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.2. Referenced Documents2.1 ASTM Standards:2E456 Terminology Relating to Quality and Stat
7、isticsE2282 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 items in a sample orpo
8、pulation 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 size.E25863.1.3 multiple-
9、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 squared deviationof a vari
10、able 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 This Standard:3.2.1 acc
11、eptable 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 will attain a prescrib
12、ed 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.4. Significance and Use4.1 This practice consid
13、ers inspection procedures that mayinvolve multiple-stage sampling, where at each stage one candecide to accept or to continue sampling, and the decision 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
14、each individual testresult, or limits on statistics based on the sample of test results,1This practice is under 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 approve
15、d Oct. 1, 2014. Published October 2014. Originallyapproved in 2009. Last previous edition approved in 2012 as E2709 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 v
16、olume information, refer to the standards Document Summary page onthe ASTM website.Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States1such as the average, standard deviation, or coefficient ofvariation (relative standard deviation).4.2 T
17、he 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 equal to a prespecified lower bound.4.2.
18、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 processdemonstration, validation of test methods, and q
19、ualification 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 it is assumed that the process isstable a
20、nd 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 drug products inthe pharmaceutical indust
21、ry 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 definition of the acceptance procedure, is
22、 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 procedureconsists of multiple stages with multiple crite
23、ria, 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 each stage in the procedure having multip
24、leacceptance 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 stage i,P(Cij) = is the probability of passi
25、ng 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 Determine the contour of the region of p
26、arameter 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 statistic or set of statistics, derive ajoi
27、nt 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, whichconsists of values of the statistic
28、s 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 will lie within acceptancelimits should
29、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 be the acceptance region and thetighter
30、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 fallwithin the acceptance limits, then there is
31、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 case are given in this section, for twosamp
32、ling 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 ()ofthepopulation. The acceptable parameter region wi
33、ll be the regionunder a curve in the half-plane where is on the horizontalaxis, on the vertical axis, such as that depicted in Fig. 1.6.3 For simple random sampling from a normal population,the method of Lindgren (5) constructs a simultaneous confi-dence region of (, ) values from the sample average
34、 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 distribution with n-1degrees of freedom, respectively. Given a confidence level(1-), choose and such that (1-) = (1-2 )(1-). Althoughthere are many cho
35、ices for and that would satisfy this3The boldface numbers in parentheses refer to a list of references at the end ofthis standard.E2709 1412equation, a reasonable choice is: 512=12 and 512=12!/2 which equally splits the overall alpha betweenestimating and . Then:PHSX2 /=nD2# Z122JPHn 2 1!s22$ 2J5 1
36、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, the maximum value of thestandard deviation
37、 (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 the sample standarddeviation s (startin
38、g 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 andfrom each of them a subsample of n items i
39、s 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 are L locations withsubsamples of n ite
40、ms, 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 (1-) = (1-2 )(1-). Anupper (1-) confidenc
41、e 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 level 1-, that a sample willpass the orig
42、inal acceptance procedure with probability at leastequal to the prespecified lower bound, values of (, ) definedby the limits 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 sta
43、ndard deviation and the standarddeviation of location means and then finding the range ofoverall sample means such that the confidence 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 Appen
44、dix X1. An acceptanceFIG. 1 Example of Acceptance Limit Contour Showing a Simultaneous Confidence Interval With 95 % and 99 % Lower Bound ContoursE2709 1413limit table is shown for a sample size of 30, but other samplesizes may be considered.7.2 An example of an evaluation of a two-stage lot accep-t
45、ance 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 eachstage using Sampling Plan 2 is giv
46、en 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(Nonmandatory Information)X1. EXAMPLE: EVAL
47、UATION 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.
48、2 Assume that the test results follow a normal distribu-tion with mean and standard deviation . Let Z denote thestandard normal variate, that is, Z is normally distributed with=0and =1.X1.3 The criterion is 95 Xi 105 for i = 1, , 5. There-fore:Ppassing test! 5 P95 2 !/,Z,105 2 !/ !#5(X1.1)For any gi
49、ven values of and , the probability of passingStage 1 can be determined.X1.4 A simultaneous confidence region for and isgenerated using the methods of Lindgren (5). See 6.3.1.X1.5 The acceptance limit table for this example wasgenerated by a computer program and is listed in Table X1.1.The table corresponds to a sample size of 30 using a 95 %confidence interval and a 95 % lower bound, and it lists theoutput showing the upper bound on
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