Effect of multiple error sources on the calibration uncertainty

Food Chemistry 177 (2015) 147–151

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Effect of multiple error sources on the calibration uncertainty Denis Badocco, Irma Lavagnini, Andrea Mondin, Paolo Pastore ⇑ Department of Chemical Sciences, University of Padua, Via Marzolo 1, 35131 Padua, Italy

a r t i c l e

i n f o

Article history: Received 2 December 2013 Received in revised form 27 August 2014 Accepted 3 January 2015 Available online 9 January 2015 Keywords: Uncertainty Calibration Variance components Inductively coupled plasma Mass spectrometry

a b s t r a c t The calibration uncertainty associated with the determination of metals at trace levels in a drinking water sample by ICP-MS was estimated when signals were affected by two error contributions, namely instrumental errors and operational condition errors. The calibration uncertainty was studied by using J concentration levels measured I times, as usual in experimental calibration procedures. The instrumental error was random in character whilst the operational error was assumed systematic at each concentration level but random among the J levels. The presence or the absence of the two error contributions was determined with an F-test between the ordinary least squares residual variance of the mean responses at each concentration and a pooled variance of the replicates. The theory was applied to the calibration of 30 elements present in a multi-standard solution and then to the analysis of boron, calcium, lithium, barium and manganese in a real drinking water sample. The need of using the proposed approach as calibration for almost all the analyzed elements resulted evident. The presence or the absence of the two error contributions was determined with an F-test between the ordinary least squares residual variance of the mean responses at each concentration and a pooled variance of the replicates. It was found that in the former instance the uncertainty determined using a two-components variance regression was greater than that obtainable from the one-variance regression. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction The estimation of the uncertainty associated with the calibration is one of the most cumbersome issues encountered both in method validation and in the determination of an unknown concentration. In fact, operational variations may occur caused either when several matrices are analyzed for the same analyte or in prolonged analytical procedures or in the presence of significant change in the temperature (Ellison, Barwick, & Farrant, 2009; Juelicher, Gowik, & Uhlig, 1998; Juelicher, Gowik, & Uhlig, 1999; Lavagnini, Fedrizzi, Versini, & Magno, 2009). Typical examples of these cases are the analysis of a doping substance searched in various animal tissues and the need of a re-calibration procedure in routine analysis prolonged for long time. These facts imply that at least two contributions of error variances may be present namely, the instrumental and the operational ones. In these cases the regression is faced with a two-component variance model (Juelicher et al., 1998; Muller & Uhlig, 2001; Searle, Casella, & McCulloch, 1992). Anyway, the uncertainty of the calibration curve is usually calculated by means of a one-variance model, the instrumental one, accounting for the total number of measurements used in ⇑ Corresponding author. Tel.: +39 0498275182; fax: +39 0498275175. E-mail address: [email protected] (P. Pastore). http://dx.doi.org/10.1016/j.foodchem.2015.01.020 0308-8146/Ó 2015 Elsevier Ltd. All rights reserved.

the construction of the curve, the number of replicates from each solution, and the concentration levels. In this experimental design, the number of measurements came from instrumental runs and the ith replicate from the jth solution is a function of the concentration level, xj, given by yij = b0 + b1xj + eij, where eij is the instrumental error. This approach was recently used for metals ICP-MS determination in food samples at trace level (Yenisoy-Karakasß, 2012). When working at very low concentration levels, however, a lot of non-instrumental error sources become relevant and may dramatically influence the analytical result requiring the two-components variance approach for the regression procedure. In fact, sample preparation, anthropic and environmental pollution, standard reagent purity, glassware quality, become more and more important and may make the non-instrumental error variance very significant. Consequently, if J solutions are prepared and then measured I times, the calibration line construction must be faced with a different approach as an additive systematic contribution affects the I replicates at the jth concentration level. This systematic error is random in character among the J concentration levels. This paper is aimed at studying the estimation of the uncertainty of the calibration curve when the relationship between response and concentration is given by yij = b0 + b1xj + gj + eij, where gj represents the non-instrumental error. The approach represents an application of the random model ANOVA procedure to

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the calibration responses in the presence of the instrumental error and additional error sources (Searle et al., 1992). The model will be applied to synthetic data, generated using the Monte Carlo method, in order to give the details of uncertainty calculation of the concentration associated with the calibration curve uncertainty. Further, the developed model will be used to determine boron, calcium, lithium, barium and manganese in real drinking water samples. Metals like boron, manganese and barium were chosen as they are present at trace levels and subjected to legal limits, whereas calcium and lithium are usually present at higher concentrations with no legal limits. Testing on the ICP-MS technique is particularly suitable owing to the intrinsic feature of the output signals which are integrals of many repeated measurements in time. The signal is usually considered as a single value but it is actually a mean value with a proper variance. 2. Materials and methods 2.1. Drinking water samples The drinking water samples were obtained from ‘‘Fonte Margherita’’ (Torrebelvicino, Vicenza, Italy). Samples were added of 3% (v/v) nitric acid to uniform the solutions to the calibration ones. 2.2. Reagents and instrumentation All reagents were of analytical grade and were used as purchased: 69% HNO3 (CAS: 7697-37-2) (ARISTARÒ), multi-element standard solution CPAchem (10 mg L1) ICP-MS calibration standard Ref. N: MS19EB.10.2N.L1. All solutions were prepared in milliQ ultrapure water obtained with a Millipore Plus System (Milan, Italy, resistivity 18.2 Mohm cm1). The ICP-MS was tuned daily using a 1 lg L1 tuning solution containing 140Ce, 7Li, 205Tl and 89 Y (Agilent Technologies, UK). A 100 lg L1 solution of 45Sc and 115 In (AristarÒ, BDH, UK) prepared in 3% (v/v) nitric acid was used as an internal standard through addition to the sample solution via a T-junction. All the elements were measured by using inductively coupled plasma coupled to a mass spectrometer (ICP-MS) Agilent Technologies 7700 ICP-MS system (Agilent Technologies International, Japan, Ltd., Tokyo, Japan). The MS detector is equipped with an octupole collision cell operating in kinetic energy discrimination mode for the removal of polyatomic interferences and argon-based interferences. The instrument was optimized daily to achieve optimum sensitivity and stability according to manufacturer recommendations. Typical operating conditions and data acquisition parameters are summarized in Table 1.

Table 1 Instrumental operative conditions for ICP-MS. Instrument

Agilent 7700 ICP-MS

RF power RF matching Plasma gas flow rate Auxiliary gas flow rate Carrier gas flow rate Make-up gas flow rate He gas flow CeO+/Ce+ Ratio(2+) 70/140 Nebuliser Spray chamber Torch Sample uptake rate Sample cone Nickel Skimmer cone nickel Sampling depth Detector mode Dwell time/mass Replicate

1550 W 1.8 V 15 L min1 Ar 1.0 L min1 Ar 1.05 L min1 Ar 0.0 L min1 Ar 4.3 mL min1 0.902% 0.944% Microflow PFA nebuliser Scott double-pass type at 2 °C Quartz glass torch 0.1 mL min1 1.0 mm aperture i.d. 0.5 mm aperture i.d. 8.5 mm Dual (pulse and analog counting) 1000 ms 9

yij ¼ b0 þ b1 xj þ gj þ eij

ð1Þ

where i ¼ 1; 2; . . . ; I denotes the measurements at the level xj . The instrumental error, eij , is assumed to have a normal distribution with zero mean and constant variance r2e : eij  Nð0; r2e Þ. The term gj takes into account random effects due to operating conditions in preparing the jth standard and assumed constant among the I replications at xj . It is assumed normally distributed with zero mean and variance r2g : gj  Nð0; r2g Þ. Moreover, the random variables eij and gj are assumed uncorrelated. Therefore the random variable yij is also normally distributed with two components of variance,   that is r2y ¼ r2g þ r2e , and yij  N b0 þ b1 xj ; r2g þ r2e . Eq. (1) corresponds to the random effects ANOVA model in regression (Searle et al., 1992). 2.4.1. Statistical analysis of the calibration experiment The b0 and b1 parameters in Eq. (1) may be estimated by the ordinary least squares method (OLS) referring to the mean j ¼ b0 þ b1 xj þ gj þ ej , where response at each concentration level, y  2 PI r ej ¼ i eij =I  N 0; Ie (Searle et al., 1992). Since the sum   2 cj ¼ gj þ ej is also normally distributed, cj  N 0; r2g þ rIe , the P j ¼ ð1=IÞ Ii¼1 yij at the concentration level xj is also average signal y j  Nðb0 þ b1 xj ; r2y Þ where r2y ¼ r2c is given by Gaussian, y j

re

j

j

2

2.3. Procedures

r2yj ¼ r2g þ

Multielement standard solutions were prepared in 3% v/v HNO3. The calibration solutions were prepared by gravimetric serial dilution from multi-element standard solutions, at six different concentrations (1 ng L1–100 mg L1). Calibration plots were obtained with an internal standard. Blank samples of ultrapure water and reagents were also prepared using the same procedures adopted for the samples. All blank levels obtained were appropriately subtracted.

The OLS estimates of the parameters b0 and b1 are obtained PJ 2 ^  ^  minimizing the sum j cj , and are given by b0 ¼ y  b1 x; PJ j xÞy ðxj  P P ^1 ¼ j¼1  ¼ ð1=JÞ Jj¼1 y j ;  b , where y x ¼ ð1=JÞ Jj¼1 xj , and Sxx ¼ Sxx PJ 2 2  j is given by the j¼1 ðxj  xÞ . The OLS estimate of the variance ry

2.4. Statistical model When replicated measurements on the same standard at the concentration level xj are made, a general relationship including the y measurement, the x concentration, and the analytical conditions at the spiked level j may be expressed in the form

I

ð2Þ

residual variance

PJ

r^ 2yj ¼

 y ^ j Þ2

j¼1 ðyj

J2

ð3Þ

2.4.2. Evidence of the experimental presence of the variance r2g The significant presence of additional errors may be checked with an F-test (Ellison et al., 2009). From Eqs. (2) and (3) the statistic

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r^ 2y F ¼ r^ 2 j e

ð4Þ

I

r2e

is considered under the hypotheses H0: r ¼ I (or rg ¼ 0Þ and H1: 2 r2yj > rIe (or r2g > 0Þ. In Eq. (4) the estimate of the instrumental error variance r2e is given by the pooled variance 2 j y

PJ PI

i¼1 ðyij

j¼1

r^ 2e ¼

2

j Þ2 y

ð5Þ

JðI  1Þ

(Draper & Smith, 1998; Searle et al., 1992). If the experimental F value is greater than the critical value

a F 1 J2;JðI1Þ ,

H1 is true and

therefore the random additional effects are statistically present at the level of significance a. An estimate of the variance r2g of 2

^ 2g ¼ r ^ 2y  r^Ie , in view of Eq. (2). the additional errors is r j

2.5. Statistical analysis Two simulated data sets were generated by using Monte Carlo method for representing a calibration with seven error free equally spaced concentration values, x, in the range 0–2.0, replicated nine times. The simulation parameters were: b0 ¼ 0; b1 ¼ 1; r2e ¼ 0:04 and r2g ¼ 0 for data set 1 and r2g ¼ 0:01 for data set 2. 3. Results and discussion 3.1. Performance of one- and two-components variance model with synthetic data Fig. 1 shows the synthetic data sets generated by the MonteCarlo method in the presence of the sole instrumental error (Fig. 1a) and instrumental and operating errors (Fig. 1b). Regression lines are also reported. The instrumental variance of data set _

2.4.3. Uncertainty of the calibration curve A crucial application of the calibration curve in chemical analysis is the inverse regression, that is, the determination of the con^1 from the mean y m  y Þ=b m of m (m P 1) centration ^ x¼ x þ ðy instrumental responses, together with its uncertainty. Following the Eurachem/Citac guide (2000), the variance of ^ x comes from ^1 . Differentiating the equation m ; y  and b the variances of y ^1 with respect to the variables y ^1 , and ^ m  y Þ=b m ; y  and b x¼ x þ ðy squaring the obtained equation, the variance of ^ x may be expressed by

1

"

r^ ^2x ¼ ^2 r^ 2e

!#  2 1 1 1 ð^x  xÞ ^ 2y 1 þ þ þr  j m I J Sxx



b1

for the two-components variance since

ð6Þ 2

r2ym ¼ r2g þ rme from Eq. (2),

1 was r2e ¼ 0:0369. The OLS regression referring to the mean ^ 2y ¼ 0:0037. Taking responses at each concentration level gave r j into account Eq. (2) the estimate of the operational variance was

r^ 2g ¼ 0:0004. Since variances are non-negative, r^ 2g can be estimated equal to zero indicating the presence of the sole instrumental error (Juelicher et al., 1998). The F-test was therefore in this case superfluous and the two regression approaches are equivalent. The instrumental error variance of the data set 2 was _

re2 ¼ 0:0373. The OLS estimate of the variance of the mean ^ 2y ¼ 0:0282 and the variance component due to responses was r j ^ 2g ¼ 0:0240. From Eq. (4) the additional errors resulted to be r experimental F value was 6.80. It was much larger than the critical one, F 0:95 5;56 ¼ 2:38 demonstrating the need of the two-components variance model although the choice of

r2g lower than r2e . The vari-

2007). The uncertainty of the discriminated concentration from the calibration curve, assuming m = I, is given by

ances of the discriminated concentration based on Eqs. (7) and (8) ^ 2^x ¼ 0:0387 and r ^ 2^x ¼ 0:00842, respectively, takx¼ x ¼ 1 were r at ^ ing into account the OLS estimate of the variance of the IJ measure^ 2y=x ¼ 0:0550. The former is about four times the latter ments, r

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 ð^x  xÞ r^ ^x ¼ ^ 1 þ þ J Sxx b1

demonstrating that the two methods are not equivalent and that the two-components variance is always preferable because more conservative.

r2y ¼

r2y

J

J

r2y

r2b^1 ¼ Sxxj (Draper & Smith, 1998; Lavagnini & Magno,

and

r^ yj

ð7Þ

The corresponding one obtained by applying the OLS regression to all IJ calibration data, that is the one-variance regression model, is

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r^ y=x 1 1 ð^x  xÞ2

r^ ^x ¼ ^ b1

I

þ

IJ

þ

ð8Þ

ISxx

^ 2y=x is the OLS residual variance (Yenisoy-Karakasß, 2012). where r

3.2. Application of one- and two-components variance model to a real drinking water sample Table 2 reports the OLS estimates of the calibration curve param^0 ; b ^1 , the residual standard deviation r ^ yj , the overall experieters b ^ e , the ratio between the uncertainties mental standard deviation r pffi ^ y Ir calculated by Eqs. (7) and (8), q ¼ r^ y=xj , and the experimental F

a

b 2

Signal

Signal

2

1

1

0

0 0

1

x

2

0

1

2

x

Fig. 1. (s) Monte Carlo generated data set. Simulation parameters: b0 ¼ 0; b1 ¼ 1; r2e ¼ 0:04, J = 7, and I = 9. (a) r2g ¼ 0:0; OLS regression line: y = 0.0584 + 0.927x; R2 = 0.912; (b) r2g ¼ 0:01; OLS regression line: y = 0.1028 + 0.9112x, R2 = 0.874.

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D. Badocco et al. / Food Chemistry 177 (2015) 147–151

Table 2 ^1 , of the OLS residual standard deviation r ^0 ; b ^ yj , of the pooled instrumental standard deviation error r ^ e , experimental value of the F OLS estimates of calibration curve parameters b statistic. The critical value of the F statistic is F 0:95 3;40 ¼ 2:84 at the level of significance 0.05. a.m.u.

Element

^0 CPS b

^1 CPS b

7 9 11 23 24 27 39 43 51 52 55 56 59 60 63 66 69 75 78 85 88 95 107 111 125 137 205 207 209 238

Li Be B Na Mg Al K Ca V Cr Mn Fe Co Ni Cu Zn Ga As Se Rb Sr Mo Ag Cd Te Ba Tl Pb Bi U

154 9 1195 329,849 20,179 7217 173,650 590 444 7918 1134 77,450 220 5484 9127 14,924 64,844 4 101 125 3787 1626 3634 241 3 178,678 4117 5963 1,348,385 8681

127 177 73 4726 1359 636 2227 7 24,297 29,443 17,026 26,978 47,619 12,503 33,611 6090 14,161 4010 302 17,640 23,500 17,347 50,147 8384 611 8529 97,808 30,140 111,833 151,237

lg1 L

values for 30 elements present in the multistandard solution at trace level. These values were obtained with ICP-MS responses using five concentration levels and nine replicates for each element in multielement standard solutions. The signals were homoscedastic in the chosen concentration calibration intervals. The experimental F values reported in the last column were larger than the

r^ yj CPS

r^ e CPS

q

F

57 292 241 214,724 1366 436 8152 209 2765 3908 2472 38,653 5105 1727 2145 28,364 35,053 5860 399 3097 2516 2609 19,790 978 26 40,610 16,409 2930 20,190 27,240

97 187 404 55,536 741 357 2012 109 566 895 896 10,602 1194 434 1120 2273 5490 1328 150 950 1655 729 4049 291 95 3214 5979 2438 27,732 9035

1.66 2.99 1.67 3.61 3.16 2.68 3.63 3.20 3.67 3.65 3.46 3.59 3.64 3.62 3.20 3.77 3.72 3.65 3.44 3.55 2.96 3.59 3.67 3.56 0.83 3.77 3.46 2.66 1.94 3.51

3.1 21.9 3.2 134.5 30.6 13.4 147.7 33.1 214.8 171.6 68.5 119.6 164.5 142.5 33.0 1401.5 366.9 175.2 63.7 95.6 20.8 115.3 215.0 101.7 0.7 1436.9 67.8 13.0 4.8 81.8

curves of Li and Te were characterized by low F values whilst Ba and Zn clearly exhibited large F values. This was associated to the dispersion of the measurements with respect to the regression line. In particular, in Fig. 2b a very good repeatability of the signal response at each concentration level was shown but the resulting mean values were located quite far from the regression line suggesting the claimed additional error. In Table 3 the discriminated values of Li, B, Ca, Ba and Mn, present in a real drinking water sample, together with the relevant ^ ^x and r ^ ^x0 from Eqs. (7) and (8), respectively. standard deviations r 0 ^ yj ; r ^ e; r ^ y=x , useful for the calcuMoreover, the standard deviations r

critical value F 0:95 3;40 ¼ 2:84 for all elements, except for Te, indicating the significant presence of additional error contributions. The ratios q are larger than 1, except for Te, indicating that Eq. (8), used instead of Eq. (7), underestimates the calibration uncertainty of the discriminated concentration. The values of the determination coefficient, R2, are larger than 0.99 except for Li, B, Na, Mg, Al, K, Ga, Ba. This is due to the fact that experimental responses of Li and B were characterized by a low sensitivity of the instrumental response whereas the concentrations of the other elements are influenced by their presence as atmospheric pollutants. Fig. 2a and b was reported as a clear experimental evidence of the one or two variance nature of the experiment. In particular, the calibration

lations are also reported. The fourth column reports the F values to ^ 2g . Finally, the evidence the significant presence of the variance r ^ ^x =r ^ ^x0 which gives an idea of last column reports the ratio Rr ¼ r 0

the discrepancy between the two considered methods. A comparison of the reported data with those presented in Table 2 points out different values of the standard deviations as a result of a different calibration interval. In particular, it must be evidenced that

a

b

6000

Ba

600000

Te

CPS

CPS

4000

2000

0 3

6

µg L-1

Zn

200000

Li 0

400000

9

0

0

25

50

75

100

µg L-1

Fig. 2. Calibration plots of (a) Li (R2 = 0.958) and Te (R2 = 0.998), (b) Ba (R2 = 0.961) and Zn (R2 = 0.991). Each concentration level was replicated nine times.

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D. Badocco et al. / Food Chemistry 177 (2015) 147–151 Table 3 Statistical overview of the analyzed real drinking water sample obtained from ‘‘Fonte Margherita’’. Element

^ x0 ðlg L1 Þ

r^ yj ðCPSÞ

r^ e ðCPSÞ

r^ y=x ðCPSÞ

F

r^ ^x0 ðlg L1 Þ

r^ ^x0 ðlg L1 Þ

Rr

Li B Ca Ba Mn

3.4 76 84,567 56 0.29

57 24 9769 4452 359

97 130 17,329 1165 309

103 127 18,419 3702 412

3.1 0.3 2.9 131 12

0.16 0.4 83 0.54 0.02

0.3 0.71 52 0.15 0.008

0.6 0.6 1.6 3.6 2.6

Calcium, calibrated at large concentration values, is characterized ^ e ¼ 17329 compared to r ^ e ¼ 109 obtained at trace level. As by r consequence, the statistics F value decreased from 33 to 2.9 clearly ^ 2g was negligible at high concentration condition. It indicating that r must be observed that when F is sufficiently large the standard deviation value of the discriminated concentration may be about four times larger (see Ba, for instance) requiring the use of Eq. (7). 4. Conclusions The calibration setup design is crucial to obtain statistically correct results. In particular, in the metal analysis at trace level by the ICP-MS technique, the calibration uncertainty must be estimated accounting for two error contributions, namely instrumental errors and operational condition errors. The former error is typically random, the latter one is systematic at each concentration level but random among the concentration levels. These two components must be therefore kept separated so that a two-component variance regression must be used. The estimation of the uncertainty of a discriminated concentration furnished larger values than those obtained with the classical one-variance regression procedure as demonstrated by Monte-Carlo simulations. The proposed theory was applied to the trace analysis of 30 elements present in a certified multi-element standard and then to determine the calibration uncertainty of some elements present in a real drinking water sample. As hypothesized, almost all the analyzed elements had to be treated with the two-variance-components approach to obtain a more robust estimate of the calibration uncertainty.

Acknowledgment The financial support of the Italian Ministry for Universities and Research (MIUR: PRIN 2010AXENJ8) is gratefully acknowledged. References Draper, N. R., & Smith, H. (1998). Applied regression analysis (3rd ed.). New York: Wiley (Chapters 1–2). Ellison, S. L. R., Barwick, V. J., & Farrant, T. J. D. (2009). Practical statistics for the analytical scientist. Cambridge: RSC Publishing (Chapter 6). Eurachem/Citac Guide CG 4 (2000). In S. L. R. Ellison, M. Rosslein, A. Williams (Eds.). Quantifying uncertainty in analytical measurement (QUAM) (2nd ed.) EURACHEM/ CITAC. Juelicher, B., Gowik, P., & Uhlig, S. (1998). Assessment of detection methods in trace analysis by means of a statistically based-in-house validation concept. Analyst, 123, 173–179. Juelicher, B., Gowik, P., & Uhlig, S. (1999). A top-down in-house validation based approach for the investigation of the measurement uncertainty using fractional factorial experiments. Analyst, 124, 537–545. Lavagnini, I., Fedrizzi, B., Versini, G., & Magno, F. (2009). Effectiveness of isotopically labelled and non-isotopically labelled internal standards in the gas chromatography/mass spectrometry analysis of sulphur compounds in wine: Use of a statistically based matrix comprehensive approach. Rapid Communications in Mass Spectrometry Reviews, 23, 1167–1172. Lavagnini, I., & Magno, F. (2007). A statistical overview on univariate calibration, inverse regression, and detection limits: Application to gas chromatography/ mass spectrometry technique. Mass Spectrometry Reviews, 26, 1–18. Muller, C. H., & Uhlig, S. (2001). Estimation of variance components with high breakdown point and high efficiency. Biometrika, 88, 353–366. Searle, S. R., Casella, G., & McCulloch, C. E. (1992). Variance components. New York: Wiley (Chapters 1–3). Yenisoy-Karakasß, S. (2012). Estimation of uncertainties of the method to determine the concentrations of Cd, Cu, Fe, Pb, Sn and Zn in tomato paste samples analysed by high resolution ICP-MS. Food Chemistry, 132, 1555–1561.