1、Designation: E2709 12E2709 14 An 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
2、 case of 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.1. Scope1.1 This practice provides a general methodology for evaluating single-stage or multiple-st
3、age acceptance procedures whichinvolve a quality characteristic measured on a numerical scale. This methodology computes, at a prescribed confidence level, alower bound on the probability of passing an acceptance procedure, using estimates of the parameters of the distribution of testresults from a
4、sampled population.1.2 For a prescribed lower probability bound, the methodology can also generate an acceptance limit table, which defines a setof test method outcomes (for example, sample averages and standard deviations) that would pass the acceptance procedure at aprescribed confidence level.1.3
5、 This approach may be used for demonstrating compliance 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 the safety concerns, if any, associated with its use. It is the responsib
6、ilityof the user of this standard to establish appropriate safety and health practices and determine the applicability of regulatorylimitations prior to use.2. Referenced Documents2.1 ASTM Standards:2E456 Terminology Relating to Quality and StatisticsE2282 Guide for Defining the Test Result of a Tes
7、t MethodE2586 Practice for Calculating and Using Basic Statistics3. Terminology3.1 DefinitionsSee Terminology E456 for a more extensive listing of terms in ASTM Committee E11 standards.3.1.1 characteristic, na property of items in a sample or population which, when measured, counted or otherwise obs
8、erved,helps to distinguish between the items. E22823.1.2 mean, nof a population, , average or expected value of a characteristic in a population, of a sampleX , sum of theobserved values in a sample divided by the sample size. E25863.1.3 multiple-stage acceptance procedure, na procedure that involve
9、s more than one stage of sampling and testing a givenquality characteristic and one or more acceptance criteria per stage.3.1.4 standard deviation, nof a population, , the square root of the average or expected value of the squared deviation of avariable from its mean of a sample, s, the square root
10、 of the sum of the squared deviations of the observed values in the sampledivided by the sample size minus 1. E25863.1.5 test method, na definitive procedure that produces a test result. E22823.2 Definitions of Terms Specific to This Standard:1 This practice is under the jurisdiction of ASTM Committ
11、ee E11 on Quality and Statistics and is the direct responsibility of Subcommittee E11.20 on Test MethodEvaluation and Quality Control.Current edition approved Nov. 1, 2012Oct. 1, 2014. Published December 2012October 2014. Originally approved in 2009. Last previous edition approved in 20112012as E270
12、9 11.E2709 12. DOI: 10.1520/E2709-12.10.1520/E2709-14.2 For referencedASTM standards, visit theASTM website, www.astm.org, or contactASTM Customer Service at serviceastm.org. For Annual Book of ASTM Standardsvolume information, refer to the standards Document Summary page on the ASTM website.This do
13、cument is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of what changes have been made to the previous version. Becauseit may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions a
14、s appropriate. In all cases only the current versionof the standard as published by ASTM is to be considered the official document.Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States13.2.1 acceptable parameter region, nthe set of values o
15、f parameters characterizing the distribution of test results for whichthe probability of passing the acceptance procedure is greater than a prescribed lower bound.3.2.2 acceptance region, nthe set of values of parameter estimates that will attain a prescribed lower bound on the probabilityof passing
16、 an acceptance procedure at a prescribed level of confidence.3.2.3 acceptance limit, nthe boundary of the acceptance region, for example, the maximum sample standard deviation testresults for a given sample mean.4. Significance and Use4.1 This practice considers inspection procedures that may involv
17、e multiple-stage sampling, where at each stage one can decideto accept or to continue sampling, and the decision to reject is deferred until the last stage.4.1.1 At each stage there are one or more acceptance criteria on the test results; for example, limits on each individual test result,or limits
18、on statistics based on the sample of test results, such as the average, standard deviation, or coefficient of variation (relativestandard deviation).4.2 The methodology in this practice defines an acceptance region for a set of test results from the sampled population such that,at a prescribed confi
19、dence level, the probability that a sample from the population will pass the acceptance procedure is greaterthan or equal to a prespecified lower bound.4.2.1 Having test results fall in the acceptance region is not equivalent to passing the acceptance procedure, but providesassurance that a sample w
20、ould pass the acceptance procedure with a specified probability.4.2.2 This information can be used for process demonstration, validation of test methods, and qualification of instruments,processes, and materials.4.2.3 This information can be used for lot release (acceptance), but the lower bound may
21、 be conservative in some cases.4.2.4 If the results are to be applied to future test results from the same process, then it is assumed that the process is stableand predictable. If this is not the case then there can be no guarantee that the probability estimates would be valid predictions offuture
22、process performance.4.3 This methodology was originally developed (1-4)3 for use in two specific quality characteristics of drug products in thepharmaceutical industry but will be applicable for acceptance procedures in all industries.4.4 Mathematical derivations would be required that are specific
23、to the individual criteria of each test.5. Methodology5.1 The process for defining the acceptance limits, starting from the definition of the acceptance procedure, is outlined in thissection. A computer program is normally required to produce the acceptable parameter region and the acceptance limits
24、.5.1.1 An expression for the exact probability of passing the acceptance procedure might be intractable when the procedureconsists of multiple stages with multiple criteria, hence a lower bound for the probability may be used.5.2 Express the probability of passing the acceptance procedure as a funct
25、ion of the parameters characterizing the distributionof the quality characteristic for items in the sampled population.5.2.1 For each stage in the procedure having multiple acceptance criteria, determine the lower bound on the probability of thatstage as a function of the probabilities of passing ea
26、ch of the criteria in the stage:PSi! 5PCi1 and Ci2 and Cim!$12(j51m 12PCij! (1)where:P(Si) = is the probability of passing stage i,P(Cij) = is the probability of passing the j-th criterion of m within the i-th stage.5.2.2 Determine the lower bound on the probability of passing a k-stage procedure as
27、 a function of probabilities of passing eachof the individual stages:P pass k 2stage procedure!$max$PS1!, PS2!, PSk!% (2)5.3 Determine the contour of the region of parameter values for which the expression for the probability of passing the givenacceptance procedure is at least equal to the required
28、 lower bound (LB) on the probability of acceptance (p). This defines theacceptable parameter region.5.4 For each value of a statistic or set of statistics, derive a joint confidence region for the distribution parameters at confidencelevel, expressed as a percentage, of 100(1-). The size of sample t
29、o be taken, n, and the statistics to be used, must be predetermined(see 5.6).3 The boldface numbers in parentheses refer to a list of references at the end of this standard.E2709 1425.5 Determine the contour of the acceptance region, which consists of values of the statistics for which the confidenc
30、e regionat level 100(1-) is entirely contained in the acceptable parameter region.The acceptance limits lie on the contour of the acceptanceregion.5.6 To select the size of sample, n, to be taken, the probability that sample statistics will lie within acceptance limits should beevaluated over a rang
31、e of values of n, for values of population parameters of practical interest, and for which probabilities ofpassing the given acceptance procedure are well above the lower bound. The larger the sample size n that is chosen, the larger willbe the acceptance region and the tighter the distribution of t
32、he statistics. Choose n so that the probability of passing acceptancelimits is greater than a predetermined value.5.7 To use the acceptance limit, sample randomly from the population. Compute statistics for the sample. If statistics fall withinthe acceptance limits, then there is 1- confidence that
33、the 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 where the quality characteristic is normally distributed. Particularinstructions for that case are given in this section, for two sampling methods, si
34、mple random and two-stage. In this standard thesesampling methods are denoted Sampling Plan 1 and Sampling Plan 2, respectively.6.2 When the characteristic is normally distributed, parameters are the mean () and standard deviation () of the population.The acceptable parameter region will be the regi
35、on under a curve in the half-plane where is on the horizontal axis, on thevertical 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 confidenceregion of (, ) values from the sample average X and the sam
36、ple standard deviation s of n test results.6.3.1 Let Zp and p2 denote percentiles of the standard normal distribution and of the chi-square distribution with n-1 degreesof freedom, respectively. Given a confidence level (1-), choose and such that (1-) = (1-2 )(1-). Although there are manychoices for
37、 and that would satisfy this equation, a reasonable choice is: 512=12 and 512=12!/2 which equally splitsthe overall alpha between estimating and . Then:PHS X 2/=nD2#Z122 JPH n 21!s22 $2 J5 122!12!5 12(3)FIG. 1 Example of Acceptance Limit Contour Showing a Simultaneous Confidence Interval With 95 % a
38、nd 99 % Lower Bound ContoursE2709 1436.3.2 The confidence region for (, ), two-sided for , one-sided for , is an inverted triangle with a minimum vertex at 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
39、thestandard deviation (or coefficient of variation) that would meet these requirements. Using a computer program that calculatesconfidence limits for and given sample mean X and standard deviation s, the acceptance limit can be derived using an iterativeloop over increasing values of the sample stan
40、dard deviation s (starting with s = 0) until the confidence limits hit the boundary ofthe acceptable parameter region, for each potential value of the sample mean.6.4 For two-stage sampling, the population is divided into primary sampling units (locations). L locations are selected and fromeach of t
41、hem a subsample of n items is taken. The variance of a single observation, 2, is the sum of between-location andwithin-location variances.6.4.1 A confidence limit for 2 is given by Graybill and Wang (6) using the between and within location mean squares fromanalysis of variance. When there are L loc
42、ations with subsamples of n items, the mean squares between locations and withinlocations, MSL and MSE, have L-1 and L(n-1) degrees of freedom respectively. Express the overall confidence level as a productof confidence levels for the population mean and standard deviation as in 6.3, so that (1-) =
43、(1-2 )(1-).An upper (1-) confidencelimit for 2 is:1/n! MSL1121/n! MSE#1$1/n! (4)L 21!/L21, 122 21!MSL#21121/n!Ln 21!/Ln21!, 122 21!MSE#2%1/2The upper (1-) confidence limit for is the square root of Eq 4. Two sided (1-2) confidence limits for are:X6Z12 =nL!(5)6.4.2 To verify, at confidence level 1-,
44、that a sample will pass the original acceptance procedure with probability at least equalto the prespecified lower bound, values of (, ) defined by the limits given in Eq 4 and Eq 5 should fall within the acceptableparameter region defined in 5.3.6.4.3 An acceptance limit table is constructed by fix
45、ing the sample within location standard deviation and the standard deviationof location means and then finding the range of overall sample means such that the confidence interval completely falls below thepre-specified lower bound.7. Examples7.1 An example of an evaluation of a single-stage lot acce
46、ptance procedure is given in Appendix X1.An acceptance limit tableis shown for a sample size of 30, but other sample sizes may be considered.7.2 An example of an evaluation of a two-stage lot acceptance procedure with one or more acceptance criterion at each stageis given in Appendix X2. An acceptan
47、ce limit table is shown for a sample size of 30.7.3 An example of an evaluation of a two-stage lot acceptance procedure with one or more acceptance criteria at each stageusing Sampling Plan 2 is given in Appendix X3. An acceptance limit table is shown for a sample size of 4 taken at each of 15locati
48、ons for a total of 60 units tested.8. Keywords8.1 acceptance limits; joint confidence regions; multiple-stage acceptance procedures; specificationsE2709 144APPENDIXES(Nonmandatory Information)X1. EXAMPLE: EVALUATION OF A SINGLE STAGE ACCEPTANCE PROCEDUREX1.1 A single-stage lot acceptance procedure i
49、s stated as follows: Sample five units at random from the lot and measure anumerical 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 distribution with mean and standard deviation . Let Z denote the standardnormal variate, that is, Z is normally distributed with = 0 and = 1.X1.3 The criterion is 95 Xi 105 for i = 1, , 5. Th
