Training and consultancy for testing laboratories.

The method traditionally practiced by most test laboratories in the estimation of measurement uncertainty is by the ISO GUM (ISO/IEC Guide 98-3) approach, which is quite tedious and time consuming to study and gather uncertainty contributions from each and every step of the test method.  An alternative way of looking at uncertainty is to attempt to study the overall performance of the analytical procedure by involving replication of the whole procedure to give a direct estimate of the uncertainty for the final test result. This is the so-called ‘top-down’ approach.

We may use the data from inter-laboratory study, in-house validation or ongoing quality control. This approach is particularly appropriate where individual effects are poorly understood in terms of their quantitative theoretical models which are capable of predicting the behavior of analytical results for particular sample types.  By this approach, it is suffice to consider reproducibility from inter-laboratory data or long-term within-laboratory precision as recommended by ISO 21748, ISO 11352 and ASTM D 6299.

However, one must be aware of that by repeatedly analyzing a given sample over several times will not be a good estimate of the uncertainty unless the following conditions are fulfilled:

  1. There must be no perceptible bias or systematic error in the procedure.  That is to say that the difference between the expected results and the true or reference value must be negligible in relation to twice of the standard deviation with 95% confidence. This condition is usually (but not always) fulfilled in analytical chemistry.
  • The replication has to explore all of the possible variations in the execution of the method by engaging different analysts on different days using different equipment on a similar sample. If not, at least all of the variations of important magnitude are considered. Such condition may not be easily met by replication under repeatability conditions (i.e. repeated testing within laboratory), because such variations would be laboratory-specific to a great extent.

The conclusion is that replicated data by a single analyst on same equipment over a short period of time are not sufficient for uncertainty estimation. If the top-down approach is to be followed, we must obtain a good estimate of the long-term precision of the analytical method.  This can be done for example, by studying the precision for a typical test method used as a QC material over a reasonable period of time. We may also use a published reproducibility standard deviation for the method in use, provided we document proof that we are able to follow the procedure closely and competently.

What is bias in measurement?

When we repeat analysis of a sample several times, we get a spread of results surrounding its average value.  This phenomenon gives rise to data precision, but provides no clue as to how close the results are to the true concentration of the analyte in the sample.

However, it is possible for a test method to produce precise results which are in very close agreement with one another but are consistently lower or higher than they should be.  How do we know that?  Well, this observation can be made when we carry out replicate analysis of a sample with a certified analyte value. In this situation, we know we have encountered a systematic error in the analysis.

The term “trueness” is generally referred to the closeness of agreement between the expectation of a test result or a measurement result and a true value or an accepted reference value. And, trueness is normally expressed in terms of bias.  Hence, bias can be evaluated by comparing the mean of measurement results and an accepted reference value, as shown in the figure below.

Therefore, bias can be evaluated by carrying out repeat analysis of a suitable material containing a known amount of the analyte (i.e. reference value) mu, and is calculated as the difference between the average of the test results and the reference value:

We often express bias in a relative form, such as a percentage:

or as a ratio when we assess ‘recovery’ in an experiment:

The revised ILAC G8 document with reference to general guidelines on decision rules to issuance of a statement of conformance to a specification or compliance to regulatory limits has been recently published in September 2019.  Being a guideline document, we can expect to be provided with various decision options for consideration but the final mode of application is entirely governed by our own decision with calculated risk in mind.  

The Section 4.2 of the document gives a series of decision rules for consideration.  In sub-section 4.2.1 which considers a binary statement (either pass or fail) for simple acceptance rule, it suggests a clear cut of test results to be given a pass or a fail without taking any risk of making a wrong decision into account, as long as the mean measured value falls inside the acceptance zone, as graphically shown in their Figure 3, whilst the reverse is also true:

In this manner, my view is that the maximum risk that the laboratory is assuming when declaring conformity to a specification limit is 50% when the test result is on the dot of the specification limit.  Would this be too high a risk for the test laboratory to take?

When guard bands (w) are used to reduce the probability of making an incorrect conformance decision by placing them between the upper and lower tolerance specification limit (TL) values so that the range between the upper and lower acceptance limits (AL) are narrower, we can simply let w = TL – AL = U where U is the expanded uncertainty of the measurement.  

By doing so, we can have one of the two situations, namely for a binary statement, see Figure 4 of the ILAC G8 reproduced below and for a non-binary statement where multiple terms may be expressed, see Figure 5 of the ILAC document.

In my opinion, the decision to give a pass for the measurement found within the acceptance zone in Figure 4 is to the full advantage of the laboratory (zero risk as long as the laboratory is confident of its measurement uncertainty, U),  but to state a clear “fail” in the case where the measurement is within the w-zone of the acceptance zone may not be received well by the customer who would expect a “pass” based on the numerical value alone, which has been done all this while.  Shouldn’t the laboratory determine and bear a certain percentage of risk by working out with the customer on its acceptable critical measurement value where a certain portion of U lies outside the upper and lower specification limits?

Similarly, the “Conditional Pass / Fail” in Figure 5 also needs further clarification and explanations with the customer after considering a certain percentage of risk to be borne for the critical measurement values to be reported by the test laboratory.  A statement to the effect that “a conditional pass / fail with 95% confidence” might be necessary to clarify the situation. 

But from a commercial point of view, the local banker clearing a shipment’s letter of credit for payment with the requirement of a certificate of analysis to certify conformance to a quality specification laid down by the overseas buyer might not appreciate such statement format and might want to hold back the payment to the local exporter until his overseas principal agrees with this.  Hence, it is advisable for the contracted laboratory service provider to explain and get written agreement with the local exporter on the decision rule in reporting conformity, so that the exporter in return can discuss such mode of reporting with the overseas buyers during the negotiation of a sales contract.

In my training workshops on decision rule for making statement of conformity after laboratory analysis of a product, some participants have found the subject of hypothesis testing rather abstract.  But in my opinion, an understanding of the significance of type I and type II error in hypothesis testing does help to formulate decision rule based on acceptable risk to be taken by the laboratory in declaring if a product tested conforms with specification.

As we know well, a hypothesis is a statement that might, or might not, be true until we put it to some statistical tests. As an analogy, a graduate studying for a Ph.D. degree always carries out research works on a certain hypothesis given by his or her supervisor. Such hypothesis may or may not be proven true at the conclusion.  Of course, a breakthrough of the research in hand means that the original hypothesis, called null hypothesis is not rejected.

In statistics, we set up the hypothesis in such as way that it is possible to calculate the probability (p) of the data, or the test statistic (such as Student’s t-tests) calculated from the data, given the hypothesis, and then to make a decision about whether this hypothesis is to be accepted (high p) or rejected (low p). 

In conformity testing, we treat the specification or regulatory limit given as the ‘true’ or certified value and our measurement value obtained is the data for us to decide whether it conforms with the specification.  Hence, our null hypothesis Ho can be put forward as that there is no real difference between the measurement and the specification. Any observed difference arises from random effects only.

To make decision rule on conformance in significance testing, a choice about the value of the probability below which the null hypothesis is rejected, and a significant difference concluded, must be made. This is the probability of making an error of judgement in the decision.

If the probability that the data are consistent with the null hypothesis Ho falls below a pre-determined low value (say, alpha = 0.05 or 0.01), then the hypothesis is rejected at that probability.  Therefore, a p<0.05 would mean that we reject Ho with 95% level of confidence (or 5% error) if the probability of the test statistic, given the truth of Ho, falls below 0.05.  In other words, if Ho were indeed correct, less than 1 in 20 repeated experiments would fall outside the limits. Hence, when we reject Ho, we conclude that there was a significant difference between the measurement and the specification limit.                 

Gone are the days when we provide a conformance statement when the measurement result is exactly on the specification value.  By doing so, we are exposed to a 50% risk of being found wrong.  This is because we either have assumed zero uncertainty in our measurement (which cannot be true) or the specification value itself has encompassed its own uncertainty which again is not likely true. 

Now, in our routine testing, we would have established the measurement uncertainty (MU) of test parameter such as contents of oil, moisture, protein, etc. Our MU as an expanded uncertainty has been evaluated by multiplying a coverage factor (normally k = 2) with the combined standard uncertainty estimated, with 95% confidence.  Assuming the MU is constant in the range of values tested, we can easily determine the critical value that is not significantly different from the specification value or regulatory limit by the use of Student’s t-test.  This is Case B in the Fig 1 below.

So, if the specification has an upper or maximum limit, any test value smaller than the critical value below the specification estimated by the Student’s t-test can be ‘safely’ claimed to be within specification (Case A).  On the other hand, any test value larger than this critical value has reduced our confidence level in claiming within specification (Case C). Do you want to claim that the test value does not meet with the specification limit although numerically it is smaller than the specification limit?   This is the dilemma that we are facing today.

The ILAC Guide G8:2009 has suggested to state “not possible to state compliance” in such situation.  Certainly, the client is not going to be pleased about it as he has used to receive your positive compliance comments even when the measurement result is exactly on the dot of the upper limit.

That is why the ISO/IEC 17025:2017 standard has required the accredited laboratory personnel to discuss his decision rule with the clients and get their written consent in the manner of reporting.

To minimize this awkward situation, one remedy is to reduce your measurement uncertainty range as much as possible, pushing the critical value nearer to the specification value. However, there is always a limit to do so because uncertainty of measurement always exists.  The critical reporting value is definitely going to be always smaller than the upper limit numerically in the above example.

Alternatively, you can discuss with the client and let him provide you his acceptance limits. In this case, your laboratory’s risk is minimized greatly as long as your reported value with its associated measurement uncertainty is well within the documented acceptance limit because your client has taken over the risk of errors in the product specification (i.e. customer risk).

Thirdly, you may want to take a certain calculated commercial risk by having the upper uncertainty limit extended into the fail zone above the upper specification limit, due to commercial reasons such as keeping good relationship with an important customer.  You may even choose to report a measurement value that is exactly on the specification limit as conformance.  However, by doing so, you are taking a 50% risk to be found err in the issued statement of conformance.  Is it worth taking such a risk? Always remember the actual meaning of measurement uncertainty (MU) which is to provide a range of values around the reported number of the test, covering the true value of the test parameter with 95% confidence.

Notes on outlier test statistics in analytical data

When an analytical method is repeated several times on a given sample, the measured values nearer to the mean (or average) of the data set tend to occur more often than those found further away from the mean value.  This is the characteristic of analytical chemistry following the normal probability distribution and the phenomenon is known to be a measure of central tendency.

However, there are times and again that we notice some extremely low or high value(s) which are visibly distant from the remainder of data.  These values can be suspected to be outliers which may be defined as observations in a set of data that appear to be inconsistent with the remainder of that set.

It is obvious that outlying values generally have an appreciable influence on calculated mean value and more influence on calculated standard deviation if they are not examined carefully and removed if necessary.

However we must remember that random variation of analysis does generate occasional values by chance. If so, these values are indeed part of the valid data and should generally be included in any statistical calculations.  However undesirable human error or other deviation in the analytical process such as instrument failure may cause outliers to appear from such faulty procedure.  Hence, it is important to have the effect of outliers minimized.

To minimize such effect, we have to find ways to identify outliers and distinguish them from chance variation. There are many outlier tests available which allow analysts to inspect suspect data and if necessary correct or remove erroneous values. These test statistics assume underlying a normal distribution and the test sample is relatively homogeneous. 

Furthermore, outlier testing needs careful consideration where the population characteristics are not known, or, worse, known to be non-normal.  For example, if the data were Poisson distributed, many valid high values might be incorrectly rejected because they appear inconsistent with a normal distribution. It is also crucial to consider whether outlying values might represent genuine features of the population.

Another approach is to use robust statistics which are not greatly affected by the presence of occasional extreme values and will still perform well when no outliers are present.

The outlier tests are aplenty for your disposal:  Dixon’s, Grubb’s, Levene’s, Cochran’s, Thompson’s, Bartlett’s,  Hartley’s, Brown-Forsythe’s, etc. They are quite simple to be applied on a set of analytical data. However, to be meaningful in the outcome, the number of data examined should be large rather than just a few.

Therefore, outlier tests are only to provide us with objective criteria or signal to investigate the cause; usually, outliers should not be removed from the data set solely because of the results of a statistical test.  Instead, the tests highlight the need to inspect the data more closely in the first instance. 

The general guidelines for acting on outlier tests on analytical data, based on the outlier testing and inspection procedure listed in ISO 5725 Part 2 Accuracy (trueness and precision) of measurement methods and results — Part 2: Basic method for the determination of repeatability and reproducibility of a standard measurement method are as follows:

  • To test at the 95% and the 99% confidence level
  • All outliers should be investigated and any errors corrected
  • Outliers significant at the 99% level may be rejected unless there is a technical reason to retain them
  • Outliers significant only at the 95% level (normally called ‘stragglers’) should be rejected only if there is an additional technical reason to do so
  • Successive testing and rejection are permissible, but not to the extent of rejecting a large proportion of the data.

The above procedure leads to results which are not so seriously biased by rejection of chance extreme values, but are rather relatively insensitive to outliers at the frequency commonly encountered in measurement work.  The application of robust statistics might be a better choice.

In statistics, the average (mean) and sample standard deviation are known as “estimators” of the population mean and standard deviation. These estimates improve as the number of data collected increases.  As we know, the use of these statistics requires data that are normally distributed, and for confidence intervals employing the standard deviation of the mean, this tends to be so.

However, real experimental data may be so distributed but often the distribution will contain data that are seriously flawed. They can be extremely low or high values. If we can identify such data and remove them from further consideration, then all is well and good.

Sometimes this is possible, but not always. This is a problem as a single rouge value can seriously upset our calculations of the mean and standard deviation.

Estimators that can tolerate a certain amount of ‘bad’ data are called robust estimators, and can be used when it is not possible to ensure that the data being processed has the correct characteristics.

For example, we can use the middle value of a set of ascending data (called median) as a robust estimator of the mean, and the range of the middle 68% of the data (called normalized interquartile range IQR) as a robust estimator of the standard deviation.

By definition, the median is the middle value of a set of data when arranged in ascending order. If there are odd number of data, then the median is the unique middle datum.  If there are an even number, then the median is the average of the middle two data. 

Median is robust, because no matter how outrageous one or more extreme values are, they are only individual values at the end of a list. Their magnitude is immaterial.

The interquartile range (IQR) is a measure of where the “middle fifty” is in a data set, i.e. the range of values that spans the middle 50% of data.  Three quarters of the IQR, known as the normalized IQR, is an estimate of the standard deviation. In other words, the interquartile range formula is the median of the first quartile Q1 subtracted from that of the third quartile Q3:

                        IQR = Q3 – Q1

A problem with the IQR is that it is unrealistic to be used to calculate for small data sets, as we must have sufficient data to define quartiles (sections of the ordered data that contain one-quarter of the data).

Another robust estimator of standard deviation is median absolute deviation (MAD). It is a fairly simple estimate that can be implemented easily in a spreadsheet. The MAD from the data set median is calculated by:

                        MAD = median (| xi – median value |i=1,2,…n)

Robust methods have their place, particularly when we must keep all the data together in, for example, an interlaboratory comparison study where an outlying result from a laboratory cannot simply be ignored.  They are less strongly affected by extreme values.

However, robust estimators are not really the best statistics, and wherever possible the statistics appropriate to the distribution of the data should be used.

So, when can we use these robust estimators?

Robust estimators can be considered to provide good estimates of the parameters for the ‘good’ data in an outlier-contaminated data set.  They are appropriate when:

  • The data are expected to be normally distributed. In here, robust statistics give answers very close to ordinary statistics
  • The data are expected to be normally distributed, but contaminated with occasional spurious values which are regarded as unrepresentative or erroneous and approximately symmetrically distributed around the population mean. Robust estimators in here are less affected by these extreme values and hence are useful.

Remember that robust estimators are not recommended where the data set shows evidence of multi-modality or shows heavy skewing, especially when it is expected to follow non-normal or skewed distributions such as binomial and Poisson with low counts, chi-squared, etc. which generate extreme values with reasonable likelihood.

A few words about Measurement Bias

In metrology, error is defined as “the result of measurement minus a given true value of the measurand”. 

What is ‘true value’?

ISO 3534-2:2006 (3.2.5) states that “Value which characterizes a quantity or quantitative characteristic perfectly defined in the conditions which exist when that quantity or quantitative characteristic is considered.”, and the Note 1 that follows suggests that this true value is a theoretical concept and generally cannot be known exactly.   

In other words, when you are asked to analyze a certain analyte concentration in a given sample, the analyte present has a value in the sample, but what we do in the experiment is only trying to determine that particular value. No matter how accurate is your method and how many repeats you have done on the sample to get an average value, we would never be 100% sure at the end that this average value is exactly the true value in the sample.  We bound to have a measurement error!

Actually in our routine analytical works, we do encounter three types of error, known as gross, random and systematic errors.

Gross errors leading to serious outcome with unacceptable measurement is committed through making serious mistakes in the analysis process, such as using a reagent titrant with wrong concentration for titration. It is so serious that there is no alternative but abandoning the experiment and making a completely fresh start. 

Such blunders however, are easily recognized if there is a robust QA/QC program in place, as the laboratory quality check samples with known or reference value (i.e. true value) will produce erratic results.

Secondly, when the analysis of a test method is repeated a large number of times, we get a set of variable data, spreading around the average value of these results.  It is interesting to see that the frequency of occurrence of data further away from the average value is getting fewer.  This is the characteristic of a random error.

There are many factors that can contribute to random error: the ability of the analyst to exactly reproduce the testing conditions, fluctuations in the environment (temperature, pressure, humidity, etc.), rounding of arithmetic calculations, electronic signals of the instrument detector, and so on.  The variation of these repeated results is referred to the precision of the method.

Systematic error, on the other hand, is a permanent deviation from the true result, no matter how many repeats of analysis would not improve the situation. It is also known as bias.

A color deficiency technician might persistently overestimate the end point in a titration, the extraction of an analyte from a sample may only be 90% efficient, or the on-line derivatization step before analysis by gas chromatography may not be complete. In each of these cases, if the results were not corrected for the problems, they would always be wrong, and always wrong by about the same amount for a particular experiment.

How do we know that we have a systematic error in our measurement?  

It can be easily estimated by measuring a reference material a large number of times.  The difference between the average of the measurements and the certified value of the reference material is the systematic error. It is important to know the sources of systematic error in an experiment and try to minimize and/or correct for them as much as possible.

If you have tried your very best and the final average result is still significantly different from the reference or true value, you have to correct the reported result by multiplying it with a certain correction factor.  If R is the recovery factor which is calculated by dividing your average test result by the reference or true value, the correction factor is 1/R.

Today, there is another statistical term in use.  It is ‘trueness’.  

The measure of truenessis usually expressed in terms of bias.

Trueness in ISO 3534-2:2006 is defined as “The closeness of agreement between the expectation of a test result or a measurement result and a true value.” whilst ISO 15195:2018 defines trueness as “Closeness of agreement between the average value obtained from a large series of results of measurements and a true value.”. The definition of ISO 15195 is quite similar to those of ISO 15971:2008 and ISO 19003:2006.  The ISO 3534-2 definition includes a note that in practice, an “accepted reference value” can be substituted for the true value.

Is there a difference between ‘accuracy’ and ‘trueness’?

The difference between ‘accuracy’ and ‘trueness’ is shown in their respective ISO definition.

ISO 3534-2:2006 (3.3.1) defines ‘accuracy’ as “closeness of agreement between a test result or measurement result and true value”, whilst the same standard in (3.2.5) defines ‘trueness’ as “closeness of agreement between the expectation of a test result or measurement result and true value”.  What does the word ‘expectation’ mean here?  It actually refers to the average of the test result, as given in the definition of ISO 15195:2018.

Hence, accuracy is a qualitative parameter whilst trueness can be quantitatively estimated through repeated analysis of a sample with certified or reference value.

References:

ISO 3534-2:2006   “Statistics – Vocabulary and symbols – Part 2: Applied statistics”

ISO 15195:2018   “Laboratory medicine – Requirements for the competence of calibration laboratories using reference measurement procedures”

In the next blog, we shall discuss how the uncertainty of bias is evaluated. It is an uncertainty component which cannot be overlooked in our measurement uncertainty evaluation, if present.

Why measurement uncertainty is important in analytical chemistry?

Conducting a laboratory analysis is to make informed decisions on the samples drawn.  The result of an analytical measurement can be deemed incomplete without a statement (or at least an implicit knowledge) of its uncertainty.  This is because we cannot make a valid decision based on the result alone, and nearly all analysis is conducted to inform a decision.

We know that the uncertainty of a result is a parameter that describes a range within which the value of the quantity being measured is expected to lie, taking into account all sources of error, with a stated degree of confidence (usually 95%).  It characterizes the extent to which the unknown value of the targeted analyte is known after measurement, taking account of the given information from the measurement.

With a knowledge of uncertainty in hand, we can make the following typical decisions based on analysis:

  • Does this particular laboratory have the capacity to perform analyses of legal and statutory significance?
  • Does this batch of pesticide formulation contain less than the maximum allowed concentration of an impurity?
  • Does this batch of animal feed contain at least the minimum required concentration of profit (protein + fat)?
  • How pure is this batch of precious metal?

The figure below shows a variety of instances affecting decisions about compliance with externally imposed limits or specifications.  The error bars can be taken as expanded uncertainties, effectively intervals containing the true value of the concentration of the analyte with 95% confidence.

We can make the following observations from the above illustration:

  1. Result A clearly indicates the test result is below the limit, as even the extremity of the uncertainty interval is below the limit,
  2. Result B is below the limit but the upper end of the uncertainty is above the limit, so we not sure if the true value is below the limit.
  3. Result C is above the limit but the lower end of the uncertainty is below the limit, so we are not sure that the true value is above.
  4. What conclusions can we draw from the equal results D and E? Both results are above the limit but, while D is clearly above the limit, E is not so because the greater uncertainty interval extends below the limit. 

In short, we have to make decisions on how to act upon results B, C and E.  What is the level of risk that can be afforded to assume the test result is in conformity with the stated specification or in compliance with the regulatory limit?

By making such a decision rule, we must be serious in the evaluation of measurement uncertainty, making sure that the uncertainty obtained is reasonable.  If not, any decision made on conformity or compliance will be meaningless.

Initial data analysis IDA

Data analysis is a systematic process examining datasets in order to draw valid conclusions about the information they contain, increasingly with the aid of specialized systems and software, leading to discovering useful information to make informed decisions to verify or disapprove some scientific or business models, theories or hypotheses.

As a researcher or laboratory analyst, we must have the drive to obtain quality data in our work. A careful plan in database design and statistical analysis with variable definitions, plausibility checks, data quality checks and ability to identifying likely errors in data and resolving data inconsistencies, etc. has to be established before embarking the full data collection.  More importantly, the plan should not be altered without agreement of the project steering team in order to reduce the extent of data dredging or hypothesis fishing leading to false positive studies.  Shortcomings in initial data analysis may result in adopting inappropriate statistical methods or making incorrect conclusions.

Our first step of initial data analysis is to check consistency and accuracy of the data, such as looking up for any outlying data. This can be visualized through plotting the data against time of data collection or other independent parameters.  This should be done before embarking on more complex analyses.

After having satisfied that the data are reasonably error-free, we should get familiar with the collected data and examine them for any consistency of data formats, number and patterns of missing data, the probability distributions of its continuous variables, etc.  For more advanced initial analysis, decisions have to be made about the way variables are used in further analyses with the aid of data analytics technologies or statistical techniques.  These variables can be studied in their raw form, transformed to some standardized format, categorized or stratified into groups for modeling.

Replication and successive dilution in constructing calibration curve

An analytical instrument generally needs to be calibrated before measurements made on prepared sample solutions, through construction of a linear regression between the analytical responses and the concentrations of the standard analyte solutions. A linear regression is favored over quadratic or exponential curve as it incurs minimum error.   

Replication

Replication in standard calibration is found to be useful if replicates are genuinely independent. The calibration precision is improved by increasing the number of replicates, n, and provides additional checks on the calibration solution preparation and on the precision of different concentrations.

The trend of its precision can be read from the variance of these calibration points. A calibration curve might be found to have roughly constant standard deviations in all these plotted points, whilst others may show a proportional increase in standard deviation in line with the increase of analyte concentration. The former behavior is known as “homoscedasticity” and the latter, “heteroscedasticity”.  

It may be noted that increasing the number of independent concentration points has actually little benefit after a certain extent. In fact, after having six calibration points, it can be shown that any further increase in the number of observations in calibration has relatively modest effect on the standard error of prediction for a predicted x value unless such number of points increases very substantially, say to 30 which of course is not practical.

Instead, independent replication at each calibration point can be recommended as a method of improving uncertainties. Indeed, independent replication is accordingly a viable method of increasing n when the best performance is desired. 

However, replication suffers from an important drawback. Many analysts incline to simply injecting a calibration standard solution twice, instead of preparing duplicate standard solutions separately for the injection. By injecting the same standard solution twice into the analytical instrument, the plotted residuals will appear in close pairs but are clearly not independent. This is essentially useless for improving precision. Worse, it artificially increases the number of freedom for simple linear regression, giving a misleading small prediction interval.

Therefore ideally replicated observations should be entirely independent, using different stock calibration solutions if at all possible. Otherwise it is best to first examine replicated injections to check for outlying differences and then to calculate the calibration based on the mean value of y for each distinct concentration.

There is one side effect of replication that may be useful. If means of replicates are taken, the distribution of errors in the mean tend to be the normal distribution as the number of replicates increases, regardless of parent distribution. The distribution of the mean of as few as 3 replicates is very close to the normal distribution even with fairly extreme departure from normality.  Averaging three or more replicates can therefore provide more accurate statistical inference in critical cases where non-normality is suspected.

Successive dilutions

A common pattern of calibration that we usually practice is doing a serial dilution, resulting in logarithmically decreasing concentrations (for example, 16, 8, 4. 2 and 1 mg/L). This is simple and has the advantage of providing a high upper calibrated level, which may be useful in analyzing routine samples that occasionally show high values.

However, this layout has several disadvantages. First, errors in dilution are multiplied at each step, increasing the volume uncertainties, and perhaps worse, increasing the risk of any undetected gross dilution error (especially if the analyst commits the cardinal sin of using one of the calibration solutions as a QC sample as well!).

Second, the highest concentration point has high leverage, affecting both the gradient and y-intercept of the line plotted; errors at the high concentration will cause potentially large variation in results.

Thirdly, departure in linearity are easier to detect with fairly even spaced points.  In general, therefore, equally spaced calibration points across the range of interest should be much preferred.