Sources of Uncertainty in Biotransformation Mechanistic

Jun 20, 2019 - However, practices to calculate and report Λ and related uncertainty are not universal, leading to the potential for misinterpretation...
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Cite This: Anal. Chem. 2019, 91, 9147−9153

Sources of Uncertainty in Biotransformation Mechanistic Interpretations and Remediation Studies using CSIA Ann Sullivan Ojeda,† Elizabeth Phillips,† Silvia A. Mancini,‡ and Barbara Sherwood Lollar*,† †

Department of Earth Sciences, The University of Toronto, 22 Russell Street, Toronto, Ontario M5S 3B1, Canada Geosyntec Consultants Inc., 243 Islington Avenue #1201, Etobicoke, Ontario M8X 1Y9, Canada

‡

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S Supporting Information *

ABSTRACT: Compound-specific isotope analysis (CSIA) is a powerful tool to understand the fate of organic contaminants. Using CSIA, the isotope ratios of multiple elements (δ13C, δ2H, δ37Cl, δ15N) can be measured for a compound. A dual-isotope plot of the changes in isotope ratios between two elements produces a slope, lambda (Λ), which can be instrumental for practitioners to identify transformation mechanisms. However, practices to calculate and report Λ and related uncertainty are not universal, leading to the potential for misinterpretations. Here, the most common methods are re-evaluated to provide the basis for a more accurate best-practice representation of Λ and its uncertainty. The popular regression technique, ordinary linear regression, can introduce mathematical bias. The York method, which incorporates error in both variables, better adapts to the wide set of data conditions observed for dual-isotope data. Importantly, the existing technique of distinguishing between Λs using the 95% confidence interval alone produces inconsistent results, whereas statistical hypothesis testing provides a more robust method to differentiate Λs. The propensity for Λ to overlap for a variety of conditions and mechanisms highlights the requirement for statistical justification when comparing data sets. Findings from this study emphasize the importance of this evaluation of best practice and provide recommendations for standardizing, calculating, and interpreting dual-isotope data.

C

ompound-specific isotope analysis (CSIA) is increasingly applied in contaminant hydrogeology to identify and quantify contaminant transformation in the field and in laboratory experiments, by measuring changes in isotope ratios (isotope fractionation).1 CSIA measures isotopic ratios of an element (e.g., carbon) in a sample expressed in Delta (δ) notation (eq 1). 13

12

δ13Csample =

13

( ) C C

−

sample 13

13

( ) 12

C C

reference

( ) 12

C C

For a reaction involving isotope fractionation of two elements, a dual-isotope plot of the simultaneous changes in the isotope ratios of each element (δ13C vs δ2H or δ13C vs δ37Cl) provides information about transformation processes from either laboratory or field data sets. The slope of a dualisotope plot (lambda, Λ)2 can be specific to a reaction mechanism. Laboratory mechanistic studies can constrain the range of Λs for a given transformation mechanism under a variety of conditions and can then be applied to field sites to identify the relevant mechanism of contaminant degradation and to optimize conditions for the relevant mechanism of transformation for more efficient remediation. The advantages of using highly resolved mechanistic information provided by two-dimensional (dual), and in some cases three-dimensional, isotope plots has led to a body of literature covering a variety of transformation processes and compound classes including benzene and toluene,3−6 MTBE,7−9 chlorinated hydrocarbons,10−15 nitrobenzenes,16,17 and micropollutants like herbicides and pesticides.18 Using the dual-isotope approach, Λ can be calculated by regression of measured data for two different isotopes7,9 or can

reference

(1)

12

where ( C/ C)reference is the isotopic ratio of an international reference standard such as Vienna-Pee Dee Beleminite (VPDB) for carbon. Isotope fractionation occurs because of differences in reactivity between molecules containing exclusively light isotopes compared with heavier isotopes of an element (12C vs 13C). One major application of CSIA is identifying and measuring the extent of contaminant transformations in situ. A benefit of CSIA compared with monitoring contaminant concentration alone, is that the degree of isotope fractionation can distinguish between physical (e.g., diffusion, sorption, volatilization) and transformation (e.g., abiotic degradation, biodegradation) processes, which greatly aids remediation efforts.1 © 2019 American Chemical Society

Received: April 10, 2019 Accepted: June 20, 2019 Published: June 20, 2019 9147

DOI: 10.1021/acs.analchem.9b01756 Anal. Chem. 2019, 91, 9147−9153

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Analytical Chemistry

where β is the estimate of the slope of the linear equation, Ui =

be approximated by calculating the ratio of enrichment factors (ε).4,7 The latter is calculated from the relationship between changes in isotope ratios and the fraction of substrate remaining as the transformation proceeds (for details, see Scott et al.19). An example of calculating Λ from ε is shown in eq 2 for carbon and hydrogen where Δδ13C is the absolute fractionation of carbon, Δδ2H is that of hydrogen. ε Δδ 2 H ≈ H εC Δδ13C

Λ=

xi − x, Vi = yi − y, Wi =

i

Wi2Ui2 W 2UV 2 − 2β 2 ∑ i i i − β( ∑ WU i i − ω(xi) ω(xi) i i

(2)

+ ∑ WUV i i i = 0 i

∑ i

, and ω(xi) and ω(yi) are

the weights of each observation. The York method is beneficial as it is an exact iterative solution, relying on no underlying assumptions if the error in the measurements is not approximated. The York method also reduces to other regression methods when appropriate.26 The variability of data compiled for dual-isotope plots is based on numerous factors, many of which influence the validity of the regression method. The range of isotope values can span a few per mil5 or hundreds per mil30 depending on the element, the isotope of interest, and the ε for a given reaction. Changes in carbon isotope values in dual-isotope plots can range from 2 to 100‰;5,31 changes in hydrogen are much larger, ranging from 10 to 250‰.4,6 Ranges for chlorine enrichment are typically smaller, from 1 to 15‰.12,14 Likewise, the analytical error associated with the measurement is specific to each isotope. Uncertainty in carbon isotope measurements using GC-IRMS is generally ±0.5‰.1,32 Reporting of uncertainty in hydrogen measurements is more variable but typically is an order of magnitude greater than carbon1,33,34 and ranges from ±5‰33,34 for BTEX compounds to ±10‰35 for chlorinated hydrocarbons using a purge-and-trap system. Reported uncertainty for chlorine isotope measurements ranges from ±0.1‰ using GC-MC-ICPMS36 to 1‰ using GC-quadrupole MS.37 The best practice for comparing individual or compiled Λ data sets for laboratory and field applications is underdeveloped in the literature to date. The popular method to distinguish between two Λs (and thereby infer different reaction mechanisms) using a dual-isotope plot is simply to compare the 95% confidence intervals calculated from OLR. Critically, such an oversimplistic approach does not incorporate the measurement uncertainty of the original data. The root of this practice is unclear, and there is no underlying statistical justification for its use. Comparisons between data sets using 95% confidence intervals have been previously shown to yield false-positive results, meaning two data sets may be statistically different yet have overlapping confidence intervals.38 By using a flawed comparison in dual-isotope plots, there is potential for misinterpreting mechanisms, and hence, it is imperative to have a community discussion about standard best practices for such data analysis. For decades, the stable isotope community has worked collaboratively to develop consensus best-practice documents to guide practitioners and ensure international comparability in reporting results through agencies such as the International Atomic Energy Agency (IAEA) and the National Institute of Standards and Technology (NIST). Prominent examples of this effort in environmental research include the EPA guide that outlines best practices for using CSIA in contaminant hydrogeology;1 an analysis of total uncertainty incorporating both accuracy and reproducibility for baseline GC-IRMS measurements;32 and identification of bias introduced during sampling for CSIA measurements.39 One substantial challenge facing dual-isotope plots is the lack of community consensus on calculating and comparing Λ with appropriate error. The objective of this paper is to address these issues. Analytical uncertainty is routinely discussed in the literature as part of method development; however, the uncertainty associated with mathematical manipulations can have a significant impact when the assumptions of the mathematical model are

The few reports that calculate Λ using ε ratios usually access published data sets to retroactively calculate Λ.4,20 In contrast, the majority of practitioners utilizing dual-isotope plots calculate Λ by regression of measured data and have generally adopted ordinary linear regression (OLR) to calculate Λ and uncertainty. OLR minimizes the sum of squares in the ydirection and thus assumes all measurement error is associated with the y-variable. The uncertainty of Λ is most frequently reported using the 95% confidence interval of the slope as first proposed by Elsner et al.,7 although consistent error reporting is far from standard practice. Many reports provide 95% confidence intervals and neglect to report the standard error (SE) of the slope,5,7,20 where the SE refers to the estimate of the standard deviation of the slope. Others report only the SE11,16 or report no error at all.21 The debate over appropriate best-fit regression methods and error reporting is widespread in the scientific literature in fields including marine biology,22 clinical chemistry,23 anthropology,24 within the geological sciences,25 and within contaminant hydrogeology.19 However, recommendations are often specific to each field. Scott et al.19 evaluated the accuracy of different data manipulations when using the Rayleigh distillation equation and found that OLR was appropriate as long as the error in the isotope value (y-direction) was sufficiently large compared with the error in the concentration measurement (xdirection). However, in dual-isotope plots, the data sets are not as standardized in range or measurement uncertainty as the Rayleigh plot. Alternatives to OLR such as reduced major-axis (RMA)24 and the York method26−28 incorporate measurement error in both the x- and y-variables and are popular regression methods for measured data sets,23,25,29 but to date, they have not been applied to isotope data sets in contaminant hydrogeology. RMA calculates the slope using the variance of the x- and yvariables to weight the error in each direction. The major assumption for RMA is that the ratio of error variance (analytical uncertainty) to variance (range) in x and y are symmetric, making its use restrictive. The unified equations for the York method28 describe the general solution for the straight-line, least-squares fit and also calculate the associated slope error. This method was first proposed as a solution for rubidium−strontium isochrons where both axes are affected by significant measurement error.26 York’s least-squares cubic equation (LSCE)26 solution can be used when errors in x and y are not correlated (eq 3): β3 ∑

ω(xi)ω(yi ) β 2ω(yi ) + ω(xi)

Wi2Vi2 ) ω(xi) (3) 9148

DOI: 10.1021/acs.analchem.9b01756 Anal. Chem. 2019, 91, 9147−9153

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Analytical Chemistry Table 1. Comparison of Λ Calculated using εH/εC, OLR, RMA, and York Methodsa method to calculate Λ

culture ID (this study)

data ref

n

εH/εC

OLR ± SE

RMA ± SE

York ± SE

nitrate-reducing A nitrate-reducing C methanogen C

5 3 3

42 9 38

15.9 ± 4.0 16.3 ± 7.1 34.5 ± 6.3

8.05 ± 0.83 14.6 ± 1.6 32.3 ± 1.9

9.25 ± 0.81 15.2 ± 1.4 34.3 ± 1.9

9.1 ± 1.0 15.5 ± 2.4 36.0 ± 4.5

The slope standard error (SE) is included for each regression, and the error for εH/εC is the mathematically propagated error from uncertainty in the numerator, εH, and the denominator, εC. a

The integrity of the modified data set was checked to ensure Λs produced are not statistically different compared to the full data set. Reporting Lambda and Uncertainty. In all cases, the value and uncertainty of lambda is reported using the NIST Good Laboratory Practices (GLP)42 and the Guide to the expression of Uncertainty in Measurement (GUM)43 authored by the Joint Committee for Guides in Metrology. Briefly, all digits are preserved during mathematical manipulations, and uncertainties are reported to two significant figures. The calculated value is reported to the same digit as the uncertainty.

inconsistent with the data. Here, several methods to calculate Λ are compared using simulated as well as previously reported data sets. The uncertainty associated with Λ and potential regression bias is discussed and compared using several different techniques including OLR, RMA, and the York method. Furthermore, the methods used to compare individual and combined Λ data sets are evaluated using modified data sets derived from previously published benzene biodegradation studies. The current approach prevalent in the literature, identifying overlapping 95% confidence intervals, is compared to using statistical tests to confidently distinguish between individual and compiled Λs with a more robust incorporation of uncertainty.

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RESULTS AND DISCUSSION Uncertainty Associated with Λ. Sample introduction techniques and instrument parameters (linearity range, split ratios) lead to uncertainty in both the accuracy and precision of isotope values.1,32−35 This analytical uncertainty can be quantified through what we refer to here as measurement uncertainty.32 Uncertainty associated with mathematical calculations or manipulations is often overlooked, but it can have a significant impact on the data when the assumptions of the mathematical model are not met. We refer to this as mathematical uncertainty. The uncertainty of Λ calculated from ε ratios is defined by propagating uncertainty from both ε calculations through the ratio calculation. Propagating error incorporates uncertainty from both isotope measurements, and so involves no mathematical uncertainty since there is no calculation bias associated with ratios. The alternative method to calculate Λ is through regression of the dual-isotope plot, using the actual dual-isotope measurements (δ13C vs δ2H or δ13C vs δ37Cl). In regression analysis, precision in the slope is quantified as the slope error, but any mathematical bias introduced by the regression method is not quantified. Many practitioners using dual-isotope plots further describe Λ error using 95% confidence intervals of the slope, where there is a 95% probability of the slope value being between an upper and lower limit. The 95% confidence interval gives a wider error margin compared to the slope error, but it too neglects to address any potential bias from the regression. As discussed later, regression bias can be significant when error is not appropriately considered and the assumptions of the regression are not met. While the most robust method should obviously integrate both measurement uncertainty and mathematical uncerainty, it is surprising how often this has not been the case. The comparison between different methods to calculate Λ for selected cases of the reworked data sets is presented in Table 1. The comparisons for all data sets in the original publications3,5 along with their original identifiers can be found in Table S1 of the SI. Calculating Λ via εH/εC gives the most imprecise value with uncertainties representing more than ±10% of the calculated value. The worst example is for nitrate-

METHODS Data Collection. Experimental data was compiled from laboratory studies involving benzene biodegradation by enriched cultures from a contaminated field site, containing carbon and hydrogen isotope measurements.3,5 Mancini et al.3 reports εC and εH associated with benzene biodegradation under nitrate-reducing, sulfate-reducing, and methanogenic conditions. In a follow-up paper, Mancini et al.5 further described how distinct slopes on a carbon−hydrogen plot could provide evidence of different initial reaction mechanisms during benzene biodegradation. In both articles, carbon and hydrogen isotope measurements are provided for each culture in the Supporting Information, allowing the reported data to be reconstructed and reworked for the purposes of this study. The uncertainty for carbon isotope measurements reported was ±0.5‰, and reported measurement uncertainty for hydrogen was ±5‰, both consistent with the EPA guide.1 Here the reported data sets are renamed for simplicity: nitratereducing A−D, methanogenic A−C, and Sulfate-Reducing A as denoted in Table 1 and Supporting Information (SI) Table S1. A carbon-chlorine data set from Wiegert et al.11 was also analyzed for this study. The δ13C and δ37Cl data was provided in the Supporting Information of the publication. Regression Methods. A best-fit line was fitted to each data set using three methods of regression: OLR, RMA, and York. All calculations were done in R using the native functions in R for OLR, the “rgr” package40 for RMA, and the IsoPlotR package for York regression.41 Pseudo code for the regressions is given in the Supporting Information. Data Set Modifications for Slope Differentiation. Differentiating between two slopes in a dual-isotope plot was tested by comparing overlapping 95% confidence intervals and using a statistical z-score (eq S1 of the SI). Data sets from Mancini et al.3,5 for the nitrate-reducing B and methanogen A cultures were modified to produce: (1) a full data set with all points preserved, and (2) a modified data set with every other point omitted. In the latter case, points were systematically omitted to investigate the influence of the number of data points (n) and the SE on the slope on the interpretations of Λ. 9149

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Analytical Chemistry reducing C, where uncertainty is ±7‰, almost 50% uncertainty in the calculated value of Λ (16‰). Although this method can be found in the literature as a way to estimate Λ2, it is fortunately infrequently used. Practitioners typically use this method to compare data to previously published results when the isotope data is not available but enrichment factors for multiple isotopes are reported.4,20 Tables 1 and S1 underscore the point that calculating lambda using εH/εC should be used only when data is not available to regress. Both accuracy and precision are significantly compromised. The overwhelming majority of papers to date calculate Λ by regression, so discussion of the most appropriate method of regression is the focus of this paper. For the reworked data sets discussed here, the slope SE was compared for regression analysis, which is shown along with Λ in Table 1. York regression generally gives a larger SE, whereas the smallest error is given by OLR. It is reasonable for regression methods like RMA and York to give a larger SE since the measurement error is accounted for in both variables. Since error in the xdirection is ignored in OLR, the relatively smaller error underestimates the total uncertainty of Λ and gives a false sense of precision. Potential Sources of Inaccuracy in Regression Methods. Potential Regression Bias. The only regression method for dual-isotope plots in the literature to date, to our knowledge, is OLR. OLR assumes that the error in x is negligible compared to that in y. Therefore, the error in the slope estimate is expressed only as a function of the error in the y-direction. Significant error in the x-direction introduces bias and results in a slope closer to zero (slope attenuation), compared to a slope with no error in x. Wehr and Saleska25 discuss the degree of bias introduced using different regression methods for isotopic mixing plots, where measurement error is inherent to both variables. In their study, a data set was simulated that varied measurement error in both x- and yvariables (δ13C, ppm of CO2) as well as the x-range. The simulated data sets generated from an idealized data set with a known slope were used to evaluate the accuracy of several bestfit regression techniques including OLR, RMA, and York. The York method agreed most closely with the idealized slope under all data conditions, leading Wehr and Saleska25 to identify the York method as the most consistent and leastbiased slope estimate. OLR consistently underestimated the modeled slope and resulted in significant bias when the x-range was small and the error in x was relatively large. The potential for similar regression bias exists in dualisotope plots and is shown by comparing Λs calculated from OLR, RMA, and York methods (Tables 1 and S1). The difference between slopes calculated using OLR, RMA, and the York method is relatively small in most cases, although the slope value is consistently lower for OLR compared with RMA and York. In addition, as noted, the precision for OLR is likely artificially small. On the basis of findings from Wehr and Saleska,25 the York method produces the least-biased slope. Therefore, in this work, we express the slope bias observed in OLR and RMA relative to the York method. There is little difference between RMA and York, except in the case of methanogen C. For this culture OLR underestimates the slope compared to York by 9%. The degree of slope attenuation observed for OLR relative to York regression is shown in Figure 1. RMA also underestimated the slope by 4.7% relative to York. This difference may be due to the fact that RMA only takes the error variance of each variable into account while

Figure 1. Dual-isotope plot of data for methanogen C shows slope attenuation for regression using OLR compared to that in York. Data from Mancini et al.3

York incorporates both the error variance and the range of each variable. The York method can also be applied in cases of extreme fractionation of hydrogen which results in nonlinear behavior30 after correcting for the reactive position and taking the natural log of the isotope ratio of both elements in the dual-plot, which linearizes the relationship. Additionally, and particularly important to field data sets, the reproducibility of a measurement can be different than the measurement uncertainty, as described in Sherwood Lollar et al.32 In this case, the larger of the two uncertainty values should be used when calculating lambda using the York method, which is capable of accommodating variable errors in x and y (commonplace in contaminant hydrogeology) and error correlation between the variables (less frequent occurrence). It is important to note that Table 1 gives a conservative example of slope bias since carbon and hydrogen dual-isotope plots exhibit conditions closest to the assumptions of OLR. In the reworked data set, OLR shows potential bias in carbon− hydrogen dual-isotope plots despite the fact that the ratio of errors (10:1) is large. Carbon-chlorine plots are more susceptible to slope bias introduced by OLR. Since the analytical uncertainty of carbon and chlorine isotope measurements is comparable (error ratio close to 1:1), ignoring the error in the x-direction using OLR could significantly increase the degree of slope attenuation. Wehr and Saleska25 demonstrated that with an error ratio of 1:1, slope attenuation increased by a factor of 40 for small ranges of x using OLR. It is also important to note that for small ranges of x, the ratio of measurement error to the measured range of x becomes increasingly important. For example, it is difficult to quantify changes in carbon isotope measurements over a range of 1‰ when the measurement uncertainty is ±0.5‰. Accordingly, the EPA recommends that changes in isotope values for carbon must be >2‰ for positive identification of biodegradation in field applications to avoid erroneous interpretations.1 In dualisotope plots, although there may be evidence for enrichment over a small isotopic range, the ratio of the measurement uncertainty to the data range cannot be ignored by the regression technique, as it is in OLR and, to a lesser extent, RMA. For conditions with small carbon enrichment scenarios for carbon−chlorine plots, Wehr and Saleska25 demonstrated that the slope attenuation could reach 12%. Regression Asymmetry. Another concern inherent in OLR is that the regression is asymmetric. For OLR, the variable with 9150

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Analytical Chemistry larger error is typically placed on the y-axis, and for carbon− hydrogen dual plots, the convention is generally followed. However, for carbon−chlorine plots, the axes are not as welldefinedthe literature shows that carbon is placed on both the x- and y-axes11,15,31 depending on the studies. In such cases, carbon−chlorine OLR slopes are not easily compared to chlorine−carbon OLR slopes because of the lack of regression symmetry. Simply put, the inverse of the OLR slope of a carbon−chlorine plot is not always equal to the OLR of a chlorine−carbon plot, making inaccurate reporting an important issue. The asymmetry of OLR is most apparent when ranges of isotopic enrichment are small, as illustrated in Figure 2 using carbon and chlorine isotope enrichment for perchloroethylene (PCE).11

The pitfalls of using 95% confidence intervals to distinguish Λs can be illustrated by the modified data set from Mancini et al.3,5 (Figure 3). The study investigated isotopic differences

Figure 3. Dual-isotope plot of the full (A) and modified (B) data set for methanogen A and nitrate-reducing B. Solid lines show the regression slope (Λ) and 95% confidence intervals are shown as dashed lines. Data from Mancini et al.3,5 Figure 2. Measurements of PCE at a field site where isotopic enrichment down gradient was interpreted as evidence for biotransformation.11 The inverse of slopes (1/slope) calculated using OLR and the York method show the differences that results when the slopes are calculated with the axes reversed.

during biodegradation of benzene under different electronaccepting conditions (sulfate-reducing, methanogenic, and nitrate-reducing). The interpretation from the original publication5 was that the two cultures, methanogenic A and nitrate-reducing B, used different reaction mechanisms (Figure 3A), which was supported by later publications.45−47 The 95% confidence intervals and the statistical tests (p < 0.001) both agree that the full data sets are distinct (Figure 3A). Using the modified data set (Figure 3B) provides a different result, with Λ= 26.71 ± 0.87 (95% C.I. = ± 1.8) and Λ = 20.0 ± 1.9 (95% C.I. = ± 5.1), respectively. The values of Λ overlap within 95% confidence intervals, but the Λs are statistically different based on the z-test for regression slopes44 (p = 0.001). In this case, relying on 95% confidence intervals alone gives a false-positive result. This result is an artifact of a smaller number of points, producing correspondingly larger confidence intervals as shown in Figure 3B. This demonstrates that mechanistic interpretations using Λ are more vulnerable to inaccurate interpretations using data sets with few data points, in addition to high ratios of error to measurement range as discussed above. The example shown uses OLR as the regression technique, although the results were similar using both RMA and York method (data not shown). The slope SE is required to perform a statistical test, but the common reporting method for Λ includes only the 95% confidence interval, making the transition to using a statistical test difficult to retroactively employ. For future analyses, studies should therefore routinely report the slope SE in addition to any other representations of error in Λ.

In contrast, the York method gives a more consistent slope irrespective of which data are plotted on the y-axis. The slope error calculated for the York method is also much more conservative than that of OLR, chiefly because the regression incorporates the ratio of errors in x and y into the slope calculation. For such a small range of both x and y, the York error is more representative of the enrichment that is slightly greater than measurement uncertainty. Based on findings from Wehr and Saleska,25 the potential bias of OLR and RMA observed in this study, as well as the symmetric nature of the regression and ability to adapt to variable uncertainty values, the York method presents the most accurate estimate of Λ calculated from dual-isotope plots compared to OLR and RMA and hence the most robust method to ensure mechanistic interpretations from dualisotope data are well justified and reproducible. Comparing Λ Values. Disadvantages of the 95% Confidence Interval. It is well-documented that the means of two data sets can have overlapping 95% confidence intervals but be statistically different at the α = 0.05 level.38 The same is true for regression slopes.44 A hypothesis test is is a more robust method to determine if means of two data sets are significantly different and has also been used to distinguish regression slopes.44 9151

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Analytical Chemistry Suggestions for Comparing Sets of Λ Values. There is a need to compare sets of Λ values between laboratories and between lab and field data, as well as comparing individual Λ values. Isotope fractionation, and therefore Λ, is mechanism specific; however, different enzymes acting with the same mechanism can also produce slightly different values of ε leading to variations in Λ.48 The ultimate goal is to use laboratory experiments to constrain a value for Λ that can be applied to interpret field data. However, the appropriate method for compiling data sets for a particular Λ has not been addressed in the literature to date. Two options exist for comparing groups of Λs. The first is to compare the absolute range of Λs for a particular mechanism, and the second is to compare the mean Λ values with appropriate uncertainty. The benefit of comparing absolute ranges of Λ values is that the experimental range is exactly represented. However, this puts inclusive emphasis on particularly high or low values of Λ, which could theoretically be outliers of the data group. For example, Figure 4 shows

6.9, is statistically different than the mean of the nitratereducing pathways, Λ= 14.5 ± 4.7, (p = 0.006). Which interpretation is correct? Using 95% confidence intervals, falsepositive results can occur when the conclusions are not based on robust best practice. Similarly, using the absolute range puts undue emphasis on the low end of Λ values for the methanogenic pathway. The rest of the data set shows much higher and more consistent values of Λ. Using the mean value for each pathway ensures that each data point is included in the analysis, but unequal weight is not placed on extreme cases. Therefore, as described here, the mean Λ with error for a particular mechanism should be considered when comparing laboratory produced dual-isotope data sets and when comparing laboratory data to field observations. This way, the magnitude and uncertainty of Λ is preserved without biasing interpretations based on extreme ends of the range. As previously discussed, the transition we recommend from OLR to the York method for regression of future dual-isotope studies would likely result in more realistic error assessment for reported slopes and thus a better estimate of the uncertainty associated with a range of Λs for reaction mechanism interpretation (see Tables 1 and S1). Conclusions and Recommendations for Dual-Isotope Plots. The increasing application of dual-isotope plots for contaminant remediation highlights the need for a community discussion on the best practices surrounding its use. Our results have identified points of potential bias that can be avoided by following several guidelines: • Λ is best calculated by regressing measured data using the York method, which incorporates uncertainty in both isotope measurements, leading to an appropriate estimation of Λ and its uncertainty. • Practitioners that calculate Λ using ε ratios should acknowledge the method has potential to produce inaccurate Λ values compared with regression analysis. • Two Λs can be distinguished on a dual-isotope plot by performing statistical z-score tests. The traditional method of comparing 95% confidence intervals may lead to inaccurate conclusions. • Sets of Λ values can be compiled by calculating the mean and SE of the data set then statistically comparing these values using the student’s t test. This set of best practices will help validate the role CSIA and dual-isotope plots play in contaminant hydrogeology.

Figure 4. Groups of Λ values for two different mechanisms (1 and 2) shown before (A) and after (B) the addition of a new data point from Bergman et al.49

compiled data for two mechanisms presumed to be different, with published lambda values from the literature.3,5 Mechanism 1 represents the nitrate-reducing cultures from Mancini et al.3,5 with Λ ranging from 8.05 ± 0.83 to 20.1 ± 2.3, while Mechanism 2 represents the methanogenic cultures from the same data set with Λ ranging from 28.46 ± 0.75 to 39.4 ± 2.2. More recently, Bergmann et al.49 added a new data point to the methanogenic data set (Figure 4B: Mechanism 2; Λ = 20 ± 2 (95% C.I.)). The added data caused the mean of Λ to shift and the 95% confidence intervals of the mean to increase for Mechanism 2, resulting in overlapping confidence intervals between the two mechanisms. The absolute range of Λs for each mechanism could be compared by using the student’s t test (eq S2 of the SI) to statistically compare the lowermost value for the methanogenic pathway (Λ= 20 ± 2 (95% C.I.)) and the uppermost value for the nitrate-reducing data set (19.4 ± 1.5 (95% C.I. = ± 3.4)). Indeed, these values are not statistically different, p > 0.05. However, using a student’s t test to compare the mean Λ for each mechanism supports the opposite conclusion. The mean of the revised data set for methanogenic pathways, Λ= 29.6 ±

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.analchem.9b01756. A table containing comparisons of all data and identifiers from original publications, statistical equations, an overview of mathematical models used, and pseudo code for calculating Λ in R (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Ann Sullivan Ojeda: 0000-0001-7875-1267 Elizabeth Phillips: 0000-0002-3412-5480 9152

DOI: 10.1021/acs.analchem.9b01756 Anal. Chem. 2019, 91, 9147−9153

Article

Analytical Chemistry

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Barbara Sherwood Lollar: 0000-0001-9758-7095 Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Funding for this research was provided by NSERC Discovery (Grant RGPIN-2018-06149) with additional funds from the NSERC Partnership Program and Canada Research Chairs. We are grateful to T. Gilevska for early comments on this manuscript. This work is dedicated to the memory of an outstanding scientist and fine colleague Dr. Derek York.

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DOI: 10.1021/acs.analchem.9b01756 Anal. Chem. 2019, 91, 9147−9153