Plans for acceptance sampling by attributes when observations are destructive

Hugalf Bernburg; Katy Klauenberg

Plans for acceptance sampling by attributes when observations are destructive

T0 review · 0 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read ISO 2859-2 sampling plans do not protect the lot that remains after destructive testing: the probability of accepting an unsatisfactory remaining lot can reach 44%, and the paper tabulates replacement plans that hold it to 10%.

desk verdict A genuinely useful applied paper on destructive acceptance sampling, with a real practical contribution in the [N-n, Ac] representation, but it overstates what the hypergeometric distribution cannot do and the 10% guarantee is prior-dependent, especially for small lots. read the letter →

arxiv 2508.05131 v1 pith:PPZK3C3W submitted 2025-08-07 stat.AP

Hugalf Bernburg , Katy Klauenberg This is my paper · ORCID

classification stat.AP MSC 62P3062F1562C10

keywords acceptancesamplingbyattributesdestructivetestingspecificconsumer'sriskBayesianstatisticshypergeometricdistributionremaininglotqualityISO2859-2referenceprior

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Acceptance-sampling standards such as ISO 2859-2 promise to cap the consumer's risk of accepting a bad lot near 10%. The paper argues that this promise breaks when the inspection destroys the sample, as in seed-germination tests or checks on installed utility meters: what matters then is the quality of the lot that remains after the sample is removed, and the hypergeometric distribution cannot express the probability of accepting a bad remaining lot. Applying ISO 2859-2 plans to destructive sampling can push the Bayesian specific consumer's risk — the posterior probability that an accepted remaining lot still fails the quality limit — to 44%. The paper designs accept-zero sampling plans that limit this risk to 10% under a uniform reference prior, and proposes a new tabular form, fixing the remaining lot size $N-n$ rather than the sample size $n$, which keeps the plans near-optimal and easy to standardize. If the paper is right, the standard should not be used for destructive testing, and its future editions should say so explicitly.

What carries the argument

Central machinery: the Bayesian specific consumer's risk (5), $P(k_{\mathrm{rem}} \ge \lceil LQ\cdot(N-n)\rceil \mid y \le Ac)$, the posterior probability that an accepted remaining lot is still unsatisfactory. It is computed from the posterior (6): the hypergeometric sampling distribution (3) for the whole lot combined with a prior, taken as uniform $P(k_{\mathrm{whole}})=1/N$, the reference prior for hypergeometric sampling. Conditioning on the observed sample sidesteps the difficulty that blocks the frequentist risk (4), whose conditioning event $k_{\mathrm{rem}} = k_{\mathrm{whole}} - Y$ involves the observed $Y$. The second piece is the square-bracket representation $[N-n, Ac]$: fixing

What would settle it

Recompute the Bayesian specific consumer's risk (5) for the ISO 2859-2 plan $n=50$, $Ac=0$, $LQ=2\%$, $N=90$ with the uniform prior: the paper predicts 44%. If an independent calculation, or a simulation of repeated destructive-sampling campaigns following ISO 2859-2, puts that risk at or below 10%, the central claim fails. A second check: compute the risk for the paper's proposed plan $[51,0]$ at the smallest lot in its range, $N=160$, with a $\beta$-binomial prior $a=1.1$, $b=1$; the paper's sensitivity analysis implies it exceeds 10%.

Watch

Extended reading notes

Core claim

Central claim: sampling plans indexed by whole-lot quality, in particular ISO 2859-2's, cannot assess the lot left after destructive sampling. The frequentist consumer's risk conditions on $k_{\mathrm{rem}} = k_{\mathrm{whole}} - Y$, which involves the sample count $Y$, so the hypergeometric distribution cannot describe it. The paper turns to the Bayesian specific consumer's risk, $P(k_{\mathrm{rem}} \ge \lceil LQ\cdot(N-n)\rceil \mid y \le Ac)$, under a uniform reference prior. ISO 2859-2 plans then exceed the intended 10% risk for small lots, peaking at 44% for plan $(50,0)$ at $N=90$ with $LQ=2\%$. The paper designs accept-zero plans holding this risk to 10%, tabulated in a new representa

Load-bearing premise

The plans cap the risk at 10% only under the uniform prior $P(k_{\mathrm{whole}})=1/N$; if the consumer's true prior differs — the sensitivity analysis (Section 4, Figures 5a-5c) shows a shift from $a=1.0$ to $a=1.1$ in a $\beta$-binomial prior can already push the risk over the limit for the smallest lot sizes — the promised protection weakens.

Editorial extensions

If this is right

If the paper is right, ISO 2859-2 plans should not be used when the sample destroys items, and future editions of the standard should state this limitation explicitly.
Destructive-sampling plans should be designed to limit the Bayesian specific consumer's risk (5); with the uniform reference prior, the required accept-zero sample sizes exceed the ISO 2859-2 sample sizes for whole lots up to $N=500$ at $LQ=2\%$.
The square-bracket representation $[N-n, Ac]$ lets standard setters tabulate destructive-sampling plans whose sample size stays within $1/LQ$ of the optimal one, even for small lots.
The tabulated plans are stable under changes of the conjugate prior: the applicable lot-size ranges shift roughly linearly, about 10 in $N$ per 0.1 change in $a$ and about $-1$ in $N$ per unit change in $b$ at $LQ=2\%$, and moving prior weight toward acceptable quality only improves consumer protection.
The same Bayesian construction and square-bracket format extend to informative priors, variable sampling, and to plans that limit producer's risk or minimize costs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The square-bracket representation likely transfers beyond destructive testing to any setting where sampled units leave the population of interest — audit sampling, food-safety testing that consumes the sample, or trials with destructive endpoints — and the Bayesian risk (5) would then define the protection guarantee there as well.
Because the 44% worst case is computed under a uniform prior, priors more pessimistic about lot quality would widen the gap between the standard's stated protection and what destructive sampling actually delivers; the failure is worst in exactly the small-lot regimes where destructive testing is common.
The reported linearity of lot-size ranges in the prior parameters suggests a simple on-the-fly recalibration: fit a beta-binomial prior to inspection history and adjust table ranges by the measured slopes, avoiding new optimization.
The tables assume the consumer adopts the same reference prior; in regulated settings (legal metrology, food safety) a standards body choosing the prior would effectively set the risk policy, a choice the paper leaves open.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read this if you care about acceptance sampling in destructive-testing settings. The paper does two useful things: it shows, with a concrete 44% peak, that ISO 2859-2 plans do not control the risk of accepting an unsatisfactory remaining lot, and it proposes a practical tabulation format — fixing the remaining lot size instead of the sample size — that makes Bayesian sampling plans efficient and suitable for standardization. That representation is the real novelty; the Bayesian specific consumer's risk is not new (Uhlig et al., Wilson and Farrow), and Herman and Robbins already noted the hypergeometric overstates confidence in small residual lots. But the systematic design of destructive-sampling plans for remaining lots, and the table format, are new and well executed. The 44% calculation checks out, and the sensitivity analysis is a credit: they explore how prior changes affect the applicable lot-size ranges and honestly report that small prior shifts can push the risk over 10% for the smallest lots.

The main soft spot is the claim that the frequentist risk 'cannot be described' by the hypergeometric distribution. That is overstated; it can be expressed as a sum of hypergeometric terms, but the conditioning event involves the sample outcome, so it is not identifiable without a prior. The paper's real point is that the standard's risk concept does not address the remaining lot, not that the math is impossible. Second, the plans' 10% guarantee is conditional on the uniform prior, which the paper calls 'the reference prior' citing Bernardo. That label is questionable — the reference prior for the underlying proportion is typically Beta(1/2,1/2) — and the sensitivity analysis shows the guarantee is not robust for small lots. The paper does acknowledge this in Section 5, but the abstract and conclusions are unqualified, which is a bit much. The lack of shipped code is minor; the computations are standard beta-binomial.

Who is this for? Statisticians and metrologists working on standards for acceptance sampling, seed testing, utility metering. It deserves a serious referee. I'd send it to peer review: the central idea is sound, the contribution is practical, and the weaknesses are caveats rather than fatal flaws. A referee should push them to soften the 'cannot be described' claim and to condition the abstract's statement on the prior. The [N-n, Ac] representation is a genuinely useful idea for standardization, and the paper is a fair and honest piece of applied work.

Referee Report

0 major / 6 minor

Summary. The paper addresses acceptance sampling by attributes when the sampled items are destroyed, so that the lot remaining after sampling, rather than the original whole lot, is the relevant quantity. It argues that ISO 2859-2 plans, designed to limit the consumer's risk for the whole lot, do not control the risk for the remaining lot. Section 2 notes that the frequentist risk in Eq. (4), which conditions on the number of nonconforming items in the remaining lot, is not directly computable from the hypergeometric distribution because the conditioning event involves the sample count Y. Section 3 defines a Bayesian specific consumer's risk in Eq. (5) under a uniform prior for kwhole, shows that an ISO plan can have a risk of 44% at N=90, and Section 4 proposes tabulated zero-acceptance plans in Table 1 using a new representation [N-n, Ac] that fixes the remaining lot size. A prior sensitivity analysis with beta-binomial priors is included.

Significance. If correct, the paper provides a practical, tabulated alternative to ISO 2859-2 for destructive testing and a simple representation that is efficient for small lots. The 44% example is concrete and checkable and usefully demonstrates that the standard's plans do not control the quality of the remaining lot under a uniform prior. The paper is transparent about restricting attention to Ac=0, about the role of the prior, and about sensitivity to prior perturbations, which strengthens reproducibility. The main weaknesses are framing rather than mathematical: the abstract overstates the frequentist impossibility, and the reference-prior label is supported only by a single citation.

minor comments (6)

[Section 3, Eq. (6)] The uniform prior is written as P(kwhole)=1/N, but kwhole takes N+1 values (0,...,N), so the normalized uniform prior is 1/(N+1). The beta-binomial with a=b=1 indeed gives 1/(N+1). Please correct the formula and the corresponding sentence. Also, the notation "P(kwhole - y | y,N,n)" is informal; it should be written as P(kwhole = krem + y | y,N,n).
[Abstract and Section 2] The statement that the hypergeometric distribution cannot describe the frequentist consumer's risk is too broad. What is shown correctly is that Eq. (4), which conditions on the random variable krem=kwhole-Y, is not a frequentist probability. For a fixed kwhole, however, the probability of accepting a remaining lot with krem >= m is a well-defined hypergeometric tail: F(min(Ac,kwhole-m); N,kwhole,n). The paper should state this distinction and restrict the claim to Eq. (4), or explain why a worst-case frequentist criterion over kwhole is rejected.
[Section 3, prior discussion] The claim that the uniform prior is the reference prior for the hypergeometric sampling distribution is load-bearing for the numerical 44% figure and for Table 1, but it is supported only by a page citation. Please provide a short derivation or exact quotation, or rephrase as "an assumed noninformative prior." It would also help to note explicitly that the hypergeometric parameter here is discrete K, so the continuous binomial reference prior Beta(1/2,1/2) is not directly applicable.
[Table 1 and Section 4] The entries "no plan" for N<50 (LQ=2%) and N<17 (LQ=20%) are not explained. For example, [5,0] at N=49 gives a risk of about 10% and [6,0] at N=16 gives about 11% under the uniform prior. Please state the selection criterion that excludes these cases, so that the table is self-contained.
[Section 4, Figure 3] The phrase "the risks ... increase only at multiples of 1/LQ" is imprecise. The risk jumps when the threshold ceil(LQ*(N-n)) changes, i.e., when LQ*(N-n) crosses an integer. Please rephrase to avoid confusion.
[Table 1 caption] The table caption should state explicitly that all plans are zero-acceptance (Ac=0) and that the risk limit is one-sided (P(krem >= ceil(LQ*(N-n)) | y,N,n) <= 10%). This is clear in the body text but should be in the caption for standalone use.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the sampling plans are constructed to limit an explicitly stated Bayesian risk and the ISO 2859-2 comparison evaluates an external benchmark.

full rationale

The paper's derivation chain is self-contained. The specific consumer's risk (Eq. 5) is defined from the posterior (Eq. 6), which follows from the hypergeometric likelihood (Eq. 3) and an explicitly stated uniform prior. Table 1 plans are constructed so that risk (5) ≤ 10%; the statement that they do so is the design specification, not a prediction from fitted data. No parameter is fitted to a subset and then reported as a prediction. The claim that ISO 2859-2 plans are ill-suited for remaining-lot assessment is supported by (i) the algebraic point that the frequentist risk (4) is not hypergeometric because the conditioning event involves Y via krem = kwhole − Y, and (ii) evaluation of risk (5) under ISO plans, which is a computation against an external standard's tables, not a restatement of the paper's inputs. The uniform prior is an explicitly subjective input, with the paper noting 'the prior may depend on the decision maker' and providing a sensitivity analysis (Section 4, Fig. 5a–5c) that quantifies how plans must change under conjugate priors, including the acknowledged limitation for the smallest lots. The 'reference prior' attribution cites Bernardo (2003), an external source, not the present authors' work; whether uniform is the correct reference prior for the hypergeometric is a technical-correctness matter, not circularity. No load-bearing step reduces by construction to its own input, and no self-citation chain is used to justify the central claims.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim about ISO 2859-2's inappropriateness rests on the definition of remaining-lot risk; the new plans rest on the chosen prior and risk threshold. All are stated and sensitivities are explored.

free parameters (3)

Prior hyperparameters a, b = a=1, b=1 (uniform prior)
Chosen by hand as a non-informative/reference prior; the sensitivity analysis shows plans shift with these.
Risk limit = 10%
Chosen as design threshold, following ISO 2859-2's consumer risk limit.
Limiting quality LQ = 2% and 20%
Example LQ values used to construct Table 1.

assumptions (4)

standard math The hypergeometric distribution governs the number of nonconforming items in a sample without replacement.
Used in eq (3) and in the posterior via eq (6).
domain assumption The uniform prior P(kwhole)=1/N is the reference prior for the hypergeometric sampling distribution.
Invoked in Section 3 to justify the prior; cited to Bernardo (2003).
domain assumption The specific consumer's risk (5) is the appropriate risk criterion for destructive acceptance sampling.
Defined in Section 3 following JCGM 106; normative choice.
standard math The beta-binomial distribution is conjugate to the hypergeometric likelihood.
Underlies the posterior computation in eq (6).

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Cite this review

Pith. "Pith review of Plans for acceptance sampling by attributes when observations are destructive." pith.science (2026). https://pith.science/paper/PPZK3C3W

@misc{pith2026250805131,
  author       = {Pith},
  title        = {Pith review of: Plans for acceptance sampling by attributes when observations are destructive},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PPZK3C3W}},
  note         = {Machine review of arXiv:2508.05131}
}

abstract

The international standard ISO 2859-2 provides plans for acceptance sampling by attributes, that ensure a defined quality level in isolated lots using the hypergeometric distribution. In destructive testing, the sample itself is damaged or changed such that the quality of an entire lot is less relevant than the quality of the lot that remains after removing the sample. Examples include assessing the germination of seeds and the conformity of in-service utility meters. This research highlights that the hypergeometric distribution cannot describe the frequentist consumer's risk of accepting a remaining lot with unsatisfactory quality. Consequently, sampling plans as those provided in ISO 2859-2 are ill-suited to assess the remaining lot when sampling destructively. In contrast, Bayesian statistics inherently infers the lot's quality after sampling. Using a reference prior, we show that sampling plans provided by ISO 2859-2 result in high specific consumer's risk for small remaining lots. The ISO 2859-2 being ill-suited, we design plans for destructive sampling that limit the (Bayesian) specific consumer's risk. To tabulate these plans in a similar way to ISO 2859-2, we propose a new representation that fixes the remaining lot size $N-n$ rather than the sample size $n$. This generalizable, concise and efficient representation is suitable for future standardization of destructive sampling.

Figures

Figures reproduced from arXiv: 2508.05131 by the authors.

**Figure 2.** Displayed are lots of different quality and size. Every point is one lot with a specific [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Specific consumer’s risk as a function of the sample size (top) and the remaining lot [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: The beta-binomial distribution for a set of different parameters is shown for a lot of [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: The minimal required lot sizes for the prespecified sampling plans [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

[1]

ISO 3534-2 Statistics – V ocabulary and symbols, Part 2: Applied statistics

ISO/TC 69/SC 1 (2006). ISO 3534-2 Statistics – V ocabulary and symbols, Part 2: Applied statistics. International Organization for Standardization (ISO). OIML TC 3/SC 4 (2017). Surveillance of utility meters in service on the basis of sampling inspections. OIML G

work page 2006
[2]

BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML (2012)

Encyclopedia of Life Support Systems (EOLSS). BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML (2012). Evaluation of measurement data — The role of measurement uncertainty in conformity assessment. Joint Committee for Guides in Metrology, JCGM 106:2012. https://doi.org/10.59161/JCGM106-2012 DELGADILLO , F., B REMER , R. and H OFFMAN , J. J. (2007). A des...

work page doi:10.59161/jcgm106-2012 2012
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Wiley publications in statistics

print ed. Wiley publications in statistics. Wiley publications in statistics, New York u.a. DYER , D. and P IERCE , R. L. (1993). On the choice of the prior distribution in hypergeometric sampling. Com- munications in Statistics-Theory and Methods 22 2125–2146. GELMAN , A. (2014). Bayesian data analysis, 3 ed. Chapman & Hall/CRC texts in statistical scien...

work page 1993
[8]

WILSON , F. (1984). Building materials evaluation handbook. Van Nostrand Reinhold Company Inc. WILSON , K. J. and F ARROW, M. (2021). Assurance for sample size determination in reliability demonstration testing. Technometrics 63 523–535

work page 1984
[20]

ISO 2859-3 Sampling procedures for inspection by attributes – Part 3: Skip-lot sam- pling procedures

ISO/TC 69/SC 5 (2005). ISO 2859-3 Sampling procedures for inspection by attributes – Part 3: Skip-lot sam- pling procedures. International Organization for Standardization (ISO). ISO/TC 69/SC 5 (2017). ISO 28590 Sampling procedures for inspection by attributes – Introduction to the ISO 2859 series of standards for sampling for inspection by attributes. In...

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Measuring Instruments Directive (2014/32/EU): Guide for generating sampling plans for statistical verification according to Annex F and F1 of MID 2014/32/EU

WELMEC (2024). Measuring Instruments Directive (2014/32/EU): Guide for generating sampling plans for statistical verification according to Annex F and F1 of MID 2014/32/EU. WELMEC European cooperation in legal metrology: Working Group

work page 2024
[740]

Analysis of longitudinal data with destructive sampling using linear mixed models

AVELLANEDA , C. A., M ELO , O. O. and C RUZ, N. A. (2024). Analysis of longitudinal data with destructive sampling using linear mixed models. arXiv preprint arXiv:2411.16153. BERNARDO , J. M. (2003). Bayesian Statistics In Probability and Statistics

work page Pith review arXiv 2024
[2465]

PHELPS , R. I. (1982). Skip-lot destructive sampling with Bayesian inference. IEEE Transactions on Reliability 31 191–193. UHLIG , S., COLSON , B., KISSLING , R., ELLIS , S., HICKS , M., VANDENBEMDEN , J., PENNECCHI , F., GÖB, R. and G OWIK , P. (2024). Acceptance Sampling Plans Based on Conformance Probability – Inspection of Lots and Processes by Attrib...

work page 1982

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

ISO 3534-2 Statistics – V ocabulary and symbols, Part 2: Applied statistics

ISO/TC 69/SC 1 (2006). ISO 3534-2 Statistics – V ocabulary and symbols, Part 2: Applied statistics. International Organization for Standardization (ISO). OIML TC 3/SC 4 (2017). Surveillance of utility meters in service on the basis of sampling inspections. OIML G

work page 2006

[2] [2]

BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML (2012)

Encyclopedia of Life Support Systems (EOLSS). BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML (2012). Evaluation of measurement data — The role of measurement uncertainty in conformity assessment. Joint Committee for Guides in Metrology, JCGM 106:2012. https://doi.org/10.59161/JCGM106-2012 DELGADILLO , F., B REMER , R. and H OFFMAN , J. J. (2007). A des...

work page doi:10.59161/jcgm106-2012 2012

[3] [6]

Wiley publications in statistics

print ed. Wiley publications in statistics. Wiley publications in statistics, New York u.a. DYER , D. and P IERCE , R. L. (1993). On the choice of the prior distribution in hypergeometric sampling. Com- munications in Statistics-Theory and Methods 22 2125–2146. GELMAN , A. (2014). Bayesian data analysis, 3 ed. Chapman & Hall/CRC texts in statistical scien...

work page 1993

[4] [8]

WILSON , F. (1984). Building materials evaluation handbook. Van Nostrand Reinhold Company Inc. WILSON , K. J. and F ARROW, M. (2021). Assurance for sample size determination in reliability demonstration testing. Technometrics 63 523–535

work page 1984

[5] [20]

ISO 2859-3 Sampling procedures for inspection by attributes – Part 3: Skip-lot sam- pling procedures

ISO/TC 69/SC 5 (2005). ISO 2859-3 Sampling procedures for inspection by attributes – Part 3: Skip-lot sam- pling procedures. International Organization for Standardization (ISO). ISO/TC 69/SC 5 (2017). ISO 28590 Sampling procedures for inspection by attributes – Introduction to the ISO 2859 series of standards for sampling for inspection by attributes. In...

work page 2005

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Measuring Instruments Directive (2014/32/EU): Guide for generating sampling plans for statistical verification according to Annex F and F1 of MID 2014/32/EU

WELMEC (2024). Measuring Instruments Directive (2014/32/EU): Guide for generating sampling plans for statistical verification according to Annex F and F1 of MID 2014/32/EU. WELMEC European cooperation in legal metrology: Working Group

work page 2024

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Analysis of longitudinal data with destructive sampling using linear mixed models

AVELLANEDA , C. A., M ELO , O. O. and C RUZ, N. A. (2024). Analysis of longitudinal data with destructive sampling using linear mixed models. arXiv preprint arXiv:2411.16153. BERNARDO , J. M. (2003). Bayesian Statistics In Probability and Statistics

work page Pith review arXiv 2024

[8] [2465]

PHELPS , R. I. (1982). Skip-lot destructive sampling with Bayesian inference. IEEE Transactions on Reliability 31 191–193. UHLIG , S., COLSON , B., KISSLING , R., ELLIS , S., HICKS , M., VANDENBEMDEN , J., PENNECCHI , F., GÖB, R. and G OWIK , P. (2024). Acceptance Sampling Plans Based on Conformance Probability – Inspection of Lots and Processes by Attrib...

work page 1982