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
Buy Standard
Standards Content (Sample)
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/TR 13195:2015(E)
©
ISO 2015
---------------------- Page: 1 ----------------------
ISO/TR 13195:2015(E)
COPYRIGHT PROTECTED DOCUMENT
© ISO 2015, Published in Switzerland
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized otherwise in any form
or by any means, electronic or mechanical, including photocopying, or posting on the internet or an intranet, without prior
written permission. Permission can be requested from either ISO at the address below or ISO’s member body in the country of
the requester.
ISO copyright office
Ch. de Blandonnet 8 • CP 401
CH-1214 Vernier, Geneva, Switzerland
Tel. +41 22 749 01 11
Fax +41 22 749 09 47
copyright@iso.org
www.iso.org
ii © ISO 2015 – All rights reserved
---------------------- Page: 2 ----------------------
ISO/TR 13195:2015(E)
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
© ISO 2015 – All rights reserved iii
---------------------- Page: 3 ----------------------
ISO/TR 13195:2015(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out
through ISO technical committees. Each member body interested in a subject for which a technical
committee has been established has the right to be represented on that committee. International
organizations, governmental and non-governmental, in liaison with ISO, also take part in the work.
ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of
electrotechnical standardization.
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
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
patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of
any patent rights identified during the development of the document will be in the Introduction and/or
on the ISO list of patent declarations received (see www.iso.org/patents).
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation on the meaning of ISO specific terms and expressions related to conformity
assessment, as well as information about ISO’s adherence to the WTO principles in the Technical
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 © ISO 2015 – All rights reserved
---------------------- Page: 4 ----------------------
ISO/TR 13195:2015(E)
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]).
© ISO 2015 – All rights reserved v
---------------------- Page: 5 ----------------------
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.)
© ISO 2015 – All rights reserved 1
---------------------- Page: 6 ----------------------
ISO/TR 13195:2015(E)
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
3
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
---------------------- Page: 7 ----------------------
ISO/TR 13195:2015(E)
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
3
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.)
© ISO 2015 – All rights reserved 3
---------------------- Page: 8 ----------------------
ISO/TR 13195:2015(E)
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
---------------------- Page: 9 ----------------------
ISO/TR 13195:2015(E)
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.)
© ISO 2015 – All rights reserved 5
---------------------- Page: 10 ----------------------
ISO/TR 13195:2015(E)
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
0
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
---------------------- Page: 11 ----------------------
ISO/TR 13195:2015(E)
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).
© ISO 2015 – All rights reserved 7
---------------------- Page: 12 ----------------------
ISO/TR 13195:2015(E)
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 repeatabil
...
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/TR 13195:2015(E)
©
ISO 2015
---------------------- Page: 1 ----------------------
ISO/TR 13195:2015(E)
COPYRIGHT PROTECTED DOCUMENT
© ISO 2015, Published in Switzerland
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized otherwise in any form
or by any means, electronic or mechanical, including photocopying, or posting on the internet or an intranet, without prior
written permission. Permission can be requested from either ISO at the address below or ISO’s member body in the country of
the requester.
ISO copyright office
Ch. de Blandonnet 8 • CP 401
CH-1214 Vernier, Geneva, Switzerland
Tel. +41 22 749 01 11
Fax +41 22 749 09 47
copyright@iso.org
www.iso.org
ii © ISO 2015 – All rights reserved
---------------------- Page: 2 ----------------------
ISO/TR 13195:2015(E)
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
© ISO 2015 – All rights reserved PROOF/ÉPREUVE iii
---------------------- Page: 3 ----------------------
ISO/TR 13195:2015(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out
through ISO technical committees. Each member body interested in a subject for which a technical
committee has been established has the right to be represented on that committee. International
organizations, governmental and non-governmental, in liaison with ISO, also take part in the work.
ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of
electrotechnical standardization.
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
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
patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of
any patent rights identified during the development of the document will be in the Introduction and/or
on the ISO list of patent declarations received (see www.iso.org/patents).
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation on the meaning of ISO specific terms and expressions related to conformity
assessment, as well as information about ISO’s adherence to the WTO principles in the Technical
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
---------------------- Page: 4 ----------------------
ISO/TR 13195:2015(E)
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]).
© ISO 2015 – All rights reserved PROOF/ÉPREUVE v
---------------------- Page: 5 ----------------------
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.)
© ISO 2015 – All rights reserved PROOF/ÉPREUVE 1
---------------------- Page: 6 ----------------------
ISO/TR 13195:2015(E)
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
3
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
---------------------- Page: 7 ----------------------
ISO/TR 13195:2015(E)
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
3
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.)
© ISO 2015 – All rights reserved PROOF/ÉPREUVE 3
---------------------- Page: 8 ----------------------
ISO/TR 13195:2015(E)
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
---------------------- Page: 9 ----------------------
ISO/TR 13195:2015(E)
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.)
© ISO 2015 – All rights reserved PROOF/ÉPREUVE 5
---------------------- Page: 10 ----------------------
ISO/TR 13195:2015(E)
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
0
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
---------------------- Page: 11 ----------------------
ISO/TR 13195:2015(E)
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).
© ISO 2015 – All rights reserved PROOF/ÉPREUVE 7
---------------------- Page: 12 ----------------------
ISO/TR 13195:2015(E)
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
...
Questions, Comments and Discussion
Ask us and Technical Secretary will try to provide an answer. You can facilitate discussion about the standard in here.