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Estimation of performance characteristics of a confirmation method for thyreostats in plasma by means of a weighted least-squares approach
Analytica Chimica Acta 592 (2007) 181–186
Estimation of performance characteristics of a confirmation method for thyreostats in plasma by means of a weighted least-squares approach P. Steliopoulos ∗ , E. Stickel Official Laboratory of Chemical and Veterinary Analysis Karlsruhe, Weißenburger Straße 3, 76187 Karlsruhe, Germany Received 23 December 2006; received in revised form 16 March 2007; accepted 16 April 2007 Available online 21 April 2007
Abstract A weighted linear regression-based approach was applied to ascertain analytical performance characteristics of a method for the quantitative determination of thyreostats in plasma by LC–MS. The weights were set equal to the reciprocal of the variances. A straightforward and practical procedure is presented that was used to model the variance of the replicate measurements as a function of concentration. Different function types were tested for their suitability. © 2007 Elsevier B.V. All rights reserved. Keywords: Weighted least-squares regression; Performance characteristics; Thyreostats
1. Introduction In laboratory practice, performance characteristics of analytical methods are usually calculated on the basis of empirical calibration data which were acquired in validation experiments. The most common evaluation approach uses a linear regression model that assumes a concentration-independent variability of the measurement error. By ordinary least-squares (OLS) fitting one obtains the coefficients of the calibration function, an estimator of the variance of the measurement error (residual variance) and an estimator for the variance of the estimation error (i.e., imprecision of the calibration function). From these parameters, the confidence interval, the prediction interval and further performance characteristics like the critical level or the limit of detection can be computed [1,2]. However, experimental errors often are heteroscedastic, i.e., the variance of the measurement error is related to the concentration. In such cases, weighted least-squares (WLS) regression is a more appropriate technique to determine the calibration line and to get realistic prediction bounds [3–5]. To perform WLS regression, it is first necessary to assign an individual weight to each concentration point. This can be achieved through various approaches. Many commercial chromatogra-
phy integration programs, for instance, offer the possibility of weighting by the inverse concentration, the inverse squared concentration, the inverse signal or the inverse squared signal. Ideally, weights should reflect the error scattering of the response and, to be more precise, should be reciprocally proportional to variance. Up to now, the acceptance of WLS regression is quite low in routine laboratories. The main purpose of this paper, therefore, is to outline the potential of this technique by describing an example case and to provide a user-friendly procedure that enables to estimate reasonable weights. In this study, the reliability of an LC–MS method for the determination of thyreostat residues in bovine plasma was assessed (thyreostats are pharmacologically active substances banned for use in food producing animals in the European Union [6–9]). Calibration data obtained from the analysis of spiked samples were processed by both OLS and WLS regression. Performance characteristics were calculated utilizing a weighted and an unweighted approach. The weights were set equal to the reciprocal of the variances and were modelled as a function of concentration. Different function types were employed and their suitability was examined.
2. Weighted least-squares model ∗
Corresponding author. Tel.: +49 721 9265451. E-mail address: [email protected] (P. Steliopoulos).
0003-2670/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.aca.2007.04.026
The concept of weighting was introduced in method calibration to account for concentration-dependent response precision.
182
P. Steliopoulos, E. Stickel / Analytica Chimica Acta 592 (2007) 181–186
The model is
The prediction interval for a given concentration x0 is defined
yij = b1 + b2 xi + εij
(1)
where yij is the response of the jth replicate at the ith concentration xi (i = 1, . . ., I denotes the concentration level, j = 1, . . ., J is the index of the replicate measurement). εij represents the measurement error. It is supposed that at each level i the error εij follows a normal distribution with mean 0 and a variance σ 2 w−1 i depending on xi . For the time being assume that the weights wi are known. Then, the estimators bˆ 1 and bˆ 2 can be calculated as bˆ 1 −1 βˆ = = (XT WX) XT WY (2) bˆ 2 with
⎛
y1
⎞
⎜y ⎟ ⎜ 2⎟ ⎟ Y =⎜ ⎜ .. ⎟ ⎝ . ⎠
(3)
yn ⎛
1
x1
⎞
⎜1 x ⎟ 2⎟ ⎜ ⎟ X=⎜ . . ⎜. . ⎟ ⎝. . ⎠ 1 xn and
⎛
w1
⎜ 0 ⎜ W =⎜ ⎜ .. ⎝ . 0
0 w2 .. . 0
(4)
···
0
⎞
0 ⎟ ⎟ ⎟ .. ⎟ . ⎠ · · · wn ··· .. .
(5)
n designates the total number of measurements (n = I × J). Note, WLS regression simply turns into OLS regression when W = diag{1, 1, . . ., 1}. The weighted residual variance is given as T
σˆ 2 =
ˆ W (Y − Xβ) ˆ (Y − Xβ) (n − 2)
(6)
By setting a certain value x0 in yˆ = bˆ 1 + bˆ 2 x one obtains an estimate yˆ 0 of the expectation value E{y0 } = b1 + b2 x0 . This estimate is afflicted with an error variance of which can be estimated as 1 −1 σˆ y2ˆ0 = σˆ 2 ( 1 x0 )(XT WX) (7) x0 The total variance of the imprecision of a prediction yˆ 0 at a certain concentration x0 consists of two components: the variance of the measurement error and the variance of the estimation 2 , can be error of yˆ 0 . An estimator of this total variance, σˆ total computed as 2 σˆ total =
σˆ 2 + σˆ y2ˆ0 wi
(8)
as prd (ˆy0 ) = yˆ 0 ± tf,1−α/2
2 σˆ total
(9)
and describes the range in which one can expect to observe with a probability of 1 − α the response of a sample that contains the analyte at a concentration x0 . tf,1−α/2 designates the 1−α/2quantile of the t-distribution with f degrees of freedom. 3. Performance characteristics 3.1. Decision limit CCα In the European Union, methods that are used in official residue control of pharmacologically active substances have to be validated in accordance with Commission Decision 2002/657/EC [10]. This Decision defines the decision limit CC␣ as: “. . .the limit at and above which it can be concluded with an error probability α that a sample is non-compliant.” In other words, CC␣ represents the measured concentration from which on a certain threshold value x0 is exceeded significantly. For substances with an established maximum residue limit (MRL), the threshold x0 equates to this MRL value. The α error probability then can be set 0.05. For banned substances x0 is zero and α = 0.01. The underlying hypotheses can be formulated as H0 :x ≤ x0 (or rather, if x0 = 0, H0 :x = x0 ) and H1 :x > x0 . x represents the unknown concentration of the analyte exhibiting the response y. The hypothesis H0 can be rejected at the significance level α in favour of the alternative H1 if
2 (10) y > yˆ 0 + tf,1−α σˆ total The critical concentration CC␣ is the corresponding x value. Hence, tf,1−α σˆ 2 CC␣ = x0 + (11) + σˆ y2ˆ0 w0 bˆ 2 This formula for the decision limit is only reasonable if yˆ 0 > 0. To take into account that yˆ 0 might be negative, Eq. (11) is changed into max {ˆy0 ; 0} − bˆ 1 tf,1−α σˆ 2 CC␣ = + + σˆ y2ˆ0 (12) w0 bˆ 2 bˆ 2 To calculate the decision limit for x0 = 0 the weight at zero is required. In this work, w(x0 = 0) was set equal to the weight assigned to the lowest spiked concentration. Note, when x0 = 0 the decision limit CC␣ corresponds to what in other fields of analytical chemistry is commonly called critical level LC [11]. Yet with the difference that the decision limit referred to in CD 2002/657/EC has to be established on the basis of experimental results that were obtained under within-laboratory reproducibility conditions.
P. Steliopoulos, E. Stickel / Analytica Chimica Acta 592 (2007) 181–186
3.2. Detection capability CCβ The official definition reads: “Detection capability means the smallest content of the substance that may be detected, identified and/or quantified in a sample with an error probability of β. In the case of substances for which no permitted limit has been established, the detection capability is the lowest concentration at which a method is able to detect truly contaminated samples with a statistical certainty of 1 − β. In the case of substances with an established permitted limit, this means that the detection capability is the concentration at which the method is able to detect permitted limit concentrations with a statistical certainty of 1 − β.” Practically, the detection capability CC is the true concentration that exceeds CC␣ with a statistical certainty of 1 − β. Thus, tf,1−β σˆ 2 (13) + σˆ y2ˆ(CC ) = CC␣ CC − w(CC ) bˆ 2 To determine CC one may plot the graph of the function f (x) =
bˆ 2 (x − CC␣ ) tf,1−β σˆ total
(x > CC␣ )
(14)
The concentration x at which the function f(x) equals to 1 is the detection capability CC . 4. Estimation of weights In the following, the procedure applied to determine the weights is described. First of all, for each concentration level, the empirical variance of replicate responses, vi , is calculated: ⎛ ⎞2 J J 1 ⎝yij − 1 yij ⎠ (15) vi = J −1 J j=1
vi = a1 ea2 xi vi =
a1 + a2 xi2
If var (vi ) = constant then
var (ψi ) ∝
dψi dvi
(22)
2 (23)
Therefore, (dψi /dvi )−2 should be adopted as a weight. In the concrete example the global weight ω is
dψi −2 ω= = v4i (24) dvi
WLS fitting provides the estimators aˆ 1 and aˆ 2 : aˆ 1 −1 = (Ξ T ΩΞ) Ξ T ΩΨ aˆ 2
with
(25)
⎞ x1−1 .. ⎟ ⎟ . ⎠
⎛
1 ⎜. Ξ=⎜ ⎝ ..
(26)
xI−1
1
Ω = diag {v41 , . . . , v4I }
(27)
and T
−1 Ψ = (v−1 1 , . . . , vI )
(28)
(18)
The weight associated with a certain concentration level is set equal to the reciprocal of the modelled variance and hence can be computed as
(19)
wi = aˆ 1 +
Consider as an example the function Eq. (16):
a2 −1 vi = a1 + xi
aˆ 2 xi
(29)
In the case of the other functions, the same procedure gives rise to the results summarized in Table 1. Fig. 1 illustrates with an example the effect of global weighting.
Transformation yields a linear equation:
Table 1 Models used to estimate weights
a2 1 = a1 + vi xi ψi = a1 + a2 ξi
analyzing the data pairs (ξ i , ψi ) by OLS regression one minimizes the sum of the squares of the residuals in ψi rather than those in vi . Thus, if the dependent variable vi is regarded as homoscedastic weighting by the reciprocal squared derivative of ψi with respect to vi should be incorporated in regression analysis of (ξ i , ψi ) (global weighting according to [12]). Explanation: Since ψi is a function of vi , it follows, applying the Gaussian law of uncertainty propagation, that
dψi 2 var (ψi ) = var (vi ) (21) dvi
j=1
Then, the variance is modelled by an appropriate curve. The following functions were used:
a2 −1 (16) vi = a1 + xi vi = a1 xia2 (17)
183
(20)
−1 where ψi = v−1 i and ξi = xi . To estimate the coefficients, ordinary least-squares fitting may be performed now. However, in
Function
Transformation
ψ
ξ
ω
ψ v = a1 xa2 v = a1 ea2 x v = a1 + a2 x2
= a1 + ax2 ln v = ln a1 + a2
1 v
1/x ln x x x2
v4 v2 v2 1
1 v
ln x ln v = ln a1 + a2 x –
ln v ln v v
184
P. Steliopoulos, E. Stickel / Analytica Chimica Acta 592 (2007) 181–186
was again shaken for further 30 s. The tube was placed in an ultrasonic bath for 10 min and centrifuged at 3500 rpm. The organic layer was removed and evaporated to dryness under a nitrogen stream. The residual material was dissolved in 500 L methanol/water (50:50) by sonicating for 10 min. Finally, the sample was centrifuged at 4000 rpm and filtered into a vial. 5.2. LC–MS
Fig. 1. Modelling the variance by fitting the transformed of a power function with (a) and without (b) global weighting.
5. Experimental 5.1. Sample extraction Samples were extracted as described in [7]. Briefly, 1 mL plasma was weighed into 15 mL polypropylene centrifuge tube. Ten microlitres 0.1 mol L−1 EDTA, 10 L 2-mercaptoethanol and 5 mL ethyl acetate were added. After shaking vigorously for 30 s, 4 g sodium sulfate were added and the sample
LC–MS analysis was performed with a Thermo Finnigan TSQ Quantum triple quadrupole mass spectrometer coupled to a Surveyor LC pump and autosampler. Chromatographic separation was achieved on a Phenomenex Prodigy C 18 5 m ODS (3) 100 A (150 mm × 3 mm) column. The mobile phase consisted of 0.1% acetic acid in water (A) and methanol (B), the flow rate was 0.3 mL min−1 . The gradient program was as follows: 0–10 min, 100–70% A; 10–25 min, 70–5% A; 25–27 min, 5–100% A. The mass spectrometer was operated in the ESI(+)MS2 SRM mode: m/z 115 [MH]+ → m/z 81 and m/z 56 (tapazole), m/z 129 [MH]+ → m/z 111 and m/z 84 (thiouracil), m/z 143 [MH]+ → m/z 126 and m/z 84 (methylthiouracil), m/z 171 [MH]+ → m/z 154 and m/z 112 (propylthiouracil), m/z 205 [MH]+ → m/z 188 and m/z 103 (phenylthiouracil). 5.3. Computations Calculations were implemented using Excel 2000 spreadsheets and Visual Basic for Applications (VBA).
Fig. 2. Modelling the variance as a function of concentration for tapazole. (a) v = (−1.33 × 10−12 + 3.60 × 10−10 /x) 1010 e0.016x , (d) v = −5.96 × 1010 + 5.15 × 107 x2 .
−1
, (b) v = 7.43 × 106 x2.37 , (c) v = 8.10 ×
P. Steliopoulos, E. Stickel / Analytica Chimica Acta 592 (2007) 181–186
185
Fig. 4. Graphical determination of the weighted detection capability CC for tapazole.
Fig. 3. Weighted and unweighted least-squares regression for tapazole (calibration lines and 98% prediction intervals; response y [area]: TIC of the daughter ions).
6. Results and discussion Seven calibration runs were carried out (J = 7), each at five concentration levels (I = 5): xi [g L−1 ] = 25, 50, 100, 150, 200. Sample series were analysed during a time period of three weeks,
for each particular run a different blank matrix was used for spiking. For all examined substances the variance of replicate measurements increased with the concentration. In the case of tapazole, the four variance models (Eqs. (16)–(19)) provided fits that were roughly comparable to each other (Fig. 2). The lowest sum of squared residuals was obtained by applying Eq. (16), hence this model was chosen to approximate the weights. For thiouracil, methylthiouracil and propylthiouracil the best fit was given by the power function (Eq. (17)), whereas for phenylthiouracil the parabola (Eq. (19)) turned out to be the most suitable model. Fig. 3 exemplarily shows the weighted and the unweighted 98% prediction interval of tapazole. As it can be seen in this figure, the weighted prediction interval describes the distribution of the measurement values much more adequately than the unweighted. A similar observation was also made for the other compounds. Table 2 lists the coefficients and the performance characteristics that were obtained using the two different least-squares approaches. Fig. 4 depicts the determination of the weighted detection capability for tapazole. In all cases, the regression coefficients are hardly affected by weighting. In contrast, the weighted performance characteristics are essentially
Table 2 Calculation of weighted and unweighted performance characteristics Ordinary least-squares
Tapazole Thiouracil Methylthiouracil Propylthiouracil Phenylthiouracil
Weight model
bˆ 1
bˆ 2
CC␣
CC
−150362 −127836 −582077 −1118009 −1167083
49064 20833 54653 197265 214915
45.5 78.6 64.7 60.8 73.1
73.9 127.0 100.8 97.7 118.4
Eq. (16) Eq. (17) Eq. (17) Eq. (17) Eq. (19)
Weighted least-squares bˆ 1
bˆ 2
CC␣
CC
−146743 −70565 −312925 −573154 −146528
49019 20097 51293 191087 203120
16.6 16.2 12.0 13.5 13.2
25.8 24.2 15.5 20.4 21.2
All concentrations are given in [g L−1 ]; x0 = 0 g L−1 , I = 5, J = 7, n = 35, α = 0.01, β = 0.05; OLS: f = n − 2 = 33, t1−α = 2.453, t1−β = 1.696; WLS: f = n − 2 − 2 = 31 (2 less than fOLS due to the 2 coefficients of the weight model), t1−α = 2.445, t1−β = 1.692.
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P. Steliopoulos, E. Stickel / Analytica Chimica Acta 592 (2007) 181–186
lower than the unweighted. According to expectation, at smaller concentrations the ordinary least-squares technique led to rather broad prediction intervals and thus to an overestimation of the performance characteristics. 7. Conclusions The dispersion of measurements often depends on the concentration. In such cases, to achieve an appropriate calibration and to construct realistic prediction bounds that allow a reliable assessment of the analytical performance, weighted least-squares regression should be applied. The procedure presented in this work appears to be a practicable way to specify the weights that are required for that purpose. References [1] K. Danzer, L.A. Currie, Guidelines for calibration in analytical chemistry, Pure and Appl. Chem. 70 (1998) 993–1014. [2] A. Hubaux, G. Vos, Anal. Chem. 42 (1970) 849–855.
[3] A. Sayago, A. Asuero, CRC Crit. Rev. Anal. Chem. 34 (2004) 133–146. [4] M.E. Zorn, R.D. Gibbons, W.C. Sonzogni, Anal. Chem. 69 (1997) 3069–3075. [5] L. Oppenheimer, T.P. Capizzi, R.M. Weppelman, H. Mehta, Anal. Chem. 55 (1983) 638–643. [6] P. Batjoens, H.F. De Brabander, K. De Wasch, J. Chromatogr. A 750 (1996) 127–132. [7] W.J. Blanchflower, P.J. Hughes, A. Cannavan, M.A. Mc Coy, D.G. Kennedy, Analyst 122 (1997) 967–972. [8] B. Le Bizec, F. Monteau, D. Maume, M.P. Montrade, C. Gade, F. Andre, Anal. Chim. Acta 340 (1997) 201–208. [9] A. Heeremans, A. Ermens, K. De Wasch, C. Van Peteghem, H.F. De Brabander, Analyst 123 (1998) 2629–2632. [10] Commission Decision of 12 August 2002 implementing Council Directive 96/23/EC concerning the performance of analytical methods and the interpretation of results (2002/657/EC), Off. J. Eur. Commun. L 221 (2002) 8–36. [11] L.A. Currie, Nomenclature in evaluation of analytical methods including detection and quantification capabilities, Pure Appl. Chem. 67 (1995) 1699–1723. [12] R. de Levie, How to Use Excel in Analytical Chemistry and in General Scientific Data Analysis, Cambridge University Press, Cambridge, 2001.
Estimation of performance characteristics of a confirmation method for thyreostats in plasma by means of a weighted least-squares approach P. Steliopoulos ∗ , E. Stickel Official Laboratory of Chemical and Veterinary Analysis Karlsruhe, Weißenburger Straße 3, 76187 Karlsruhe, Germany Received 23 December 2006; received in revised form 16 March 2007; accepted 16 April 2007 Available online 21 April 2007
Abstract A weighted linear regression-based approach was applied to ascertain analytical performance characteristics of a method for the quantitative determination of thyreostats in plasma by LC–MS. The weights were set equal to the reciprocal of the variances. A straightforward and practical procedure is presented that was used to model the variance of the replicate measurements as a function of concentration. Different function types were tested for their suitability. © 2007 Elsevier B.V. All rights reserved. Keywords: Weighted least-squares regression; Performance characteristics; Thyreostats
1. Introduction In laboratory practice, performance characteristics of analytical methods are usually calculated on the basis of empirical calibration data which were acquired in validation experiments. The most common evaluation approach uses a linear regression model that assumes a concentration-independent variability of the measurement error. By ordinary least-squares (OLS) fitting one obtains the coefficients of the calibration function, an estimator of the variance of the measurement error (residual variance) and an estimator for the variance of the estimation error (i.e., imprecision of the calibration function). From these parameters, the confidence interval, the prediction interval and further performance characteristics like the critical level or the limit of detection can be computed [1,2]. However, experimental errors often are heteroscedastic, i.e., the variance of the measurement error is related to the concentration. In such cases, weighted least-squares (WLS) regression is a more appropriate technique to determine the calibration line and to get realistic prediction bounds [3–5]. To perform WLS regression, it is first necessary to assign an individual weight to each concentration point. This can be achieved through various approaches. Many commercial chromatogra-
phy integration programs, for instance, offer the possibility of weighting by the inverse concentration, the inverse squared concentration, the inverse signal or the inverse squared signal. Ideally, weights should reflect the error scattering of the response and, to be more precise, should be reciprocally proportional to variance. Up to now, the acceptance of WLS regression is quite low in routine laboratories. The main purpose of this paper, therefore, is to outline the potential of this technique by describing an example case and to provide a user-friendly procedure that enables to estimate reasonable weights. In this study, the reliability of an LC–MS method for the determination of thyreostat residues in bovine plasma was assessed (thyreostats are pharmacologically active substances banned for use in food producing animals in the European Union [6–9]). Calibration data obtained from the analysis of spiked samples were processed by both OLS and WLS regression. Performance characteristics were calculated utilizing a weighted and an unweighted approach. The weights were set equal to the reciprocal of the variances and were modelled as a function of concentration. Different function types were employed and their suitability was examined.
2. Weighted least-squares model ∗
Corresponding author. Tel.: +49 721 9265451. E-mail address: [email protected] (P. Steliopoulos).
0003-2670/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.aca.2007.04.026
The concept of weighting was introduced in method calibration to account for concentration-dependent response precision.
182
P. Steliopoulos, E. Stickel / Analytica Chimica Acta 592 (2007) 181–186
The model is
The prediction interval for a given concentration x0 is defined
yij = b1 + b2 xi + εij
(1)
where yij is the response of the jth replicate at the ith concentration xi (i = 1, . . ., I denotes the concentration level, j = 1, . . ., J is the index of the replicate measurement). εij represents the measurement error. It is supposed that at each level i the error εij follows a normal distribution with mean 0 and a variance σ 2 w−1 i depending on xi . For the time being assume that the weights wi are known. Then, the estimators bˆ 1 and bˆ 2 can be calculated as bˆ 1 −1 βˆ = = (XT WX) XT WY (2) bˆ 2 with
⎛
y1
⎞
⎜y ⎟ ⎜ 2⎟ ⎟ Y =⎜ ⎜ .. ⎟ ⎝ . ⎠
(3)
yn ⎛
1
x1
⎞
⎜1 x ⎟ 2⎟ ⎜ ⎟ X=⎜ . . ⎜. . ⎟ ⎝. . ⎠ 1 xn and
⎛
w1
⎜ 0 ⎜ W =⎜ ⎜ .. ⎝ . 0
0 w2 .. . 0
(4)
···
0
⎞
0 ⎟ ⎟ ⎟ .. ⎟ . ⎠ · · · wn ··· .. .
(5)
n designates the total number of measurements (n = I × J). Note, WLS regression simply turns into OLS regression when W = diag{1, 1, . . ., 1}. The weighted residual variance is given as T
σˆ 2 =
ˆ W (Y − Xβ) ˆ (Y − Xβ) (n − 2)
(6)
By setting a certain value x0 in yˆ = bˆ 1 + bˆ 2 x one obtains an estimate yˆ 0 of the expectation value E{y0 } = b1 + b2 x0 . This estimate is afflicted with an error variance of which can be estimated as 1 −1 σˆ y2ˆ0 = σˆ 2 ( 1 x0 )(XT WX) (7) x0 The total variance of the imprecision of a prediction yˆ 0 at a certain concentration x0 consists of two components: the variance of the measurement error and the variance of the estimation 2 , can be error of yˆ 0 . An estimator of this total variance, σˆ total computed as 2 σˆ total =
σˆ 2 + σˆ y2ˆ0 wi
(8)
as prd (ˆy0 ) = yˆ 0 ± tf,1−α/2
2 σˆ total
(9)
and describes the range in which one can expect to observe with a probability of 1 − α the response of a sample that contains the analyte at a concentration x0 . tf,1−α/2 designates the 1−α/2quantile of the t-distribution with f degrees of freedom. 3. Performance characteristics 3.1. Decision limit CCα In the European Union, methods that are used in official residue control of pharmacologically active substances have to be validated in accordance with Commission Decision 2002/657/EC [10]. This Decision defines the decision limit CC␣ as: “. . .the limit at and above which it can be concluded with an error probability α that a sample is non-compliant.” In other words, CC␣ represents the measured concentration from which on a certain threshold value x0 is exceeded significantly. For substances with an established maximum residue limit (MRL), the threshold x0 equates to this MRL value. The α error probability then can be set 0.05. For banned substances x0 is zero and α = 0.01. The underlying hypotheses can be formulated as H0 :x ≤ x0 (or rather, if x0 = 0, H0 :x = x0 ) and H1 :x > x0 . x represents the unknown concentration of the analyte exhibiting the response y. The hypothesis H0 can be rejected at the significance level α in favour of the alternative H1 if
2 (10) y > yˆ 0 + tf,1−α σˆ total The critical concentration CC␣ is the corresponding x value. Hence, tf,1−α σˆ 2 CC␣ = x0 + (11) + σˆ y2ˆ0 w0 bˆ 2 This formula for the decision limit is only reasonable if yˆ 0 > 0. To take into account that yˆ 0 might be negative, Eq. (11) is changed into max {ˆy0 ; 0} − bˆ 1 tf,1−α σˆ 2 CC␣ = + + σˆ y2ˆ0 (12) w0 bˆ 2 bˆ 2 To calculate the decision limit for x0 = 0 the weight at zero is required. In this work, w(x0 = 0) was set equal to the weight assigned to the lowest spiked concentration. Note, when x0 = 0 the decision limit CC␣ corresponds to what in other fields of analytical chemistry is commonly called critical level LC [11]. Yet with the difference that the decision limit referred to in CD 2002/657/EC has to be established on the basis of experimental results that were obtained under within-laboratory reproducibility conditions.
P. Steliopoulos, E. Stickel / Analytica Chimica Acta 592 (2007) 181–186
3.2. Detection capability CCβ The official definition reads: “Detection capability means the smallest content of the substance that may be detected, identified and/or quantified in a sample with an error probability of β. In the case of substances for which no permitted limit has been established, the detection capability is the lowest concentration at which a method is able to detect truly contaminated samples with a statistical certainty of 1 − β. In the case of substances with an established permitted limit, this means that the detection capability is the concentration at which the method is able to detect permitted limit concentrations with a statistical certainty of 1 − β.” Practically, the detection capability CC is the true concentration that exceeds CC␣ with a statistical certainty of 1 − β. Thus, tf,1−β σˆ 2 (13) + σˆ y2ˆ(CC ) = CC␣ CC − w(CC ) bˆ 2 To determine CC one may plot the graph of the function f (x) =
bˆ 2 (x − CC␣ ) tf,1−β σˆ total
(x > CC␣ )
(14)
The concentration x at which the function f(x) equals to 1 is the detection capability CC . 4. Estimation of weights In the following, the procedure applied to determine the weights is described. First of all, for each concentration level, the empirical variance of replicate responses, vi , is calculated: ⎛ ⎞2 J J 1 ⎝yij − 1 yij ⎠ (15) vi = J −1 J j=1
vi = a1 ea2 xi vi =
a1 + a2 xi2
If var (vi ) = constant then
var (ψi ) ∝
dψi dvi
(22)
2 (23)
Therefore, (dψi /dvi )−2 should be adopted as a weight. In the concrete example the global weight ω is
dψi −2 ω= = v4i (24) dvi
WLS fitting provides the estimators aˆ 1 and aˆ 2 : aˆ 1 −1 = (Ξ T ΩΞ) Ξ T ΩΨ aˆ 2
with
(25)
⎞ x1−1 .. ⎟ ⎟ . ⎠
⎛
1 ⎜. Ξ=⎜ ⎝ ..
(26)
xI−1
1
Ω = diag {v41 , . . . , v4I }
(27)
and T
−1 Ψ = (v−1 1 , . . . , vI )
(28)
(18)
The weight associated with a certain concentration level is set equal to the reciprocal of the modelled variance and hence can be computed as
(19)
wi = aˆ 1 +
Consider as an example the function Eq. (16):
a2 −1 vi = a1 + xi
aˆ 2 xi
(29)
In the case of the other functions, the same procedure gives rise to the results summarized in Table 1. Fig. 1 illustrates with an example the effect of global weighting.
Transformation yields a linear equation:
Table 1 Models used to estimate weights
a2 1 = a1 + vi xi ψi = a1 + a2 ξi
analyzing the data pairs (ξ i , ψi ) by OLS regression one minimizes the sum of the squares of the residuals in ψi rather than those in vi . Thus, if the dependent variable vi is regarded as homoscedastic weighting by the reciprocal squared derivative of ψi with respect to vi should be incorporated in regression analysis of (ξ i , ψi ) (global weighting according to [12]). Explanation: Since ψi is a function of vi , it follows, applying the Gaussian law of uncertainty propagation, that
dψi 2 var (ψi ) = var (vi ) (21) dvi
j=1
Then, the variance is modelled by an appropriate curve. The following functions were used:
a2 −1 (16) vi = a1 + xi vi = a1 xia2 (17)
183
(20)
−1 where ψi = v−1 i and ξi = xi . To estimate the coefficients, ordinary least-squares fitting may be performed now. However, in
Function
Transformation
ψ
ξ
ω
ψ v = a1 xa2 v = a1 ea2 x v = a1 + a2 x2
= a1 + ax2 ln v = ln a1 + a2
1 v
1/x ln x x x2
v4 v2 v2 1
1 v
ln x ln v = ln a1 + a2 x –
ln v ln v v
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was again shaken for further 30 s. The tube was placed in an ultrasonic bath for 10 min and centrifuged at 3500 rpm. The organic layer was removed and evaporated to dryness under a nitrogen stream. The residual material was dissolved in 500 L methanol/water (50:50) by sonicating for 10 min. Finally, the sample was centrifuged at 4000 rpm and filtered into a vial. 5.2. LC–MS
Fig. 1. Modelling the variance by fitting the transformed of a power function with (a) and without (b) global weighting.
5. Experimental 5.1. Sample extraction Samples were extracted as described in [7]. Briefly, 1 mL plasma was weighed into 15 mL polypropylene centrifuge tube. Ten microlitres 0.1 mol L−1 EDTA, 10 L 2-mercaptoethanol and 5 mL ethyl acetate were added. After shaking vigorously for 30 s, 4 g sodium sulfate were added and the sample
LC–MS analysis was performed with a Thermo Finnigan TSQ Quantum triple quadrupole mass spectrometer coupled to a Surveyor LC pump and autosampler. Chromatographic separation was achieved on a Phenomenex Prodigy C 18 5 m ODS (3) 100 A (150 mm × 3 mm) column. The mobile phase consisted of 0.1% acetic acid in water (A) and methanol (B), the flow rate was 0.3 mL min−1 . The gradient program was as follows: 0–10 min, 100–70% A; 10–25 min, 70–5% A; 25–27 min, 5–100% A. The mass spectrometer was operated in the ESI(+)MS2 SRM mode: m/z 115 [MH]+ → m/z 81 and m/z 56 (tapazole), m/z 129 [MH]+ → m/z 111 and m/z 84 (thiouracil), m/z 143 [MH]+ → m/z 126 and m/z 84 (methylthiouracil), m/z 171 [MH]+ → m/z 154 and m/z 112 (propylthiouracil), m/z 205 [MH]+ → m/z 188 and m/z 103 (phenylthiouracil). 5.3. Computations Calculations were implemented using Excel 2000 spreadsheets and Visual Basic for Applications (VBA).
Fig. 2. Modelling the variance as a function of concentration for tapazole. (a) v = (−1.33 × 10−12 + 3.60 × 10−10 /x) 1010 e0.016x , (d) v = −5.96 × 1010 + 5.15 × 107 x2 .
−1
, (b) v = 7.43 × 106 x2.37 , (c) v = 8.10 ×
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185
Fig. 4. Graphical determination of the weighted detection capability CC for tapazole.
Fig. 3. Weighted and unweighted least-squares regression for tapazole (calibration lines and 98% prediction intervals; response y [area]: TIC of the daughter ions).
6. Results and discussion Seven calibration runs were carried out (J = 7), each at five concentration levels (I = 5): xi [g L−1 ] = 25, 50, 100, 150, 200. Sample series were analysed during a time period of three weeks,
for each particular run a different blank matrix was used for spiking. For all examined substances the variance of replicate measurements increased with the concentration. In the case of tapazole, the four variance models (Eqs. (16)–(19)) provided fits that were roughly comparable to each other (Fig. 2). The lowest sum of squared residuals was obtained by applying Eq. (16), hence this model was chosen to approximate the weights. For thiouracil, methylthiouracil and propylthiouracil the best fit was given by the power function (Eq. (17)), whereas for phenylthiouracil the parabola (Eq. (19)) turned out to be the most suitable model. Fig. 3 exemplarily shows the weighted and the unweighted 98% prediction interval of tapazole. As it can be seen in this figure, the weighted prediction interval describes the distribution of the measurement values much more adequately than the unweighted. A similar observation was also made for the other compounds. Table 2 lists the coefficients and the performance characteristics that were obtained using the two different least-squares approaches. Fig. 4 depicts the determination of the weighted detection capability for tapazole. In all cases, the regression coefficients are hardly affected by weighting. In contrast, the weighted performance characteristics are essentially
Table 2 Calculation of weighted and unweighted performance characteristics Ordinary least-squares
Tapazole Thiouracil Methylthiouracil Propylthiouracil Phenylthiouracil
Weight model
bˆ 1
bˆ 2
CC␣
CC
−150362 −127836 −582077 −1118009 −1167083
49064 20833 54653 197265 214915
45.5 78.6 64.7 60.8 73.1
73.9 127.0 100.8 97.7 118.4
Eq. (16) Eq. (17) Eq. (17) Eq. (17) Eq. (19)
Weighted least-squares bˆ 1
bˆ 2
CC␣
CC
−146743 −70565 −312925 −573154 −146528
49019 20097 51293 191087 203120
16.6 16.2 12.0 13.5 13.2
25.8 24.2 15.5 20.4 21.2
All concentrations are given in [g L−1 ]; x0 = 0 g L−1 , I = 5, J = 7, n = 35, α = 0.01, β = 0.05; OLS: f = n − 2 = 33, t1−α = 2.453, t1−β = 1.696; WLS: f = n − 2 − 2 = 31 (2 less than fOLS due to the 2 coefficients of the weight model), t1−α = 2.445, t1−β = 1.692.
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lower than the unweighted. According to expectation, at smaller concentrations the ordinary least-squares technique led to rather broad prediction intervals and thus to an overestimation of the performance characteristics. 7. Conclusions The dispersion of measurements often depends on the concentration. In such cases, to achieve an appropriate calibration and to construct realistic prediction bounds that allow a reliable assessment of the analytical performance, weighted least-squares regression should be applied. The procedure presented in this work appears to be a practicable way to specify the weights that are required for that purpose. References [1] K. Danzer, L.A. Currie, Guidelines for calibration in analytical chemistry, Pure and Appl. Chem. 70 (1998) 993–1014. [2] A. Hubaux, G. Vos, Anal. Chem. 42 (1970) 849–855.
[3] A. Sayago, A. Asuero, CRC Crit. Rev. Anal. Chem. 34 (2004) 133–146. [4] M.E. Zorn, R.D. Gibbons, W.C. Sonzogni, Anal. Chem. 69 (1997) 3069–3075. [5] L. Oppenheimer, T.P. Capizzi, R.M. Weppelman, H. Mehta, Anal. Chem. 55 (1983) 638–643. [6] P. Batjoens, H.F. De Brabander, K. De Wasch, J. Chromatogr. A 750 (1996) 127–132. [7] W.J. Blanchflower, P.J. Hughes, A. Cannavan, M.A. Mc Coy, D.G. Kennedy, Analyst 122 (1997) 967–972. [8] B. Le Bizec, F. Monteau, D. Maume, M.P. Montrade, C. Gade, F. Andre, Anal. Chim. Acta 340 (1997) 201–208. [9] A. Heeremans, A. Ermens, K. De Wasch, C. Van Peteghem, H.F. De Brabander, Analyst 123 (1998) 2629–2632. [10] Commission Decision of 12 August 2002 implementing Council Directive 96/23/EC concerning the performance of analytical methods and the interpretation of results (2002/657/EC), Off. J. Eur. Commun. L 221 (2002) 8–36. [11] L.A. Currie, Nomenclature in evaluation of analytical methods including detection and quantification capabilities, Pure Appl. Chem. 67 (1995) 1699–1723. [12] R. de Levie, How to Use Excel in Analytical Chemistry and in General Scientific Data Analysis, Cambridge University Press, Cambridge, 2001.