Designation: E2709 19 An American National Standard Standard Practice for Demonstrating Capability to Comply with an Acceptance Procedure 1 This standard is issued under the ﬁxed designation E2709; the number immediately following the designation indicates the year of original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A superscript epsilon (´) indicates an editorial change since the last revision or reapproval. 1. Scope 1.1 This practice provides a general methodology for evalu- ating single-stage or multiple-stage acceptance procedures which involve a quality characteristic measured on a numerical scale. This methodology computes, at a prescribed conﬁdence level, a lower bound on the probability of passing an accep- tance procedure, using estimates of the parameters of the distribution 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, which deﬁnes a set of test method outcomes (for example, sample averages and standard deviations) that would pass the accep- 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 the safety concerns, if any, associated with its use. It is the responsibility of the user of this standard to establish appro- priate safety, health, and environmental practices and deter- mine the applicability of regulatory limitations prior to use. 1.6 This international standard was developed in accor- dance with internationally recognized principles on standard- ization established in the Decision on Principles for the Development of International Standards, Guides and Recom- mendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee. 2. Referenced Documents 2.1 ASTM Standards: 2 E456 Terminology Relating to Quality and Statistics E2282 Guide for Deﬁning the Test Result of a Test Method E2586 Practice for Calculating and Using Basic Statistics 3. Terminology 3.1 Deﬁnitions—See Terminology E456 for a more exten- sive listing of terms in ASTM Committee E11 standards. 3.1.1 characteristic, n—a property of items in a sample or population which, when measured, counted or otherwise observed, helps to distinguish between the items. E2282 3.1.2 mean, n—of a population, µ, average or expected value of a characteristic in a population, of a sample X ¯ , sum of the observed values in a sample divided by the sample size. E2586 3.1.3 multiple-stage acceptance procedure, n—a procedure that involves more than one stage of sampling and testing a givenqualitycharacteristicandoneormoreacceptancecriteria per stage. 3.1.4 standard deviation, n—of a population, σ, the square root of the average or expected value of the squared deviation of a variable from its mean – of a sample, s, the square root of the sum of the squared deviations of the observed values in the sample divided by the sample size minus 1. E2586 3.1.5 test method, n—a deﬁnitive procedure that produces a test result. E2282 3.2 Deﬁnitions of Terms Speciﬁc to This Standard: 3.2.1 acceptable parameter region, n—the set of values of parameters characterizing the distribution of test results for which the probability of passing the acceptance procedure is greater than a prescribed lower bound. 3.2.2 acceptance region, n—the set of values of parameter estimates that will attain a prescribed lower bound on the probability of passing an acceptance procedure at a prescribed level of conﬁdence. 3.2.3 acceptance limit, n—the boundary of the acceptance region, for example, the maximum sample standard deviation test results for a given sample mean. 4. Signicance and Use 4.1 This practice considers inspection procedures that may involve multiple-stage sampling, where at each stage one can 1 This practice is under the jurisdiction ofASTM Committee E11 on Quality and Statistics and is the direct responsibility of Subcommittee E11.20 on Test Method Evaluation and Quality Control. Current edition approved April 1, 2019. Published April 2019. Originally approved in 2009. Last previous edition approved in 2014 as E2709 – 14 ɛ 1 . DOI: 10.1520/E2709-19. 2 For referenced ASTM standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at

[email protected] For Annual Book of ASTM Standards volume information, refer to the standard’s Document Summary page on the ASTM website. Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee. 1decide to accept or to continue sampling, and the decision to reject is deferred until the last stage. 4.1.1 Ateachstagethereareoneormoreacceptancecriteria on the test results; for example, limits on each individual test result, or limits on statistics based on the sample of test results, such as the average, standard deviation, or coefficient of variation (relative standard deviation). 4.2 The methodology in this practice deﬁnes an acceptance regionforasetoftestresultsfromthesampledpopulationsuch that, at a prescribed conﬁdence level, the probability that a sample from the population will pass the acceptance procedure is greater than 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 provides assurance that a sample would 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),butthelowerboundmaybeconservativeinsome 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 stable and predictable. If this is not the case then there can be no guarantee that the probability estimates would be valid predictions of future process performance. 4.3 This methodology was originally developed (1-4) 3 for use in two speciﬁc quality characteristics of drug products in the pharmaceutical industry but will be applicable for accep- tance procedures in all industries. 4.4 Mathematical derivations would be required that are speciﬁc to the individual criteria of each test. 5. Methodology 5.1 The process for deﬁning the acceptance limits, starting from the deﬁnition of the acceptance procedure, is outlined in this section. A computer program is normally required to produce the acceptable parameter region and the acceptance limits. 5.1.1 An expression for the exact probability of passing the acceptance procedure might be intractable when the procedure consists 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 pro- cedure as a function of the parameters characterizing the distribution of 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 prob- ability of that stage as a function of the probabilities of passing each of the criteria in the stage: P~S i ! 5 P~C i1 and C i2 … and C im !$ 1 2 (j51 m ~1 2 P~C ij !! (1) where: P(S i ) = is the probability of passing stage i, P(C ij ) = 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 a function of probabilities of passing each of the individual stages: P ~pass k 2 stage procedure!$ max$P~S 1 !, P~S 2 !, … , P~S k !% (2) 5.3 Determine the contour of the region of parameter values for which the expression for the probability of passing the given acceptance procedure is at least equal to the required lower bound (LB) on the probability of acceptance (p). This deﬁnes the acceptable parameter region. Since the acceptance parameter region is a lower bound, it should be compared to the simulated probability of passing the acceptance procedure. 5.4 For each value of a statistic or set of statistics, derive a joint conﬁdence region for the distribution parameters at conﬁdence level, expressed as a percentage, of 100(1-α). The size of sample to be taken, n, and the statistics to be used, must be predetermined (see 5.6). 5.5 Determine the contour of the acceptance region, which consists of values of the statistics for which the conﬁdence region at level 100(1-α) is entirely contained in the acceptable parameter region. The acceptance limits lie on the contour of the acceptance region. 5.6 To select the size of sample, n, to be taken, the probability that sample statistics will lie within acceptance limits should be evaluated over a range of values of n, for values of population parameters of practical interest, and for which probabilities of passing the given acceptance procedure are well above the lower bound. The larger the sample size n that is chosen, the larger will be the acceptance region and the tighter the distribution of the statistics. Choose n so that the probability of passing acceptance limits 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 within the acceptance limits, then there is 1-α conﬁdence that the probability of acceptance is at least p. 6. Procedures for Sampling from a Normal Distribution 6.1 An important class of procedures is for the case where the quality characteristic is normally distributed. Particular instructions for that case are given in this section, for two sampling methods, simple random and two-stage. In this standard, these sampling methods are denoted Sampling Plan 1 and Sampling Plan 2, respectively. 6.2 When the characteristic is normally distributed, param- eters are the mean (µ) and standard deviation (σ)o ft h e population. The acceptable parameter region will be the region under a curve in the half-plane where µ is on the horizontal axis, σ 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 (µ, σ) values from the sample average X ¯ and the sample standard deviation s of n test results. 3 The boldface numbers in parentheses refer to a list of references at the end of this standard. E2709 19 26.3.1 Let Z p and χ p 2 denote percentiles of the standard normal distribution and of the chi-square distribution with n-1 degrees of freedom, respectively. Given a conﬁdence level (1-α), choose δ and ε such that (1-α) = (1-2δ)(1-ε). Although there are many choices for δ and ε that would satisfy this equation, a reasonable choice is: ε512=12α and δ5~12=12α!/2 which equally splits the overall alpha be- tween estimating µ and σ. Then: P HS X ¯ 2 µ σ/=n D 2 # Z 12δ 2 J P H ~n 2 1!s 2 σ 2 $ χ ε 2 J 5 ~1 2 2δ!~1 2 ε! 5 1 2 α (3) 6.3.2 The conﬁdence region for (µ, σ), two-sided for µ, one-sided for σ, 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 the standard deviation (or coefficient of variation) that would meet these requirements. Using a computer program that calculates conﬁdence limits for µ and σ given sample mean X ¯ and standard deviation s, the acceptance limit can be derived using an iterative loop over increasing values of the sample standard deviation s (starting with s = 0) until the conﬁdence limits hit the boundary of the acceptable parameter region, for each potential value of the sample mean. 6.4 For two-stage sampling, the population is divided into primarysamplingunits(locations).Llocationsareselectedand from each of them a subsample of n items is taken. The variance of a single observation, σ 2 , is the sum of between- location and within-location variances. 6.4.1 A conﬁdence limit for σ 2 is given by Graybill and Wang (6) using the between and within location mean squares from analysis of variance. When there are L locations with subsamplesof nitems,themeansquaresbetweenlocationsand within locations, MS L and MS E , have L-1 and L(n-1) degrees of freedom respectively. Express the overall conﬁdence level as a product of conﬁdence levels for the population mean and standard deviation as in 6.3, so that (1-α) = (1-2δ)(1-ε). An upper (1-ε) conﬁdence limit for σ 2 is: @~1/n! MS L 1~1 21/n! MS E #

[email protected]~1/n! (4) ~~L 2 1!/χ L21, 12ϵ 2 2 1 ! MS L # 2

[email protected]~1 21/n! ~ L~n 2 1!/χ L~n21!,1 2ϵ 2 2 1 ! MS E # 2 % 1/2 The upper (1-ε) conﬁdence limit for σ is the square root of Eq 4. Two sided (1-2δ) conﬁdence limits for µ are: X ¯ 6Z 12δ σ =~nL! (5) 6.4.2 To verify, at conﬁdence level 1-α, that a sample will pass the original acceptance procedure with probability at least equal to the prespeciﬁed lower bound, values of (µ, σ) deﬁned by the limits given in Eq 4 and Eq 5 should fall within the acceptable parameter region deﬁned in 5.3. 6.4.3 An acceptance limit table is constructed by ﬁxing the sample within location standard deviation and the standard deviation of location means and then ﬁnding the range of FIG. 1 Example of Acceptance Limit Contour Showing a Simultaneous Condence Interval With 95 % and 99 % Lower Bound Contours E2709 19 3overall sample means such that the conﬁdence interval com- pletely falls below the pre-speciﬁed lower bound. 7. Examples 7.1 An example of an evaluation of a single-stage lot acceptance procedure is given in Appendix X1.An acceptance limit table is 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 accep- tance procedure with one or more acceptance criterion at each stage is given in Appendix X2. An acceptance limit table is shown 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 each stage using Sampling Plan 2 is given in Appendix X3.An acceptance limit table is shown for a sample size of 4 taken at each of 15 locations for a total of 60 units tested. 8. Keywords 8.1 acceptance limits; joint conﬁdence regions; multiple- stage acceptance procedures; speciﬁcations APPENDIXES (Nonmandatory Information) X1. EXAMPLE: EV ALUATION OF A SINGLE STAGE ACCEPTANCE PROCEDURE X1.