1、Designation: E2709 141An American National StandardStandard Practice forDemonstrating Capability to Comply with an AcceptanceProcedure1This standard is issued under the fixed designation E2709; the number immediately following the designation indicates the year oforiginal adoption or, in the case of
2、 revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon () indicates an editorial change since the last revision or reapproval.1NOTEEditorial corrections made throughout in March 2015.1. Scope1.1 This practice provides a general metho
3、dology for evalu-ating single-stage or multiple-stage acceptance procedureswhich involve a quality characteristic measured on a numericalscale. This methodology computes, at a prescribed confidencelevel, a lower bound on the probability of passing an accep-tance procedure, using estimates of the par
4、ameters of thedistribution of test results from a sampled population.1.2 For a prescribed lower probability bound, the method-ology can also generate an acceptance limit table, whichdefines a set of test method outcomes (for example, sampleaverages and standard deviations) that would pass the accep-
5、tance procedure at a prescribed confidence level.1.3 This approach may be used for demonstrating compli-ance with in-process, validation, or lot-release specifications.1.4 The system of units for this practice is not specified.1.5 This standard does not purport to address all of thesafety concerns,
6、if any, associated with its use. It is theresponsibility of the user of this standard to establish appro-priate safety and health practices and determine the applica-bility of regulatory limitations prior to use.1.6 This international standard was developed in accor-dance with internationally recogn
7、ized principles on standard-ization established in the Decision on Principles for theDevelopment of International Standards, Guides and Recom-mendations issued by the World Trade Organization TechnicalBarriers to Trade (TBT) Committee.2. Referenced Documents2.1 ASTM Standards:2E456 Terminology Relat
8、ing to Quality and StatisticsE2282 Guide for Defining the Test Result of a Test MethodE2586 Practice for Calculating and Using Basic Statistics3. Terminology3.1 DefinitionsSee Terminology E456 for a more exten-sive listing of terms in ASTM Committee E11 standards.3.1.1 characteristic, na property of
9、 items in a sample orpopulation which, when measured, counted or otherwiseobserved, helps to distinguish between the items. E22823.1.2 mean, nof a population, , average or expectedvalue of a characteristic in a population, of a sample X, sum ofthe observed values in a sample divided by the sample si
10、ze.E25863.1.3 multiple-stage acceptance procedure, na procedurethat involves more than one stage of sampling and testing agiven quality characteristic and one or more acceptance criteriaper stage.3.1.4 standard deviation, nof a population, , the squareroot of the average or expected value of the squ
11、ared deviationof a variable from its mean of a sample, s, the square root ofthe sum of the squared deviations of the observed values in thesample divided by the sample size minus 1. E25863.1.5 test method, na definitive procedure that produces atest result. E22823.2 Definitions of Terms Specific to
12、This Standard:3.2.1 acceptable parameter region, nthe set of values ofparameters characterizing the distribution of test results forwhich the probability of passing the acceptance procedure isgreater than a prescribed lower bound.3.2.2 acceptance region, nthe set of values of parameterestimates that
13、 will attain a prescribed lower bound on theprobability of passing an acceptance procedure at a prescribedlevel of confidence.3.2.3 acceptance limit, nthe boundary of the acceptanceregion, for example, the maximum sample standard deviationtest results for a given sample mean.1This practice is under
14、the jurisdiction of ASTM Committee E11 on Quality andStatistics and is the direct responsibility of Subcommittee E11.20 on Test MethodEvaluation and Quality Control.Current edition approved Oct. 1, 2014. Published October 2014. Originallyapproved in 2009. Last previous edition approved in 2012 as E2
15、709 12. DOI:10.1520/E2709-14E01.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at serviceastm.org. For Annual Book of ASTMStandards volume information, refer to the standards Document Summary page onthe ASTM website.Copyright ASTM International,
16、 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United StatesThis international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for theDevelopment of International Standards, Guides an
17、d Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.14. Significance and Use4.1 This practice considers inspection procedures that mayinvolve multiple-stage sampling, where at each stage one candecide to accept or to continue sampling, and the decisio
18、n toreject is deferred until the last stage.4.1.1 At each stage there are one or more acceptance criteriaon the test results; for example, limits on each individual testresult, or limits on statistics based on the sample of test results,such as the average, standard deviation, or coefficient ofvaria
19、tion (relative standard deviation).4.2 The methodology in this practice defines an acceptanceregion for a set of test results from the sampled population suchthat, at a prescribed confidence level, the probability that asample from the population will pass the acceptance procedureis greater than or
20、equal to a prespecified lower bound.4.2.1 Having test results fall in the acceptance region is notequivalent to passing the acceptance procedure, but providesassurance that a sample would pass the acceptance procedurewith a specified probability.4.2.2 This information can be used for processdemonstr
21、ation, validation of test methods, and qualification ofinstruments, processes, and materials.4.2.3 This information can be used for lot release(acceptance), but the lower bound may be conservative in somecases.4.2.4 If the results are to be applied to future test resultsfrom the same process, then i
22、t is assumed that the process isstable and predictable. If this is not the case then there can beno guarantee that the probability estimates would be validpredictions of future process performance.4.3 This methodology was originally developed (1-4)3foruse in two specific quality characteristics of d
23、rug products inthe pharmaceutical industry but will be applicable for accep-tance procedures in all industries.4.4 Mathematical derivations would be required that arespecific to the individual criteria of each test.5. Methodology5.1 The process for defining the acceptance limits, startingfrom the de
24、finition of the acceptance procedure, is outlined inthis section. A computer program is normally required toproduce the acceptable parameter region and the acceptancelimits.5.1.1 An expression for the exact probability of passing theacceptance procedure might be intractable when the procedureconsist
25、s of multiple stages with multiple criteria, hence a lowerbound for the probability may be used.5.2 Express the probability of passing the acceptance pro-cedure as a function of the parameters characterizing thedistribution of the quality characteristic for items in thesampled population.5.2.1 For e
26、ach stage in the procedure having multipleacceptance criteria, determine the lower bound on the prob-ability of that stage as a function of the probabilities of passingeach of the criteria in the stage:PSi! 5 PCi1and Ci2 and Cim! $1 2(j51m1 2 PCij! (1)where:P(Si) = is the probability of passing stag
27、e i,P(Cij) = is the probability of passing the j-th criterion of mwithin the i-th stage.5.2.2 Determine the lower bound on the probability ofpassing a k-stage procedure as a function of probabilities ofpassing each of the individual stages:P pass k 2 stage procedure! $max$PS1!, PS2!, , PSk!% (2)5.3
28、Determine the contour of the region of parameter valuesfor which the expression for the probability of passing thegiven acceptance procedure is at least equal to the requiredlower bound (LB) on the probability of acceptance (p). Thisdefines the acceptable parameter region.5.4 For each value of a sta
29、tistic or set of statistics, derive ajoint confidence region for the distribution parameters atconfidence level, expressed as a percentage, of 100(1-). Thesize of sample to be taken, n, and the statistics to be used, mustbe predetermined (see 5.6).5.5 Determine the contour of the acceptance region,
30、whichconsists of values of the statistics for which the confidenceregion at level 100(1-) is entirely contained in the acceptableparameter region. The acceptance limits lie on the contour ofthe acceptance region.5.6 To select the size of sample, n, to be taken, theprobability that sample statistics
31、will lie within acceptancelimits should be evaluated over a range of values of n, forvalues of population parameters of practical interest, and forwhich probabilities of passing the given acceptance procedureare well above the lower bound. The larger the sample size nthat is chosen, the larger will
32、be the acceptance region and thetighter the distribution of the statistics. Choose n so that theprobability of passing acceptance limits is greater than apredetermined value.5.7 To use the acceptance limit, sample randomly from thepopulation. Compute statistics for the sample. If statistics fallwith
33、in the acceptance limits, then there is 1- confidence thatthe probability of acceptance is at least p.6. Procedures for Sampling from a Normal Distribution6.1 An important class of procedures is for the case wherethe quality characteristic is normally distributed. Particularinstructions for that cas
34、e are given in this section, for twosampling methods, simple random and two-stage. In thisstandard these sampling methods are denoted Sampling Plan 1and Sampling Plan 2, respectively.6.2 When the characteristic is normally distributed, param-eters are the mean () and standard deviation ()ofthepopula
35、tion. The acceptable parameter region will be the regionunder a curve in the half-plane where is on the horizontalaxis, on the vertical axis, such as that depicted in Fig. 1.3The boldface numbers in parentheses refer to a list of references at the end ofthis standard.E2709 14126.3 For simple random
36、sampling from a normal population,the method of Lindgren (5) constructs a simultaneous confi-dence region of (, ) values from the sample average Xandthe sample standard deviation s of n test results.6.3.1 Let Zpand p2denote percentiles of the standardnormal distribution and of the chi-square distrib
37、ution with n-1degrees of freedom, respectively. Given a confidence level(1-), choose and such that (1-) = (1-2)(1-). Althoughthere are many choices for and that would satisfy thisequation, a reasonable choice is: 512=12 and512=12!/2 which equally splits the overall alpha be-tween estimating and . Th
38、en:PHSX2 /=nD2# Z122JPHn 2 1!s22$ 2J5 1 2 2!1 2 !5 1 2 (3)6.3.2 The confidence region for (, ), two-sided for ,one-sided for , is an inverted triangle with a minimum vertexat X,0!, as depicted in Fig. 1.6.3.3 The acceptance limit takes the form of a table giving,for each value of the sample mean, th
39、e maximum value of thestandard deviation (or coefficient of variation) that would meetthese requirements. Using a computer program that calculatesconfidence limits for and given sample mean Xandstandard deviation s, the acceptance limit can be derived usingan iterative loop over increasing values of
40、 the sample standarddeviation s (starting with s = 0) until the confidence limits hitthe boundary of the acceptable parameter region, for eachpotential value of the sample mean.6.4 For two-stage sampling, the population is divided intoprimary sampling units (locations). Llocations are selected andfr
41、om each of them a subsample of n items is taken. Thevariance of a single observation, 2, is the sum of between-location and within-location variances.6.4.1 A confidence limit for 2is given by Graybill andWang (6) using the between and within location mean squaresfrom analysis of variance. When there
42、 are L locations withsubsamples of n items, the mean squares between locations andwithin locations, MSLand MSE, have L-1 and L(n-1) degreesof freedom respectively. Express the overall confidence levelas a product of confidence levels for the population mean andstandard deviation as in 6.3, so that (
43、1-) = (1-2)(1-). Anupper (1-) confidence limit for 2is:1/n! MSL11 2 1/n! MSE#1$1/n! (4)L 2 1!/L21, 1222 1!MSL#211 2 1/n!Ln 2 1!/Ln21!,1222 1!MSE#2%1/2The upper (1-) confidence limit for is the square root ofEq 4. Two sided (1-2) confidence limits for are:X6Z12=nL!(5)6.4.2 To verify, at confidence le
44、vel 1-, that a sample willpass the original acceptance procedure with probability at leastequal to the prespecified lower bound, values of (, ) definedFIG. 1 Example of Acceptance Limit Contour Showing a Simultaneous Confidence Interval With 95 % and 99 % Lower Bound ContoursE2709 1413by the limits
45、given in Eq 4 and Eq 5 should fall within theacceptable parameter region defined in 5.3.6.4.3 An acceptance limit table is constructed by fixing thesample within location standard deviation and the standarddeviation of location means and then finding the range ofoverall sample means such that the co
46、nfidence interval com-pletely falls below the pre-specified lower bound.7. Examples7.1 An example of an evaluation of a single-stage lotacceptance procedure is given in Appendix X1. An acceptancelimit table is shown for a sample size of 30, but other samplesizes may be considered.7.2 An example of a
47、n evaluation of a two-stage lot accep-tance procedure with one or more acceptance criterion at eachstage is given in Appendix X2. An acceptance limit table isshown for a sample size of 30.7.3 An example of an evaluation of a two-stage lot accep-tance procedure with one or more acceptance criteria at
48、 eachstage using Sampling Plan 2 is given in Appendix X3.Anacceptance limit table is shown for a sample size of 4 taken ateach of 15 locations for a total of 60 units tested.8. Keywords8.1 acceptance limits; joint confidence regions; multiple-stage acceptance procedures; specificationsAPPENDIXES(Non
49、mandatory Information)X1. EXAMPLE: EVALUATION OF A SINGLE STAGE ACCEPTANCE PROCEDUREX1.1 A single-stage lot acceptance procedure is stated asfollows: Sample five units at random from the lot and measurea numerical quality characteristic (Xi) of each unit. Criterion:Pass if all 5 individual units are between 95 and 105;otherwise, fail.X1.2 Assume that the test results follow a normal distribu-tion with mean and standard deviation . Let Z denote thestandard
