1、 ISO 2016 Statistical methods for implementation of Six Sigma Selected illustrations of contingency table analysis Mthodes statistiques pour limplmentation de Six Sigma Exemples slectionns dapplication de lanalyse de tableau de contingence TECHNICAL REPORT ISO/TR 16705 Reference number ISO/TR 16705:
2、2016(E) First edition 2016-08-15 ISO/TR 16705:2016(E)ii ISO 2016 All rights reserved COPYRIGHT PROTECTED DOCUMENT ISO 2016, 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, elect
3、ronic 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 ISOs member body in the country of the requester. ISO copyright office Ch. de Blandonnet 8 CP 401 CH-1214 Vern
4、ier, Geneva, Switzerland Tel. +41 22 749 01 11 Fax +41 22 749 09 47 copyrightiso.org www.iso.org ISO/TR 16705:2016(E)Foreword iv Introduction v 1 Scope . 1 2 Normative references 1 3 T erms and definitions . 1 4 Symbols and abbreviated terms . 2 5 General description of contingency table analysis .
5、2 5.1 Overview of the structure of contingency table analysis . 2 5.2 Overall objectives of contingency table analysis 3 5.3 List attributes of interest 3 5.4 State a null hypothesis 3 5.5 Sampling plan. 3 5.6 Process and analyse data . 4 5.6.1 Chi-squared test 4 5.6.2 Linear trend test . 6 5.6.3 Co
6、rrespondence analysis 6 5.7 Conclusions 7 6 Description of Annexes A through D 7 Annex A (informative) Distribution of number of technical issues found after product r elease t o the field. 8 Annex B (informative) Peoples perception about contented life .15 Annex C (informative) Customer satisfactio
7、n research on a brand of beer 20 Annex D (informative) Proportions of nonconforming parts of production lines .26 Bibliography .31 ISO 2016 All rights reserved iii Contents Page ISO/TR 16705:2016(E) Foreword ISO (the International Organization for Standardization) is a worldwide federation of nation
8、al 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. Internatio
9、nal 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
10、 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
11、.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
12、 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 express
13、ions related to conformit y assessment, as well as information about ISOs adherence to the World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT) see the following URL: www.iso.org/iso/foreword.html. The committee responsible for this document is ISO/TC 69, Applications o
14、f statistical methods, Subcommittee SC 7, Applications of statistical and related techniques for the implementation of Six Sigma.iv ISO 2016 All rights reserved ISO/TR 16705:2016(E) Introduction The Six Sigma and international statistical standards communities share a philosophy of continuous improv
15、ement and many analytical 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, an
16、d statistical software usage have inhibited exchanges, synergy, and mutual appreciation between the two groups. The present document takes one specific statistical tool (Contingency Table Analysis), develops the topic somewhat generically (in the spirit of International Standards), then illustrates
17、it through the use of several detailed and distinct applications. The generic description focuses on the commonalities across studies designed to assess the association of categorical variables. The Annexes containing illustrations do not only follow the basic framework, but also identify the nuance
18、s and peculiarities in the specific applications. Each example will offer at least one “winkle” to the problem, which is generally the case for real Six Sigma and other fields application. ISO 2016 All rights reserved v Statistical methods for implementation of Six Sigma Selected illustrations of co
19、ntingency table analysis 1 Scope This document describes the necessary steps for contingency table analysis and the method to analyse the relation between categorical variables (including nominal variables and ordinal variables). This document provides examples of contingency table analysis. Several
20、 illustrations from different fields with different emphasis suggest the procedures of contingency table analysis using different software applications. In this document, only two-dimensional contingency tables are considered. 2 Normative references There are no normative references in this document
21、. 3 Terms a nd definiti ons For the purposes of this document, the terms and definitions given in ISO 3534-1 and ISO 3534-2 and the following apply. ISO and IEC maintain terminological databases for use in standardization at the following addresses: IEC Electropedia: available at http:/www.electrope
22、dia.org/ ISO Online browsing platform: available at http:/www.iso.org/obp 3.1 categorical variable variable with the measurement scale consisting of a set of categories 3.2 nominal data variable with a nominal scale of measurement SOURCE: ISO 3534-2:2006, 1.1.6 3.3 ordinal data variable with an ordi
23、nal scale of measurement SOURCE: ISO 3534-2:2006, 1.1.7 3.4 contingency table tabular representation of categorical data, which shows frequencies for particular combinations of values of two or more discrete random variables Note 1 to entry: A table that cross-classifies two variables is called a “t
24、wo-way contingency table;” the one that cross-classifies three variables is called a “three-way contingency table.” A two-way table with r rows and c columns is also named “r c table.” EXAMPLE Let n items be classified by categorical variables X and Y with levels X 1 , X 2and Y 1 , Y 2 , respectivel
25、y. The number of items with both attribute X iand Y jis n ij . Then, a 2 2 table is as follows. TECHNICAL REPORT ISO/TR 16705:2016(E) ISO 2016 All rights reserved 1 ISO/TR 16705:2016(E) Table 1 2 2 contingency table Variable X Variable Y Y 1 Y 2 X 1 n 11 n 12 X 2 n 21 n 22 3.5 p-value probability of
26、 observing the observed test statistic value or any other value at least as unfavorable to the null hypothesis SOURCE: ISO 3534-1:2006, 1.49 4 Symbols and abbreviated terms H 0 null hypothesis H a alternative hypothesis 2 Chi-square statistic G 2 likelihood-ratio statistic n total number of cell cou
27、nt r c table contingency table with r rows and c columns DF degree of freedom 5 General description of contingency table analysis 5.1 Overview of the structure of contingency table analysis This document provides general guidelines on the design, conduct, and analysis of contingency table analysis a
28、nd illustrates the steps with distinct applications given in Annexes A through D. Each of these examples follows the basic structure given in Table 2. Table 2 Basic steps for contingency table analysis 1 State the overall objective 2 List attributes of interest 3 State a null hypothesis 4 Sampling p
29、lan 5 Process and analyse data 6 Accept or reject the null hypothesis (Conclusions) Contingency table analysis is used to assess the association of two or more categorical variables. This document focuses on two-way contingency table analysis, which only considers the relation of two categorical var
30、iables. Particular methods for three or more categorical variables analysis are not included in this document. The steps given in Table 1 provide general techniques and procedures for contingency table analysis. Each of the six steps is explained in general in 5.2 to 5.7.2 ISO 2016 All rights reserv
31、ed ISO/TR 16705:2016(E) 5.2 Overall objectives of contingency table analysis Contingency table analysis can be employed in Six Sigma 1)projects in the “Analyse” phase of DMAIC methodologies, and often used in sampling survey, social science and medical research, etc. Apart from the usual statistical
32、 methods focusing on continuous variables, contingency table analysis mainly handles the categorical data, including nominal data and ordinal data. In the case that the observed value is the frequency of certain combinations of several objective conditions, but not the continuous value from the equi
33、pment, the contingency table analysis is needed. The primary motivation of this method is to test the association of categorical variables, including the following situations: a) to assess whether an observed frequency distribution differs from a theoretical distribution; b) to assess the independen
34、ce of two categorical variables; c) to assess the homogeneity of several distributions of same type; d) to assess the trend association of observations on ordinal variables; e) to assess extensive association between levels of categorical variables. 5.3 List attributes of interest This document cons
35、iders the association of two categorical variables based on the observed frequency of the characteristic corresponding to combinations of different levels of attributes of interest. If the association between quantitative variable and categorical variable is of interest (e.g. cup size versus surface
36、 decoration), it is necessary to divide quantitative data into ordinal classes (e.g. small, medium, large). 5.4 State a null hypothesis This document is to determine whether row variable and column variable are independent. The null hypothesis for Chi-square test is H 0 : the row variable and column
37、 variable are independent; and the alternative hypothesis is H a : the row variable and column variable are not independent. 5.5 Sampling plan In the sampling plan for contingency table analysis, variables and the levels should be determined first. For two-way contingency tables, there are four poss
38、ible sampling plans to generate the tables. a) The total number of cell count n is not fixed. b) The total number of cell count n is fixed, but none of the total rows or columns are fixed. c) The total number of cell count n is fixed, and either the row marginal totals or the column marginal totals
39、are fixed; d) The total number of cell count n is fixed, and both row marginal totals and the column marginal totals are fixed. 1) Six Sigma is the trademark of a product supplied by Motorola, Inc. This information is given for the convenience of users of this document and does not constitute an end
40、orsement by ISO of the product named. Equivalent products may be used if they can be shown to lead to the same results. ISO 2016 All rights reserved 3 ISO/TR 16705:2016(E) The aforementioned four sampling plans correspond to different purposes of categorical data analysis. Case a) is a random sampli
41、ng, that all frequency numbers are independent. For example, the number of customers entering a supermarket during the day is a random variable. The customers are divided into four classes based on their gender and whether they are shopping or not (male/shopping, male/no shopping, female/shopping, f
42、emale/no shopping). These four numbers form a contingency table. Case b) is applicable to a sampling survey where the sample size is fixed. Case c) is usually an analysis of a comparative analysis. For example, when conducting a research on the relationship of lung cancer and smoking, a group of pat
43、ients with lung cancer and a group of healthy people with similar age, gender, and other physical condition are chosen for the research. The total number of people in each group is fixed. Case d) is another test of attribute agreement analysis, usually used to test whether the results from two measu
44、rement systems are consistent with each other. For attribute agreement analysis, one can refer to ISO 14468. The calculated statistics of the test of independence for the first three cases are the same. Randomization is very important when sampling for experiments. The observations in each cell are
45、made on a random sample. When it is inconvenient or difficult to attain adequate samples, one should pay close attention to any confounding factors that may affect the results of the analysis. Table 3 shows a two-way contingency table with r levels of variable X and c levels of Y. The observed frequ
46、ency of each combination of the two variables is n ij(i =1, r, j=1,c). Table 3 Layout of a generic r c contingency table analysis Variable X Variable Y Y 1 Y 2 Y c X 1 n 11 n 12 n 1c X 2 n 21 n 22 n 2c X r n r1 n r2 n rc 5.6 Process and analyse data 5.6.1 Chi-squared test Chi-square ( 2 ) test is th
47、e most fundamental tool for contingency table analysis to test independence of variables. It is commonly used to compare observed data with some expected data according to a specific test purpose. For a one-dimension contingency table, which has only one categorical variable with two or more levels,
48、 Chi-square test, usually called “goodness-of-fit test,” can be used to assess whether the observed data classified by levels follow an theoretical distribution. For a two-dimensional contingency table, r c table, Chi-square test can be used to evaluate whether two categorical variables are independ
49、ent. It can test the homogeneity of distributions with same type, which is also called “homogeneity test.” Chi-square test is defined to evaluate the distance of the observed data from the expected data. The formula for calculating Chi-square statistic is: 2 2 (1) whereo is the observed frequency data;e is the expected frequency data.4 ISO 2016 All rights reserved ISO/TR 16705:2016(E) The formula is the sum of the squared difference between observed and expected frequency, d
