1. Summary of the attribute sampling standards

Many companies use sampling by attribute standards to control the presence of nonconformities in their production. The most widely applied standards for this purpose are ISO 2859-1 and ANSI/ASQ Z1.4-2003. They are generally used for incoming inspection – e.g., of packaging components or materials – or for final inspection or release inspection. 

In the life sciences, this standard is often applied following a 100% automated inspection of parenteral, ophthalmic, veterinary units, and similar products. In this context, only the compliant units resulting from automatic sorting are subjected to statistical inspection according to the selected inspection level. United States Pharmacopeia <1790> also requires a 100% inspection of all units in a lot and specifies that when part of a lot is recirculated, it must be considered as an independent lot, subject to separate statistical inspection. Only if conformity is established can the original lot and the sub- lot be combined again to form a single lot.

Sampling by attribute consists of selecting a sample of fixed size and counting whether the number of nonconform units within that sample exceeds the acceptance criterion. The sample size is determined in the standards according to lot size, level of control (S1 to S4, I, II or III), inspection severity (normal, tightened, reduced) and type of sampling (single, double, multiple).

The criteria to be applied depend on the chosen Acceptance Quality Limits (AQL): tables in the standards indicate the maximum number of nonconform units that can be accepted, as well as the lowest number of nonconform units considered unacceptable. Several AQLs are generally set for minor, major, and critical non-conformities. AQL standards are applied to homogeneous lots (lots composed of items of a single type, grade, class, size, and composition, manufactured under uniform conditions at essentially the same time).

These standards are mainly intended for the monitoring of operations in which continuous series of lots are produced and inspected one after another. Indeed, rules for changing severity are mandatory to apply when there are too many rejected lots: For example, if 2 out of 5 or fewer consecutive lots are not accepted, the inspection level must be switched to tightened. 

Normal inspection can only resume after 5 consecutive lots have been accepted at the tightened level.

Understanding how a sampling plan behaves across a range of possible nonconform rates is therefore essential. Attribute-sampling standards address this need through operating-characteristic (OC), or “efficiency,” curves, which quantify the probability that a lot with a given nonconformity level will be accepted or rejected. These curves enable to verify that the chosen AQLs and inspection severities provide the desired balance between consumer protection and inspection effort.

2. The Operating Characteristic (OC) curves

The 1999 version of ISO 2859-1 provided numerous tables in the annex that were illustrated with operating characteristic curves. The 2026 revision of ISO 2859-1 now provides a standardised procedure to calculate these curves. 

To illustrate how operating characteristic curves work, let us consider a level-II single sampling inspection of lots containing 30,000 units.  According to Table 1 of ISO 2859-1, the code letter “M” is applied to this  lot size and inspection level, corresponding to a sample size of 315 units under normal inspection. With an AQL of 0.65, the acceptance criterion is calculated as 5 and the rejection criterion as 6. In other words, the lot is accepted if the sample contains 5 or fewer nonconform units; if it contains 6 or more, it is rejected.

The OC curve can be built using binomial distribution, as described in Appendix E.2 of ISO 2859-1. The probability of accepting a lot that contains a proportion (p) of nonconform units is calculated according to the following formula : =BINOM.DIST(5 ; 315 ;  p ; TRUE) * 100 and gives the following OC curve: (Figure 1)

The X-axis represents the submitted quality of the products, meaning the theoretical nonconformity rate within the lots. The Y axis indicates the probability of accepting a lot with that specific nonconformity proportion. 

By applying the AQL of 0.65 % the following probabilities are calculated:

• At a nonconformity rate of 1.0%, 90% of the lots will be accepted.
• At a nonconformity rate of 2.0%, 40% of the lots will be accepted.
• At a nonconformity rate of 3.0%, 9% of the lots will be accepted.

3. What about using AQL standards for process validation?

Visual inspection validation is required for any new process or product, and each time a process change occurs. As with any critical operation, the process must be validated by testing a sufficiently large number of units. This ensures that the equipment can consistently and reliably detect nonconformities and that the overall inspection process performs adequately. 

At present, neither prevailing standards nor regulatory guidelines provide an explicit methodology for determining the appropriate sample size for such validation activities. Some companies therefore apply a “worst-case” AQL sampling approach: To minimize risk, they select the highest level of control (level III) along with the tightened inspection severity. Is it a risky approach?

Using the previously described example, the table below contrasts routine attribute sampling with those for the alternative approach sometimes used by certain companies. 

The following graphic compares the two sampling methods:(Figure 2)

 As expected, the approach with 500 units is more stringent. However, is it sufficient? If three consecutive lots pass inspection, can we truly conclude that the process is under control?

Two hypotheses have been analyzed to evaluate the scenario:

– Hypothesis 1: the production quality is not satisfactory: p=1% > 0.65%

In routine (315 units, A=5 & R=6), the probability of acceptance is 90.1%⁽¹⁾

When the AQL sample is 500 units with A=5 & R=6, the probability of acceptance is reduced to 61.6%⁽²⁾ The probability of acceptance of three consecutive AQL samples of 500 units is reduced even further to 23%⁽³⁾. There is a one in four probability that a process producing 1% nonconformities will still pass a three lot validation however, such a nonconformity rate exceeds the AQL and cannot be considered acceptable for validation.

– Hypothesis 2: the production quality is the worst tolerable quality level: p=0.65%

In routine (315 units), the probability of acceptance is 98.2%⁽⁴⁾.

When the AQL sample is 500 units with A=5 & R=6, the probability of acceptance is 89.0%⁽⁵⁾. The probability of acceptance of three consecutive AQL samples of 500 units is reduced to 70%⁽⁶⁾. Therefore, there is a high probability of concluding that the process is under control, even when the production quality is the worst tolerable quality level. However, accepting a quality level at the AQL carries substantial risk; it is strongly recommended that production quality be better than the AQL, otherwise there is no safety margin.

4. How to determine the sample size?

The same approach can be applied, using relevant parameters regarding the acceptable level of risk during validation. The parameters to be defined are:

• the AQL level that will be used for routine inspection, 
• the number of lots you plan to test during validation, 
• the consumer risk is defined as the nonconformity rate at which  a lot should be rejected in more than x% of cases. This rate is generally defined for a single lot, but can be defined for all validation lots.
• the producer risk is defined as the nonconformity rate at which a lot should be accepted in more than y% of the cases. This rate is generally defined for a single lot, but can be defined for all validation lots.

Once these elements are defined, binomial distribution can be used to determine the appropriate sampling size and acceptance criteria.

For example, consider the following assumptions: 

• 3 lots will be tested during validation, 
• The AQL level for routine inspection will be 0.65, 
• Consumer risk for validation: 
• If the nonconformity rate reaches 1.0 %, the probability of accepting an individual lot is less than 10%.
• If the nonconformity rate reaches 0.7 %, the probability of accepting all three lots is less than 10% (at least one lot not accepted in 90% of the cases).
• Producer risk for validation: For a nonconformity rate of 0.3%, the probability of accepting all three lots is 95%

The probability of accepting each individual lot can be calculated, using different possible nonconformity rates. These values can then be combined to determine the probability of accepting all three validation lots. 

Probability of acceptance for each individual lot: Figure 3

Probability of acceptance of the three validation lots: Figure 4

If the sample size of 1500 units is chosen with a maximum acceptance criterion of 9 non-conforming units, then:

• At a nonconformity rate of 0.3% (half the AQL of 0.65%), the probability of accepting all 3 lots is 95%.
• At a nonconformity rate of 0.7% (that is, slightly higher than 0.65%), the probability that all 3 lots pass the validation is less than 10%.
• At a nonconformity rate of 1.0%, the probability that each lot passes the validation is lower than 10%.

Determining suitable parameters is inherently complex. The values used for validation depend on the criticality of the nonconformities, the process performance, and the level of risk each company is willing to accept. Using larger sample sizes helps to establish justified criteria that improve the reliability of validation and reduce the risk of undetected issues in process control during validation.

Conclusion

AQL based sampling plans are widely used for routine inspection. However, they are not directly suitable for process validation as they do not ensure that the process will consistently meet routine acceptance criteria.

During the operational qualification of the line, it is necessary to manually inspect the compliant units resulting from automated inspection, according to a predetermined severity. When sample sizes and criteria are chosen based on producer risk and consumer risk via binomial distribution, the resulting strategy can be scientifically justified. Manufacturers can thus align validation requirements with the expected performance of the visual inspection process, ensuring that the process complies with acceptance criteria when used in routine production. Moreover, it is recommended that 100% of the rejected units be manually characterized by operators qualified, in order to confirm detection and to identify the nonconformities. When the number of rejected units is too high, a sampling strategy should also be applied if appropriate and justified.

As part of the Quality Process Performance (QPP) or Continued Process Verification (CPV) process, it is recommended to start production with increased control (level III control, or enhanced or adapted control according to the manufacturer’s strategy). These initial phases of process monitoring help to consolidate the start-up. 

Once performance has been demonstrated and remains stable, accepted units of routine lots can then be inspected with a simple level II control. It is recommended to characterize the rejected units in order to identify the existing nonconformities as well as false rejections. This characterization allows to monitor the evolution of nonconformities, including process degradation, and emergence of new nonconformities, and also helps to detect potential deviations in the visual inspection process. When the number of rejected units is excessive, a sampling strategy can be applied, if appropriate. The limits for rejected units should be based on historical data of rejection and should be consistent with AQL controls and the performance of the inspection system. 

The use of the binomial distribution during validation provides a rigorous and scientifically sound method for demonstrating process reliability. Furthermore, on account of its ability to provide objective documentation of process performance, this approach can be used in other contexts, such as investigations, comparability studies, or continuous improvement initiatives.

References

• 1. Microsoft Excel formula: 0.901 =BINOM.DIST(5 ; 315 ; 0,01 ; TRUE) 
• 2. Microsoft Excel formula: 0.616 =BINOM.DIST(5 ; 500 ; 0,01 ; TRUE).
• 3. 0.23 = 0.616 x 0.616 x 0.616
• 4. Microsoft Excel formula: 0.982 =BINOM.DIST(5 ; 315 ; 0,0065 ; TRUE).
• 5. Microsoft Excel formula: 0.890 =BINOM.DIST(5 ; 500 ; 0, 0065; TRUE).
• 6. 0.70 = 0.89 x 0.89 x 0.89