Uncertainty evaluation in normalization of isotope delta measurement results against international reference materials
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Abstract
Isotope delta measurements are normalized against international reference standards. Although multipoint normalization is becoming a standard practice, the existing uncertainty evaluation practices are either undocumented or are incomplete. For multipoint normalization, we present errorsinvariables 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 multilaboratory 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 modelIntroduction
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 largescale 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
At the Beijing IUPAC General Assembly in 2005, the Commission on Isotopic Abundances and Atomic Weights (CIAAW) recommended that δ(^{13}C) values of all carbonbearing 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 agreedupon 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 scaledefining materials, one has to rely on a variety of secondary international reference materials.
Calibration against secondary reference materials
In addition to scaledefining 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 IUPACCIAAW along with the agreedupon 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}(^{13}C) = − 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
Sources of uncertainty
There are three sources of uncertainty that should be considered in the multipoint 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.
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.
Singlelaboratory results
Calibration function
Uncertainty assessment
Multipoint 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 errorsinvariables regression using criterion S _{2}:
for k = 1…K iterations (typically K = 10 000 or higher)
 (i)Simulate values for u(δ _{ i }) aswhere χ ^{2}(v) is a sample drawn from chisquared distribution with v degreesoffreedom.$$\begin{array}{@{}rcl@{}} u_{k}(\delta_{i}) = u(\delta_{i}) \sqrt{\frac{v_{\delta,i}}{\chi^{2}(v_{\delta,i})}} \end{array} $$
 (ii)
Simulate values for δ _{ i } as random draws from normal distribution with mean δ _{ i } and uncertainty u _{ k }(δ _{ i }), \(\delta _{k,i} \sim \mathrm {N}(\delta _{i},{u^{2}_{k}}(\delta _{i}))\).
 (iii)Simulate values for u(d _{ i }) aswhere χ ^{2}(v) is a sample drawn from chisquared distribution with v degreesoffreedom.$$\begin{array}{@{}rcl@{}} u_{k}(d_{i}) = u(d_{i}) \sqrt{\frac{v_{d,i}}{\chi^{2}(v_{d,i})}} \end{array} $$
 (iv)
Simulate values for d _{ i } as random draws from normal distribution with mean d _{ i } and uncertainty u _{ k }(d _{ i }), \(d_{k,i} \sim \mathrm {N}(d_{i},{u^{2}_{k}}(d_{i}))\).
 (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 generalpurpose optimization function optim.
 (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}).
 (vii)
Obtain value δ _{ k,X} from Eq. 6.
Combining multilaboratory results
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 withinstudy variances u ^{2}(x _{ j }) are known beforehand whereas the betweenstudy 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 noniterative consensus estimators [16]. In addition, Bayesian approaches using noninformative 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 largescale 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 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)
 (i)
Simulate values for u(δ _{ i })
 (ii)
Simulate values for δ _{ i }
 (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”.
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.
Singlelaboratory results for NIST SRM 350b
As a part of a larger study, measurements of carbon13 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 (IAEACH6), polyethylene (IAEACH7), caffeine (IAEA600), 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.
Example dataset for the carbon13 delta measurements in NIST SRM 350b benzoic acid at NRC
Material, X  d _{X,S}(^{13}C)  N  δ _{X,VPDB}(^{13}C) 

SRM 350b  + 12.235(22) ‰  10  (measurand) 
IAEACH6  + 30.458(27) ‰  3  − 10.449(33) ‰ 
IAEACH7  + 8.141(21) ‰  3  − 32.151(50) ‰ 
IAEA600  + 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) ‰ 
Results of carbon13 delta measurements in NIST SRM 350b at NRC (using data from Table 1)
Uncertainty evaluation method  δ _{VPDB}(^{13}C) 

GUM  − 28.211(14) ‰ 
IUPAC  − 28.210(26) ‰ 
S _{0}  − 28.223(29) ‰ 
S _{1}  − 28.206(29) ‰ 
S _{2}  − 28.217(34) ‰ 
If, however, the calibration is established using errorsinvariables 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}(^{13}C) = − 28.206(29) ‰. Furthermore, the modified errorsinvariables regression which accounts for the fact that all measurement uncertainty estimates are based on small degreesoffreedom (S _{2}, Eq. 10) gives a more conservative uncertainty estimate of δ _{VPDB}(^{13}C) = − 28.217(34) ‰ as shown in Table 2.
In summary, while the average estimates of δ _{VPDB}(^{13}C) 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.
Multilaboratory results for NRC BEET1
Example dataset for the carbon13 delta measurements relevant to the NRC sugar isotope delta reference material BEET1
Material, X  Lab.  d _{X,S}(^{13}C)  N 

BEET1  A  − 26.028(85) ‰  12 
BEET1  B  − 26.054(59) ‰  12 
BEET1  C  − 26.037(37) ‰  12 
BEET1  D  − 25.987(73) ‰  12 
IAEACH6  A  − 10.418(79) ‰  4 
IAEACH6  B  − 10.450(14) ‰  4 
IAEACH6  C  − 10.455(66) ‰  4 
IAEACH6  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 
Fitting these data to the multivariate metaanalysis model of random laboratory effects (Eq. 12) provides with a consensus value δ _{VPDB}(^{13}C) = − 25.992(47) ‰ for BEET1 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}(^{13}C) = − 25.995(34) ‰. Hence, it is important to evaluate the possible correlations between the laboratories in order to avoid unrealistic (incorrect) uncertainty assessments.
 (i)Data

results from each laboratory, δ _{ j } (j = 1…4)

covariances of the laboratory results, V

 (ii)Priors

noninformative prior for the consensus isotope delta value, α ∼N (0,10^{5})

noninformative prior for the betweenlaboratory variance term, τ ^{− 2} ∼G (10^{− 4},10^{− 4}), using gamma distribution

 (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})
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 multipoint normalization algorithms. Contrary to traditional methods, the errorsinvariables 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 multilaboratory data reduction routines is desirable as it enables to interpret mutually inconsistent observed interlaboratory 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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