1、 Reference number ISO/TR 29901:2007(E) ISO 2007TECHNICAL REPORT ISO/TR 29901 First edition 2007-11-01 Selected illustrations of full factorial experiments with four factors Illustrations choisies de plans dexprience factoriels complets quatre facteurs ISO/TR 29901:2007(E) PDF disclaimer This PDF fil
2、e may contain embedded typefaces. In accordance with Adobes licensing policy, this file may be printed or viewed but shall not be edited unless the typefaces which are embedded are licensed to and installed on the computer performing the editing. In downloading this file, parties accept therein the
3、responsibility of not infringing Adobes licensing policy. The ISO Central Secretariat accepts no liability in this area. Adobe is a trademark of Adobe Systems Incorporated. Details of the software products used to create this PDF file can be found in the General Info relative to the file; the PDF-cr
4、eation parameters were optimized for printing. Every care has been taken to ensure that the file is suitable for use by ISO member bodies. In the unlikely event that a problem relating to it is found, please inform the Central Secretariat at the address given below. COPYRIGHT PROTECTED DOCUMENT ISO
5、2007 All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying and microfilm, without permission in writing from either ISO at the address below or ISOs member body in the c
6、ountry of the requester. ISO copyright office Case postale 56 CH-1211 Geneva 20 Tel. + 41 22 749 01 11 Fax + 41 22 749 09 47 E-mail copyrightiso.org Web www.iso.org Published in Switzerland ii ISO 2007 All rights reservedISO/TR 29901:2007(E) ISO 2007 All rights reserved iii Contents Page Foreword iv
7、 Introduction v 1 Scope . 1 2 Normative references . 1 3 Terms and definitions. 2 4 Symbols and abbreviated terms . 4 5 Generic description of full factorial designs . 4 5.1 Overview of the structure of the four factor examples in Annexes A through E. 4 5.2 Overall objective(s) of the experiment 4 5
8、.3 Response variable(s) 5 5.4 Factors affecting the response(s). 5 5.5 “Full” factorial design 5 5.6 Analyse the results Numerical summaries and graphical displays 6 5.7 Present the results 7 5.8 Perform confirmation runs 7 6 Description of Annexes A through E 7 6.1 Comparing and contrasting the exa
9、mples. 7 6.2 Experiment summaries 7 Annex A (informative) Solder bar experiment . 8 Annex B (informative) Direct mail marketing campaign. 17 Annex C (informative) Button tactility experiment 25 Annex D (informative) Optimizing a customer PVC formulation. 34 Annex E (informative) Genetic algorithms f
10、or DNA sequencing experiment. 44 Bibliography . 52 ISO/TR 29901:2007(E) iv ISO 2007 All rights reservedForeword ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies (ISO member bodies). The work of preparing International Standards is normal
11、ly 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
12、 work. ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization. International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2. The main task of technical committees is to prepare I
13、nternational Standards. Draft International Standards adopted by the technical committees are circulated to the member bodies for voting. Publication as an International Standard requires approval by at least 75 % of the member bodies casting a vote. In exceptional circumstances, when a technical co
14、mmittee has collected data of a different kind from that which is normally published as an International Standard (“state of the art”, for example), it may decide by a simple majority vote of its participating members to publish a Technical Report. A Technical Report is entirely informative in natur
15、e and does not have to be reviewed until the data it provides are considered to be no longer valid or useful. 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
16、 rights. ISO/TR 29901 was prepared by Technical Committee ISO/TC 69, Applications of statistical methods. ISO/TR 29901:2007(E) ISO 2007 All rights reserved v Introduction The Six Sigma and international statistical standards communities share a philosophy of continuous improvement and many analytica
17、l tools. The Six Sigma community tends to adopt a pragmatic approach driven by time and resource constraints. The statistical standards community arrives at rigorous documents through long-term international consensus. The disparities in time pressures, mathematical rigor and statistical software us
18、age have inhibited exchanges, synergy and mutual appreciation between the two groups. The present document takes one specific statistical tool (full factorial designs with four factors, 2 4 designs) and develops the topic somewhat generically (in the spirit of International Standards) but then illus
19、trates it through the use of five detailed and distinct applications. The generic description focuses on the commonalities across 2 4 designs. These commonalities hold more generally for arbitrary numbers of factors, but a value of four was chosen for this Technical Report. The annexes containing th
20、e five 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 Six Sigma applications. It is thus hoped that practitioners can identify with
21、at least one of the five examples, if only to remind them of the basic material on factorial designs that was encountered during their Six Sigma training. Each of the five examples is developed and analysed using statistical software of current vintage. The explanations throughout are devoid of math
22、ematical detail such material can be readily obtained from the many design and analysis of experiments textbooks available (such as those given in the Bibliography). TECHNICAL REPORT ISO/TR 29901:2007(E) ISO 2007 All rights reserved 1 Selected illustrations of full factorial experiments with four fa
23、ctors 1 Scope This Technical Report describes the steps necessary to specify, to use and to analyse 2 4full factorial designs through illustration, with five distinct applications of this methodology. Depending on the application, a number of factors other than four may be considered in the experime
24、nt. NOTE 1 Each of these five illustrations is similar in that sufficient resources were available to implement the design. Other commonalities among the five examples are noted (e.g. study objective, two levels for factors, response variable(s), factors effecting the response). The individual illus
25、trations have some salient features that are distinct such as presence/absence of repetitions, centre points, interactions, or different types of response variables. Each illustration takes place in a different environment such as marketing, software, manufacturing, telecommunications and chemical p
26、rocessing. NOTE 2 For the purposes of this Technical Report, the selection of four factors with two levels (aside from centre points) was made in advance. Furthermore, the detailed use of response surface designs as a follow-up or augmentation of the existing designs was excluded from this Technical
27、 Report, although their use is noted in some of the illustrations. Likewise, Taguchi designs and blocking designs were not included. NOTE 3 Full factorial experiments are often employed by individuals (so-called “black belts” or “green belts”) associated with Six Sigma methods. Six Sigma methods are
28、 concerned with problem solving and continuous improvement. A full factorial experiment with four factors is one of many tools available to Six Sigma practitioners, but hitherto has not been addressed in detail in ISO International Standards. 2 Normative references The following referenced documents
29、 are indispensable for the application of this document. For dated references, only the edition cited applies. For undated references, the latest edition of the referenced document (including any amendments) applies. ISO 3534-1:2006, Statistics Vocabulary and symbols Part 1: General statistical term
30、s and terms used in probability ISO 3534-2, Statistics Vocabulary and symbols Part 2: Applied statistics ISO 3534-3:1999, Statistics Vocabulary and symbols Part 3: Design of experiments ISO/TR 29901:2007(E) 2 ISO 2007 All rights reserved3 Terms and definitions For the purposes of this document, the
31、terms and definitions in ISO 3534-1, ISO 3534-2, ISO 3534-3 and the following apply. 3.1 analysis of variance ANOVA technique which subdivides the total variation of a response variable into meaningful components associated with specific sources of variation NOTE Adapted from ISO 3534-3:1999, defini
32、tion 3.4. 3.2 binomial distribution discrete distribution having the probability mass function ! () ( 1) !( )! x nx n PX x p p xnx = where x = 0, 1, , n and with indexing parameters n = 1, 2, , and 0 p 1. NOTE Adapted from ISO 3534-1:2006, definition 2.46. 3.3 block collection of experimental units
33、more homogeneous than the full set of experimental units NOTE Adapted from ISO 3534-3:1999, definition 1.11. 3.4 centre point vector of factor level settings of the form (a 1 , a 2 , ., a k ), where all a iequal 0, as notation for the coded levels of the factors NOTE Adapted from ISO 3534-3:1999, de
34、finition 1.36. 3.5 design matrix matrix with rows representing individual treatments (possibly transformed according to the assumed model) which can be extended by deduced levels of other functions of factor levels (interactions, quadratic terms, etc.) but are dependent upon the assumed model NOTE A
35、dapted from ISO 3534-3:1999, definition 2.7.1. 3.6 factor predictor variable that is varied with the intent of assessing its effect on the response variable NOTE Adapted from ISO 3534-3:1999, definition 1.5. 3.7 full factorial experiment experiment consisting of all possible treatments formed from t
36、wo or more factors, each being studied at two or more levels NOTE Adapted from ISO 3534-3:1999, definition 2.1. ISO/TR 29901:2007(E) ISO 2007 All rights reserved 3 3.8 interaction effect for which the apparent influence of one factor on the response variable depends upon one or more other factors NO
37、TE Adapted from ISO 3534-3:1999, definition 1.17. 3.9 level potential setting, value or assignment of a factor NOTE Adapted from ISO 3534-3:1999, definition 1.6. 3.10 normal distribution continuous distribution having the probability density function 2 2 () 2 1 () e 2 x fx = where 0 NOTE Adapted fro
38、m ISO 3534-1:2006, definition 2.50. 3.11 predictor variable variable that can contribute to the explanation of the outcome of an experiment NOTE Adapted from ISO 3534-3:1999, definition 1.3. 3.12 randomization process used to assign treatments to experimental units so that each experimental unit has
39、 an equal chance of being assigned a particular treatment NOTE Adapted from ISO 3534-3:1999, definition 1.29. 3.13 replication performance of an experiment more than once for a given set of predictor variables NOTE Adapted from ISO 3534-3:1999, definition 1.27. 3.14 split-plot design design in which
40、 a group of experimental units (plot) to which the same level assigned to the principal factor is subdivided (split) so as to study one or more additional principal factors within each level of that factor NOTE Adapted from ISO 3534-3:1999, definition 2.3.6. ISO/TR 29901:2007(E) 4 ISO 2007 All right
41、s reserved4 Symbols and abbreviated terms The symbols and abbreviated terms used in this Technical Report are as follows: y Response variable A, B, C, D Factors AB, AC, AD, BC, BD, CD 2-way interactions ABC, ACD, BCD 3-way interactions ABCD 4-way interactions +1/ 1 High and low settings 2 4Four fact
42、ors each with two levels Standard deviation 5 Generic description of full factorial designs 5.1 Overview of the structure of the four factor examples in Annexes A through E This Technical Report provides general guidelines on the design, conduct and analyses of two-level full factorial designs and i
43、llustrates the steps with five distinct applications given in Annexes A through E. Each of these five examples follows the basic structure given in Table 1. The steps given in Table 1 apply to design and analysis of experiments in general, although this Technical Report focuses on 2 4full factorial
44、designs. Each of the seven steps is explained in general below. Specific explanations of the substance of these steps is provided in the examples in Annexes A through E. Table 1 Basic steps in experimental design 1 State the overall objective(s) of the experiment 2 Describe the response variable(s)
45、3 List the factors that might affect the response(s) 4 Select a “full” factorial design 5 Analyse the results Numerical summaries and graphical displays 6 Present the results 7 Perform a confirmation run 5.2 Overall objective(s) of the experiment Experiments are conducted for a variety of reasons. T
46、he primary motivation 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. There may be secondary objectives which could be addressed with the full factorial experiment. The ultimate outcome of
47、 the experiment could be to take immediate action on factor levels or to obtain a predictive model, both of which dictate some elements of the analyses. ISO/TR 29901:2007(E) ISO 2007 All rights reserved 5 5.3 Response variable(s) Associated with the objective of an experiment is a measurable outcome
48、 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). The response variable (denoted here by the variable y) should be intimately (if not directly) related to the objecti
49、ve of the experiment. For some situations, there may be multiple characteristics of interest to be considered, although there typically is a primary response variable associated with the experiment. In other cases, multiple responses must be considered; however, for purposes of this document, a single response is considered in each example. 5.4 Factors affecting the response(s) The response variable likely depends in some unknown way on a variety of conditions that occur or could be set in the course o
