ASTM E2709-19(2023)
(Practice)Standard Practice for Demonstrating Capability to Comply with an Acceptance Procedure
Standard Practice for Demonstrating Capability to Comply with an Acceptance Procedure
ABSTRACT
This practice provides a general methodology for evaluating 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 acceptance procedure, using estimates of the parameters of the distribution of test results from a sampled population.
SIGNIFICANCE AND USE
4.1 This practice considers inspection procedures that may involve multiple-stage sampling, where at each stage one can decide to 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 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 region for a set of test results from the sampled population such 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), but the lower bound may 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 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 acceptance procedures in all industries.
4.4 Mathematical derivations would be required that are specific to the individual criteria of each test.
SCOPE
1.1 This practice provides a general methodology for evaluating 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 acceptance 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 methodology 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 acceptance procedure at a prescribed confidence level.
1.3 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 responsibility of the user of this standard to establish appropriate safety, health, and environmental practices and determine the applicability of regulatory limitations prior to use.
1.6 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.
General Information
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Standards Content (Sample)
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.
Designation: E2709 − 19 (Reapproved 2023) An American National Standard
Standard Practice for
Demonstrating Capability to Comply with an Acceptance
Procedure
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 E2282 Guide for Defining the Test Result of a Test Method
E2586 Practice for Calculating and Using Basic Statistics
1.1 This practice provides a general methodology for evalu-
ating single-stage or multiple-stage acceptance procedures
3. Terminology
which involve a quality characteristic measured on a numerical
scale. This methodology computes, at a prescribed confidence 3.1 Definitions—See Terminology E456 for a more exten-
level, a lower bound on the probability of passing an accep-
sive listing of terms in ASTM Committee E11 standards.
tance procedure, using estimates of the parameters of the
3.1.1 characteristic, n—a property of items in a sample or
distribution of test results from a sampled population.
population which, when measured, counted or otherwise
observed, helps to distinguish between the items. E2282
1.2 For a prescribed lower probability bound, the method-
ology can also generate an acceptance limit table, which 3.1.2 mean, n—of a population, μ, average or expected
¯
defines a set of test method outcomes (for example, sample
value of a characteristic in a population, of a sample X, sum of
averages and standard deviations) that would pass the accep-
the observed values in a sample divided by the sample size.
tance procedure at a prescribed confidence level.
E2586
1.3 This approach may be used for demonstrating compli-
3.1.3 multiple-stage acceptance procedure, n—a procedure
ance with in-process, validation, or lot-release specifications. that involves more than one stage of sampling and testing a
given quality characteristic and one or more acceptance criteria
1.4 The system of units for this practice is not specified.
per stage.
1.5 This standard does not purport to address all of the
3.1.4 standard deviation, n—of a population, σ, the square
safety concerns, if any, associated with its use. It is the
root of the average or expected value of the squared deviation
responsibility of the user of this standard to establish appro-
of a variable from its mean – of a sample, s, the square root of
priate safety, health, and environmental practices and deter-
the sum of the squared deviations of the observed values in the
mine the applicability of regulatory limitations prior to use.
sample divided by the sample size minus 1. E2586
1.6 This international standard was developed in accor-
dance with internationally recognized principles on standard- 3.1.5 test method, n—a definitive procedure that produces a
ization established in the Decision on Principles for the test result. E2282
Development of International Standards, Guides and Recom-
3.2 Definitions of Terms Specific to This Standard:
mendations issued by the World Trade Organization Technical
3.2.1 acceptable parameter region, n—the set of values of
Barriers to Trade (TBT) Committee.
parameters characterizing the distribution of test results for
which the probability of passing the acceptance procedure is
2. Referenced Documents
greater than a prescribed lower bound.
2.1 ASTM Standards:
3.2.2 acceptance region, n—the set of values of parameter
E456 Terminology Relating to Quality and Statistics
estimates that will attain a prescribed lower bound on the
probability of passing an acceptance procedure at a prescribed
level of confidence.
This practice is under the jurisdiction of ASTM Committee E11 on Quality and
3.2.3 acceptance limit, n—the boundary of the acceptance
Statistics and is the direct responsibility of Subcommittee E11.20 on Test Method
Evaluation and Quality Control.
region, for example, the maximum sample standard deviation
Current edition approved April 1, 2023. Published April 2023. Originally
test results for a given sample mean.
approved in 2009. Last previous edition approved in 2019 as E2709 – 19. DOI:
10.1520/E2709-19R23.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or 4. Significance and Use
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
4.1 This practice considers inspection procedures that may
Standards volume information, refer to the standard’s Document Summary page on
the ASTM website. involve multiple-stage sampling, where at each stage one can
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E2709 − 19 (2023)
decide to accept or to continue sampling, and the decision to where:
reject is deferred until the last stage.
P(S ) = is the probability of passing stage i,
i
4.1.1 At each stage there are one or more acceptance criteria
P(C ) = is the probability of passing the j-th criterion of m
ij
on the test results; for example, limits on each individual test
within the i-th stage.
result, or limits on statistics based on the sample of test results,
5.2.2 Determine the lower bound on the probability of
such as the average, standard deviation, or coefficient of
passing a k-stage procedure as a function of probabilities of
variation (relative standard deviation).
passing each of the individual stages:
4.2 The methodology in this practice defines an acceptance
P ~pass k 2 stage procedure! $ max$P~S !, P~S !, … , P~S !% (2)
1 2 k
region for a set of test results from the sampled population such
5.3 Determine the contour of the region of parameter values
that, at a prescribed confidence level, the probability that a
for which the expression for the probability of passing the
sample from the population will pass the acceptance procedure
given acceptance procedure is at least equal to the required
is greater than or equal to a prespecified lower bound.
lower bound (LB) on the probability of acceptance (p). This
4.2.1 Having test results fall in the acceptance region is not
defines the acceptable parameter region. Since the acceptance
equivalent to passing the acceptance procedure, but provides
parameter region is a lower bound, it should be compared to
assurance that a sample would pass the acceptance procedure
the simulated probability of passing the acceptance procedure.
with a specified probability.
4.2.2 This information can be used for process 5.4 For each value of a statistic or set of statistics, derive a
demonstration, validation of test methods, and qualification of
joint confidence region for the distribution parameters at
instruments, processes, and materials. confidence level, expressed as a percentage, of 100(1-α). The
4.2.3 This information can be used for lot release
size of sample to be taken, n, and the statistics to be used, must
(acceptance), but the lower bound may be conservative in some be predetermined (see 5.6).
cases.
5.5 Determine the contour of the acceptance region, which
4.2.4 If the results are to be applied to future test results
consists of values of the statistics for which the confidence
from the same process, then it is assumed that the process is
region at level 100(1-α) is entirely contained in the acceptable
stable and predictable. If this is not the case then there can be
parameter region. The acceptance limits lie on the contour of
no guarantee that the probability estimates would be valid
the acceptance region.
predictions of future process performance.
5.6 To select the size of sample, n, to be taken, the
4.3 This methodology was originally developed (1-4) for
probability that sample statistics will lie within acceptance
use in two specific quality characteristics of drug products in
limits should be evaluated over a range of values of n, for
the pharmaceutical industry but will be applicable for accep-
values of population parameters of practical interest, and for
tance procedures in all industries.
which probabilities of passing the given acceptance procedure
are well above the lower bound. The larger the sample size n
4.4 Mathematical derivations would be required that are
that is chosen, the larger will be the acceptance region and the
specific to the individual criteria of each test.
tighter the distribution of the statistics. Choose n so that the
5. Methodology
probability of passing acceptance limits is greater than a
predetermined value.
5.1 The process for defining the acceptance limits, starting
from the definition of the acceptance procedure, is outlined in 5.7 To use the acceptance limit, sample randomly from the
this section. A computer program is normally required to
population. Compute statistics for the sample. If statistics fall
produce the acceptable parameter region and the acceptance within the acceptance limits, then there is 1-α confidence that
limits.
the probability of acceptance is at least p.
5.1.1 An expression for the exact probability of passing the
6. Procedures for Sampling from a Normal Distribution
acceptance procedure might be intractable when the procedure
6.1 An important class of procedures is for the case where
consists of multiple stages with multiple criteria, hence a lower
the quality characteristic is normally distributed. Particular
bound for the probability may be used.
instructions for that case are given in this section, for two
5.2 Express the probability of passing the acceptance pro-
sampling methods, simple random and two-stage. In this
cedure as a function of the parameters characterizing the
standard, these sampling methods are denoted Sampling Plan 1
distribution of the quality characteristic for items in the
and Sampling Plan 2, respectively.
sampled population.
6.2 When the characteristic is normally distributed, param-
5.2.1 For each stage in the procedure having multiple
eters are the mean (μ) and standard deviation (σ) of the
acceptance criteria, determine the lower bound on the prob-
population. The acceptable parameter region will be the region
ability of that stage as a function of the probabilities of passing
under a curve in the half-plane where μ is on the horizontal
each of the criteria in the stage:
axis, σ on the vertical axis, such as that depicted in Fig. 1.
m
P~S ! 5 P~C and C … and C ! $ 1 2 ~1 2 P~C !! (1)
i i1 i2 im (j51 ij
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 boldface numbers in parentheses refer to a list of references at the end of
this standard. the sample standard deviation s of n test results.
E2709 − 19 (2023)
FIG. 1 Example of Acceptance Limit Contour Showing a Simultaneous Confidence Interval With 95 % and 99 % Lower Bound Contours
2 2
6.3.1 Let Z and χ denote percentiles of the standard variance of a single observation, σ , is the sum of between-
p p
normal distribution and of the chi-square distribution with n-1 location and within-location variances.
degrees of freedom, respectively. Given a confidence level
6.4.1 A confidence limit for σ is given by Graybill and
(1-α), choose δ and ε such that (1-α) = (1-2δ)(1-ε). Although
Wang (6) using the between and within location mean squares
there are many choices for δ and ε that would satisfy this
from analysis of variance. When there are L locations with
equation, a reasonable choice is: ε512=12α and
subsamples of n items, the mean squares between locations and
δ5~12=12α!/2 which equally splits the overall alpha be-
within locations, MS and MS , have L-1 and L(n-1) degrees
L E
tween estimating μ and σ. Then:
of freedom respectively. Express the overall confidence level
as a product of confidence levels for the population mean and
¯ 2
X 2 μ ~n 2 1!s
2 2
standard deviation as in 6.3, so that (1-α) = (1-2δ)(1-ε). An
P # Z P $ χ
H 2 J
HS D 12δJ ε
σ
σ/=n
upper (1-ε) confidence limit for σ is:
(3)
5 ~1 2 2δ!~1 2 ε!
@~1/n! MS 1~1 2 1/n! MS #1$@~1/n! (4)
L E
5 1 2 α
2 2
~L 2 1!/χ 2 1 MS # 1@~1 2 1/n!
~ !
L21, 12ϵ L
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
2 2 1/2
L~n 2 1!/χ 2 1 MS # %
~ !
L n21 , 12ϵ E
~ !
¯
at ~X, 0!, as depicted in Fig. 1.
The upper (1-ε) confidence limit for σ is the square root of
6.3.3 The acceptance limit takes the form of a table giving,
Eq 4. Two sided (1-2δ) confidence limits for μ are:
for each value of the sample mean, the maximum value of the
standard deviation (or coefficient of variation) that would meet
σ
¯
X6Z (5)
12δ
these requirements. Using a computer program that calculates
= nL
~ !
¯
confidence limits for μ and σ given sample mean X and
standard deviation s, the acceptance limit can be derived using 6.4.2 To verify, at confidence level 1-α, that a sample will
an iterative loop over increasing values of the sample standard
pass the original acceptance procedure with probability at least
deviation s (starting with s = 0) until the confidence limits hit
equal to the prespecified lower bound, values of (μ, σ) defined
the boundary of the acceptable parameter region, for each
by the limits given in Eq 4 and Eq 5 should fall within the
potential value of the sample mean.
acceptable parameter region defined in 5.3.
6.4 For two-stage sampling, the population is divided into 6.4.3 An acceptance limit table is constructed by fixing the
sample within location standard deviation and the standard
primary sampling units (locations). L locations are selected and
from each of them a subsample of n items is taken. The deviation of location means and then finding the range of
E2709 − 19 (2023)
overall sample means such that the confidence interval com- 7.3 An example of an evaluation of a two-stage lot accep-
pletely falls below the pre-specified lower bound. tance procedure with one or more acceptance criteria at each
stage using Sampling Plan 2 is given in Appendix X3. An
7. Examples
acceptance limit table is shown for a sample size of 4 taken at
7.1 An example of an evaluation of a single-stage lot
each of 15 locations for a total of 60 units tested.
acceptance procedure is given in Appendix X1. An acceptance
limit table is shown for a sample size of 30, but other sample
8. Keywords
sizes may be considered.
8.1 acceptance limits; joint confidence regions; multiple-
7.2 An example of an evaluation of a two-stage lot accep-
stage accept
...
NOTICE: This standard has either been superseded and replaced by a new version or withdrawn.
Contact ASTM International (www.astm.org) for the latest information
Designation: E2709 − 19 (Reapproved 2023) An American National Standard
Standard Practice for
Demonstrating Capability to Comply with an Acceptance
Procedure
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 E2282 Guide for Defining the Test Result of a Test Method
E2586 Practice for Calculating and Using Basic Statistics
1.1 This practice provides a general methodology for evalu-
ating single-stage or multiple-stage acceptance procedures
3. Terminology
which involve a quality characteristic measured on a numerical
scale. This methodology computes, at a prescribed confidence
3.1 Definitions—See Terminology E456 for a more exten-
level, a lower bound on the probability of passing an accep- sive listing of terms in ASTM Committee E11 standards.
tance procedure, using estimates of the parameters of the
3.1.1 characteristic, n—a property of items in a sample or
distribution of test results from a sampled population.
population which, when measured, counted or otherwise
observed, helps to distinguish between the items. E2282
1.2 For a prescribed lower probability bound, the method-
ology can also generate an acceptance limit table, which 3.1.2 mean, n—of a population, µ, average or expected
¯
defines a set of test method outcomes (for example, sample
value of a characteristic in a population, of a sample X, sum of
averages and standard deviations) that would pass the accep-
the observed values in a sample divided by the sample size.
tance procedure at a prescribed confidence level.
E2586
1.3 This approach may be used for demonstrating compli- 3.1.3 multiple-stage acceptance procedure, n—a procedure
ance with in-process, validation, or lot-release specifications.
that involves more than one stage of sampling and testing a
given quality characteristic and one or more acceptance criteria
1.4 The system of units for this practice is not specified.
per stage.
1.5 This standard does not purport to address all of the
3.1.4 standard deviation, n—of a population, σ, the square
safety concerns, if any, associated with its use. It is the
root of the average or expected value of the squared deviation
responsibility of the user of this standard to establish appro-
of a variable from its mean – of a sample, s, the square root of
priate safety, health, and environmental practices and deter-
the sum of the squared deviations of the observed values in the
mine the applicability of regulatory limitations prior to use.
sample divided by the sample size minus 1. E2586
1.6 This international standard was developed in accor-
dance with internationally recognized principles on standard- 3.1.5 test method, n—a definitive procedure that produces a
ization established in the Decision on Principles for the test result. E2282
Development of International Standards, Guides and Recom-
3.2 Definitions of Terms Specific to This Standard:
mendations issued by the World Trade Organization Technical
3.2.1 acceptable parameter region, n—the set of values of
Barriers to Trade (TBT) Committee.
parameters characterizing the distribution of test results for
which the probability of passing the acceptance procedure is
2. Referenced Documents
greater than a prescribed lower bound.
2.1 ASTM Standards:
3.2.2 acceptance region, n—the set of values of parameter
E456 Terminology Relating to Quality and Statistics
estimates that will attain a prescribed lower bound on the
probability of passing an acceptance procedure at a prescribed
level of confidence.
This practice is under the jurisdiction of ASTM Committee E11 on Quality and
3.2.3 acceptance limit, n—the boundary of the acceptance
Statistics and is the direct responsibility of Subcommittee E11.20 on Test Method
Evaluation and Quality Control. region, for example, the maximum sample standard deviation
Current edition approved April 1, 2023. Published April 2023. Originally
test results for a given sample mean.
approved in 2009. Last previous edition approved in 2019 as E2709 – 19. DOI:
10.1520/E2709-19R23.
4. Significance and Use
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
4.1 This practice considers inspection procedures that may
Standards volume information, refer to the standard’s Document Summary page on
the ASTM website. involve multiple-stage sampling, where at each stage one can
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E2709 − 19 (2023)
decide to accept or to continue sampling, and the decision to where:
reject is deferred until the last stage.
P(S ) = is the probability of passing stage i,
i
4.1.1 At each stage there are one or more acceptance criteria
P(C ) = is the probability of passing the j-th criterion of m
ij
on the test results; for example, limits on each individual test
within the i-th stage.
result, or limits on statistics based on the sample of test results,
5.2.2 Determine the lower bound on the probability of
such as the average, standard deviation, or coefficient of
passing a k-stage procedure as a function of probabilities of
variation (relative standard deviation).
passing each of the individual stages:
4.2 The methodology in this practice defines an acceptance
P pass k 2 stage procedure $ max P S , P S , … , P S (2)
~ ! $ ~ ! ~ ! ~ !%
1 2 k
region for a set of test results from the sampled population such
5.3 Determine the contour of the region of parameter values
that, at a prescribed confidence level, the probability that a
for which the expression for the probability of passing the
sample from the population will pass the acceptance procedure
given acceptance procedure is at least equal to the required
is greater than or equal to a prespecified lower bound.
lower bound (LB) on the probability of acceptance (p). This
4.2.1 Having test results fall in the acceptance region is not
defines the acceptable parameter region. Since the acceptance
equivalent to passing the acceptance procedure, but provides
parameter region is a lower bound, it should be compared to
assurance that a sample would pass the acceptance procedure
the simulated probability of passing the acceptance procedure.
with a specified probability.
4.2.2 This information can be used for process
5.4 For each value of a statistic or set of statistics, derive a
demonstration, validation of test methods, and qualification of joint confidence region for the distribution parameters at
instruments, processes, and materials.
confidence level, expressed as a percentage, of 100(1-α). The
4.2.3 This information can be used for lot release size of sample to be taken, n, and the statistics to be used, must
(acceptance), but the lower bound may be conservative in some be predetermined (see 5.6).
cases.
5.5 Determine the contour of the acceptance region, which
4.2.4 If the results are to be applied to future test results
consists of values of the statistics for which the confidence
from the same process, then it is assumed that the process is
region at level 100(1-α) is entirely contained in the acceptable
stable and predictable. If this is not the case then there can be
parameter region. The acceptance limits lie on the contour of
no guarantee that the probability estimates would be valid
the acceptance region.
predictions of future process performance.
5.6 To select the size of sample, n, to be taken, the
4.3 This methodology was originally developed (1-4) for
probability that sample statistics will lie within acceptance
use in two specific quality characteristics of drug products in
limits should be evaluated over a range of values of n, for
the pharmaceutical industry but will be applicable for accep-
values of population parameters of practical interest, and for
tance procedures in all industries.
which probabilities of passing the given acceptance procedure
are well above the lower bound. The larger the sample size n
4.4 Mathematical derivations would be required that are
that is chosen, the larger will be the acceptance region and the
specific to the individual criteria of each test.
tighter the distribution of the statistics. Choose n so that the
5. Methodology
probability of passing acceptance limits is greater than a
predetermined value.
5.1 The process for defining the acceptance limits, starting
from the definition of the acceptance procedure, is outlined in
5.7 To use the acceptance limit, sample randomly from the
this section. A computer program is normally required to population. Compute statistics for the sample. If statistics fall
produce the acceptable parameter region and the acceptance
within the acceptance limits, then there is 1-α confidence that
limits.
the probability of acceptance is at least p.
5.1.1 An expression for the exact probability of passing the
6. Procedures for Sampling from a Normal Distribution
acceptance procedure might be intractable when the procedure
6.1 An important class of procedures is for the case where
consists of multiple stages with multiple criteria, hence a lower
the quality characteristic is normally distributed. Particular
bound for the probability may be used.
instructions for that case are given in this section, for two
5.2 Express the probability of passing the acceptance pro-
sampling methods, simple random and two-stage. In this
cedure as a function of the parameters characterizing the
standard, these sampling methods are denoted Sampling Plan 1
distribution of the quality characteristic for items in the
and Sampling Plan 2, respectively.
sampled population.
6.2 When the characteristic is normally distributed, param-
5.2.1 For each stage in the procedure having multiple
eters are the mean (µ) and standard deviation (σ) of the
acceptance criteria, determine the lower bound on the prob-
population. The acceptable parameter region will be the region
ability of that stage as a function of the probabilities of passing
under a curve in the half-plane where µ is on the horizontal
each of the criteria in the stage:
axis, σ on the vertical axis, such as that depicted in Fig. 1.
m
P S 5 P C and C … and C $ 1 2 1 2 P C (1)
~ ! ~ ! ~ ~ !!
i i1 i2 im (j51 ij
6.3 For simple random sampling from a normal population,
the method of Lindgren (5) constructs a simultaneous confi-
¯
3 dence region of (µ, σ) values from the sample average X and
The boldface numbers in parentheses refer to a list of references at the end of
this standard. the sample standard deviation s of n test results.
E2709 − 19 (2023)
FIG. 1 Example of Acceptance Limit Contour Showing a Simultaneous Confidence Interval With 95 % and 99 % Lower Bound Contours
2 2
6.3.1 Let Z and χ denote percentiles of the standard variance of a single observation, σ , is the sum of between-
p p
normal distribution and of the chi-square distribution with n-1
location and within-location variances.
degrees of freedom, respectively. Given a confidence level
6.4.1 A confidence limit for σ is given by Graybill and
(1-α), choose δ and ε such that (1-α) = (1-2δ)(1-ε). Although
Wang (6) using the between and within location mean squares
there are many choices for δ and ε that would satisfy this
from analysis of variance. When there are L locations with
equation, a reasonable choice is: ε512=12α and
subsamples of n items, the mean squares between locations and
δ5~12=12α!/2 which equally splits the overall alpha be-
within locations, MS and MS , have L-1 and L(n-1) degrees
L E
tween estimating µ and σ. Then:
of freedom respectively. Express the overall confidence level
as a product of confidence levels for the population mean and
¯
X 2 µ n 2 1 s
~ !
2 2
standard deviation as in 6.3, so that (1-α) = (1-2δ)(1-ε). An
P # Z P $ χ
H J
HS D 12δJ 2 ε
σ 2
σ/=n
upper (1-ε) confidence limit for σ is:
(3)
5 1 2 2δ 1 2 ε
~ !~ !
1/n MS 1 1 2 1/n MS 1 1/n (4)
@~ ! ~ ! # $@~ !
L E
5 1 2 α
2 2
L 2 1 /χ 2 1 MS # 1 1 2 1/n
~ ! @~ !
~ L21, 12ϵ ! L
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
2 2 1/2
L n 2 1 /χ 2 1 MS # %
~ !
~ !
L~n21!, 12ϵ E
¯
at ~X, 0!, as depicted in Fig. 1.
The upper (1-ε) confidence limit for σ is the square root of
6.3.3 The acceptance limit takes the form of a table giving,
Eq 4. Two sided (1-2δ) confidence limits for µ are:
for each value of the sample mean, the maximum value of the
standard deviation (or coefficient of variation) that would meet
σ
¯
X6Z (5)
these requirements. Using a computer program that calculates 12δ
=
~nL!
¯
confidence limits for µ and σ given sample mean X and
standard deviation s, the acceptance limit can be derived using
6.4.2 To verify, at confidence level 1-α, that a sample will
an iterative loop over increasing values of the sample standard
pass the original acceptance procedure with probability at least
deviation s (starting with s = 0) until the confidence limits hit
equal to the prespecified lower bound, values of (µ, σ) defined
the boundary of the acceptable parameter region, for each
by the limits given in Eq 4 and Eq 5 should fall within the
potential value of the sample mean.
acceptable parameter region defined in 5.3.
6.4.3 An acceptance limit table is constructed by fixing the
6.4 For two-stage sampling, the population is divided into
primary sampling units (locations). L locations are selected and sample within location standard deviation and the standard
from each of them a subsample of n items is taken. The deviation of location means and then finding the range of
E2709 − 19 (2023)
overall sample means such that the confidence interval com- 7.3 An example of an evaluation of a two-stage lot accep-
pletely falls below the pre-specified lower bound.
tance procedure with one or more acceptance criteria at each
stage using Sampling Plan 2 is given in Appendix X3. An
7. Examples
acceptance limit table is shown for a sample size of 4 taken at
7.1 An example of an evaluation of a single-stage lot
each of 15 locations for a total of 60 units tested.
acceptance procedure is given in Appendix X1. An acceptance
limit table is shown for a sample size of 30, but other sample
8. Keywords
sizes may be considered.
8.1 acceptance limits; joint confidence regions; multiple-
7.2 An example of an evaluation of a two-stage lot accep-
stage acceptance procedures; specifications
tance procedure with one or more acce
...
This document 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. Because
it may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current version
of the standard as published by ASTM is to be considered the official document.
Designation: E2709 − 19 E2709 − 19 (Reapproved 2023) An American National Standard
Standard Practice for
Demonstrating Capability to Comply with an Acceptance
Procedure
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 evaluating 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 acceptance 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 methodology 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 acceptance procedure at a
prescribed confidence level.
1.3 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 responsibility
of the user of this standard to establish appropriate safety, health, and environmental practices and determine the applicability of
regulatory limitations prior to use.
1.6 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.
2. Referenced Documents
2.1 ASTM Standards:
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 extensive listing of terms in ASTM Committee E11 standards.
This practice is under the jurisdiction of ASTM 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, 2019April 1, 2023. Published April 2019April 2023. Originally approved in 2009. Last previous edition approved in 20142019 as
ɛ1
E2709 – 14E2709 – 19. . DOI: 10.1520/E2709-19.10.1520/E2709-19R23.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at service@astm.org. 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
E2709 − 19 (2023)
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 sampleX, 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 given
quality characteristic and one or more acceptance criteria 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. Significance and Use
4.1 This practice considers inspection procedures that may involve multiple-stage sampling, where at each stage one can decide
to 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 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 region for a set of test results from the sampled population such 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), but the lower bound may 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 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) for use in two specific quality characteristics of drug products in the
pharmaceutical industry but will be applicable for acceptance procedures in all industries.
The boldface numbers in parentheses refer to a list of references at the end of this standard.
E2709 − 19 (2023)
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 procedure 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 probability of that
stage as a function of the probabilities of passing each of the criteria in the stage:
m
P S 5 P C and C … and C $ 12 12 P C (1)
~ ! ~ ! ~ ~ !!
i i1 i2 im (j51 ij
where:
P(S ) = is the probability of passing stage i,
i
P(C ) = is the probability of passing the j-th criterion of m within the i-th stage.
ij
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 !, P~S !, … , P~S !% (2)
1 2 k
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.
E2709 − 19 (2023)
6.2 When the characteristic is normally distributed, parameters are the mean (μ) and standard deviation (σ) of the 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 confidence
¯
region of (μ, σ) values from the sample average X and the sample standard deviation s of n test results.
6.3.1 Let Z and χ denote percentiles of the standard normal distribution and of the chi-square distribution with n-1 degrees of
p p
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 between estimating μ and σ. Then:
¯ 2
X 2 μ n 2 1 s
~ !
2 2
P # Z P $χ
H J
HS D 12δJ 2 ε
σ
σ/=n
(3)
5 12 2δ 12ε
~ !~ !
5 12α
¯
6.3.2 The confidence region for (μ, σ), two-sided for μ, one-sided for σ, is an inverted triangle with a minimum vertex at ~X, 0!,
as depicted in Fig. 1.
6.3.3 The acceptance limit takes the form of a table giving, for each value of the sample mean, the maximum value of the standard
deviation (or coefficient of variation) that would meet these requirements. Using a computer program that calculates confidence
¯
limits for μ and σ given sample mean X and standard deviation s, the acceptance limit can be derived using an iterative loop over
increasing values of the sample standard deviation s (starting with s = 0) until the confidence limits hit the boundary of the
acceptable parameter region, for each potential value of the sample mean.
FIG. 1 Example of Acceptance Limit Contour Showing a Simultaneous Confidence Interval With 95 % and 99 % Lower Bound Contours
E2709 − 19 (2023)
6.4 For two-stage sampling, the population is divided into primary sampling units (locations). L locations are selected and from
each of them a subsample of n items is taken. The variance of a single observation, σ , is the sum of between-location and
within-location variances.
6.4.1 A confidence limit for σ is given by Graybill and Wang (6) using the between and within location mean squares from
analysis of variance. When there are L locations with subsamples of n items, the mean squares between locations and within
locations, MS and MS , have L-1 and L(n-1) degrees of freedom respectively. Express the overall confidence level as a product
L E
of confidence levels for the population mean and standard deviation as in 6.3, so that (1-α) = (1-2δ)(1-ε). An upper (1-ε) confidence
limit for σ is:
@~1/n! MS 1~12 1/n! MS #1$@~1/n! (4)
L E
2 2
~L 2 1!/χ 2 1 MS # 1@~12 1/n!
~ !
L21, 12ϵ L
2 2 1/2
L n 2 1 /χ 2 1 MS # %
~ !
~ !
L~n21!, 12ϵ E
The upper (1-ε) confidence limit for σ is the square root of Eq 4. Two sided (1-2δ) confidence limits for μ are:
σ
¯
X6Z (5)
12δ
=
~nL!
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 overall sample means such that the confidence interval completely 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 acceptance procedure with one or more acceptance criterion at each stage is
given in Appendix X2. An acceptance limit table is shown for a sample si
...
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