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Analytical and Bioanalytical Chemistry

, Volume 410, Issue 3, pp 1061–1069 | Cite as

Uncertainty evaluation in normalization of isotope delta measurement results against international reference materials

  • Juris Meija
  • Michelle M. G. Chartrand
Research Paper
Part of the following topical collections:
  1. ABCs 16th Anniversary.

Abstract

Isotope delta measurements are normalized against international reference standards. Although multi-point normalization is becoming a standard practice, the existing uncertainty evaluation practices are either undocumented or are incomplete. For multi-point normalization, we present errors-in-variables regression models for explicit accounting of the measurement uncertainty of the international standards along with the uncertainty that is attributed to their assigned values. This manuscript presents framework to account for the uncertainty that arises due to a small number of replicate measurements and discusses multi-laboratory data reduction while accounting for inevitable correlations between the laboratories due to the use of identical reference materials for calibration. Both frequentist and Bayesian methods of uncertainty analysis are discussed.

Keywords

Isotope delta Normalization Uncertainty evaluation Random effects model 

Introduction

In 2015, the International Committee for Weights and Measures (CIPM) noted that assigned values for isotope delta replacement materials should be done through a formal internationally vetted procedure that assures the continued comparability of delta value measurements. A recent large-scale international isotope delta intercomparison exercise [1] has shown that the statistical analysis of isotope delta materials can be indeed difficult and that more guidance on this matter seems necessary.

The current viewpoint is that normalization based on two or more international standards is the best method for isotope delta analysis [2]. Despite this, the existing uncertainty evaluation practices are largely undocumented or incomplete [3]. In this vein, we present mathematical framework for linear normalization of isotope delta measurement results against secondary international reference materials with exclusive emphasis to the uncertainty evaluation. This article does not address the various prerequisite quality control measures that analysts must undertake during the data generation itself such as the uncertainties due to sample preparation, drift and blank correction [4, 5].

General aspects

Definitions and nomenclature

Isotope delta is the relative difference of the isotope amount ratios of an element between two materials:
$$ \delta_{\mathrm{A},\mathrm{B}}(^{13}\mathrm{C}) = R_{\mathrm{A}}/R_{\mathrm{B}}-1 $$
(1)
The material A is the sample and the material B is the reference standard. The above notation is variably shortened to δ B(13C), δ(13C)B, δ 13CB, or simply δ 13C. The numerical values of isotope deltas are almost universally reported in parts per thousand (‰) which often leads to appearance of extraneous multiplying factors (0.001 or 1000) in the quantity equations involving isotope deltas [6].

At the Beijing IUPAC General Assembly in 2005, the Commission on Isotopic Abundances and Atomic Weights (CIAAW) recommended that δ(13C) values of all carbon-bearing materials be measured and expressed relative to the VPDB on a scale normalized by assigning consensus values of − 46.6 ‰ to LSVEC lithium carbonate and + 1.95 ‰ to NBS 19 calcium carbonate, and authors should clearly state so in their reports[7]. Thus, the measurement results for any given sample are expressed against an agreed-upon scale which sets the numerical value of carbon isotope deltas to these two international reference materials.

In practice, however, not everyone employs LSVEC or NBS 19 in their daily measurements. One reason for this is that other secondary reference materials are sometimes desirable in order to better match the nature of the calibrators and samples (principle of identical treatment [4]). For example, hair reference materials might be more desirable than carbonates when analyzing human hair. Another reason for using secondary isotopic reference materials is out of practical necessity since NBS 19 (also known as the ‘toilet seat’ limestone) is no longer available.

Since isotope delta measurements cannot always be performed directly against the scale-defining materials, one has to rely on a variety of secondary international reference materials.

Calibration against secondary reference materials

In addition to scale-defining primary isotope delta reference materials, there are a number of natural or synthetic compounds which have been carefully calibrated against the primary calibration materials. The list of such materials is published and maintained by IUPAC-CIAAW along with the agreed-upon isotope delta values [8].

The characterization of the secondary isotope delta reference materials is not owned by the institutions that disseminate the physical materials. For this reason, several property values might be available for the same material. As an example, IUPAC lists published delta values from δ VPDB(13C) = − 30.03 to − 29.40 ‰ for NBS 22 oil [8]. In addition, providers of this reference material give markedly different uncertainty estimates: − 30.03(6) k= 1 ‰ (USGS), − 30.03(9) k= 2 ‰ (NIST) and − 30.031(43) k= 1 ‰ (IAEA). Consequently, when reporting isotope delta measurement results, it is essential that the values assigned to primary and secondary materials are given along the measurement results.

Calibration algorithms

There are several algorithms which convert measured isotope delta values of a sample to the calibrated international scale: single-, two-, and multi-point linear normalization [9]. A two-point calibration, for example, invokes two reference standards whose isotope deltas are measured against an in-house internal standard. The observed instrumental indications (d) are then plotted against the assigned known isotope delta values (δ) for the primary standards and the linear relationship between these data serves as the calibration function:
$$ \delta_{\mathrm{X}} = \delta_{1} + \frac{\delta_{2} - \delta_{1}}{d_{2} - d_{1}}(d_{\mathrm{X}} - d_{1}) $$
(2)
Some authors prefer the model equation of the two-point calibration to be expressed in a different, yet equivalent, mathematical form [10]:
$$ \delta_{\mathrm{X}} = \frac{(n d_{\mathrm{X}} + 1)(\delta_{1} + 1)}{(n d_{1} + 1)} - 1 $$
(3)
where
$$ n = \frac{\delta_{1} - \delta_{2}}{d_{1} - d_{2} - \delta_{1} d_{2} + \delta_{2} d_{1}} $$
(4)
Uncertainty evaluation of isotope delta measurement results with two-point calibration can be done by applying the law of uncertainty propagation to the above measurement models.
Two-point calibration can be naturally extended to multi-point calibration. In this case, however, one no longer can write down a physical measurement model as in Eq. 2. Rather, the measurement model becomes a statistical model relating the observed isotope delta values from N standards, d i (i = 1…N), and the known isotope delta values of these standards, δ i , in a linear relationship:
$$ d_{i} = a + b \cdot \delta_{i} + e_{i} $$
(5)
The residual error is typically modeled as normal random variable, e i ∼N(0,σ 2). Equation 5 is a statistical model, hence it must be solved for the unknown parameters a and b (and σ) using statistical methods which we discuss later. The isotope delta for a given sample, δ X, is then obtained from the uncalibrated measured value, d X, by solving the above calibration equation:
$$ \delta_{\mathrm{X}} = \frac{d_{\mathrm{X}} - a}{b} $$
(6)

Sources of uncertainty

Multi-point calibration of isotope delta measurement results against international reference materials is summarized visually in Fig. 1.
Fig. 1

Sources of uncertainty in the isotope delta measurements using multi-point calibration

There are three sources of uncertainty that should be considered in the multi-point calibration of isotope delta measurement results: (1) the uncertainty due to measurement of isotope delta of the sample material, (2) uncertainty due to isotope delta measurement of the primary reference materials, and (3) the uncertainty associated with the isotope delta values of the reference materials used.

Ordinary linear regression is frequently employed to obtain calibration coefficients and their uncertainties. However, such an approach neglects the uncertainty that is inherent to the isotope delta values of the reference materials used. To account for this, one can employ the errors-in-variables statistical model
$$ d_{i} = a + b (\delta_{i} + u_{i}) + e_{i} $$
(7)
which explicitly reflects the fact that isotope delta values of the standards, δ i , themselves are accompanied with uncertainties, u i ∼N(0,u 2(δ i )).

This statistical model can be solved using generalized distance regression in conjunction with the Monte Carlo uncertainty evaluation. In addition to the above sources of uncertainty, we also consider the possible biases in the results obtained by the various laboratories using random effects statistical models.

Single-laboratory results

Calibration function

Typically, the best multi-point calibration line is obtained using least squares fitting where one assumes that the values of the independent variable are known without error. In isotope delta measurements, however, both variables d i and δ i have non-negligible uncertainties and the calibration line must therefore be estimated using statistical methods that accommodate this. Such cases can be modeled with the errors-in-variables regression method as described, for example, in ISO 6143 [11]. The weighted least squares procedure minimizes the sum of S 0 for all values of i:
$$ S_{0}(a,b)_{i} = \frac{(d_{i} - a - b \delta_{i})^{2}}{u^{2}(d_{i})} $$
(8)
In contrast, the errors-in-variables regression involves minimizing the sum of S 1 for all values of i with respect to regression parameters a, b and the unknown ‘true’ values of δ i , \(\hat {\delta }_{i}\):
$$ S_{1}(a,b,\hat{\delta}_{i})_{i} = \frac{(d_{i} - a - b \hat{\delta}_{i})^{2}}{u^{2}(d_{i})} + \frac{(\delta_{i} - \hat{\delta}_{i})^{2}}{u^{2}(\delta_{i})} $$
(9)
The uncertainties u(d i ) and u(δ i ) are assumed to be known. Regression based on minimizing the sum S 1,i is also known as the generalized Deming regression (available in R using function deming from the package of the same name).
Guenther and Possolo have put forward a modified equation for errors-in-variables regression which recognizes the fact that the uncertainties of d i are typically based on a small number of degrees-of-freedom [12]. This additional source of uncertainty can be accounted for in the errors-in-variables regression by replacing the elements of normal probability density function with Student’s t distribution. Subsequently, one minimizes the sum of S 2 for all values of i:
$$\begin{array}{@{}rcl@{}} S_{2}(a,b,\hat{\delta}_{i})_{i} &=& (v_{d,i} + 1) \log \left( 1+ \frac{(d_{i} - a - b \hat{\delta}_{i})^{2}}{v_{d,i} u^{2}(d_{i})} \right) \\ &&+ (v_{\delta,i}+ 1) \log \left( 1+\frac{(\delta_{i} - \hat{\delta}_{i})^{2}}{v_{\delta,i} u^{2}(\delta_{i})} \right) \end{array} $$
(10)
where v d,i and v δ,i are the degrees-of-freedom associated with u 2(d i ) and u 2(δ i ), respectively. If unavailable, the degrees-of-freedom that are associated with the uncertainties of the international reference materials can be taken as a sufficiently large number, say v δ,i = 100.

Uncertainty assessment

Multi-point calibration of isotope deltas typically involves measurements made of three to six international reference materials. While the parameters of linear regression (intercept and slope) can be obtained using the various methods described earlier, the uncertainty of the regression parameters is best estimated using the Monte Carlo method [13]. This consists of applying perturbations to the values of all quantities consistent with the available data. Below is a general description of the Monte Carlo method for evaluating the uncertainty of the isotope delta measurement results with errors-in-variables regression using criterion S 2:

for k = 1…K iterations (typically K = 10 000 or higher)

  1. (i)
    Simulate values for u(δ i ) as
    $$\begin{array}{@{}rcl@{}} u_{k}(\delta_{i}) = u(\delta_{i}) \sqrt{\frac{v_{\delta,i}}{\chi^{2}(v_{\delta,i})}} \end{array} $$
    where χ 2(v) is a sample drawn from chi-squared distribution with v degrees-of-freedom.
     
  2. (ii)

    Simulate values for δ i as random draws from normal distribution with mean δ i and uncertainty u k (δ i ), \(\delta _{k,i} \sim \mathrm {N}(\delta _{i},{u^{2}_{k}}(\delta _{i}))\).

     
  3. (iii)
    Simulate values for u(d i ) as
    $$\begin{array}{@{}rcl@{}} u_{k}(d_{i}) = u(d_{i}) \sqrt{\frac{v_{d,i}}{\chi^{2}(v_{d,i})}} \end{array} $$
    where χ 2(v) is a sample drawn from chi-squared distribution with v degrees-of-freedom.
     
  4. (iv)

    Simulate values for d i as random draws from normal distribution with mean d i and uncertainty u k (d i ), \(d_{k,i} \sim \mathrm {N}(d_{i},{u^{2}_{k}}(d_{i}))\).

     
  5. (v)

    Minimize the sum of S 2 in Eq. 10 for all values of i to obtain estimates a k and b k . In R, this can be done using the general-purpose optimization function optim.

     
  6. (vi)

    Simulate the value for the measurement of the sample X, d k,X, similar to how it was done in the preceding steps. As above, this involves simulating the uncertainty u k (d X).

     
  7. (vii)

    Obtain value δ k,X from Eq. 6.

     
The uncertainty assessments for δ X are produced from the obtained set of simulated measurement results δ X which can be conveniently visualized as histograms.

Combining multi-laboratory results

Isotope delta results obtained by numerous laboratories for the same material are often compared in order to assign a consensus value. The data reduction problem can be summarized as follows: given a number of measured values x j (j = 1…M), along with their standard uncertainties u j , the goal is to combine these values into a single best estimate. To achieve this, a statistical model is needed to codify the relationship between the data. For this purpose, we adopt a random laboratory effects model for the reported data which represents each data point as an additive superposition of three effects [14, 15]:
$$ x_{j} = \mu + \lambda_{j} + \varepsilon_{j} $$
(11)
Here, x j is the reported isotope delta value by j th laboratory, μ is the consensus value of all laboratory results, λ j denotes the effect of the j th laboratory, which is modeled as a normal random variable with mean 0 and variance τ 2 , λ j ∼N (0,τ 2), and ε j denotes measurement error, ε j ∼N (0,u 2(x j )).

The random effects model captures any additional uncertainty, τ, when the reported isotope deltas are more scattered than their values of u(x j ) would intimate. In other words, we accept that the stated uncertainties do not represent complete knowledge. The within-study variances u 2(x j ) are known beforehand whereas the between-study variance τ 2 and the consensus value μ must both be estimated from the data.

In practice, consensus isotope delta values can be obtained from the reported laboratory results and their uncertainties using a variety of methods. Albeit not without its shortcomings, the DerSimonian and Laird method is a popular choice among simple and non-iterative consensus estimators [16]. In addition, Bayesian approaches using non-informative reference priors are gaining widespread acceptance [16, 17].

Correlations between the laboratory results

Often, the laboratory results are not independent because some laboratories might employ identical international reference standards for calibration. As an example, during the recent large-scale certification campaign of several carbon isotope delta materials [1], USGS61 caffeine was measured by nine laboratories most of which relied on four common isotope delta reference materials: LSVEC, NBS19, USGS40, and USGS41. Since the uncertainty of these reference materials is a significant component in the overall uncertainty budget, the results from these laboratories cannot be treated as independent.

The between-laboratory correlations can be accounted for in the random effects model (Eq. 11) as follows:
$$ \mathbf{x} \sim \mathrm{N}(\mu\mathbf{1},\mathbf{V}+\tau^{2} \mathbf{I}) $$
(12)
where V is the covariance matrix of x j which contains the quoted uncertainties and covariances. The diagonal elements of this matrix equal the squared standard uncertainties and the off-diagonal elements are determined as V i j = cor(x i ,x j )u(x i )u(x j ) where cor(x i ,x j ) is the correlation between the results x i and x j .

The laboratory results are represented in Eq. 12 in a form of a vector x, and I is the identity matrix (of size M × M). In programming language R, the model in Eq. 12 can be fitted to the data using several approaches. As an example, with the restricted maximum likelihood method one can use the command rma.mv(X, V, method=”reml”), where the function rma.mv is defined in the R package metafor, X is a vector of isotope delta measurement results, x = {x 1,…,x M }, and V is their covariance matrix.

Determination of correlations between the laboratory results

Correlations between the reported laboratory results can be evaluated using the Monte Carlo method as described in section “Uncertainty assessment” with few modifications. Instead of simulating a result from a single laboratory, one now has to parallelize such calculations across all laboratories with the sole exception of the parts that perturb the values of u(δ i ) and δ i .

for k = 1…K iterations (typically K = 10 000 or higher)

  1. (i)

    Simulate values for u(δ i )

     
  2. (ii)

    Simulate values for δ i

     
  3. (iii)

    For each laboratory j = 1…M, obtain the value of δ k,X which we denote here as δ k,j . This follows the steps (iii)–(vii) of section “Uncertainty assessment”.

     
The outcome of the above Monte Carlo simulation is a K × M matrix containing simulated values of δ X. This provides the estimate of the covariance between the results of any two laboratories necessary to construct matrix V.

Examples

In order to illustrate the methods described above and the impact of various assumptions that can be taken during the data reduction and interpretation, we provide two detailed examples.

Single-laboratory results for NIST SRM 350b

As a part of a larger study, measurements of carbon-13 isotope delta in NIST SRM 350b benzoic acid primary (acidimetric) standard were performed at NRC in 2017. For this purpose, four internationally recognized stable isotope reference materials were selected for calibration: sucrose (IAEA-CH-6), polyethylene (IAEA-CH-7), caffeine (IAEA-600), and glutamic acid (USGS40) [8] along with two additional reference materials available from the US Geological Survey: caffeine (USGS62) and glycine (USGS65) [1]. The relevant measurement results and the assigned isotope delta values for the six reference materials are listed in Table 1.

Table 1

Example dataset for the carbon-13 delta measurements in NIST SRM 350b benzoic acid at NRC

Material, X

d X,S(13C)

N

δ X,VPDB(13C)

SRM 350b

+ 12.235(22) ‰

10

(measurand)

IAEA-CH-6

+ 30.458(27) ‰

3

− 10.449(33) ‰

IAEA-CH-7

+ 8.141(21) ‰

3

− 32.151(50) ‰

IAEA-600

+ 12.729(12) ‰

3

− 27.771(43) ‰

USGS40

+ 14.128(30) ‰

3

− 26.39(4) ‰

USGS62

+ 26.040(46) ‰

3

− 14.79(4) ‰

USGS65

+ 20.355(17) ‰

3

− 20.29(4) ‰

Standard deviation from N replicate measurements is given in the parenthesis. For example, 12.235(22) ‰refers to a standard deviation of 0.022 ‰. d X,S(13C) is the measurement result of carbon-13 isotope delta in sample X against in-house reference gas S. Full dataset is given in the Supplementary Information.

A simple approach to establish the isotope delta value for SRM 350b is to perform ordinary least squares fit from the eighteen (6 × 3) pairs of data {d X,S(13C),δ X,VPDB(13C)} from the international standards with disregard of the uncertainties assigned to δ X,VPDB(13C). The resulting intercept and the slope are then used in Eq. 6, together with the measured value d X,S for SRM 350b to obtain a value of δ VPDB(13C) = − 28.211 ‰ for SRM 350b. The uncertainty propagation calculations are done from Eq. 6 by taking into account the correlation between the intercept and the slope, and by using \(0.022/\sqrt {10} = 0.007\) ‰ as the uncertainty for the mean value of d X,S for SRM 350b. The uncertainty propagation of Eq. 6 is done according to the Guide to the Expression of Uncertainty in Measurement (GUM) [18]:
$$ u^{2}(\delta_{\mathrm{X}}) = \mathbf{C} \mathbf{U} \mathbf{C}^{T} $$
(13)
where C is the vector of partial derivatives,
$$ \mathbf{C} = \left[ \begin{array}{lll} \frac{\partial \delta_{\mathrm{X}}}{\partial a} & \frac{\partial \delta_{\mathrm{X}}}{\partial b} & \frac{ \partial \delta_{\mathrm{X}}}{\partial d_{\mathrm{X}}} \end{array} \right] = \left[\begin{array}{lll} \frac{-1}{b} & \frac{a-d_{\mathrm{X}}}{b^{2}} & \frac{1}{b} \end{array} \right] $$
(14)
and U is the covariance matrix,
$$ \mathbf{U} = \left[ \begin{array}{lll} u^{2}(a) & u(a,b) & 0 \\ u(a,b) & u^{2}(b) & 0 \\ 0 & 0 & u^{2}(d_{\mathrm{X}}) \end{array} \right] $$
(15)
In this example, a = 41.213(29), b = 1.0272(13), u(a,b) = 3.454 × 10− 5 which leads to a standard uncertainty of 0.014 ‰ for δ VPDB(13C) (labelled as the GUM approach in Table 2). Uncertainty calculations such as the one described above can be facilitated with the NIST Uncertainty Machine [19].
Table 2

Results of carbon-13 delta measurements in NIST SRM 350b at NRC (using data from Table 1)

Uncertainty evaluation method

δ VPDB(13C)

GUM

− 28.211(14) ‰

IUPAC

− 28.210(26) ‰

S 0

− 28.223(29) ‰

S 1

− 28.206(29) ‰

S 2

− 28.217(34) ‰

In contrast to the above calculation, IUPAC recommends using the following expression for calculating the uncertainty of a predicted mean from N repetitions when using ordinary linear regression[20]:
$$ u(\delta) = \frac{s_{\mathrm{R}}}{b} \sqrt{\frac{1}{N}+\frac{1}{M} + \frac{(d - \bar{d})^{2}}{b^{2} {\sum}_{j} (\delta_{j} - \bar{\delta})^{2}}} $$
(16)
where \(\bar {\delta }\) and \(\bar {d}\) are the mean values of all (M) regression points and s R is the residual standard error,
$$ s_{\mathrm{R}} = \sqrt{\frac{{\sum}_{j} (d_{j} - a - b \!\cdot\! \delta_{j})^{2}}{M-2}} $$
(17)
In the example given in Table 1, M = 18 and N = 10 from which u(δ) = 0.026 ‰. A similar uncertainty estimate is obtained from weighted ordinary regression with best fit criterion S 0 (see Table 2).

If, however, the calibration is established using errors-in-variables regression which accounts for uncertainties in both the predictor and outcome variables (criterion S 1, Eq. 9), we obtain similar uncertainty estimate but noticeably different mean value: δ VPDB(13C) = − 28.206(29) ‰. Furthermore, the modified errors-in-variables regression which accounts for the fact that all measurement uncertainty estimates are based on small degrees-of-freedom (S 2, Eq. 10) gives a more conservative uncertainty estimate of δ VPDB(13C) = − 28.217(34) ‰ as shown in Table 2.

In summary, while the average estimates of δ VPDB(13C) from all methods considered herein are similar, but not identical, the same cannot be said about the uncertainty estimates. Note that the ordinary least squares (weighted or not) does not provide unbiased parameter estimates when the predictor variable (x axis) has measurement uncertainty [21]. Generally speaking, uncertainty in the predictor variable leads to negatively biased estimates of the calibration slope and, in turn, biased value for the isotope deltas which is an aditional reason to explicitly account all uncertainty sources.

Multi-laboratory results for NRC BEET-1

A variability in carbon isotope ratio of approx. 180 ‰ is observed in naturally occurring terrestrial materials [22]. Sugar beet has δ VPDB(13C) ≈− 28 ‰, whereas sugar cane has δ VPDB(13C) ≈− 14 ‰. Given that sugar cane is one of the major sources of added sugar in processed food, isotope delta standards can enable tracking of added sugars [23]. For these reasons, NRC has recently produced a suite of isotope delta sugar reference materials aided by an interlaboratory comparison study. We use a small portion of this dataset to illustrate the data handling from multi-laboratory results as summarized in Table 3. Although Table 3 lists the reported calibrated isotope delta measurement results from four laboratories, for the purposes of this example we treat these results as non-calibrated indications, d X,S(13C) instead of δ X,VPDB(13C).
Table 3

Example dataset for the carbon-13 delta measurements relevant to the NRC sugar isotope delta reference material BEET-1

Material, X

Lab.

d X,S(13C)

N

BEET-1

A

− 26.028(85) ‰

12

BEET-1

B

− 26.054(59) ‰

12

BEET-1

C

− 26.037(37) ‰

12

BEET-1

D

− 25.987(73) ‰

12

IAEA-CH-6

A

− 10.418(79) ‰

4

IAEA-CH-6

B

− 10.450(14) ‰

4

IAEA-CH-6

C

− 10.455(66) ‰

4

IAEA-CH-6

D

− 10.465(71) ‰

4

USGS40

A

− 26.385(73) ‰

4

USGS40

B

− 26.430(71) ‰

4

USGS40

C

− 26.455(66) ‰

4

USGS40

D

− 26.402(61) ‰

4

USGS62

A

− 14.840(92) ‰

4

USGS62

B

− 14.797(13) ‰

4

USGS62

C

− 14.846(30) ‰

4

USGS62

D

− 14.675(182) ‰

4

Standard deviation from N replicate measurements is given in the parenthesis

Isotope delta values for the three reference materials (IAEA-CH-6, USGS40, USGS62) are given in Table 1

Following the general procedure outlined in section “Determination of correlations between the laboratory results”, we obtain estimates of δ VPDB(13C) for BEET-1 by each of the four laboratories. This involves a three-point calibration using errors-in-variables regression with criterion S 2. The following results were obtained:
$$\begin{array}{@{}rcl@{}} \begin{array}{llll} \mathrm{A}& -26.022(78)~\textperthousand & \mathrm{B} & -26.017(72)~\textperthousand \\ \mathrm{C}& -25.965(63)~\textperthousand & \mathrm{D} & -25.981(66)~\textperthousand \end{array} \end{array} $$
(18)
In addition, the following correlations are obtained between the results of the four laboratories:
$$ \text{cor}(\delta_{i},\delta_{j}) = \begin{array}{llll} 1 & 0.28 & 0.31 & 0.32 \\ 0.28 & 1 & 0.37 & 0.30 \\ 0.31 & 0.37 & 1 & 0.34 \\ 0.32 & 0.30 & 0.34 & 1 \end{array} $$
(19)

Fitting these data to the multivariate meta-analysis model of random laboratory effects (Eq. 12) provides with a consensus value δ VPDB(13C) = − 25.992(47) ‰ for BEET-1 from the example dataset. In R, this can be achieved by using code metafor::rma.mv(x,V) where x is the vector of mean laboratory results (Eq. 19) and V is their covariance matrix.

Dismissing correlations between the laboratories has the effect of lowering the uncertainty of the consensus value to δ VPDB(13C) = − 25.995(34) ‰. Hence, it is important to evaluate the possible correlations between the laboratories in order to avoid unrealistic (incorrect) uncertainty assessments.

Establishing consensus estimates from correlated datasets is not unique to this study. Bodnar et al. describe a similar problem in determining fundamental constants from various studies using Bayesian methods [17]. Analysis of the NRC BEET-1 data can also be performed using Bayesian method of uncertainty evaluation. For this purpose, one can adopt the following Bayesian model:
  1. (i)
    Data
    • results from each laboratory, δ j (j = 1…4)

    • covariances of the laboratory results, V

     
  2. (ii)
    Priors
    • non-informative prior for the consensus isotope delta value, α ∼N (0,105)

    • non-informative prior for the between-laboratory variance term, τ − 2 ∼G (10− 4,10− 4), using gamma distribution

     
  3. (iii)

    Likelihood (data generation model)

     

laboratory results are like outcomes of multivariate normal distribution, δ j ∼N (μ j ,V) with hierarchical model for the laboratory effects, μ j ∼N (α,τ 2)

The model parameters (consensus isotope delta value and the between-laboratory variance) can be obtained using the Markov chain Monte Carlo method [24]. Alternatively, the above model (with slightly different specification of the non-informative priors according to the currently favored principles of Berger and Bernardo) can be solved analytically [17]. Using the Markov Chain Monte Carlo method, we obtain the consensus value δ VPDB(13C) = − 25.991(56) ‰ for BEET-1 from the example dataset. For this purpose, we use openBUGS software from R as shown in the Listing 1. The posterior probability densities for the consensus isotope delta and the between-laboratory uncertainty are both shown in Fig. 2.
Fig. 2

Posterior distribution of isotope delta consensus value and the between-laboratory uncertainty for the dataset summarized in Eqs. 18 and 19. Presented are the calculations when the correlations between the laboratory results are properly taken into account (red lines) and ignored (black line)

Listing 1

Implementation of Markov Chain Monte Carlo method for Bayesian inference of the interlaboratory random effects model with regard to possible correlations between the laboratory results in R programming language with openBUGS software.

Conclusions

This manuscript outlines a data reduction framework for normalizing isotope delta measurement results applicable for two- and multi-point normalization algorithms. Contrary to traditional methods, the errors-in-variables statistical model utilized herein is adopted with explicit accounting for the uncertainty in estimating the standard deviation due to the small number of replicate measurements. The computational complexities are largely reduced by the use of Monte Carlo method for uncertainty evaluation which, to witt, relies on iterative perturbation of all input data according to their respective uncertainties. The adoption of random effects models in multi-laboratory data reduction routines is desirable as it enables to interpret mutually inconsistent observed inter-laboratory results to be consistent with a common consensus value. In addition, Bayesian paradigm might be preferred over the classical statistical methods as it obviates the need for asymptotic arguments when dealing with small sample sizes.

Notes

Acknowledgments

The authors have benefited from discussions with Blaza Toman, National Institute of Standards and Technology, USA.

Compliance with Ethical Standards

Conflict of interests

The authors declare that they have no conflict of interest.

Supplementary material

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216_2017_659_MOESM2_ESM.xlsm (71 kb)
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© Crown 2017

Authors and Affiliations

  1. 1.Measurement Science and StandardsNational Research Council CanadaOttawaCanada

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