Statistics for the Quality Control Chemistry Laboratory / Edition 1

by Eamonn Mullins

Overview

Statistical methods are essential tools for analysts, particularly those working in Quality Control Laboratories. This book provides a sound introduction to their use in analytical chemistry, without requiring a strong mathematical background. It emphasises simple graphical methods of data analysis, such as control charts, which are also a fundamental requirement in laboratory accreditation. A large part of the book is concerned with the design and analysis of laboratory experiments, including sample size determination. Practical case studies and many real databases from both QC laboratories and the research literature, are used to illustrate the ideas in action.

The aim of Statistics for the Quality Control Chemistry Laboratory is to give the reader a strong grasp of the concept of statistical variation in laboratory data and of the value of simple statistical ideas and methods in thinking about and manipulation of such data. It will be invaluable to analysts working in QC laboratories in industry, hospitals and public health, and will also be welcomed as a textbook for aspiring analysts in colleges and universities.

Product Details

ISBN-13:	9780854046713
Publisher:	RSC
Publication date:	09/01/2003
Pages:	474
Product dimensions:	6.15(w) x 9.20(h) x (d)

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Statistics for the Quality Control Chemistry Laboratory

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By Eamonn Mullins

The Royal Society of Chemistry

Copyright © 2003 The Royal Society of Chemistry
All rights reserved.
ISBN: 978-1-84755-193-1

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CHAPTER 1

Variability in Analytical Measurements

1.1 INTRODUCTION

All measurements are subject to measurement error. By error is meant the difference between the observed value and the true value of the quantity being measured. Since true values are invariably unknown, the exact magnitude of the error involved in an analytical result is also invariably unknown. It is possible, however, to estimate the likely magnitude of such errors by careful study of the properties of the analytical system. The term 'analytical system' refers to everything that impinges on a measurement: the method, the equipment, the reagents, the analyst, the laboratory environments, etc. It is fair comment that a measurement is of no value unless there is attached to it, either explicitly or implicitly, some estimate of the probable error involved. The Analytical Methods Committee of the Royal Society of Chemistry has taken a very clear position on this question in saying that 'analytical results must be accompanied by an explicit quantitative statement of uncertainty, if any definite meaning is to be attached to them or an informed interpretation made. If this requirement cannot be fulfilled, there are strong grounds for questioning whether analysis should be undertaken at all.' It gives a simple example which illustrates the necessity for measures of uncertainty: 'Suppose there is a requirement that a material must not contain more that 10 µg g-1 of a particular constituent. A manufacturer analyses a batch and obtains a result of 9 µg g-1. If the uncertainty ... in the results is 0.1 µg g-1(i.e., the true result falls within the range 8.9–9.1 µg g-1 with a high probability) then it can be accepted that the limit is not exceeded. If, in contrast, the uncertainty is 2 µg g-1 there can be no such assurance. The 'meaning' or information content of the measurement thus depends on the uncertainty associated with it."

In recent years the question of uncertainty in measurements has been of major concern to the measurement community in general, and has stimulated much activity among analytical chemists. This has been driven, in part, by the requirement by laboratory accreditation bodies for explicit statements of the quality of the analytical measurements being produced by accredited laboratories. The term 'uncertainty' has been given a specific technical meaning, viz., "a parameter associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand" and guidelines have been developed to help chemists estimate uncertainties. Uncertainty will be discussed as a technical concept in Chapter 8, but the remainder of the book may be read as a discussion of how to describe and analyze uncertainty in its usual sense of indicating a lack of certainty regarding the trueness of measured values.

Consultancy experience suggests that the importance of measurement error is often not fully appreciated. In pharmaceutical manufacturing, for instance, crucially important management decisions can be made on the basis of a single measurement or on the average of a small number of measurements, often poorly determined. Examples of such measurements would be: the purity of a raw material on the basis of which the conversion rate of a process is to be determined; the strength of an intermediate product, which will determine the dilution of the next stage of production; the potency of a final product, which will determine the amount of product to be filled into vials intended to contain a given strength of product. In the latter case lack of confidence in the quality of the measurement may lead to overfill of the vials in order to ensure conformance to statutory or customer requirements. The problem in all of these cases is not one of product variability, though this may be present also, but rather the uncertainty in a decision criterion (i.e., a test result) due to measurement variability. Since the costs of improving such measurements will generally be small compared to the costs of poor production decisions, it is clear that the economic consequences of measurement error are, as yet, not fully appreciated.

In industrial experimentation measurement error can be vitally important also. Studies are often carried out to investigate proposed process changes which it is hoped will result in small but economically important yield improvements. In such situations the ability of the experimenter to detect small yield improvements will be influenced in an important way by the number of measurements made. If the measurement error is relatively large and the number of measurements small, the effect of the process change may be missed. The determination of appropriate sample sizes for experiments is an important topic which will be discussed in Chapter 4.

This chapter begins with an example illustrating the effects of random measurement error on the results produced by a stable analytical system. The terms bias and precision are introduced to describe the nature of measurement errors; this is done in the context of an inter-laboratory study. The Normal curve is discussed as the most important statistical model for describing and quantifying analytical error. Various measures of the magnitude of analytical error are then considered: these include the standard deviation, the relative standard deviation or coefficient of variation, repeatability and reproducibility. Finally, the effect on the precision of an analytical result of averaging several measurements is discussed.

1.2 AN EXAMPLE OF MEASUREMENT VARIABILITY

Figure 1.1 is a histogram based on 119 determinations of the potency of a quality control material used in monitoring the stability of an analytical system; this system is used for measuring the potency of batches of a pharmaceutical product. The measurements were made by high performance liquid chromatography (HPLC). Here the heights of the bars represent the frequencies of values in each of the potency intervals on the horizontal axis. Very often the areas of the bars (height and area are equivalent when the bases are the same) are drawn to represent the relative frequencies, i.e., the fractions of the total number of data points that fall into each interval. In such a case the total area is one – this will be discussed again later.

The results are distributed in a roughly symmetrical bell-shaped form, with most values clustered around the centre and few values very far from the centre in either direction. This shape is often seen in laboratory data and is typical of the type of variation that results from the combined effects of a myriad of small influences. It can be shown mathematically that the cumulative effect on a system of very many small chance perturbations will be to produce a Normal distribution for the measured responses. A Normal curve is symmetrical and bell-shaped and its exact shape can be varied to fit data from many different contexts; consequently, it is the statistical model most often used to describe laboratory data. The Normal model will be discussed later.

It is clear that this HPLC measurement process is subject to some variation. While the average result obtained is 96.5%, individual measurements range from 95.1 to 97.9%. Errors of this magnitude would be significant in characterizing the potency of valuable pharmaceutical product and, undoubtedly, decisions should not be based on single measurements of batch potency in this case. We will return to this point later.

1.3 DESCRIBING MEASUREMENT ERROR

When it comes to assessing the likely magnitude of the analytical error in any test result, it is useful to distinguish between the types of error that may occur. These are often described as systematic and random errors, though the distinction between them is not always as clearcut as the words might imply. The two types of variability are often clearly seen in inter-laboratory studies such as, for example, that shown in Figure 1.2. These results come from an international study, viz., IMEP – the International Measurement Evaluation Programmecarried out by the European Union's Institute for Reference Materials and Measurements (IRMM). Laboratories from fifteen countries each measured the amounts of ten different elements in a synthetic water whose certified value was determined jointly by IRMM and NIST (the United States' National Institute for Standards and Technology). The shaded area in Figure 1.2 gives the error bounds on the certified value for lead, while the error bars attached to each result represent each laboratory's assessment of the uncertainty of its result. The laboratories employed a wide range of analytical techniques, ranging from atomic absorption spectrometry to inductively coupled plasma spectrometry, with a mass spectrometer detector; different techniques gave results of similar quality.

The extent of the inter-laboratory variation is shocking, especially when it is realized that the data were produced by highly reputable laboratories. Less than half of the error bars intersect the shaded area in the graph, indicating that very many laboratories are over-optimistic about the levels of uncertainty in the results they produce. This picture is a convincing argument for laboratory accreditation, proficiency testing schemes and international cooperation, such as IMEP, to help laboratories produce consistent results.

Figure 1.2 illustrates the need to distinguish between different sources of variation. The error bars that bracket each laboratory's result clearly reflect mainly within-laboratory variation, i.e., that which was evident in Figure 1.1. The differences between laboratories are much greater than would be expected if only the random within-laboratory variation were present. The terminology used to distinguish between the two sources of error will be introduced in the context of a schematic inter-laboratory study.

1.3.1 A Schematic Inter-laboratory Study

Suppose a company with a number of manufacturing plants wants to introduce a new (fast) method of measuring a particular quality characteristic of a common product. An inter-laboratory trial is run in which test material from one batch of product is sent to five laboratories where the test portions are each measured five times. The standard (slow and expensive) method measures the quality characteristic of the material as 100 units; this will be taken as the true value. The results of the trial are shown schematically below (Figure 1.3).

It is clear that laboratories differ in terms of both their averages (they are 'biased' relative to each other and some are biased relative to the true value) and the spread of their measurements (they have different 'precision').

Bias. Even though individual measurements are subject to chance variation it is clearly desirable that an analytical system (instrument, analyst, procedure, reagents, etc.) should, on average, give the true value. If this does not happen then the analytical system is said to be biased. The bias is the difference between the true value and the long-run average value obtained by the analytical system. Since true values are never available, strictly it is only possible to talk about the relative bias between laboratories, analysts or analytical systems. While certified reference materials provide 'true values', even they are subject to some uncertainty. The most that can be said about them is that they are determined to a much higher precision than is required by the current analysis and, where their characteristics are determined by different methods, they are likely to be more free of any method bias that may be present in any one analytical method. As such, they are often treated as if they provide true values.

In our example, laboratories B and C give average results close to 100 so they show no evidence of bias. On the other hand, laboratory A is clearly biased downwards while D is biased upwards. Laboratory E is not so easy to describe. Four of the measurements are clustered around 100 but there is one extremely high value (an 'outlier'). This looks like it was subject to a gross error, i.e., one which was not part of the stable system of chance measurement errors that would lead to a bell-shaped distribution. It could also be simply a recording error. If the reason for the outlier can be identified then laboratory E can probably be safely assumed to be unbiased, otherwise not.

Biases between two laboratories, two analytical methods or two analysts arise because of persistent differences in, for example, procedures, materials or equipment. Thus, two analysts might obtain consistently different results while carrying out the same assay because one allows 5 minutes for a solution to come to room temperature while the other always allows half an hour. The problem here is that the method is inadequately specified: a phrase such as 'allow the solution to come to room temperature' is too vague. Since bias is persistent there is no advantage to be gained from repeating the analysis under the same conditions: bias does not average out. Irrespective of the number of replicate analyses carried out in laboratory A, in our schematic example above, the average result will still be too low.

Precision. Even if an analytical system is unbiased it may still produce poor results because the system of chance causes affecting it may lead to a very wide distribution of measurement errors. The width of this distribution is what determines the precision of the analytical system.

Although laboratory A is biased downwards it has, nevertheless, good precision. Laboratory C, on the other hand, has poor precision, even though it is unbiased. D is poor on both characteristics.

The precision of an analytical system is determined by the myriad of small influences on the final result. These include impurities in the various reagents involved in the analysis, slight variations in environmental conditions, fluctuations in electrical quantities affecting instrumentation, minor differences between weights or volumes in making up solutions, minor differences between concentrations of solvents, flow rates, flow pressure, constitution and age of chromatographic columns, stirring rates and times, etc. If the influences are more or less random then they are equally likely to push the result upwards or downwards and the bell shaped distribution of Figure 1.1 will be seen if sufficient data are observed. Because the influences are random they can be averaged out: if a large number of measurements is made positive random errors will tend to cancel negative ones and the average result will be more likely to be closer to the true value (assuming no bias) than individual results would be.

1.4 SOURCES OF ANALYTICAL VARIABILITY

Figure 1.4 represents the results of a brainstorming exercise on the sources of variability affecting the results of a HPLC assay of an in-process pharmaceutical material. The exercise was conducted during an in-house short course and took about fifteen minutes to complete – it is not, by any means, an exhaustive analysis. The diagram is known as a 'Cause and Effect', 'Fishbone' or 'Ishikawa' diagram (after the Japanese quality guru who invented it). The sixth 'bone', would be labelled 'measurements' when tackling production problems; since the subject of analysis here is a measurement system, this is omitted. Use of the diagram in brainstorming focuses attention, in turn, on each of the major sources of variability.

The content of the diagram should be self-explanatory and will not be elaborated upon further. While some of the identified causes will be specific to particular types of assay, the vast majority will be relevant to virtually all chromatographic analyses. The diagram is presented as it emerged from the brainstorming session: no attempt has been made to clean it up. Arguments could be made to move individual items from one bone to another, but as it stands the diagram illustrates very clearly, the large number of possible sources of variability in a routine assay.

Diagrams such as Figure 1.4 are a useful starting point when troubleshooting problems that arise in an analytical system.

1.5 MEASURING PRECISION

Before discussing how to quantify the precision of an analytical system the properties of the Normal curve, which is the standard model for measurement error, will be considered. This model implicitly underlies many of the commonly used measures of precision and so an under- standing of its properties is essential to understanding measures of precision.

The Normal Curve. If Figure 1.1 were based on several thousand observations, instead of only 119, the bars of the histogram could be made very narrow indeed and its overall shape might then be expected to approximate the smooth curve shown below as Figure 1.5. This curve may be considered a 'model' for the random variation in the measurement process. The area under the curve corresponding to any two points on the horizontal axis represents the relative frequency of test results between these two values – thus, the total area under the curve is one.

(Continues...)

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Excerpted from Statistics for the Quality Control Chemistry Laboratory by Eamonn Mullins. Copyright © 2003 The Royal Society of Chemistry. Excerpted by permission of The Royal Society of Chemistry.
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Table of Contents

Variability in Analytical Measurements;
Control Charts in the Analytical Laboratory;
Some Important Statistical Ideas;
Simple Comparative Studies;
Studying Complex Systems;
Fitting Equations to Data;
The Design and Analysis of Laboratory Studies Re-visited;
Assessing Measurement Quality.

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Meets the demands of an introduction to statistics applied to quality control. The text can be used for courses or for self-teaching.

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Meets the demands of an introduction to statistics applied to quality control. The text can be used for courses or for self-teaching.

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