1、Designation: E 2709 09Standard Practice forDemonstrating Capability to Comply with a Lot AcceptanceProcedure1This standard is issued under the fixed designation E 2709; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of la
2、st 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 evalu-ating single-stage or multiple-stage lot acceptance procedureswhi
3、ch involve a quality characteristic measured on a numericalscale. This methodology computes, at a prescribed confidencelevel, a lower bound on the probability of passing a lotacceptance procedure, using estimates of the parameters of thedistribution of test results from the lot.1.2 For a prescribed
4、lower probability bound, the method-ology can also generate an acceptance limit table, whichdefines a set of test method outcomes (e.g., sample averagesand standard deviations) that would pass the multiple-stageprocedure at a prescribed confidence level.1.3 This approach may be used for demonstratin
5、g 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, if any, associated with its use. It is theresponsibility of the user of this standard to establ
6、ish appro-priate safety and health practices and determine the applica-bility of regulatory limitations prior to use.2. Referenced Documents2.1 ASTM Standards:2E 456 Terminology Relating to Quality and StatisticsE 2234 Practice for Sampling a Stream of Product byAttributes Indexed by AQLE 2281 Pract
7、ice for Process and Measurement CapabilityIndicesE 2282 Guide for Defining the Test Result of a Test MethodE 2586 Practice for Calculating and Using Basic StatisticsE 2587 Practice for Use of Control Charts in StatisticalProcess Control3. Terminology3.1 DefinitionsSee Terminology E 456 for a more ex
8、ten-sive listing of terms in ASTM Committee E11 standards.3.1.1 characteristic, na property of items in a sample orpopulation which, when measured, counted or otherwise ob-served, helps to distinguish between the items. E 22823.1.2 mean, n of a population, , average or expectedvalue of a characteris
9、tic in a population, of a sample X , sum ofthe observed values in a sample divided by the sample size.E 25863.1.3 standard deviation, nof a population, s, the squareroot of the average or expected value of the squared deviationof a variable from its mean of a sample, s, the square root ofthe sum of
10、the squared deviations of the observed values in thesample divided by the sample size minus 1. E 25863.1.4 test method, na definitive procedure that produces atest result. E 22823.2 Definitions of Terms Specific to This Standard:3.2.1 acceptable parameter region, nthe set of values ofparameters char
11、acterizing the distribution of test results forwhich the probability of passing the lot acceptance procedureis greater than a prescribed lower bound.3.2.2 acceptance region, nthe set of values of parameterestimates that will attain a prescribed lower bound on theprobability of passing a lot acceptan
12、ce procedure at a pre-scribed level of confidence.3.2.3 acceptance limit, nthe boundary of the acceptanceregion, e.g., the maximum sample standard deviation testresults for a given sample mean.3.2.4 multiple-stage lot acceptance procedure, na proce-dure for accepting a lot that involves more than on
13、e stage ofsampling and testing a given quality characteristic and one ormore acceptance criteria per stage.4. Significance and Use4.1 Lot acceptance procedures are used in industry forinspecting quality characteristics of raw materials, in-processproduct, and finished product. These procedures, toge
14、ther withprocess controls, comprise a quality control program. Foradditional information on process control see Practice E 22811This practice is under the jurisdiction ofASTM Committee E11 on Quality andStatistics and is the direct responsibility of Subcommittee E11.30 on StatisticalQuality Control.
15、Current edition approved Aug. 1, 2009. Published September 2009.2For 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 websi
16、te.1Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.dealing with process capability evaluation and Practice E 2587dealing with the use of control charts in statistical processcontrol.4.1.1 Lot inspection procedures classify quality ch
17、aracteris-tics as either attributes (measured on discrete scales such aspercent defective) or variables (measured on continuous scalessuch as length, weight, or concentration).4.1.2 Operating characteristic curves, which plot the rela-tionship of the lot acceptance probability versus the true lotper
18、cent defective, are used to evaluate the discriminatorypower of a given lot inspection procedure, or acceptancesampling plan, and are discussed in Practice E 2234.4.2 This practice considers inspection procedures that mayinvolve multiple-stage sampling, where at each stage one candecide to accept th
19、e lot or to continue sampling, and thedecision to reject the lot is deferred until the last stage.4.2.1 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 t
20、he average, standard deviation, or coefficient ofvariation (relative standard deviation).4.3 The methodology in this practice defines an acceptanceregion for a set of test results from the lot such that, at aprescribed confidence level, the probability that a sample fromthe lot will pass the origina
21、l lot acceptance procedure is greaterthan or equal to a prespecified lower bound.4.3.1 Having test results fall in the acceptance region is notequivalent to passing the original lot acceptance procedure, butprovides assurance that a sample would pass the lot acceptanceprocedure with a specified prob
22、ability.4.3.2 This information can be used for process demonstra-tion or validation.4.3.3 This information can be used for lot release (accep-tance), but the lower bound may be conservative in some cases.4.3.4 If the results are to be applied to test results fromfuture lots from the same process, th
23、en it is assumed that theprocess is in a state of statistical control (see 4.1). If this is notthe case then there can be no guarantee that the probabilityestimates would be valid predictions of future process perfor-mance.4.4 This methodology was originally developed by J. S.Bergum (1-4) for use in
24、 two specific quality characteristics ofdrug products in the pharmaceutical industry: content unifor-mity and dissolution, as respectively defined in chapters and of the United States Pharmacopeia (5).4.5 Mathematical derivations would be required that arespecific to the individual criteria of each
25、test.5. Methodology5.1 The process for defining the acceptance limits, startingfrom the definition of the original lot acceptance procedure, isoutlined. A computer program is normally required to producethe acceptable parameter region and acceptance limits.5.1.1 An important class of procedures is f
26、or the case wherethe quality characteristic is normally distributed. Particularinstructions for that case are given in this section.5.2 Express the probability of passing the given lot accep-tance procedure as a function of parameters characterizing thedistribution of the quality characteristic for
27、items in the lot.5.2.1 When the characteristic is normally distributed, pa-rameters are the mean () and standard deviation (s) of the lot.5.2.2 An expression for the exact probability of passing thelot acceptance procedure may be intractable.Alower bound forthe probability may be used. For multiple
28、stage tests, thefollowing lower bounds on the probability of passing theprocedure as a function of probabilities of passing stages, andon the probability of passing a stage having multiple criteria asa function of the probabilities of passing the criteria, may beuseful (4).P pass k stage procedure!
29、$max $PS1!, PS2!,.,PSk!% (1)where:P(Si) = is the probability of passing stage i, evaluatedregardless of whether previous stages pass or not.PSi! 5 PCi1and Ci2.and Cim! $1(mj511PCij! (2)where:P(Cij) = is the probability of passing the j-th criterion of mwithin the i-th stage.5.3 Determine the contour
30、 of the region of parameter valuesfor which the expression for the probability of passing thegiven lot acceptance procedure is at least equal to the requiredlower bound (LB) on the probability of acceptance (p). Thisdefines the region of acceptable parameters.5.3.1 For a normally distributed populat
31、ion, this will be aregion under a curve in the half-plane where is on thehorizontal axis, s on the vertical axis, such as that depicted inFig. 1.5.4 For each value of a statistic or set of statistics, derive ajoint confidence region (confidence coefficient 1-a) for thedistribution parameters. The si
32、ze of sample to be taken, n, andthe statistics to be used, must be predetermined.5.4.1 For a normally distributed lot, the method of Lindgren(6) constructs a simultaneous confidence region of (, s)values from the sample average X and the sample standarddeviation s from a set of n test results. Let Z
33、pand xp2denotepercentiles of the standard normal distribution and of thechi-square distribution with n-1 degrees of freedom, respec-tively. Given a confidence level 100(1-a), choose d and e suchthat (1-a) = (1-2d )(1-e). The valuese51=1aandd51=1a!/2meet this condition. ThenPHSX s/=nD2#Z21 dJPHns2s2#
34、x21eJ5 12d!1e! 5 1a!(3)The region for (, s), two-sided for , one-sided for s,isaninverted triangle with a minimum vertex at ( X ,0), as depictedin Fig. 1.5.5 Determine the contour of the acceptance region, whichconsists of values of the statistics ( X , s) for which theconfidence region (confidence
35、level 1-a) is entirely containedin the acceptable parameter region.This is the acceptance limit.5.5.1 For a normally distributed characteristic, the accep-tance limit takes the form of a table giving, for each value ofE2709092the sample mean, the minimum and/or maximum value of thestandard deviation
36、 (or coefficient of variation) that would meetthese requirements.5.5.2 Using a computer program that calculates confidencelimits for and s given sample mean X and standard deviations, the acceptance limit for a normally distributed characteristiccan be derived using an iterative loop over increasing
37、 values ofthe sample standard deviation s (starting withs=0)until theconfidence limits hit the boundary of the acceptable parameterregion, for each potential value of the sample mean.5.6 To use the acceptance limit, sample randomly from thebatch or lot. Evaluate statistics for the sample. If statist
38、ics fallwithin the acceptance limit, then there is 1-a confidence thatthe probability of acceptance is at least p.6. Examples6.1 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 samples
39、izes may be considered.6.2 An example of an 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. Keywords7.1 acceptance limits; acceptance sampling inspection;multi
40、ple-stage lot acceptance procedures; simultaneous confi-dence regions; specificationsFIG. 1 Acceptance Limit Contour Showing a Simultaneous Confidence Interval With 95% and 99% Lower Bound ContoursE2709093APPENDIXESX1. Example of a Single Stage Acceptance ProcedureX1.1 A single -stage lot acceptance
41、 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 stan
42、dard deviation s. Let Z denote thestandard normal variate, i.e., Z is normally distributed with =0 and s =1.X1.3 The criterion is 95 # Xi#105 for i = 1, 5.Therefore:P(Passing Test) = P(95 )/s Z (105 )/s )5For any given values of and s, the probability of passingStage 1 can be determined.X1.4 A simul
43、taneous confidence region for and s isgenerated using the methods of Lindgren (6). See 5.4.1X1.5 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 boun
44、d, and it lists theoutput showing the upper bound on the sample standarddeviation for sample means between 97 and 103.X1.6 ASASt program for the generation of the acceptancetable follows. See Fig. X1.1.TABLE X1.1 Acceptance Limit Table (5% Confidence Interval/95% Coverage)Mean Standard Deviation96.0
45、 0.27397.0 0.54698.0 0.81999.0 1.092100.0 1.350101.0 1.092102.0 0.819103.0 0.546104.0 0.273E2709094FIG. X1.1 SAS ProgramE2709095X2. Example of a Multiple-Stage Acceptance ProcedureX2.1 A multiple-stage lot acceptance procedure is stated asfollows:Stage 1: Sample five units at random from the lot and
46、measure a numerical quality characteristic (Xi) of each unit.Criterion: Pass if all 5 individual units are between 95 and 105;otherwise go to Stage 2.Stage 2: Randomly sample five additional units from the lotand measure a numerical quality characteristic (Xi) of eachunit. Criteria: Pass if the aver
47、age of the 10 test results isbetween 97 and 103 and all 10 individual results are between90 and 110; otherwise fail.X2.2 To obtain the lower bound on the probability ofacceptance, assume that the test results follow a normaldistribution with mean and standard deviation s. Let Zdenote the standard no
48、rmal variate, i.e., Z is normally distrib-uted with=0ands = 1. Let Sidenote the event that the testresults meet the acceptance criteria for Stage i, and let Cijdenote the event that the jthcriterion for the ith stage is met.X2.2.1 For Stage 1 the criterion C11is 95#Xi#105 for i =1,5.Therefore:P(S1)
49、= P(C11) = P(95 )/s Z (105 )/s)5For any given values of and s, theprobability of passingstage 1 can be determined.X2.2.2 For Stage 2 the criterionC21is 97#X#103, where X is the average of the 10 test results, and the criterionC22is 90 # Xi#110fori=1,10.Therefore:PC21! 5 P =10 97 !/s,Z , =10103 !/s!PC22! 5 P90 !/s,Z , 110 !/s!#10PS2! $PC21! 1 PC22! 1 (X2.1)For any given values of and s, the probability of passingstage 2 can be determined.X2.3 The Lower Bound (LB) = Max
