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Sampling in Pharmaceutical and Chemical Industries

  1. Henrik Melgaard

Published Online: 15 MAR 2008

DOI: 10.1002/9780470061572.eqr151

Encyclopedia of Statistics in Quality and Reliability

Encyclopedia of Statistics in Quality and Reliability

How to Cite

Melgaard, H. 2008. Sampling in Pharmaceutical and Chemical Industries. Encyclopedia of Statistics in Quality and Reliability. IV.

Author Information

  1. Novo Nordisk A/S, Allé, Bagsværd, Denmark

Publication History

  1. Published Online: 15 MAR 2008

1 Introduction

  1. Top of page
  2. Introduction
  3. Sampling under Measurement Uncertainty
  4. Bulk Sampling
  5. References

There are some characteristics of the pharmaceutical and chemical industries that make a difference on how sampling inspections are performed. The product characteristics are given by biological, chemical, and physical measurements. It is often related to batch production. At least for the pharmaceutical industry they are often highly regulated by government authorities with high demands on quality standards and level of documentation.

The measurement methods being used include both slow and expensive off-line methods and fast and cheap on-line measurements. The trend in these industries is that more of the fast and cheap on-line methods become available and to some extent replace the off-line methods. This also means that larger sampling and data evaluation becomes an integrated part of sampling inspection and batch release.

The US Food and Drug Administration (FDA) has recently launched a “process analytical technology (PAT) initiative” supporting this development, see [1]. A desired goal of the PAT framework is to design and develop processes that can consistently ensure a predefined quality at the end of the manufacturing process. Such procedures would be consistent with the basic tenet of quality by design and could reduce risks to quality and regulatory concerns while improving efficiency. Gains in quality, safety, and/or efficiency vary depending on the product and are likely to come from the following actions:

  • reducing production cycle times by using on-line measurements and controls;

  • preventing rejects, scrap, and reprocessing;

  • considering the possibility of real-time release;

  • increasing automation to improve operator safety and reduce human error;

  • facilitating continuous processing to improve efficiency and manage variability

    • using small-scale equipment (to eliminate certain scale-up issues) and dedicated manufacturing facilities,

    • improving energy and material use and increasing capacity.

There are obviously large benefits to gain from this development for the customers in terms of quality, safety, and cost. There will still be a need to perform off-line measurements and classical sampling inspection and release testing in the pharmaceutical and chemical industries as there may be no alternative.

2 Sampling under Measurement Uncertainty

  1. Top of page
  2. Introduction
  3. Sampling under Measurement Uncertainty
  4. Bulk Sampling
  5. References

Sampling of discrete items in lots for inspection is mainly used in the medical device part of the industry. Figure 1 is an example of a person with diabetes injecting himself with a prefilled insulin pen. The example illustrates the critical environment for certain medical products, hence the need for documented high quality products. This is why, for this part of the industry, there may be a specific need for taking measurement uncertainty into consideration.

thumbnail image

Figure 1. Person with diabetes injecting himself with a prefilled insulin pen

Sampling plans for inspection by attributes are given in the ISO standard series ISO 2859 [2]; and similar plans for inspection by variables are given by the ISO standard series ISO 3951 [3]. It is a general idea—at least among statisticians—that sampling by variables is a more efficient procedure for acceptance sampling, than sampling by attributes. This idea is supported by the fact that under a correct model, a parametric estimator of the fraction nonconforming is more efficient than the nonparametric estimator based upon the crude count of nonconforming items in the sample.

In industrial praxis, an important issue is the presence of measurement errors. In the measurement situation, the ideal situation is of course that measurement errors are negligible compared to the tolerance interval of the item to be measured. This is not always the case, though. For medical devices there are often high requirements on certain critical parameters, e.g. the dosing accuracy. This again leads to high requirements on the individual components in the tolerance stack up. Both the ISO standards mentioned before [2, 3] consider the case of negligible measurement errors. Therefore, in the following, we shall discuss the implications of using acceptance sampling by variables under measurement error.

The theory for acceptance sampling by variables under a normal distribution dates back to the papers by Lieberman and Resnikoff [4] and [5]. The current ISO standard with sampling plans, ISO 3951 [3], is based upon this original idea of basing the test upon a minimum variance, unbiased estimate of the fraction nonconforming in the process. A description of the theory may be found in [6].

In [7], Owen and Chou, consider the effect of measurement error and a constant offset error on the operating characteristic (OC) curves (see Single Sampling by Attributes and by Variables) of the one-sided plans. In [8], Thyregod and Melgaard, however, extend these considerations to the double-sided plans in the case of measurement error and a random “laboratory” bias.

We shall assume that items in a lot are produced from an in-control process. Let X denote the quantity of interest. We shall assume that the distribution of X over the items in a lot may be described by independent and identically distributed (i.i.d.) random variables that follow a normal distribution with mean E(X) = μ and V(X) = σ2. It should be noted, that in pharmaceutical and chemical processes the assumption of independent items in a lot often must be challenged. Simple graphical methods such as plotting the items in the order of production will often reveal correlations that must be taken into account.

Let the specification limits for X be upper limit, U, and lower limit, L, respectively. The fraction of nonconforming items, p, is the fraction of items below L or above U. In the related article Variables Sampling under Measurement Error the case of high-frequency measurement error is discussed in detail.

2.1 High-Frequency Measurement Error

Assume that the quantity of interest cannot be measured without measurement error.

Let Y = Y(X) denote the measurement result for an item with value X. Assume that

  • equation image(1)

where E1, …, En are i.i.d. and normally distributed with E(Ei) = 0 and equation image. Let, as usual,

  • equation image(2)


  • equation image(3)

Then Y follows a normal distribution with E(Y) = μ and equation image, and equation image follows a equation image distribution with f = n − 1, and Y and equation image are independent. For simplicity we will use the OC curves, corresponding to the one-sided plans, which are similar to the ones given in ISO 3951. These will be close to the double specification limits case, when σ ≪ σmax. It is shown in [8] that by using the same sampling plan from ISO 3951—but in this case random measurement error is present—according to the model (1) the OC curve is given by

  • equation image(4)

where equation image. One thing to remark is that the OC curve in the case of random measurement noise is always to the left of the noise-free OC curve, which means that in the case of high-frequency measurement noise; there is a higher chance of rejecting the batch. An example from ISO 3951 is given by Figure 2.

thumbnail image

Figure 2. OC curves under high-frequency measurement error, AQL = 6.5, code K, n = 50, σe = 0.2σ, the dashed curve is for the model without measurement error

2.2 Low-Frequency Measurement Error

Assume instead that the major part of the measurement error is a random “laboratory” or instrument bias that affects all measurements in the same manner, i.e.

  • equation image(5)

where B varies from lot to lot as i.i.d. and normally distributed with E(B) = 0 and equation image. Under this assumption we have that Y follows a normal distribution with E(Y) = μ and equation image, and equation image follows a χ2(f)/f distribution with f = n − 1, and Y and equation image are independent. Thus, in this case the uncertainty of the estimate, Y, of the position, is heavily influenced by the measurement uncertainty, but the estimate, equation image of the process spread is not affected.

The OC curve for this model is then given by

  • equation image(6)

where equation image. If we let

  • equation image(7)

in the equation for L2(p) we see, that the equation is very close to the noise-free one, but with a sample size of n*. The only difference is that the degrees of freedom in the t distribution are not changed. This means, that approximately the OC curve in case of a systematic measurement error is similar to the OC curve of the noise-free case, but with a reduced sample size.

In the example given by Figure 3, we have a sample size of n = 50 and a measurement uncertainty of σB = 0.2σ. Approximately this corresponds to the OC curve of the noise-free case with a sample size of

  • equation image(8)
thumbnail image

Figure 3. OC curves under low-frequency measurement error, AQL = 6.5, code K, n = 50, σB = 0.2σ, the dashed curve is for the model without measurement error

This fact is also revealed from Figure 3. It is important to notice the difference between the noise-free case and the situations with high-frequency as well as low-frequency noise. Especially for testing in the pharmaceutical and chemical industries the measurement uncertainty is not negligible. This requires that the measurement method is well documented, including quantification of the high- and low-frequency measurement error. Since the influence of a low-frequency measurement error can be large, as shown in the previous example, steps will normally be taken to minimize this effect, e.g. by calibrating between sampling of lots.

3 Bulk Sampling

  1. Top of page
  2. Introduction
  3. Sampling under Measurement Uncertainty
  4. Bulk Sampling
  5. References

3.1 Purpose and Procedure of Bulk Sampling

Chemical analyses of bulk materials as powder or liquids are often found in the pharmaceutical and chemical industries by its nature. Sampling can be used for the inspection of raw materials, intermediate products, or final product release. In this case a lot is a definite quantity of bulk material. The quality of the lot is measured by a single suitable quality indicator, e.g. the content of active ingredient. It is assumed that the mean quality of the lot is to be determined and that this is the factor used for determining the acceptability of the lot. In this case, acceptance sampling plans and procedures for the inspection of bulk materials are found in ISO 10725 [9].

A number of increments (smaller volumes), k, of the same size are taken randomly from the lot. These increments are mixed into a gross sample. From this gross sample a laboratory sample is prepared and a number of samples, m, is taken from this laboratory sample and analyzed individually. The results, xi, i = 1, …, m, are combined into a single mean value that is representative of the lot. The mean value of the analyzed samples is used as the estimate of the mean quality of the lot. The mean is calculated as

  • equation image(9)

The uncertainty (variance) of this mean value is given by

  • equation image(10)


  • equation image is the variance of the increments from the bulk due to variations between containers and within containers,

  • equation image is the variance associated with the preparation of the laboratory sample taken from the gross sample,

  • equation image is the variance describing the uncertainty from preparing and analyzing the individual samples for analyses taken from the laboratory sample.

The model above is a simple statistical model describing the bulk sampling situation. To have a “statistical rationale” for the choice of k, the number of containers to sample from, and the choice of m, the number of chemical analyses to perform, some estimates of the variance components must be found, e.g. from designed statistical experiments, see [9].

3.2 Retesting of Bulk Materials

Retesting is when new laboratory samples are taken from the gross sample of a lot to be analyzed because of suspicious analytical results. For specific chemical or biochemical laboratory analysis, a large number of steps are involved and the analyses are not always fully automated. In these cases there is a possibility of occasional (human) errors affecting the results. Therefore, there are well-described procedures for good manufacturing practice (GMP) including procedures for handling out-of-specification (OOS) measurement results, see [10]. In the case of discrete items handled in the section titled “Sampling under Measurement Uncertainty”, the ISO standards have built-in procedures on how to handle the situation of OOS results and batch rejection.

The procedure [10], applies to laboratory testing. It applies to the situation described previously where the mean quality of the lot is of interest. It does not apply if the purpose of the analyses is to measure uniformity of a lot, e.g., content uniformity, release profile, and powder blend.

To identify the cause of the OOS result, statistical hypothesis testing may be relevant. Hypothesis testing may consist of repetition of the test procedure or part of the procedure, or of experiments designed specifically with the purpose of identifying an analytical problem. Below is an example of retesting.

3.2.1 Use of t-Test to Compare OOS Results and Retest Results

The t-test is appropriate in a retest situation where an OOS result is compared to a group of results obtained specifically, to compare it to the earlier result, as described by Søren Andersen [11]. The test statistic is given by

  • equation image(11)

where mretest and sretest are the mean and standard deviation of the retest results and m is the number of retests. The retest samples should be representative of the lot being sampled, i.e., use the gross sample from the previous test or create a new gross sample based on increments randomly taken from the lot. The calculated test statistic, T, is compared to the critical value of the t-test, given e.g., in [11]; if T is greater than the 5% critical value, the OOS is considered a statistical outlier. If the OOS result is considered a statistical outlier, we have a strong indication that the result is due to a laboratory error. A qualified person should then be able to evaluate the necessary steps to eventually release the batch.


  1. Top of page
  2. Introduction
  3. Sampling under Measurement Uncertainty
  4. Bulk Sampling
  5. References
  • 1
    FDA PAT initiative:, 2005.
  • 2
    ISO 2859, series Sampling Procedures for Inspection by Attributes, International Organization for Standardization, 1992–2007.
  • 3
    ISO 3951. (1989). Sampling Procedures and Charts for Inspection by Variables for Percent Nonconforming, International Organization for Standardization, Geneva.
  • 4
    Resnikoff, G. J. (1952). A new two-sided acceptance region for sampling by variables, Technical Report No. 8, Applied Mathematics and Statistics Laboratory, Stanford University, Stanford.
  • 5
    Lieberman, G. J. & Resnikoff, G. J. (1955). Sampling plans for inspection by variables, Journal of the American Statistical Association 50, 457516.
  • 6
    Schilling, E. G. (1982). Acceptance Sampling in Quality Control, Marcel Dekker, New York.
  • 7
    Owen, D. B. & Chou, Y.-M. (1983). Effect of measurement error and instrument bias on operating characteristics for variables sampling plans, Journal of Quality Technology 15, 107117.
  • 8
    Melgaard, H. & Thyregod, P. (2001). Acceptance sampling by variables under measurement uncertainty, in Frontiers in Statistical Quality Control, H.-J. Lenz & P. Th. Wilrich, eds, Physica-Verlag, Heidelberg, Vol. 6, pp. 4757.
  • 9
    ISO 10725. (2000). Acceptance Sampling Plans and Procedures for the Inspection of Bulk Materials, International Organization for Standardization.
  • 10
    Guidance for Industry Investigating Out-of-Specification (OOS) Test Results for Pharmaceutical Production, Center for Drug Evaluation and Research (CDER) and U.S. Food and Drug Administration, 2006.
  • 11
    Andersen, S. (2004). An Alternative to the ESD Approach for Determining Sample Size and Test for Out-of-Specification Situations, Pharmaceutical Technology, May 2004.