All measurements are subject to error. When a reportable value is derived from a measurement or series of measurements, this value is only an estimate of the “true” value and has a range around it associated with how confident one is that the true value lies within it. Traditionally in the pharmaceutical industry, a range is selected corresponding to 95% confidence (1).
Reportable value data quality
The quality of a reportable value or an analytical result depends upon the size of the confidence interval. The smaller the confidence interval is, the more confident one is in relying on one’s reportable value or analytical result. Unfortunately, also for historical reasons relating primarily to physical metrological considerations, the International Organization on Standardization (ISO) uses the term “measurement uncertainty” (MU) for the same concept (2).
One difference between the ISO MU approach and the International Conference on Harmonization (ICH) Q2(R1) and United States Pharmacopeia (USP) approaches is that in the latter, the effects of imprecision and bias are considered separately (3). It should be noted, however, that the USP General Chapter <1225>, “Validation of Compendial Procedures,” and related General Chapters <1224>, “Transfer of Analytical Procedures,” and <1226>, “Verification of Compendial Procedures,” are under revision at present (4-6).
USP General Chapter <1010>, Analytical Data--Interpretation and Treatment, clearly states that accuracy has a different meaning from ISO (7). The USP states, “In ISO, accuracy combines the concepts of unbiasedness (termed trueness) and precision,” and USP further defines a conventional 95% confidence interval around the mean of:
is the standard error of the mean and is called the standard uncertainty in ISO.
is called the coverage factor.
is called the expanded uncertainty in ISO.
Another difference is the way in which the standard deviation (s) is calculated. The ISO approach is by means of a calculated error budget (8), whereas the ICH Q2(R1) relies upon information derived from an experimentally designed analytical trial (3). Theoretically, these two approaches should yield similar results. In practice, however, this is not always the case. ISO also uses a different nomenclature from ICH. What would usually be called the analytical measurement or result is called in ISO the measurand. This measurand is the particular quantity subject to measurement and is related to the measured analytical response function by means of an equation in the same way as an analytical result.
Concept of an error budget
The idea behind an error budget is that if all sources of error are known, it is possible to calculate an estimate of the uncertainty of the measurand or reportable value based upon converting all the errors to standard deviations and then combining the variances. If all the error processes are independent, then an error budget can be defined in five steps:
- Define all the process elements involved and their interrelationships
- Define the measurand in terms of these process elements
- Identify all error sources and group them as required
- Estimate their individual contributions and convert them to standard deviations and combine them to produce an overall estimate of standard deviation
- Estimate the overall uncertainty using an appropriate coverage factor as described previously.
Figure 1 shows the error budget process diagrammatically.
An example of a simple error budget for a standard solution. The error budget approach may seem rather daunting, but a simple example of the preparation of a standard solution will make things clearer. This example is a common task in the laboratory, but few calculate how good their standard solutions are.
The reference standard purchased has a certified purity of 99.46 ± 0.25. Approximately 100 mg of this reference standard is weighed, by difference, accurately using a five-place analytical balance. The reference standard is dissolved in water, and a solution is made up to the mark with water in a Grade A 100.0 mL capacity volumetric flask at ambient laboratory temperature. It is assumed that the laboratory temperature is controlled but may vary between 16 °C and 24 °C. The first step is to draw a flow diagram of the analytical process used to prepare the standard solution. This diagram is shown in Figure 2.
Identify the measurand. In this instance, the measurand (C) is the concentration of the reference material in the standard solution in mg l-1 and is defined by the equation:
where m is the mass of reference material in mg. P is the purity as a mass fraction of the standard, and V is the volume of the volumetric flask in mL.
Identify the error sources. Based upon the analytical process flow (see Figure 2), one can now identify three main areas of error, namely, the reference stand itself, the weighing process and the solution, and the final volume of the solution. It is helpful to use a Ishikawa diagram to aid the identification and grouping of error sources. For this example, the Ishikawa diagram is shown in Figure 3. In Figure 3, the possible sources of error are shown for each of the three groups. In this example, it is assumed that the reference standard is sufficiently homogeneous to ignore any error contribution and is freely and easily soluble in water.
Note that the volume of the solution has three distinct uncertainty components that need to be taken into account:
- The uncertainty in the marked calibration volume of the volumetric flask itself at 20 °C
- The difference between the calibration temperature of the flask and the temperature at which the solution was prepared
- The uncertainty associated with filling the flask to the calibration mark.
Not all error contributions are of equal importance. To find out which error contributions are of importance, however, it is essential to convert all errors to standard deviations (8).
Processes to convert specifications, ranges, and measurement data into a standard deviation. The easiest method to evaluate the standard deviation is by the statistical analysis of series of observations and assume the normal distribution. In the example, this method would be used in determining the uncertainty of filling the volumetric flask to the mark. This direct determination is known as a Type A uncertainty.
Type B uncertainties are derived from two approaches:
- Converting certificate ranges where there is no knowledge of the shape of the distribution so the rectangular distribution is assumed. For a range of ± a, the corresponding estimate for the standard deviation would be:
In the example, the uncertainty in the purity of ± 0.25 would be converted using the rectangular distribution.
- If it is more likely that the value lies closer to the central value, then the triangular distribution is assumed. For a range of ± a, the corresponding estimate for the standard deviation would be:
In the example, the uncertainty in the grade A volumetric flask of ± 0.10 would be converted using the triangular distribution.
Uncertainty contributions in the example. Now we can proceed to quantify all the uncertainties in our analytical process in the following manner:
Reference standard uncertainty, uP. Using the rectangular distribution we have:
Note that the purity and it uncertainty have been converted to mass fractions.
Weighing uncertainty, um. Using the balance manufacturer’s data (Type A) we have:
Note that our actual value of weighed material was 100.28mg.
Volumetric uncertainty (uV. ). Here we have three different contributions to uV:
- The flask itself using the triangular distribution:
- The temperature effect assuming the coefficient of expansion of water of 0.00021 °C-1 and assuming the rectangular distribution:
and the Type A uncertainty associated with the filling of the flask to the calibration mark. This was determined by the filling repeatability for assuming a normal distribution; uvr=0.02 mL.
One can now combine these three standard deviations to arrive at the overall volumetric uncertainty:
Finalizing the error budget. Now that all uncertainties have been converted into standard deviations, they can be combined to produce an uncertainty for the measurand C as shown in Table I and using the variance combination equation:
It is important to note that the uncertainty contribution from the reference standard is greater than either the weighing or the volumetric errors.
Expression of confidence: calculating the reportable value and its uncertainty. The concentration of the reference standard solution is directly available from the measurand equation:
The uncertainty in the measurand uc and the expanded uncertainty U are now readily available.
The coverage factor of k=2 corresponds to a confidence of 95.45%.
Based upon this expanded uncertainty, we calculate that we have confidence that the standard solution uncertainty is approximately 0.34%.
This article covered some of the basics of error budgets and carried out a calculation of an expanded uncertainty for a standard solution. The expanded uncertainty is small (0.34%) and is dominated by the contribution from the reference standard itself. The more complex the analytical procedure, however, the more expanded uncertainties will build.
In regulated laboratories, such as the Official Medicines Control Laboratories in Europe, it is a prerequisite that analytical tests are performed under a properly functioning quality system, which means that:
- All balances and volumetric glassware are under regular control
- Official reference substances or in-house reference substances are properly qualified and stored
- Instruments are regularly calibrated
- Equipment is regularly requalified
- Laboratory technicians are (re-)qualified.
The uncertainties due to these sources are under control and are assumed to contribute little to the total uncertainty of the test result (9).
1. L. Torbeck, Pharm. Tech., 34 (7) (2010).
2. See, for example, NIST Reference on Constants, Units and Uncertainty; http://physics.nist.gov/cuu/Uncertainty/index.html, accessed Aug. 12, 2013.
3. ICH, Q2(R1) Harmonised Tripartite Guideline, Validation of Analytical Procedures: Text And Methodology (2005).
4. USP, General Chapter <1224>, “Transfer of Analytical Procedures,” United States Pharmacopeia, 36 (US Pharmacopeial Convention, Rockville, Md, 2013).
5. USP, General Chapter <1225>, “Validation of Compendial Proceduures,” United States Pharmacopeia, 36 (US Pharmacopeial Convention, Rockville, Md, 2013).
6. USP, General Chapter <1226>, “Verification of Compendial Procedures,” United States Pharmacopeia, 36 (US Pharmacopeial Convention, Rockville, Md, 2013).
7. USP, General Chapter <1010>, United States Pharmacopeia, 36 (US Pharmacopeial Convention, Rockville, Md, 2013).
8. S.L.R. Ellison and A Williams (Eds), Eurachem/CITAC guide: Quantifying Uncertainty in Analytical Measurement, Third edition, (2012) ISBN 978-0-948926-30-3, available from www.eurachem.org/index.php/publications/guides/quam, accessed Aug. 12, 2013.
9. PA/PH/OMCL (05) 49 DEF CORR-- OMCL Guideline on Uncertainty of Measurement (for compliance testing) (2007), www.edqm.eu/en/EDQM-Downloads-527.html, accessed Aug. 12, 2013.
About the Author
Christopher Burgess, PhD, is an analytical scientist at Burgess Analytical Consultancy Limited ‘Rose Rae,’ The Lendings, Startforth, Barnard Castle, Co Durham, DL12 9AB, UK; +44-(0)1833-637446; [email protected]; www.burgessconsultancy.com