Matrix mismatch and its estimation in single-lab studies

Introduction

Laboratory bias can vary considerably – even among laboratories using one and the same standardized analytical method. This variation can be explained by differences with respect to reagents, staff competence, equipment, etc., see Section 0.3 in ISO 5725-3.¹ One important source of between-laboratory variation is matrix mismatch. From the chemical analyst’s point of view, matrix mismatch arises when the matrix of the test sample (the substance or material in which the analyte of interest is present) differs from the calibration or standard matrix.^2–4 Such differences may result in chemical or physical interferences that affect the measurement. In particular: if the calibration standards do not adequately represent the sample matrix, the calibration curve may not accurately capture the relationship between the analyte concentration and the instrument response, resulting in a bias.⁵ In the literature, matrix mismatch is also referred to by other terms such as matrix effects.

Figure 1 illustrates how matrix mismatch arises in connection with calibration curves. The underlying data were generated with spreadsheet software for the purpose of illustration.¹⁷ Three calibration curves are shown for the determination of a given analyte via mass spectrometry. The top calibration curve (black) corresponds to results obtained from a standard solution. The second curve (blue) corresponds to results obtained from spiked samples for a given matrix (matrix 1). Finally, the lowest curve (green) corresponds to results obtained for another given matrix (matrix 2). As can be seen, while the calibration curve for matrix 1 lies very close to the standard solution curve, there are considerable suppression effects for the second matrix, resulting in a considerable negative bias. For instance, if the standard solution calibration curve is used for the determination of analyte concentration in a sample with matrix 2 and a peak intensity of 6,000,000 is obtained, analyte concentration will be determined to be 52.7 ng/ml, whereas the correct concentration should be 100 ng/ml.

Figure 1. Matrix mismatch arising from calibration curves in mass spectrometry.

Three calibration curves are shown: one obtained on the basis of standard solutions (black curve), and two “matrix calibration” curves (blue and green). If the standard solution calibration curve is used to measure analyte concentration in a sample with matrix 2, measurement results will display a considerable negative bias.

Some analytical methods have a scope which includes many different matrices (such methods are sometimes referred to as “horizontal” methods). For instance, a cold vapor atomic fluorescence spectrometry method for the determination of mercury content in foodstuffs will be applied to matrices such as carrots, broccoli, spinach, etc. see method BVL L00.00-19-7 in the official collection of methods of analysis according to § 64 of the German Food and Feed Act.⁶ For such methods, a complete characterization of method performance should include a measure of matrix mismatch.

In this paper, the focus is on the quantification of matrix mismatch effects and their contribution to between-laboratory variation by means of the statistical evaluation of data from method validation studies. In such data sets, matrix mismatch manifests itself as the variation of bias across matrices, and this will serve as a metrological definition of matrix mismatch in the following.

Bias is the difference between the mean value (across test results) and the “true value”. Strictly speaking, bias is a measure of systematic error, see definition 3.3.2 in ISO 3534-2.⁷ However, calculating individual bias values – e.g. for a given laboratory and a given matrix – often offers little practical use. Rather, the aim is to determine the degree to which bias varies across the matrices lying within the method’s scope. In other words, the aim is to characterize matrix mismatch as a random effect.

While matrix-mismatch effects may be observed between different “types of matrices” such as beef or pork, they may also be observed within one type of matrix, e.g. between different types of beef. Depending on the method and its scope, it may be expedient to divide the population of different matrices in different matrix groups, whereby all the matrices in a given group can be expected to interact in a similar manner with the analyte. Matrix mismatch can then be characterized in terms of the variation of bias across matrix groups, see Jülicher et al. (1998).^8,9

Approaches for the characterization of matrix mismatch were described in the European Union CD 657 (2002)¹⁰ – revised and published in 2021 as CIR 808.¹¹ As far as authors are aware, a characterization of matrix mismatch is not required in any other method validation standard or guideline. The alternative approach described in the CD 657 and CIR 808 is implemented in InterVAL.¹²

In the following, the relationship between matrix mismatch and precision will be discussed and a relatively simple approach for the estimation of matrix mismatch by means of a single-lab study will be presented.

Matrix mismatch and precision

Matrix mismatch, considered as variation of bias across matrices, has two different components: variation of laboratory bias and variation of method bias. These two components have a different relation to precision, as will now be discussed.

Precision is defined in terms of the degree of agreement between test results obtained under given conditions on the basis of identical samples, see ISO 3534-2.⁷ Thus, in theory, it seems reasonable to conclude that matrix mismatch – considered as variation of bias across matrices, i.e. across different types of samples – is not a component of precision. Notwithstanding, in practice, matrix mismatch and precision are often conflated. The reason is the following.

The basic design for interlaboratory method validation studies – often referred to as collaborative studies – conducted in order to estimate method precision is described in ISO 5725-2¹³ and allows the estimation of two effects: laboratory bias and repeatability. If only one matrix is represented in the collaborative study – as is typically the case, albeit at different concentrations levels – the variation of bias from matrix to matrix cannot be characterized. If part of the observed between-laboratory variation is caused by matrix mismatch – for instance, if the calibration procedure interacts differently with the matrix under consideration in each laboratory – then matrix-mismatch effects will be included in the estimate of the reproducibility standard deviation. In summary: if the basic design from ISO 5725-2 is applied, there is no way to extricate the matrix-mismatch component from the reproducibility estimate, and matrix mismatch and precision are thus conflated.

For this reason, it is proposed to take a pragmatic approach and to include matrix mismatch (to be specific: the variation of laboratory bias across matrices) among the components of precision.

In order to shed further light on the relation between precision and matrix mismatch, the following scenario is considered:

Table 1 shows recovery values (generated with spreadsheet software for the purpose of illustration¹⁷) corresponding to the above scenario.

As can be seen in Table 1, laboratory bias depends on the matrix. For instance, Lab 2 displays a positive bias for Matrix A, but a negative bias for Matrix C. Conversely, Lab 9 displays a negative bias for Matrix A but a positive bias for Matrix C. Even when a consistently positive or negative bias across matrices is observed (e.g. consistent positive bias for Lab 1), variation of bias across matrices is nonetheless present (the bias for Matrix A is larger than for Matrix B). The variation of laboratory bias across matrices can be modelled as a random effect for the interaction of laboratory and matrix. The corresponding standard deviation is denoted $s_{\mathit{\text{matrix}},\mathit{lab}}$.

The mean recovery values across laboratories [1] are provided in Table 2.

These mean recovery values allow a characterization of the variation of method bias across matrices. This component of matrix mismatch is a measure of the dispersion of the matrix-specific mean values (with due consideration of the mean values’ statistical uncertainty). The corresponding standard deviation is denoted $s_{\mathit{\text{matrix}},\mathit{\text{method}}}$. Unlike the laboratory bias component of matrix mismatch, $s_{\mathit{\text{matrix}},\mathit{\text{method}}}$ cannot be calculated if only one matrix is represented in the collaborative study. It is straightforward to distinguish precision components from $s_{\mathit{\text{matrix}},\mathit{\text{method}}}$.

In summary, in this scenario, the following precision and matrix-mismatch parameters can be calculated:

These different variance components are best estimated by means of a linear model such as the following:

$y_{\mathit{ijk}}$ denotes The test result for matrix $i$, laboratory $j$ and replicate $k$

${\mathrm{\mu }}_{\mathrm{i}}$ denotes The mean value (across laboratories) for matrix $\mathrm{i}$

$a_j$ denotes The laboratory effect for lab $j$, modelled as a random effect with standard deviation (SD) $s_L$ (in the absence of matrix mismatch, we have $s_R^2=s_L^2+s_r^2$)

$b_{ij}$ denotes The interaction effect for matrix $i$ and lab $j$, modelled as a random effect with SD $s_{\mathit{\text{matrix}},\mathit{lab}}$

$e_{\mathit{ijk}}$ denotes The repeatability effect for matrix $i$, lab $j$ and replicate $k$, modelled as a random effect with SD $s_r$

For further information regarding mixed linear models, the reader is referred to Searke SR et al.¹⁴

Figure 2 illustrates the relationship between matrix mismatch and precision.

Figure 2. Relationship between matrix mismatch and precision.

The following notation is used: $s_R$ denotes the reproducibility standard deviation, $s_r$ denotes repeatability standard deviation, $s_L$ denotes the between-laboratory standard deviation, $s_{L,\mathit{\text{nonmatrix}}}$ denotes the component of between-laboratory variation consisting of sources of laboratory bias other than matrix mismatch, $s_{\mathit{\text{matrix}}}$ denotes the matrix-mismatch standard deviation consisting of (a) the variation of lab bias across matrices $s_{\mathit{\text{matrix}},\mathit{lab}}$ and (b) the variation of method bias across matrices $s_{\mathit{\text{matrix}},\mathit{\text{method}}}$. The latter is not a component of $s_R$.

If the samples sent to the laboratories in the collaborative study are not “true” samples, the various estimates of precision and matrix mismatch may be affected. In particular, if sample material is (a) homogenous (b) sent to the laboratories in the form of test portions requiring no further sample preparation steps, then the reproducibility standard deviation may be underestimated in the sense that observed variation during testing with “true” samples may be much larger.

It should be noted that while sample heterogeneity [2] will impact repeatability, it does not impact laboratory bias. Thus, while sample heterogeneity may cause differences in the repeatability standard deviations for different matrices, it will have no effect on the between-laboratory standard deviation.

By contrast, sample preparation can impact both repeatability and lab bias. Indeed, since the sample preparation steps must be performed separately for each replicate, it is clear they will contribute to repeatability. Moreover, there may be differences between the sample preparation procedures from laboratory to laboratory, resulting in a contribution to lab bias, and inflating the between-laboratory standard deviation. Finally, if there are interactions between the different sample preparation procedures and the matrices, sample preparation effects can also contribute to the variation of laboratory bias across matrices, i.e. to matrix mismatch.

Estimation of matrix mismatch by means of a single-lab study

A simple design for the evaluation of matrix mismatch in an in-house validation study will now be described. This design can be applied for the estimation of the variation of a given laboratory’s bias across matrices. It is only applicable if spiking is possible. Accordingly, in this design, the materials from which the different samples are collected must not contain the analyte. This will ensure that, upon spiking, the different samples can be considered to have identical analyte concentration levels.

In this design, test results are obtained on the basis of 12 samples – representing 12 different matrices. For each sample, duplicate measurements are performed. In this manner, variation between the samples (matrix mismatch) can be distinguished from variation within each matrix (repeatability). As explained above, matrix mismatch is modelled as a random effect, and the result is a standard deviation characterizing variation across all the samples consistent with the specification of the measurand.

Table 3 provides an example of data which could be obtained in such a single-lab experiment. These data were generated using spreadsheet software for the purpose of illustration.¹⁷

Values for the matrix-mismatch standard deviation $s_{\mathit{\text{matrix}}}$ and the repeatability standard deviation $s_r$ are calculated by means of one-way analysis of variance (ANOVA). For an introduction to ANOVA, the reader is referred to Sahai H et al.¹⁵

The following notation is introduced: the samples are indexed $i=1,\dots ,m$ (in this example, $m=12$); the replicates within each sample are indexed $j=1,n$ (in this example, $n=2$); and the individual measurement results are denoted $x_{\mathit{ij}}$.

First, compute the overall mean value $\overline{x}$, and the sample-specific mean values ${\overline{x}}_i$. Then compute the between-sample sum of squares:

The repeatability standard deviation $s_r$ is then obtained as

(If the value under the square root sign is negative, then $s_{\mathit{\text{matrix}}}=0$.)

Table 4 and Table 5 illustrate the ANOVA calculations by applying equations 2-3 to the data in Table 3.

Using the above equations, the following precision estimates are obtained:

In this example, matrix-mismatch effects can thus be considered non-negligible.

More sophisticated single-lab designs are described in Jülicher et al. (1998)^8,9 and ISO TS 23471.¹⁶

Conclusions

In method validation studies, one of the aims is a characterization of method performance, often in terms of trueness and precision. For horizontal methods – or any method whose scope includes several types of matrices – this should involve obtaining an estimate of matrix mismatch. Considered as a source of metrological variation, matrix mismatch consists of two components: variation of laboratory bias and variation of method bias – whereby the term variation is meant here as variation across matrices. Each component will typically be expressed as a standard deviation: $s_{\mathit{\text{matrix}},\mathit{lab}}$ and $s_{\mathit{\text{matrix}},\mathit{\text{method}}}$. In the statistical model for precision experiments described in ISO 5725-2, there is no matrix mismatch term. Indeed, a characterization of matrix mismatch is not possible if only one matrix is represented in the collaborative study, or if the samples sent to the laboratories do not reflect the properties of the matrices of “true” samples. If matrix mismatch was not estimated in the validation study, a subsequent single-lab study can be conducted. A relatively simple design was described, but more sophisticated designs may present various advantages. If spiking is not possible, the laboratory bias component of matrix mismatch $s_{\mathit{\text{matrix}},\mathit{lab}}$ can be estimated by means of a collaborative study with “real” samples (i.e. samples representing the matrices in the method’s scope). For the estimation of the method bias component of matrix mismatch $s_{\mathit{\text{matrix}},\mathit{\text{method}}}$, a collaborative study with “real” samples and (certified) reference values is required. It should be noted that proper estimation of the latter term may require due consideration of the statistical uncertainty of the matrix-specific mean values (across laboratories). The estimation of matrix mismatch by means of collaborative studies will be discussed in a subsequent article.