Christopher Burgess, PhD, is managing director, Burgess Analytical Consultancy Limited, Barnard Castle, Co Durham, UK.

Christopher Burgess

Many may wonder at the number of analytical chemists who have faced challenges from management who demand answers from analytical results without explanation of their statistical significance. For the analytical chemist, however, results are never just about getting an answer. For any analytical procedure’s reportable value, there is only a most probable estimate of the true value with an associated interval of confidence. Unfortunately, this fact is not always readily appreciated by management. Perhaps they would prefer to have precise ‘lies’ than the inexact ‘truth.’

So, what might be the root cause of this problem?One suggestion is that some members of management don’t feel comfortable with the idea of uncertainty at all. This attitude probably stems from a fundamental lack of understanding that all measurements and their derived values are subject to error, and that this error is normal (usually). I hope that I may be excused the pun!

Suppose that management can be convinced of the reality of error and uncertainty in analytical measurements. Hence, there will be a necessary requirement to minimize or remove as much error as economically feasible. What can be done about it? As a first step, the potential sources of error within analytical processes must be identified and their individual contributions to that error must be quantified, as illustrated in 

 Error sources and types for analytical procedures.

Gross errors are such that only abandonment of the analysis and a fresh start is an adequate cure.For example, if a portion of the sample is spilt or an instrument has been improperly set up, then no statistical methodology can be applied.Random errors occur on either side of the reported result and affect precision (repeatability, intermediate precision, and reproducibility).This type of error may be thought of as the noise element in the measurement.Systematic errors relate to all the results being biased in one direction, either too high or too low, and hence affect accuracy (closeness to the true value).Random and systematic errors are amenable to statistical evaluation.

Process mapping as an aid to error evaluation

It has long been standard practice to draw process maps for many business processes. However, this is strangely not so common for laboratory analyses. If one can draw a process, one gain an objective understanding of the technical and procedural steps involved. Process mapping tools have long been available since the early days of Six Sigma evaluations, including flow charts and the associated Ishikawa diagrams, for example.

These tools and techniques are less well applied in the laboratory, which is unfortunate. Since the rise of data integrity concerns by regulators and the necessity to assure demonstrable control in analysis, however, their use is increasing.

The principle of starting simple to understand the process is a good one. Learning to walk before attempting running is advisable. This article illustrates the mapping concept with a simple manual volumetric titration involving the standardization of a sodium hydroxide (NaOH) solution with a certified reference material (CRM) of potassium hydrogen phthalate (KHP).

First, a basic process flow can be drawn, as shown in 

. The question then must be asked: is this sufficient for the required purpose? Ideally, process would need to be drawn in more detail to identify the possible error sources and possible error types. A more detailed process flow has been drawn in 

 and expands the basic process flow from 

 Basic process flow for the manual titration. C

is concentration of the sodium hydroxide solution.

 Error source mapping and process flow for a simple manual titration. CRM is certified reference material. NaOH is sodium hydroxide.

With the details now clear, it is reasonable to suppose that the manual operations conducted by the analyst pose the greatest potential for error. The assumption can be made that no gross errors have occured and that the proper precautions have been taken to protect the NaOH solution from possible contamination with carbon dioxide (CO2), and that the reference standard is 100% pure.

Consideration of the manual procedure clearly shows that there could be systematic errors by the analyst in the burette readings as well, because the analyst determines the endpoint visually.In addition, there could be systematic errors in the burette itself, despite the burette being Grade A certified. Classical volumetric analysis requires analyst skill and experience, which is not as readily available as it was in the days before instrumental analysis became widely available.

, one can calculate the concentration of the NaOH solution (CNaOH) as 0.1019 mol L

. However, one must ask, how accurate is this manual result, and what confidence can be ascribed to its value?

The answer to these questions will be explored in the latter part of this article.

Automated titrimetry is the way forward, isn’t it?

Autotitrators are now commonly available in many laboratories and should be used to mitigate the manual operational risks of analyst systematic error. For the purpose of this article, it is assumed that management is convinced by the argument for automated titrimetry and has thus condoned the purchase of an autotitator.

 can be redrawn to produce an automated version, which is now shown in 

. As the concept of uncertainty is discussed later in this article using International Organization for Standardization error budget concepts, a new term must be introduced here to denote the output from an analytical process; that term is the measurand (1). This term simply means the particular quantity to be measured; in this instance, the quantity to be measured is the C

 Basic process flow for the automated titration.

Assuming that the instrument has been properly qualified, the titration process can be automated to minimize the titration procedural errors. Note that the word used here is “minimize” not “remove.” The process now relies on an autotitator with a digitally controlled piston burette under software control. As will be demonstrated, the use of this autotitrator makes the estimation of uncertainty in the measurand a straightforward process. This particular process was chosen because it was used as a worked example, A2, in the Quantifying Uncertainty in Analytical Measurement (QUAM:2012.P1) (2) for the classical error budget approach that will be described later.

Having drawn the automated process flow, an Ishikawa (Fishbone) diagram can be used in 

 to examine in more detail the four process elements that contribute to the uncertainty, namely:

the molecular weight of the KHP CRM itself

the volume of the NaOH titrant required to neutralize the weight of the KHP CRM.

 Ishikawa diagram for the automated titration and the measurand. KHP is potassium hydrogen phthalate. CRM is certified reference material. NaOH is sodium hydroxide.

As this is an exercise, the uncertainties in the atomic masses of carbon, hydrogen, oxygen, and potassium will be considered, and the uncertainty of the molecular weight of KHP will be calculated. As will become apparent, this error source would not need to be considered routinely; however, without these data, there will be no basis for ignoring the error source in the future. With the setup in place, it will now be possible to estimate the uncertainty in the measurand. An error budget must first be constructed from the Ishikawa diagram, however.

The basics of the measurement uncertainty approach and error budgets have been described in previous papers (3,4), and the detailed formal calculation approach is described in another paper (2). However, it is best to visualize the components as a structured model as shown in 

, the reason being that the visualization makes clear the interrelationships between the error sources needed when performing the calculations. 

 Error budget structure for the automated titration. NaOH is sodium hydroxide. SD is standard deviation. KHP is potassium hydrogen phthalate.

There are two distinct calculation methods available. The first method requires approximations of the error distributions and an Excel spreadsheet to give a single value as described in the EURACHEM/CITAC Guide (2), and the second method does not require approximations and produces a series distribution of values; the second method does, however, require simulation software (5). The starting point is the same for both these methods.

The equations to be used are facilitated if 

 is updated to reflect the autotitator process and the uncertainty contribution data are added, as shown in 

 Error source mapping and process flow for the automated titration with uncertainty data. KHP is potassium hydrogen phthalate. NaOH is sodium hydroxide.

For example, the corrected volume of the titrant, V

, is a function of the accuracy of the autotitator, the temperature of the laboratory, and the measured volume. The accuracy of the autotitator is known to be ±0.03 mL. It is assumed that this range is distributed as a triangular distribution because it is more likely to have a value nearer the center.

The laboratory temperature is controlled between 17 °C and 23 °C, a range that has a uniform or rectangular distribution. In the first calculation approach (2), these ranges are converted to standard deviations, using approximations, by dividing the values using factors of √6 and √3. The basis for these approximation factors is described in a technical appendix (2).

The overall uncertainty of the measurand using the method fully described in the EURACHEM/CITAC Guide (2) is given as 

This equation may be mathematically elegant, but it is time consuming to carry out and not really user friendly for use in laboratories.

However, there is another, more visual way. The second method, which uses Monte Carlo Simulation (MCS), will be described in more detail later in this article, where a comparison of the results from these two approaches will be given. The classical calculation, illustrated in 

, gives a single value of the uncertainties and approximates the distributions. MCS yields many values that are then used to calculate the required values and associated uncertainties. This method is much quicker and easier to perform but requires large amounts of computing power and a specialist software application.

MCS statistical modeling has been around for more than 70 years (6), but because it requires large amounts of computational power, it remained a specialist area for approximately 60 years or so. With the advent of cloud computing and user-friendly application software, however, MCS modeling is now readily achievable without the need for programming knowledge. In this paper, Minitab Workspace software (5) is used for the automated titration example.

Having identified 12 inputs (X) for the error budget, one can set up the contributions using drop down menus for distributions from 

 and then write the output equations. The 12 inputs are shown in 

. The distribution type can be a fixed value (constant) as in the case of the observed mass of KHP and the observed volume of titrant. Visually, the distribution shape is automatically given based on the selected distribution. 

 Inputs for the Monte Carlo Simulation of the automated titration. KHP is potassium hydrogen phthalate. NaOH is sodium hydroxide.

Based on these inputs, simple equations can be written for the five outputs (Y) needed to calculate the uncertainty of the measurand only using the X input variables already defined in 

 Output equations required for the Monte Carlo Simulation of the automated titration (note that the temperature correction for volume name is truncated in the display). KHP is potassium hydrogen phthalate. NaOH is sodium hydroxide.

Before the simulation is run, there is a dendrogram display of the relationships between the 12 (X) inputs in blue and the five (Y) output equations in orange, which should be checked for model correctness and for completeness, as shown in 

Monte Carlo Simulation model relationships for the automated titration. NaOH is sodium hydroxide. KHP is potassium hydrogen phthalate.

The purity of the KHP input has been included in the mass of KHP output, and the RFP factor has been included in the volume output for computational convenience; these inputs yield the same numerical answers. However, the structure in 

 is necessary when the comparison of the two approaches is considered.

Now the MCS model can be run. How many iterations (N) are needed to get reliable outputs?Usually, 10 to 50 thousand iterations are sufficient; however, Minitab Workspace currently allows up to one million (note that this would not be practical in Excel).

N = 500,000 was selected for this example. How long does it take to do the calculations half a million times, do the statistics, and plot the output files in the Cloud? The answer to that question is not clear because much of the time depends on the speed of the Internet connection. For this example, the time taken was about 10 seconds, which was quick enough, and another few seconds to download and export the file to PowerPoint. The total time to set up the X inputs and Y outputs was approximately 30 minutes. The output (see 

) gave a value for the measurand of 0.1021, which was close to that of the manual value of 0.1019. When a 95% confidence range is calculated from the modeling, the results come out to 0.1019–0.1023 by both approaches. 

 Measurand output from Minitab Workspace. NaOH is sodium hydroxide.

Before looking at a comparison between the two approaches in more detail, it is interesting to take note of the uncertainty of the molecular weight of the KHP, while keeping in mind that the atomic masses of carbon, hydrogen, oxygen, and potassium from the EURACHEM/CITAC Guide were inputted, and that those atomic masses were taken from International Union of Pure and Applied Chemistry values at the time with their uncertainties model as a rectangular distribution. The molecular weight of KHP, as defined by the first Y output equation in 

, was calculated 500,000 times from values selected at random from the defined rectangular distributions.The result is shown in 

 Molecular weight for potassium hydrogen phthalate (KHP) from the Monte Carlo Simulation model.

Therefore, the relative standard deviation in the molecular weight of KHP due to uncertainties in the atomic masses is 0.0019%. Now that this value is known, it won’t be necessary to include it next time.

The results from the classical QUAM calculation approach and the MCS approach are shown in 

. Whilst there are some small individual differences, which would be expected, the uncertainty values for the measurand are comfortingly close. However, the real power of using a modeling approach is the comparison of the sizes of the error sources. This comparison is graphically shown in 

 Numerical results from the two measurement uncertainty calculation approaches. MCS is Monte Carlo Simulation. KHP is potassium hydrogen phthalate. NaOH is sodium hydroxide.

 Error contributions as calculated by the two approaches.

Pictures speak louder than words! Both approaches are comparable and provide the same information: having gotten rid of the analyst contributions, the remainder is dominated by the autotitator contributions for accuracy and precision of the titrant volume. At 95% confidence, the uncertainty for the molar concentration of the NaOH solution is 0.20%, which is, for most purposes, good enough.

Analytical chemists need to be comfortable with the error structure of their procedures and need to be able to communicate measurement uncertainty to management. To do this effectively and efficiently, they should consider employing the tools of process mapping and error budget calculation approaches as described.Data integrity and data quality are increasingly subject to regulatory scrutiny, and scientifically sound and defendable approaches are thus required.

International Vocabulary of Metrology—Basic and General Concepts and Associated Terms (VIM)

Quantifying Uncertainty in Analytical Measurement

.
3. Burgess, C. The Basics of Measurement Uncertainty in Pharma Analysis. 

 (9).
4. Burgess, C. Measurement Uncertainty without the Math. 

 (2), pp. 36–40.
5. Minitab. Minitab Workspace. www.minitab.com (accessed on Sept. 14, 2022).
6. Metropolis, N.; Ulam, S. The Monte Carlo Method

Articles

How confident are you in your lab results? This paper explores this question and, by way of a worked example, provides a methodology to answer it.

Never Mind the Statistics; Just Tell Me What the Answer Is! 

Weighing samples and standards is a fundamental analytical operation. It is such a familiar laboratory operation, usually digitally displayed and captured, that one tends to intuitively believe that the numbers from balances are error free, or at least have such a small uncertainty that any errors will be negligible. On many occasions, this may well be true but not always. Modern electronic balances rely on a load cell. As force is applied to the loadcell or transducer by calibrated masses, the response is measured. The responses from the loadcell are transformed into a mass calibration function supported by digital averaging, smoothing, and rounding firmware, which leads to the displayed mass on the balance. The purpose of this article is to look at the uncertainties in terms of their source, and provide one best practice to avoid errors in weighing by difference caused by buoyancy and drift.

Chris Burgess is managing director, Burgess Analytical Consultancy Limited, Barnard Castle, Co Durham, UK.


Volume 46, Number 5
May 2022
Pages: 48–51

When referring to this article, please cite it as C. Burgess, "Weighing Isn’t an Uncertain Process, Is It?," 

This article provides best practice to avoid errors in weighing by difference caused by buoyancy and drift.

Weighing Isn’t an Uncertain Process, Is It?

Regulations require drug manufacturers determine the detection limit (DL) and quantitation limit (QL) of impurities in APIs and drug products. There are multiple ways of doing so, some of which are described in the International Conference on Harmonisation (ICH) guideline, ICH Q2 (R1) 

Validation of Analytical Procedures: Text and Methodology

. Primarily, these methods are based upon the statistical uncertainties from a linear least squares regression calibration curve. However, different calculation methods give different answers. The following describes an example and a recommended best practice method.


Volume 45, Number 11
November 2021
Pages: 54–59

When referring to this article, please cite it as C. Burgess, “Detecting and Determining Quantitation Limits for Impurities,” 

Different methods give different answers when calculating limits for impurities. The prediction interval method may be the best option.

Detecting and Determining Quantitation Limits for Impurities

	Analytical people tend not to worry too much about the nature of the distribution of their data.  Statisticians, on the other hand do worry.  Some 10 years ago, Rebecca Elliott, a senior statistician with Eli Lilly, gave an excellent presentation on some ways users drive statisticians crazy (1).  This column is about just one method to prevent that happening, as we do need statisticians occasionally. 

	The central limit theorem (CLT)-actually theorems as there are more than one-is at the heart of much statistical methodology. The CLT is a lifesaver for analytical data, be it continuous or discrete.  The good news is that the part that must be understood is very simple.  The really good news is that if the CLT did not exist, many familiar statistical methods would not be valid.  The first lesson for analytical people to learn is that, from a statistical point of view, there is a world of difference between individual or replicate data from a sample and means of those data.

	It is visually apparent, therefore, that to use methods based on the assumption of normality would be invalid in many instances. Fortunately, the CLT comes to our aid.

	It is the mathematical model (shape) of data population which determines the applicability (suitability) of the statistical methodology.  A summary of some commonly met distributions are shown in 

	The simple part of the CLT is that for any sample of N independent determinations, the means of n values tend to a normal distribution irrespective of the underlying population distribution. In addition, the overall or grand mean tends to the population mean. Note that the bigger n is, the more this is true. Unlikely though this seems, some calculations can be performed in Microsoft Excel to demonstrate this property. This is not well described in books on statistics, but 

Basic Statistics and Pharmaceutical Statistical Applications

 does describe this well, albeit somewhat hidden away in the 800 pages (3). The following is one way to visualize the value of the CLT.

	Let us consider a small data set of 15 integer numbers far away from a normal distribution, namely 8,8,8,9, 9,9,10,10,10,11,11,11,12,12,12, which are shown as a histogram in 

	Let us take the decision that n is to be two and we know that there will be 105 means of two combinations of these 15 numbers. In Excel, this would be calculated using the combination formula;=COMBIN(15,2). Sadly, there is no function currently in Excel to list all of these 105 combinations, but fortunately there is a macro available on the Internet to do just that (4). Once you have these combinations listed, then you can calculate the means. The result is shown in 

	Looking good so far; so what about n=3? In Excel, this is calculated from the formula =COMBIN(15,3), which gives 455 combinations. Modifying the macro works well and generates the 455 combinations listing. We proceed as before by calculating the means of 3 resulting in the histogram shown in 

	The approximation to the normal distribution becomes even more apparent. The normal approximation becomes good when n is in the region of 25 to 30. This was demonstrated in an earlier column using the t distribution (5). This particular calculation cannot be performed in Excel because if one used N=50 data points and n=30, one would have 47,129,212,243,960 combinations, which would take, at one calculated mean per second, just under 1.5 million years to achieve even if Excel allowed that many rows.

	The central limit theorem concerning the distribution of means allows one to justify the assumption of the normal distribution so that we can use many of the statistical formulae that require normality on our datasets. As long as mean and individual data are clearly differentiated, we will help our statistician colleagues remain sane.

	1. R. Elliott, “Product Specifications: How to Drive a Statistician Crazy,” Pharmaceutical Statistics 2008: Confronting Controversy Conference, Arlington VA, 2008.
	2. A. Damodaran, Probabilistic Approaches: Scenario Analysis, Decision Trees And Simulations, people.stern.nyu.edu/adamodar/pdfiles/papers/probabilistic.pdf, Figure 6A.15: Distributional Choices.
	3. JE De Muth, 

, 3rd edition, page 123 (CRC Press, 2014).
	4. Allen Wyatt, Listing Combinations, 


	Vol. 43, No. 10
	October 2019
	Pages: 62–64

	When referring to this article, please cite it as C. Burgess, "Distribution of Data: The Central Limit Theorem," 

Using the central limit theorem concerning the distribution of means allows one to justify the assumption of the normal distribution. 

Distribution of Data: The Central Limit Theorem

	Performance metrics for processes are an area of much regulatory interest currently. There isn’t always a readily available clear definition of what is needed, however, and guidance from regulators is not always consistent. This column goes back to the basics that were first set out by Shewhart (1) nearly 90 years ago and relates them to some modern process performance and capability indices.

	Definitions are important in providing a consistent nomenclature, and the global International Organization for Standardization (ISO) standard 3534-2:2006 (2) will be used.

	All process measurement results are subject to variations that come from a variety of sources as was seen in the previous Statistical Solutions column (3). However, there are only two types as defined by Shewhart, namely common cause variation and special cause variation.

	Common cause variation is the inherent noise in a process over time due to random effects and hence predictable within statistically derived limits.

	By definition, a process that contains only common cause variation is said to be in statistical control.

	Special cause variation occurs because of specific circumstances that are not always present manifesting themselves by, for example, a shift or drift in the process mean or excessive noise. If a process contains special cause variation, it is unstable from a statistical point of view, and the overall variation observed contains both common and special cause components. Control charts are designed to detect the presence of special causes of variation. The normal distribution is characterized by two parameters: a measure of location (the arithmetic mean or average) and a measure of dispersion (the standard deviation). An unstable process means that both of these parameters could be or are changing in an uncontrolled manner (

	The task is to bring these two parameters into a state of statistical control. This would entail ensuring that the mean and the standard deviations were not varying significantly. This ideal situation is illustrated in 

. This process would then be said to be under statistical control (i.e., no special cause variation and stable common cause variation). In this state, the process is amenable to the tools of statistical process control (SPC). However, a stable process may not be statistically capable of meeting the specification limits. 

 illustrates this, showing that the red process, albeit stable, is incapable. The desired state is to arrive at the blue capable state.

	Capability is assessed using a family of quality metrics or indices called process performance and capability indices.

	Quality metrics for process performance and capability

	There are a variety of performance indices for processes in regular use. However, in this column, only four will be discussed, P

	The definition and meaning of these four will be defined later. Of these four, only two have any practical relevance, P

. The other two are of theoretical interest as they do not occur in practice other than by chance. Process performance. P

, a process performance index, relates to the output performance of a process, irrespective if it is in control or not, with the specification assuming that the long-term mean will be on the target for the product (an unbiased process).

	The index is defined as a ratio of the difference between the upper and lower specification limits (called the specified tolerance in ISO) and the 99.73% probability of a value lying within ±3 standard deviations from the target (called the reference interval in ISO).

	Hence, it can be said that this index would represent what the customer actually receives from the overall process (see 

 , is calculated from the usual formula for a sample standard deviation.

	Where X is the mean of the N data points.

 of 1.33 or more would indicate a highly capable process. A value of less than 1 would indicate an incapable process that would lead to out-of-specification (OOS) results.

	As it is highly improbable that processes are unbiased, a practical process performance index would need to take this bias into account when assessing process performance. This is done by calculating the upper and lower process performance indices P

 using not the target but the actual observed mean to calculate them from:

	Hence the process performance index, Ppk is given by the smaller of the two values above.

	Process capability. Process capability refers to the performance of the process when it is operating under statistical control. Two capability indices are usually computed: C

 in a similar way as was described with P

 measures the potential capability in the process, if the process was centered, while C

 measures the actual capability in a process, which is off-center or biased. If a process is centered, then C

	The critical thing to note is that while the formulae for P

 look very similar, the standard deviation used to calculate the reference interval for Cpk  is not S

 is the within batch standard deviation (called the within sub-group standard deviation in ISO) not the overall process standard deviation. It is usually estimated from a Shewhart mean and range control chart using the formula:

 are 0.5 to 1 for incapable processes, 1 to 2 for capable processes and >2 for highly capable processes.

	A word of caution is necessary in interpreting C

 analysis requires a normal underlying distribution and a demonstrated state of statistical process control. When reporting a C

 value, a 95% or 99% confidence interval should always be reported because this takes into account the sample size used in the calculation (5,6).

	Sadly, this is usually missing. The confidence interval is extremely important because it is not always recognized that, for reasonably small confidence intervals around C

  values, the number of data points needs to be large. 

 shows that to have a 95% confidence interval in C

  of 1.33 ±10% requires in excess of 200 data points. One commonly used approximation formula (5) for the confidence interval is:

 values for comparison of performance needs to be interpreted with great care when n is small.

	It has been shown how to differentiate between process performance and process capability. Equations, however, are not normally as clear as an example.

 shows data from 157 batches of a product with a target of 7.0 and upper and lower specification limits ± 1.5. The data are nicely normally distributed as can be seen from the normal probability plot, but the long-term mean of 7.4 is biased high. However, the process capability C

 is excellent at 1.31 and even with the bias would be unlikely to produce OOS results due to common cause variation (red curve).

	Unfortunately, the process suffers from considerable special cause variation, the dashed black curve, with P

  being an unacceptable 0.74 because the overall batch standard deviation (St ) is 0.49, whereas the within batch standard deviation (S

 is much smaller at 0.28. Note that if we could remove the mean bias, P

 would be a more acceptable 1.02. However, it would require a root cause investigation and process change(s) to remove some of the special cause variation(s) to approach a truly capable process.

Economic Control of Quality of Manufactured Product

	2. ISO 3534-2:2006; Statistics-Vocabulary and symbols-Part 2: Applied statistics

 GHTF/SG3/N99-10:2004 (Edition 2) Annex A, Statistical methods and tools for process validation, www.imdrf.org/documents/doc-ghtfsg3.asp.

	5. ASTM E2281-15, Standard Practice for Process and Measurement Capability Indices.


	Vol. 42, No. 10
	October 2018
	Pages: 64-67

	When referring to this article, please cite it as C. Burgess, "Process Stability, Performance, and Capability; What is the Difference?," 

This article applies the basics of stability, performance, and capability to modern process performance and capability indices.

Process Stability, Performance, and Capability; What is the Difference?

	In February 2005, Lynn Torbeck wrote “In Defence of USP Singlet Testing,” in which he discussed the US Pharmacopeial Convention’s (USP) philosophy of singlet testing and the clear differentiation between standards and specifications (1). Later that year, Tim Schofield, David Leblond, and Stan Altan offered comments and an alternative viewpoint (2). Much has changed over the past 12 years, although many people in industry still regard the USP standard as a specification for the release of product to the market. This article reviews these changes based on the latest USP General Notices (3) and provides a view on the meaning of ‘singlet testing’. 

	Torbeck emphasized the important philosophical difference between a standard and a specification, and he quoted from the 

) 28 (2005) General Notices (1), “Compendial standards define what is an acceptable article and give test procedures that demonstrates that the article is in compliance. These standards apply at any time in the life of the article from production to consumption.” 

	Furthermore, these General Notices defined the testing, which constitutes a singlet determination. 

	“Thus, when tested from the viewpoint of commercial or regulatory compliance, any specimen tested directed in the monograph for that article shall comply…,” and “Tests and assays in this Pharmacopeia prescribe operations on a single specimen, that is, the singlet determination which is the minimum sample on which the attributes of a compendial article should be measured.”

	It is this last sentence that has given rise to much debate and dissention. Schofield 

 took this to mean that “Every lot of a pharmaceutical products with possibly millions of dosage units per lot must be made by processes that guaranteed that every dosage unit tested by the filed analytical method at any time before expiry conforms with the stated USP standard” (2).

	Clearly, this guarantee is unattainable from both a statistical and practical viewpoint and appears to contradict the uniformity of dosage requirements, which they point out. However, what is clear is that USP monographs are standards and not specifications. The current version of the USP General Notices and Requirements doesn’t mention the singlet determination and clearly states in 3.10 Applicability of Standards (3): “Standards for an article recognized in the compendia (

) are expressed in the article’s monograph, applicable general chapters and General Notices.” Hence, there is a strict hierarchy for compliance in the order; monograph, General Chapters and General Notices: “The standards in the relevant monograph, General Chapters and General Notices’ apply at all times in the life of the article from production to expiration” (3). This has not changed from 2005. 

	“It is also noted that the manufacturer’s specifications and manufacturing practices (e.g., quality by design, process analytical technology, and real-time release testing initiatives), generally are followed to ensure that the article will comply with compendial standards until its expiration date, when stored as directed” (3). This is a significant change from 2005. This quotation addresses the valid concern of Scofield et al. in 2005 that a singlet determination may discourage a full scientific understanding of a manufacturing process and interfere with the objective of providing quality product to patient.

	USP standards are legal benchmarks of acceptable quality or performance and must be met with some acceptable degree of confidence (probability). In testing of a sample, there is no such thing as an exact value. There is only an estimate of the true value with an associated confidence or tolerance interval.

	Specifications, on the other hand, apply to reportable results and are decision limits. They must be met or failure occurs. There is no probability level associated with specifications. 

	What is not disputed is that all acceptance criteria in any monograph must be met to achieve compliance (3). However, the basis for the acceptance criteria often is disputed. USP 4.10.20 in the General Notices says, “The acceptance criteria allow for analytical error, for unavoidable variations in manufacturing and compounding, and for deterioration to an extent considered acceptable under practical conditions …” and requires in addition that, “An official product shall be formulated with the intent to provide 100% of the quantity of each ingredient declared on the label” (3). 

	The purpose of compliance testing is to demonstrate that the sample taken from the lot meets the acceptance criteria for the specific monograph. General Notices 3.10 states, “… in all cases, statements about whether the compendial standard is met apply only to the units tested” (3).

	This statement leaves no doubt that compliance with the monograph relates only to the sample tested and not to the batch or lot. Therefore, it is scientifically unsound to use the acceptance criteria of the standard as the release specification for a lot. The General Notices go on to state, “Repeats, replicates, statistical rejection of outliers, or extrapolations of results to larger populations, as well as the necessity and appropriate frequency of batch testing, are neither specified nor prescribed by the compendia; such decisions are based on the objectives of the testing. Frequency of testing and sampling are left to the preferences or direction of those performing the compliance testing …” (3).

	In 7.10, Interpretation of Requirements, the reportable value is defined as: “The reportable value, which often is a summary value for several individual determinations, is compared with the acceptance criteria. The reportable value is the end result of a completed measurement procedure, as documented” (3). 

	In addition, regarding multiple dosage unit testing, 

 states, “Some tests, such as those for Dissolution and Uniformity of Dosage units, require multiple dosage units in conjunction with a decision scheme. These tests, albeit using a number of dosage units, are in fact one determination” (3).

	The statement regarding the singlet determination was removed from the General Notices in 

	The USP requirements for sampling remains very weak. The only mention is in 6.60 Units necessary to complete a test, “Unless otherwise specified, a sufficient number of units to ensure a suitable analytical result shall be taken” (3).

	This review of changes to the meaning of singlet testing raises the following points:

		It is important to understand the difference between a standard and a specification

		A singlet determination did not mean testing must only be on one sample to determine compliance. It was the minimum sample size.

		The degree of testing replication necessary to arrive at a reportable value must be determined and justified by the manufacturer or tester in a documented procedure.

		Monograph acceptance criteria relate to the samples (units) tested for compliance and not the manufactured lot. It is therefore scientifically unsound to use the acceptance criteria of the standard as the release specification for a lot.

		The monograph acceptance criteria assume that the amount of specified material is targeted at 100% of claim.

		Sampling is not addressed well in the General Notices and only General Chapter <1010>, which is non-mandatory, considers it (4).

		The manufacturer needs to have in place risk-based in-house specifications designed to take into account sampling, analytical, and manufacturing variation with the objective of ensuring with a high degree of probability that any sample take from any manufactured lot would meet the acceptance criteria given in the monograph. Such a risk-based approach is available based on the ISO guard band principle (5, 6) and references therein.

 28 (2) 105-106 (2005).
	2. T. Schofield, D. Leblond, and S. Altan

 First Supplement, General Notices (USP, 2017) (valid from Aug. 1, 2017).
	4. USP, 

, First Supplement, General Chapter <1010>, Analytical Data--Interpretation And Treatment, (USP, 2017) (valid from Aug. 1, 2017).
	5. C. Burgess, 


	Vol. 41, No. 10
	October 2017
	Pages: 82–83, 92

	When referring to this article, please cite it as C. Burgess, "Singlet Determination Revisited," 

Is there a difference between a specification and a standard?