs
4.1 Overview of the structure of the examples in Annexes A to D
This Technical Report provides general guidelines on the design, conduct and analysis of central
composite designs consisting of a specified number of two-level factors, and illustrates the steps with
four distinct applications given in the annexes. Each of the four examples in Annexes A through D
follows the basic structure as given in Table 1.
Table 1 — Basic steps in CCD design
1 Overall objective(s) of experiment
2 Description of the response variable(s)
3 Identification of factors affecting the response(s)
4 Selection of levels for each factor
5 Identification of measurement systems
Layout plan of the CCD (depending upon which main effects and two factor
6 interactions are to be studied) with “randomization” principle (if these
are physical runs)
7 Analyse the results – numerical summaries and graphical displays
8 Present the results
9 Perform confirmation run
4.2 Overall objective(s) of a response surface experiment
Experiments may be conducted for a variety of reasons. Therefore, the primary objective(s) for the
experiment should be clearly stated and agreed to by all parties involved in the design, conduct, analysis
and implications of the experimental effort.
The main goal of response surface experiments is to create a model of the relationship between the
factors and the response in order to explore optimum operating conditions. This involves choosing a
design which allows the fitting of a quadratic function as the systematic part of the model. The Central
Composite Design (CCD) can achieve this and this design has been popular since its introduction in the
[1]
first paper on response surface methods in 1951.
Although the fundamental method for fitting first order (linear) or second order (quadratic) function
of the predictor variables to the response is regression, the focus is not on the individual regression
coefficients but on the regression function, the response surface, as a whole. This emphasis is reflected
in the name Response Surface Methodology. Strong arguments in favour of this approach are given on
pages 508-509 of Reference [2].
Typically, the primary goal for the experiment is to find optimal operating conditions based on the
estimated response surface, this could involve doing several experiments, using the results of one
experiment to provide direction for what to do next. This next action could be to focus the experiment
around a different set of conditions, or to collect more data in the current experimental region in order
to fit a higher-order model or confirm what seemed to be the conclusion.
The CCD is an appropriate name because three types of design points can be identified after a coding
of the factor levels: centre points (2.10), factorial points (2.9) and star points (2.11), and those design
points are indeed centred at the origin of the design space after the coding of the factor levels (2.7).
Response surface experiments traditionally involve a small number of continuous factors. Some
software packages have an upper limit of 8 factors. Response surface experiments are typically used
when the investigators already know which factors are important. One way to obtain this knowledge
is to apply a screening experiment, for example a fractional factorial experiment as explained in
[11]
ISO/TR 12845.
4.3 Description of the response variable(s)
Associated with the objective of an experiment is a continuous outcome or performance measure. A
response of interest could involve maximization (larger is better), minimization (smaller is better) or
meet a target value (be close to a specified value), but, in all cases, that task is one of optimization.
The response variable (denoted by the variable y) should be closely related to the objective of the
experiment. For some situations, there are more than one variable of interest to be considered, although,
typically, only a primary response variable will be associated with the experiment. In other cases,
multiple responses should be considered. In case of multiple responses, the approach taken in response
surface methodology is to analyse and optimize each response separately. The fitted response surfaces
will then be studied to find settings that meet the requirements of all the responses. The example in
Annex C has three responses.
4.4 Identification of measurement systems
Assessment of repeatability and reproducibility of the measurement systems for factors and responses
should be done prior to designing the experiment.
4.5 Identification of factors affecting the response(s)
Response surface experiments are usually not done in isolation. They rely on prior knowledge
concerning important influential variables on the selected response. If this knowledge is not available,
it is necessary to conduct a different type of experiment to identify the factors affecting the response.
During the final selection of factors, attention shall be paid to the ability to set the levels of each
individual factor independently of the other factors.
4.6 Selection of levels for each factor
There are two aspects to the selection of factors. One is selecting the experimental region which is the
multidimensional range of interest for the factors selected. The other is the exact selection of the factor
levels in such a way that the design has desirable properties. The first one requires subject matter
knowledge as to the impact of factors on the response. The second one is more straightforward once the
factors and the type of design to be used are known. The second one is further discussed in this Clause.
The response surface methods considered in this Technical Report are about the second order centre
k k−p
models using the CCD. The CCD is an augmentation of 2 factorial experiments (or 2 fractional
factorial experiments). In addition to the two factorial levels that are used in the (fractional) factorial
experiments, the user selects three additional levels, one centre level which is the average of the two
factorial levels, and two extreme levels which are chosen symmetrically around the centre level and
typically outside the range of the two factorial levels.
When the experimenter selects the levels of each factor he will be thinking in terms of the operational
levels of a factor, the exact setting of a temperature, for example. But when studying the properties of
the design and also when analysing the data from the design coded levels of the design are used.
8 PROOF/ÉPREUVE © ISO 2015 – All rights reserved
If the actual (or operational) levels are l < l < l < l < l then the middle level l shall be the average of
1 2 3 4 5 3
the lowest level l and the highest level l , and, furthermore, l shall be the average of the intermediate
1 5 3
levels l and l . The form of the coding operation can be expressed as Formula (1):
2 4
actual value−l
coded value= (1)
C
where C is half the distance from l to l . The coded value of the upper extreme level, l , will be denoted
2 4 5
by α, and the coded value of the lower extreme level, l , will be denoted by −α. It is very important to
note that the value of α is the same for all the factors of the design. Thus, the coded levels of all the
factors are (−α, −1, 0, 1, + α).
An alternative coding is sometimes applied in the computations in software programs. The form of the
coding operation can be expressed as Formula (2):
original value−l
coded value= (2)
M
where M is half the distance from the lowest level l to the highest level l .or, equivalently, the distance
1 5
from l to l This coding will be referred to as software coding in this Technical Report.
3 5.
When the levels of the factors have been chosen, the levels of the individual factors have to be
combined to define the runs of the experiment. A CCD has three types of experimental runs: factorial,
centre and axial ones.
4.6.1 Factorial runs
A factorial run is a setting of all k factors to coded levels either −1 or +1. The factorial runs are the runs
k−p k
used in 2 fractional factorial experiments or 2 factorial experiments.
Written as a k-dimensional vector in coded levels, the factorial run has the form (±1, ±1,…, ±1,…, ±1).
Considered as points in k-dimensional space, the factorial runs are the vertices of a cube and the
factorial runs are for this reason also called cube points.
k
There are 2 different factorial runs with k factors.
4.6.2 Star runs
The star runs are those where one of the factors has its coded levels either − α or α and the remaining
factors are at their coded level 0.
Written as a k-dimensional vector in coded levels, the star run has the form (0, 0,…, ±α,…, 0), having −α
th
or α on the i position and 0 on all other positions.
Viewed as points in k-dimensional space, the star runs are located on the coordinate axes, and for this
reason, the star runs are also called axial runs.
k
There are 2 different star runs with k factors.
4.6.3 Centre run
The centre run is the one where all the factors are on their coded level 0. Written as a k-dimensional
vector, it is the point (0, 0,…, 0).
There is only one centre run but the centre run may be replicated in a CCD. One reason for replicating
the centre point is to get an estimate of pure error which can be used to check the fit of the model.
4.7 Layout plan of the CCD with randomization principle
In a report, the full description of the design and the observed responses should be given. In addition to
reporting the levels of the factors, as described in 4.6, this includes reporting the following:
— the number of replications of the three types of design points;
— the number of blocks;
— the randomization.
If the design has only a few factors, a table where each run is represented as a row is useful for this
purpose. It is easier to grasp the design if coded levels are used. The randomization can be explained
by including a column giving the order in which the runs have been performed. If blocking is applied, a
column can similarly be added to explain the allocation of runs to blocks.
As an illustration, Table 3 provides a basic layout of a CCD for 2 factors with full factorial design in serial
order. While there is a standard order for factorial designs there is no such thing for a CCD, so the term
serial order is used for a way to write the runs in a CCD in such a way that the design is easily recognized.
The serial order used in Table 3 and throughout this Technical Report lists the factorial runs in standard
order, first, followed by the star runs and, finally, the centre runs. Each row of the table represents one
set of experimental conditions that when run will produce a value of the response variable y. The two
factors are designated as A and B. In this case, only one response variable is shown, but more columns
should be added if more than one response variable is studied. In this case, it is readily seen that the
design has five centre runs replicated. The last column shows the run order, i.e. the order in which the
experimental conditions have been applied. This experiment has been performed in one block.
Table 3 — Layout of a generic Central Composite Design
Serial order A B y Run order
1 −1 −1 y 6
2 1 −1 y 4
3 −1 1 y 13
4 1 1 y 7
5 −1,41 0 y 12
6 1,41 0 y 8
7 0 −1,41 y 9
8 0 1,41 y 5
9 0 0 y 2
10 0 0 y 10
11 0 0 y 1
12 0 0 y 11
13 0 0 y 3
4.8 Analyse the results — Numerical summaries and graphical displays
The second order model that can be fitted with the data from a CCD is a regression model and the
usual output and graphs from a regression analysis are reported. This includes the table of estimated
regression coefficients and associated t-statistics and p-values. A special feature of RSM regression is
ANOVA tables that exploit the structure of the model.
For the design in Table 3, one such ANOVA table is shown in Table 4.
10 PROOF/ÉPREUVE © ISO 2015 – All rights reserved
Table 4 — ANOVA table for Central Composite Design
Response: y
Df Sum Sq Mean Sq F-value Pr(>F)
FO(x1, x2) 2 ddd,dd ddd,dd ddd,dd d,dd
TWI(x1, x2) 1 ddd,dd ddd,dd ddd,dd d,dd
PQ(x1, x2) 2 ddd,dd ddd,dd ddd,dd d,dd
Residuals 7 ddd,dd ddd,dd
Lack of fit 3 ddd,dd ddd,dd ddd,dd d,dd
Pure error 4 ddd,dd ddd,dd
Table 4 is given by most, if not all, statistical packages for RSM analysis. The names of the rows and the
columns may differ between statistical packages, but they are easily understood. Here, FO, TWI and PQ
are short for First Order, Two Way Interaction and Pure Quadratic. Examples are given in Tables A.5,
B.6 to B.7, and D.5.
In this case, there is no data, so the table is filled in with ddd,dd or d,dd except for the degrees of
freedom which are the correct ones for the design in Table 3. The four F-tests are of interest. Read from
the bottom of the table, the first one is the ratio of the mean squares for Lack of fit and Pure error and is
a check of the fit of the model. The second F-test is in the PQ line and it tests the need for pure quadratic
terms in the model; in this case the hypothesis that β = β = 0. It is the ratio of the mean square in the
11 22
PQ line to the mean square in the Residuals line of the table.
Another ANOVA table of potential interest tests for each predictor variable, the hypotheses that it is not
needed in the model. For the example with two predictor variables in Table 3, this ANOVA table has two
F-tests: One for the hypothesis that β = β = β = 0, and one for the hypothesis that β = β = β = 0.
1 12 11 2 12 22
Note that β , β , and β , for example, are the coefficients of all the terms in the model that involve x .
1 12 11 1
Examples can be found in Tables B.5 and C.7.
When looking at analyses of response surface experiments using different software, it is possible to
identify two different approaches. One approach uses all the tools of regression analysis to find a model
where only significant terms are included in the estimated systematic part of the model. The other
approach focuses on the response surface and only tests the hypotheses described in the two clauses
above, i.e. tests whether a linear model can be used instead of a quadratic model or tests whether a
factor can be considered to be essentially inert. In this Technical Report, the second approach is taken
and therefore individual non-significant terms are not removed from the model. Strong arguments in
favour of the second approach are given on pages 508-509 in Reference [2].
4.9 Present the results
Of course, one presentation of the result of the analysis is to give the estimated systematic part of
the model. Although the systematic part of the model is indispensable for calculating the predicted
response for various settings, it is not very useful for understanding the nature of the response surface.
In Annex E, four second order polynomials in two variables are given in Formulae (E.3), (E.4), (E.5)
and (E.6). The four polynomials look very similar. They have the same distribution of signs among the
coefficients and the coefficients are very similar. But the perspective and contour plots in Figures E.1 to
E.4 show that the polynomials represent four very different response surfaces.
For two and three predictor variables, it is possible to get an understanding of the response surface
from contour plots, but a more formal analysis called canonical analysis is always useful, because it
gives a precise characterization of the surface that gives an understanding of the response surface even
when the number of predictor variables is larger than 3.
Canonical analysis is a method of rewriting a fitted quadratic function of the predictor variables in a
form which can be more easily understood. This is achieved by a rotation of the coordinate axes which
removes all cross-product terms. This may be followed by a change of origin to remove first order terms
as well. Details are given in E.3.
Software packages for response surface methodology report the result of the canonical analysis. The
result has the following parts:
— stationary point and predicted value at stationary point of the response surface;
— eigenvalues;
— eigenvectors.
The stationary point is the point where the fitted response surface has a maximum, a minimum or a
saddle point, and the eigenvalues tell exactly which one is the case. Associated with each eigenvalue
is an eigenvector. Moving away from the stationary point in the positive or negative direction of the
eigenvector the response will decrease if the eigenvalue is negative, and it will increase if the eigenvalue
is positive, so
— if all the eigenvalues are negative, the stationary point is a maximum,
— if all the eigenvalues are positive, the stationary point is a minimum, and
— if some eigenvalues are negative and some are positive, the stationary point is a saddle point or a
minimax.
The situation is further complicated by the fact that one can only hope to approximate the response by
the fitted second order model in the experimental region, so it should also be considered whether the
stationary point is inside or outside the experimental region.
The situation is only simple, if the object is to maximize the response and if the stationary point is
a maximum located inside the experimental region, or, alternatively, if the object is to minimize the
response and if the stationary point is a minimum located inside the experimental region, and then
the optimal settings is the stationary point. In all other cases, the optimal settings need to be found
towards the boundary of the experimental region. The proprietary software packages provide a variety
of tool for this purpose.
4.10 Perform confirmation run
After determining the optimal combination of levels of the factors and predicting the value of response
variable for these factor levels, it is recommended that the user performs confirmation runs at the
chosen settting to check whether the new observations confirm the predictions of the experiment.
If the computer output gives the standard deviation (std.dev) of a single observation and the standard
error (std.err) of the predicted value at the optimum, then the following formulas apply for
a) a 95% confidence interval for the mean at the optimum: ˆyt s±⋅()v td.err and
0,975
ˆ
b) a 95% prediction interval for a new observation at the optimum: yt±⋅()v std.devs+ td.err .
0,975
Here, v denotes the degrees of freedom for error.
The 95 % prediction interval is particularly useful for it explains what can be expected from future
observations; if the model is good, 95 % of the observations will fall inside the 95 % prediction interval.
5 Description of Annexes A through D
5.1 Comparing and contrasting the examples
Four distinct examples of response surface designs are illustrated in Annexes A to D. Each of these
...
TECHNICAL ISO/TR
REPORT 13195
First edition
2015-12-15
Selected illustrations of response
surface method — Central composite
design
Illustrations choisies de méthodologie à surface de réponse — Plans
composites centrés
Reference number
©
ISO 2015
© ISO 2015, Published in Switzerland
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ii © ISO 2015 – All rights reserved
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Terms and definitions . 1
3 Symbols and abbreviated terms . 6
3.1 Symbols . 6
3.2 Abbreviated terms . 6
4 Generic descriptions of central composite designs . 7
4.1 Overview of the structure of the examples in Annexes A to D . 7
4.2 Overall objective(s) of a response surface experiment . 7
4.3 Description of the response variable(s) . 8
4.4 Identification of measurement systems . 8
4.5 Identification of factors affecting the response(s) . 8
4.6 Selection of levels for each factor . 8
4.6.1 Factorial runs . 9
4.6.2 Star runs . 9
4.6.3 Centre run . 9
4.7 Layout plan of the CCD with randomization principle .10
4.8 Analyse the results — Numerical summaries and graphical displays .10
4.9 Present the results .11
4.10 Perform confirmation run .12
5 Description of Annexes A through D .12
5.1 Comparing and contrasting the examples .12
5.2 Experiment summaries .13
Annex A (informative) Effects of fertilizer ingredients on the yield of a crop .14
Annex B (informative) Optimization of the button tactility using central composite design .28
Annex C (informative) Semiconductor die deposition process optimization .41
Annex D (informative) Process yield-optimization of a palladium-copper catalysed C-C-
bond formation .52
Annex E (informative) Background on response surface designs .70
Bibliography .80
Foreword
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bodies (ISO member bodies). The work of preparing International Standards is normally carried out
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The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for the
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editorial rules of the ISO/IEC Directives, Part 2 (see www.iso.org/directives).
Attention is drawn to the possibility that some of the elements of this document may be the subject of
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The committee responsible for this document is ISO/TC 69, Applications of statistical methods,
Subcommittee SC 7, Applications of statistical and related techniques for the implementation of Six Sigma.
iv © ISO 2015 – All rights reserved
Introduction
The present Technical Report takes one specific statistical tool (Central Composite Designs in Response
Surface Methodology) and develops the topic somewhat generically (in the spirit of International
Standards) but then illustrates it through the use of four detailed and distinct applications. The generic
description focuses on the Central Composite Designs.
The annexes containing the four illustrations follow the basic framework but also identify the nuances
and peculiarities in the specific applications. Each example offers at least one “wrinkle” to the problem,
which is generally the case for real applications. It is hoped that practitioners can identify with at least
one of the four examples, if only to remind them of the basic material on response surface method that
was encountered during their training.
Each of the four examples is developed and analysed using statistical software of current vintage. The
explanations throughout are devoid of mathematical detail—such material can be readily obtained from
the many design and analysis of experiments textbooks (such as those given in References [1] to [7]).
TECHNICAL REPORT ISO/TR 13195:2015(E)
Selected illustrations of response surface method —
Central composite design
1 Scope
This Technical Report describes the steps necessary to understand the scope of Response Surface
Methodology (RSM) and the method to analyse data collected using Central Composite Designs (CCD)
through illustration with four distinct applications of this methodology.
Response surface methodology (RSM) is used in order to investigate a relation between the response
and the set of quantitative predictor variables or factors. Especially after specifying the vital few
controllable factors, RSM is used in order to find the factor setting which optimizes the response.
2 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
2.1
experiment
purposive investigation of a system through selective adjustment of controllable conditions and
allocation of resources
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.1. (The notes are not reproduced here.)
2.2
response variable
variable representing the outcome of an experiment (2.1)
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.3. (Except for NOTE 3 the notes are not
reproduced here.)
Note 2 to entry: A common synonym is “output variable”.
Note 3 to entry: The response variable is likely to be influenced by one or more predictor variables (2.3), the
nature of which can be useful in controlling or optimizing the response variable.
2.3
predictor variable
variable that can contribute to the explanation of the outcome of an experiment (2.1)
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.4. (The notes are not reproduced here.)
Note 2 to entry: Natural predictor variables are expressed in natural units of measurement such as degrees
Celcius (°C) or grams per liter, for example. In RSM work, it is convenient to transform the natural variables to
coded variables which are dimensionless variables, symmetric around zero and all with the same spread.
2.4
model
formalized representation of outcomes of an experiment (2.1)
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.2. (The notes and examples are not reproduced
here except for NOTE 2 which is NOTE 1 in ISO 3534-3.)
Note 2 to entry: The model consists of three parts. The first part is the response variable (2.2) that is being
modelled. The second part is the deterministic or the systematic part of the model that includes predictor
variable(s) (2.3). Finally, the third part is the residual error (2.12) that can involve pure random error (2.13)
and misspecification error (2.14). The model applies for the experiment as a whole and for separate outcomes
denoted with subscripts. The model is a mathematical description that relates the response variable to predictor
variables and includes associated assumptions. Outcomes refer to recorded or measured observations of the
response variable.
Note 3 to entry: In some areas the term transfer function is used for the systematic part of the model.
EXAMPLE In the models considered in response surface methodology the deterministic or systematic part
are polynomials in the predictor variables. A second order model with two predictor variables is written as
2 2
yx =+ββ ++ββxx xx++ββ x +ε
0111 22 12 12 11 22 2
where ε is the random error. The associated assumptions on the random error could be either that individual
random errors are uncorrelated with constant variance or independent and normally distributed. The
deterministic part of the model is the second degree polynomial in the predictor variables x and x
1 2
2 2
E yx=+ ββ ++ββxx xx++ββ x
0111 22 12 12 11 22 2
which explains the mean (Ey) of the response variable as a function of the predictor variables.
2.5
factor
feature under examination as a potential cause of variation
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.5. (The notes are not reproduced here.)
Note 2 to entry: Generally the symbol k is used to indicate the number of factors in the experiment.
2.6
factor level
setting, value or assignment of a factor (2.5)
Note 1 to entry: Adapted from ISO 3534-3:2013, definition (3.1.12). (The notes are not reproduced here.)
2.7
coding of factor levels
one-to-one relabelling of factor levels
Note 1 to entry: The coding of factor levels facilitates the identification of the design and the properties of the design.
Note 2 to entry: In response surface experiments the actual (or natural or operational) levels are relabelled such
that the coded levels are numeric and symmetric around 0.
Note 3 to entry: A two-level factor is usually coded to have coded levels −1 and +1. A factorial design where all
factors are two-level factors can be coded such that all runs are represented as factorial runs (2.9).
Note 4 to entry: In central composite designs numeric (or continuous) factors with five levels are considered,
except for the face-centred central composite deigns, where only three levels are needed, see note 6 to 2.7. If the
actual (or natural or operational) levels are l < l < l < l < l then the middle level l shall be the average of the
1 2 3 4 5 3
lowest level l and the highest level l , and, furthermore, l shall be the average of the intermediate levels l and l .
1 5 3 2 4
The form of the coding operation can be expressed as
actual value− l
coded value=
C
where C is half the distance from l to l . With this coding of the factors each run (2.8) of a central composite
2 4
design can be identified as either a factorial point (2.9), a centre point (2.10), or an star point (2.11). This is the
coding used in textbooks for discussing central composite designs.
2 © ISO 2015 – All rights reserved
Note 5 to entry: An alternative coding is sometimes applied in the computations in software programs. The form
of the coding operation can be expressed as
actual value−l
coded value=
M
where M is half the distance from the lowest level l to the highest level l . This coding will be referred to as
1 5
software coding in this Technical Report.
Note 6 to entry: In the face-centred CCD, only three levels of each factor are needed, so l = l < l < l = l , and l
1 2 3 4 5 3
shall be the average of the lowest level l and the highest level l . This design could be of interest if it is difficult to
1 5
select five levels of the factors. For the face-centred CCD, the possible coded values of a factor are only −1, 0, 1.The
face-centred CCD is not rotatable, see 2.18.
Note 7 to entry: A class of designs that can be used to fit second order models and only require three equidistant
levels of each factor are Box-Behnken designs. Box-Behnken designs are not central composite designs and are
therefore not treated in this Technical Report. But they may be a useful alternative, if only three equidistant
levels of each factor can be used, see References [5], [2] and [7].
2.8
run
experimental treatment
specific settings of every factor (2.5) used on a particular experimental unit
(2.15)
Note 1 to entry: Ultimately, the impact of the factors will be captured through their representation in the
predictor variables (2.3) and the extent to which the model matches the outcome of the experiment (2.1).
EXAMPLE Consider a chemical process experiment (2.1) in which a high yield is the objective and the
predictor variables are temperature, duration, and concentration of a catalyst. A run could be a setting of
temperature of 350 °C, 30 min duration and 10 % concentration of the catalyst, assuming that all of these settings
are possible and permissible.
Note 2 to entry: Adapted from ISO 3534-3:2013, definition 3.1.13.
2.9
factorial point
factorial run
cube point
cube run
vector of factor level settings of the form (a , a , ., a ), where each a equals −1 or +1 as a notation for
1 2 k i
the coded levels of the factors
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.37. (The notes are not reproduced here.)
2.10
centre point
centre run
vector of factor level settings of the form (a , a , ., a ), where all a equal 0, as notation for the coded
1 2 k i
levels of the factors
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.38. (The notes are not reproduced here.)
2.11
star point
axial point
star run
axial run
vector of factor level (2.7) settings of the form (a , a , …, a ), where one a equals α or −α and the other
1 2 k i
a ’s equal 0, as notation for the coded levels of the factors (2.6)
i
Note 1 to entry: For a k factor experiment, this process yields 2k-star points of the form: (±α, 0, …, 0), (0, ±α, 0, …,
0), …, (0, 0, …, ±α).
Note 2 to entry: Star points are added to the design in order to estimate a quadratic response surface.
Note 3 to entry: Special values of α give a nice geometric structure. For a k factor experiment, if α = k then the
factorial points and the star points are all on the sphere with radius k . This design is therefore called a
spherical CCD. If α = 1, the star points are on the faces of the unit cube and the design is a face-centred CCD.
2.12
residual error
error term
random variable representing the difference between the response variable (2.3) and its prediction
based on an assumed model (2.4)
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.6. (The notes are not reproduced here.)
2.13
pure random error
pure error
part of the residual error (2.12) associated with replicated observations
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.9. (The notes are not reproduced here.)
2.14
misspecification error
part of the residual error (2.12) not accounted for by pure random error (2.13)
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.9. (The notes are not reproduced here.)
2.15
experimental unit
basic unit of the experimental material
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.24. (The notes are not reproduced here.)
2.16
designed experiment
experiment (2.1) with an explicit objective and structure of implementation
Note 1 to entry: The purpose of a properly designed experiment is to provide the most efficient and economical
method of reaching valid and relevant conclusions from the experiment.
Note 2 to entry: Associated with a designed experiment is an experimental design (2.17) that includes the response
variable (2.2) or variables and the experimental treatments (2.8) with prescribed factor levels (2.6). A class of
models that relates the response variable to the predictor variables could also be envisaged.
Note 3 to entry: Adapted from ISO 3534-3:2013, definition 3.1.27.
2.17
experimental design
assignment of experimental treatments (2.7) to each experimental unit (2.15)
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.28. (The notes are not reproduced here.)
4 © ISO 2015 – All rights reserved
2.18
rotatability
characteristic of a designed experiment (2.16) for which the response variable (2.2) that is predicted
from a fitted model (2.4) has the same variance at all equal distances from the centre of the design
Note 1 to entry: A design is rotatable if the variance of the predicted response at any point x depends only on
the distance of x from the centre point (2.10). A design with this property can be rotated around its centre point
without changing the prediction variance at x.
Note 2 to entry: Rotatability is a desirable property for response surface designs (2.25).
Note 3 to entry: Rotatability of a central composite design is obtained setting α equal to the fourth root of the
number of factorial points, i.e
14/
α = ()n
F
where n denotes the number of factorial points in a CCD.
F
Note 4 to entry: The definition and notes 1 and 2 are adapted from ISO 3534-3:2013, definition 3.1.40.
2.19
interaction
influence of one factor (2.6) on one or more other factors’ impact on the response variable (2.2)
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.17. (The notes are not reproduced here.)
2.20
factorial experiment
designed experiment (2.16) with one or more factors (2.5) and with at least two levels applied for one
of the factors
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.2.1. (The notes are not reproduced here.)
2.21
full factorial experiment
factorial experiment (2.12) consisting of all possible combinations of the levels of the factors (2.6)
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.2.2. (The notes are not reproduced here.)
2.22
fractional factorial experiment
factorial experiment (2.12) consisting of a subset of the full factorial experiment (2.21)
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.2.3. (The notes are not reproduced here.)
2.23
randomization
process used to assign treatments to experimental units so that each experimental unit has an equal
chance of being assigned a particular treatment
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.26. (The notes are not reproduced here.)
2.24
replication
performance of an experiment more than once for a given set of predictor variables
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.35. (The notes are not reproduced here.)
2.25
response surface design
designed experiment (2.16) that identifies a subset of factors (2.5) to be optimized
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.2.19. (The notes are not reproduced here.)
2.26
analysis of variance
ANOVA
technique which subdivides the total variation of a response variable (2.2) into components associated
with defined sources of variation
Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.3.8. (The notes are not reproduced here.)
3 Symbols and abbreviated terms
3.1 Symbols
y Response variable
Predicted response variable
ˆy
Predicted response variable at the stationary point
ˆ
y
S
x Stationary point of fitted response surface
S
D Distance of stationary point to the design centre
S
A, B, C, D Factors
k Number of factors
k
2 Number of runs in a full factorial experiment with k factors all having two levels
k−p −p
2 Number of runs in a fractional factorial experiment with k factors and fraction 2
n Number of factorial points in a CCD
F
n Number of star points in a CCD
S
n Number of centre points in a CCD
a , b , l Levels of factors
i i i
+1, −1 High and low coded factorial levels
−α, α Axial levels of coded factors
σ Standard deviation
3.2 Abbreviated terms
ANOVA analysis of variance
CCD central composite design
DOE design of experiments
6 © ISO 2015 – All rights reserved
RSM response surface methodology
R&R repeatability and reproducibility
4 Generic descriptions of central composite designs
4.1 Overview of the structure of the examples in Annexes A to D
This Technical Report provides general guidelines on the design, conduct and analysis of central
composite designs consisting of a specified number of two-level factors, and illustrates the steps with
four distinct applications given in the annexes. Each of the four examples in Annexes A through D
follows the basic structure as given in Table 1.
Table 1 — Basic steps in CCD design
1 Overall objective(s) of experiment
2 Description of the response variable(s)
3 Identification of factors affecting the response(s)
4 Selection of levels for each factor
5 Identification of measurement systems
Layout plan of the CCD (depending upon which main effects and two factor
6 interactions are to be studied) with “randomization” principle (if these
are physical runs)
7 Analyse the results – numerical summaries and graphical displays
8 Present the results
9 Perform confirmation run
4.2 Overall objective(s) of a response surface experiment
Experiments may be conducted for a variety of reasons. Therefore, the primary objective(s) for the
experiment should be clearly stated and agreed to by all parties involved in the design, conduct, analysis
and implications of the experimental effort.
The main goal of response surface experiments is to create a model of the relationship between the
factors and the response in order to explore optimum operating conditions. This involves choosing a
design which allows the fitting of a quadratic function as the systematic part of the model. The Central
Composite Design (CCD) can achieve this and this design has been popular since its introduction in the
[1]
first paper on response surface methods in 1951.
Although the fundamental method for fitting first order (linear) or second order (quadratic) function
of the predictor variables to the response is regression, the focus is not on the individual regression
coefficients but on the regression function, the response surface, as a whole. This emphasis is reflected
in the name Response Surface Methodology. Strong arguments in favour of this approach are given on
pages 508-509 of Reference [2].
Typically, the primary goal for the experiment is to find optimal operating conditions based on the
estimated response surface, this could involve doing several experiments, using the results of one
experiment to provide direction for what to do next. This next action could be to focus the experiment
around a different set of conditions, or to collect more data in the current experimental region in order
to fit a higher-order model or confirm what seemed to be the conclusion.
The CCD is an appropriate name because three types of design points can be identified after a coding
of the factor levels: centre points (2.10), factorial points (2.9) and star points (2.11), and those design
points are indeed centred at the origin of the design space after the coding of the factor levels (2.7).
Response surface experiments traditionally involve a small number of continuous factors. Some
software packages have an upper limit of 8 factors. Response surface experiments are typically used
when the investigators already know which factors are important. One way to obtain this knowledge
is to apply a screening experiment, for example a fractional factorial experiment as explained in
[11]
ISO/TR 12845.
4.3 Description of the response variable(s)
Associated with the objective of an experiment is a continuous outcome or performance measure. A
response of interest could involve maximization (larger is better), minimization (smaller is better) or
meet a target value (be close to a specified value), but, in all cases, that task is one of optimization.
The response variable (denoted by the variable y) should be closely related to the objective of the
experiment. For some situations, there are more than one variable of interest to be considered, although,
typically, only a primary response variable will be associated with the experiment. In other cases,
multiple responses should be considered. In case of multiple responses, the approach taken in response
surface methodology is to analyse and optimize each response separately. The fitted response surfaces
will then be studied to find settings that meet the requirements of all the responses. The example in
Annex C has three responses.
4.4 Identification of measurement systems
Assessment of repeatability and reproducibility of the measurement systems for factors and responses
should be done prior to designing the experiment.
4.5 Identification of factors affecting the response(s)
Response surface experiments are usually not done in isolation. They rely on prior knowledge
concerning important influential variables on the selected response. If this knowledge is not available,
it is necessary to conduct a different type of experiment to identify the factors affecting the response.
During the final selection of factors, attention shall be paid to the ability to set the levels of each
individual factor independently of the other factors.
4.6 Selection of levels for each factor
There are two aspects to the selection of factors. One is selecting the experimental region which is the
multidimensional range of interest for the factors selected. The other is the exact selection of the factor
levels in such a way that the design has desirable properties. The first one requires subject matter
knowledge as to the impact of factors on the response. The second one is more straightforward once the
factors and the type of design to be used are known. The second one is further discussed in this Clause.
The response surface methods considered in this Technical Report are about the second order centre
k k−p
models using the CCD. The CCD is an augmentation of 2 factorial experiments (or 2 fractional
factorial experiments). In addition to the two factorial levels that are used in the (fractional) factorial
experiments, the user selects three additional levels, one centre level which is the average of the two
factorial levels, and two extreme levels which are chosen symmetrically around the centre level and
typically outside the range of the two factorial levels.
When the experimenter selects the levels of each factor he will be thinking in terms of the operational
levels of a factor, the exact setting of a temperature, for example. But when studying the properties of
the design and also when analysing the data from the design coded levels of the design are used.
8 © ISO 2015 – All rights reserved
If the actual (or operational) levels are l < l < l < l < l then the middle level l shall be the average of
1 2 3 4 5 3
the lowest level l and the highest level l , and, furthermore, l shall be the average of the intermediate
1 5 3
levels l and l . The form of the coding operation can be expressed as Formula (1):
2 4
actual value−l
coded value= (1)
C
where C is half the distance from l to l . The coded value of the upper extreme level, l , will be denoted
2 4 5
by α, and the coded value of the lower extreme level, l , will be denoted by −α. It is very important to
note that the value of α is the same for all the factors of the design. Thus, the coded levels of all the
factors are (−α, −1, 0, 1, + α).
An alternative coding is sometimes applied in the computations in software programs. The form of the
coding operation can be expressed as Formula (2):
original value−l
coded value= (2)
M
where M is half the distance from the lowest level l to the highest level l .or, equivalently, the distance
1 5
from l to l This coding will be referred to as software coding in this Technical Report.
3 5.
When the levels of the factors have been chosen, the levels of the individual factors have to be
combined to define the runs of the experiment. A CCD has three types of experimental runs: factorial,
centre and axial ones.
4.6.1 Factorial runs
A factorial run is a setting of all k factors to coded levels either −1 or +1. The factorial runs are the runs
k−p k
used in 2 fractional factorial experiments or 2 factorial experiments.
Written as a k-dimensional vector in coded levels, the factorial run has the form (±1, ±1,…, ±1,…, ±1).
Considered as points in k-dimensional space, the factorial runs are the vertices of a cube and the
factorial runs are for this reason also called cube points.
k
There are 2 different factorial runs with k factors.
4.6.2 Star runs
The star runs are those where one of the factors has its coded levels either −α or α and the remaining
factors are at their coded level 0.
Written as a k-dimensional vector in coded levels, the star run has the form (0, 0,…, ±α,…, 0), having −α
th
or α on the i position and 0 on all other positions.
Viewed as points in k-dimensional space, the star runs are located on the coordinate axes, and for this
reason, the star runs are also called axial runs.
k
There are 2 different star runs with k factors.
4.6.3 Centre run
The centre run is the one where all the factors are on their coded level 0. Written as a k-dimensional
vector, it is the point (0, 0,…, 0).
There is only one centre run but the centre run may be replicated in a CCD. One reason for replicating
the centre point is to get an estimate of pure error which can be used to check the fit of the model.
4.7 Layout plan of the CCD with randomization principle
In a report, the full description of the design and the observed responses should be given. In addition to
reporting the levels of the factors, as described in 4.6, this includes reporting the following:
— the number of replications of the three types of design points;
— the number of blocks;
— the randomization.
If the design has only a few factors, a table where each run is represented as a row is useful for this
purpose. It is easier to grasp the design if coded levels are used. The randomization can be explained
by including a column giving the order in which the runs have been performed. If blocking is applied, a
column can similarly be added to explain the allocation of runs to blocks.
As an illustration, Table 3 provides a basic layout of a CCD for 2 factors with full factorial design in serial
order. While there is a standard order for factorial designs there is no such thing for a CCD, so the term
serial order is used for a way to write the runs in a CCD in such a way that the design is easily recognized.
The serial order used in Table 3 and throughout this Technical Report lists the factorial runs in standard
order, first, followed by the star runs and, finally, the centre runs. Each row of the table represents one
set of experimental conditions that when run will produce a value of the response variable y. The two
factors are designated as A and B. In this case, only one response variable is shown, but more columns
should be added if more than one response variable is studied. In this case, it is readily seen that the
design has five centre runs replicated. The last column shows the run order, i.e. the order in which the
experimental conditions have been applied. This experiment has been performed in one block.
Table 3 — Layout of a generic Central Composite Design
Serial order A B y Run order
1 −1 −1 y 6
2 1 −1 y 4
3 −1 1 y 13
4 1 1 y 7
5 −1,41 0 y 12
6 1,41 0 y 8
7 0 −1,41 y 9
8 0 1,41 y 5
9 0 0 y 2
10 0 0 y 10
11 0 0 y 1
12 0 0 y 11
13 0 0 y 3
4.8 Analyse the results — Numerical summaries and graphical displays
The second order model that can be fitted with the data from a CCD is a regression model and the
usual output and graphs from a regression analysis are reported. This includes the table of estimated
regression coefficients and associated t-statistics and p-values. A special feature of RSM regression is
ANOVA tables that exploit the structure of the model.
For the design in Table 3, one such ANOVA table is shown in Table 4.
10 © ISO 2015 – All rights reserved
Table 4 — ANOVA table for Central Composite Design
Response: y
Df Sum Sq Mean Sq F-value Pr(>F)
FO(x1, x2) 2 ddd,dd ddd,dd ddd,dd d,dd
TWI(x1, x2) 1 ddd,dd ddd,dd ddd,dd d,dd
PQ(x1, x2) 2 ddd,dd ddd,dd ddd,dd d,dd
Residuals 7 ddd,dd ddd,dd
Lack of fit 3 ddd,dd ddd,dd ddd,dd d,dd
Pure error 4 ddd,dd ddd,dd
Table 4 is given by most, if not all, statistical packages for RSM analysis. The names of the rows and the
columns may differ between statistical packages, but they are easily understood. Here, FO, TWI and PQ
are short for First Order, Two Way Interaction and Pure Quadratic. Examples are given in Tables A.5,
B.6 to B.7, and D.5.
In this case, there is no data, so the table is filled in with ddd,dd or d,dd except for the degrees of
freedom which are the correct ones for the design in Table 3. The four F-tests are of interest. Read from
the bottom of the table, the first one is the ratio of the mean squares for Lack of fit and Pure error and is
a check of the fit of the model. The second F-test is in the PQ line and it tests the need for pure quadratic
terms in the model; in this case the hypothesis that β = β = 0. It is the ratio of the mean square in the
11 22
PQ line to the mean square in the Residuals line of the table.
Another ANOVA table of potential interest tests for each predictor variable, the hypotheses that it is not
needed in the model. For the example with two predictor variables in Table 3, this ANOVA table has two
F-tests: One for the hypothesis that β = β = β = 0, and one for the hypothesis that β = β = β = 0.
1 12 11 2 12 22
Note that β , β , and β , for example, are the coefficients of all the terms in the model that involve x .
1 12 11 1
Examples can be found in Tables B.5 and C.7.
When looking at analyses of response surface experiments using different software, it is possible to
identify two different approaches. One approach uses all the tools of regression analysis to find a model
where only significant terms are included in the estimated systematic part of the model. The other
approach focuses on the response surface and only tests the hypotheses described in the two clauses
above, i.e. tests whether a linear model can be used instead of a quadratic model or tests whether a
factor can be considered to be essentially inert. In this Technical Report, the second approach is taken
and therefore individual non-significant terms are not removed from the model. Strong arguments in
favour of the second approach are given on pages 508-509 in Reference [2].
4.9 Present the results
Of course, one presentation of the result of the analysis is to give the estimated systematic part of
the model. Although the systematic part of the model is indispensable for calculating the predicted
response for various settings, it is not very useful for understanding the nature of the response surface.
In Annex E, four second order polynomials in two variables are given in Formulae (E.3), (E.4), (E.5)
and (E.6). The four polynomials look very similar. They have the same distribution of signs among the
coefficients and the coefficients are very similar. But the perspective and contour plots in Figures E.1 to
E.4 show that the polynomials represent four very different response surfaces.
For two and three predictor variables, it is possible to get an understanding of the response surface
from contour plots, but a more formal analysis called canonical analysis is always useful, because it
gives a precise characterization of the surface that gives an understanding of the response surface even
when the number of predictor variables is larger than 3.
Canonical analysis is a method of rewriting a fitted quadratic function of the predictor variables in a
form which can be more easily understood. This is achieved by a rotation of the coordinate axes which
removes all cross-product terms. This may be followed by a change of origin to remove first order terms
as well. Details are given in E.3.
Software packages for response surface methodology report the result of the canonical analysis. The
result has the following parts:
— stationary point and predicted value at stationary point of the response surface;
— eigenvalues;
— eigenvectors.
The stationary point is the point where the fitted response surface has a maximum, a minimum or a
saddle point, and the eigenvalues tell exactly which one is the case. Associated with each eigenvalue
is an eigenvector. Moving away from the stationary point in the positive or negative direction of the
eigenvector the response will decrease if the eigenvalue is negative, and it will increase if the eigenvalue
is positive, so
— if all the eigenvalues are negative, the stationary point is a maximum,
— if all the eigenvalues are positive, the stationary point is a minimum, and
— if some eigenvalues are negative and some are positive, the stationary point is a saddle point or a
minimax.
The situation is further complicated by the fact that one can only hope to approximate the response by
the fitted second order model in the experimental region, so it should also be considered whether the
stationary point is inside or outside the experimental region.
The situation is only simple, if the object is to maximize the response and if the stationary point is
a maximum located inside the experimental region, or, alternatively, if the object is to minimize the
response and if the stationary point is a minimum located inside the experimental region, and then
the optimal settings is the stationary point. In all other cases, the optimal settings need to be found
towards the boundary of the experimental region. The proprietary software packages provide a variety
of tool for this purpose.
4.10 Perform confirmation run
After determining the optimal combination of levels of the factors and predicting the value of response
variable for these factor levels, it is recommended that the user performs confirmation runs at the
chosen settting to check whether the new observations confirm the predictions of the experiment.
If the computer output gives the standard deviation (std.dev) of a single observation and the standard
error (std.err) of the predicted value at the optimum, then the following formulas apply for
a) a 95% confidence interval for the mean at the optimum: ˆyt s±⋅()v td.err and
0,975
ˆ
b) a 95% prediction interval for a new observation at the optimum: yt±⋅()v std.devs+ td.err .
0,975
Here, v denotes the degrees of freedom for error.
The 95 % prediction interval is particularly useful for it explains what can be expected from future
observations; if the model is good, 95 % of the observations will fall inside the 95 % prediction interval.
5 Description of Annexes A through D
5.1 Comparing and contrasting the examples
Four distinct examples of response surface designs are illustrated in Annexes A to D. Each of these
examples follows the same general template as given in Table 1.
12 © ISO 2015 – All rights reserved
5.2
...
Frequently Asked Questions
ISO/TR 13195:2015 is a technical report published by the International Organization for Standardization (ISO). Its full title is "Selected illustrations of response surface method - Central composite design". This standard covers: ISO/TR 13195:2015 describes the steps necessary to understand the scope of Response Surface Methodology (RSM) and the method to analyse data collected using Central Composite Designs (CCD) through illustration with four distinct applications of this methodology. Response surface methodology (RSM) is used in order to investigate a relation between the response and the set of quantitative predictor variables or factors. Especially after specifying the vital few controllable factors, RSM is used in order to find the factor setting which optimizes the response.
ISO/TR 13195:2015 describes the steps necessary to understand the scope of Response Surface Methodology (RSM) and the method to analyse data collected using Central Composite Designs (CCD) through illustration with four distinct applications of this methodology. Response surface methodology (RSM) is used in order to investigate a relation between the response and the set of quantitative predictor variables or factors. Especially after specifying the vital few controllable factors, RSM is used in order to find the factor setting which optimizes the response.
ISO/TR 13195:2015 is classified under the following ICS (International Classification for Standards) categories: 03.120.30 - Application of statistical methods. The ICS classification helps identify the subject area and facilitates finding related standards.
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