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Analytica Chimica Acta 657 (2010) 136–146

Contents lists available at ScienceDirect

Analytica Chimica Acta

journa l homepage: www.e lsev ier .com/ locate /aca

multivariate multianalyte screening method for sulfonamidesn milk based on front-face fluorescence spectroscopy

. Rodrígueza, M.C. Ortiza,∗, L.A. Sarabiab, A. Herreroa

Department of Chemistry, Faculty of Sciences, University of Burgos, Pza. Misael Banuelos s/n, 09001 Burgos, SpainDepartment of Mathematics and Computation, Faculty of Sciences, University of Burgos, Pza. Misael Banuelos s/n, 09001 Burgos, Spain

r t i c l e i n f o

rticle history:eceived 11 August 2009eceived in revised form9 September 2009ccepted 21 October 2009vailable online 27 October 2009

eywords:ulfonamides in milk

a b s t r a c t

Screening methods are used to detect the presence of a substance or class of substances at the levelof interest and are specifically designed to avoid false compliant results. They should allow the run-ning of a high number of samples per day at a low cost under routine conditions. In this work, a rapidand simple method for the screening of six sulfonamides (sulfadiazine, SD; sulfamerazine, SMR; sul-famethazine, SMT; sulfachloropyridazine, SCP; sulfathiazole, STZ and sulfamethoxazole, SMO) in milksamples is proposed and assessed according to the criteria required by the European Regulation, Deci-sion 2002/657/EC. The method is based on modelling front-face fluorescence emission spectra by meansof partial least squares class modelling (PLS-CM). The milk samples are pre-treated with a single easy

ront-face fluorescence spectroscopyultivariate analytical sensitivity

artial least squarescreening methodecision 2002/657/EC

step of derivatization with fluorescamine.After confirming that the method has equal analytical sensitivity for all the six sulfonamides, it is estab-

lished that the multivariate analytical sensitivity at 100 �g L−1 is 37.5 �g L−1 when analysing a mixtureof six sulfonamides added to different brands of milk and measured in different days. In addition, themethod is applied to samples from 11 commercial brands of milk. For ˇ = 0.05, threshold value estab-lished by the Decision 2002/657/EC for this method, the probability of false non-compliance, ˛, is equal

able s

to 0.17, allowing the suit

. Introduction

Antibiotics are widely used as feed additives for treating dis-ases and/or promoting growth in food-producing animals, whichan lead to the contamination of foodstuffs with the risk of unde-irable health effects [1]. So regulatory authorities have establishedaximum residue limits (MRLs) for a number of antibacterial

gents in different target tissues [2,3], although worldwide harmo-ized legislation does not exit. In the case of sulfonamide group,uropean Union establishes that the combined total residues ofll substances within it should not exceed 100 �g kg−1 in milk [2].n the last few years the percentage of non-compliant results forntibacterial agents has increased [4].

Screening methods are used to detect the presence of a sub-tance or class of substances at the level of interest and arepecifically designed to avoid false compliant results [5]. They

hould allow the running of a high number of samples per day atlow cost under routine conditions. So, the availability of suitable

creening tests is a key element in the detection of these residues.everal bibliographic references deal with the topic of characteris-

∗ Corresponding author. Tel.: +34 947258800; fax: +34 947258831.E-mail address: mcortiz@ubu.es (M.C. Ortiz).

003-2670/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.aca.2009.10.048

creening of these six sulfonamides.© 2009 Elsevier B.V. All rights reserved.

ing and validating the performance of screening methods [6,7]. Fourapproaches for estimating the performance of screening methodsare discussed in ref. [8], and in ref. [9] authors propose the appli-cation of logistic regression to interlaboratory and intralaboratorystudies of screening methods performance. In all these cases, zeroorder signals are used.

The analysis of sulfonamides usually implies HPLC with massspectrometry detection [10], method that is highly advisable forconfirmation analysis but not for screening purposes of large num-ber of samples since it is too expensive and time-consuming[11].

For screening of sulfonamides in milk, analytical methods[12] mainly include microbial inhibitor assays, enzyme-linkedimmunosorbent assays (ELISA) and biosensor immunoassays.Microbiological procedures are easy to perform and with highcapacity but they usually involve extended incubation time and,in some cases, have insufficient sensitivity to certain substances,such is the case of STAR protocol [13]. This five-plate test does notdetect effectively among other sulfonamides, SMT and STZ (sen-

sitivities between 600 and 1000 �g L−1). This method is unable tofulfil for these sulfonamides the criterion, ˇ (probability of falsecompliance) less than 0.05, demanded by Decision 2002/657/EC[5]. Other works, based on TTC (2,3,5-triphenyoltetrazolium chlo-ride reduction) test [14] and Premi® test [15], report sensitivity

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possible threshold value tv. The risk curve is the plot of ˇ versus ˛probabilities ([29], p. 413). Both two risks are inter-related, and it is

N. Rodríguez et al. / Analytica

alues below the MRL for sulfonamides in milk, but no estimationf ˇ is provided.

Immunochemical methods are widely used for screening ofntibiotic residues in milk [11]. In ref. [16] a semi-quantitativemmunological method (BIACORE method) for detecting 8 sul-onamides in bovine milk is developed and validated accordingo Decision 2002/657/EC [5]. In ref. [17], PASA working withutomated chip-based ELISA and a chemiluminescence reactiononitored by a CCD camera is proposed for the analysis of two

ulfonamides and other antibiotics in milk. No estimation of ˇ isrovided in this work, and not very successful results for SMT areound. The disadvantages of these methods are the usually sig-ificant development time and often being too specific to allowffective identification of multi-analytes.

Also physical-chemical methods, such as molecular fluores-ence spectrophotometry can be used to screen samples since itses relatively inexpensive and rugged instrumentation. In ref.18], this technique, combined with class-modelling based on par-ial least squares regression (PLS-CM), is successfully applied as acreening method to detect SD, SMR and SMT in milk. The method isased on PLS regressions with a binary response to decide whetherhe sum of concentrations of the three sulfonamides exceeds the

RL. That work shows that PLS-CM is more efficient for this taskhan a quantitative regression. In this methodology, each class

odel is characterised by the probability of false non-compliance˛) and the probability of false compliance (ˇ). Other chemometricools, such as discriminant analysis based on PLS (PLS-DA), coulde used, but the evaluation of false non-compliance and false com-liance probabilities cannot be estimated. As an example, this lasttatistical technique is coupled to front-face fluorescence spec-rophotometry in ref. [19] to characterize ice-cream formulations.

In front-face fluorescence spectrophotometry the irradiationngle may vary between 30◦ and 60◦ to avoid specular reflectionn the perpendicular direction where emission fluorescence is col-ected. Front-face fluorescence is recorded directly on turbid orolid intact samples, such as milk, to rapidly and non-destructivelyetermine some constituents of interest; which makes this spectro-copic technique really adequate for food analysis [20] especiallyn combination of multivariate chemometric tools [21]. This tech-ique has been used, in combination with principal componentsegression (PCR) to determine lactulose and furosine [22] andarboxymethyllysine, a Maillard reaction product indicative ofevere heating [23], and in association to PLS regression to pre-ict nutritional parameters in heat treated infant formula models24].

The present work is organized as follows. In Section 2 the appli-ation of a PLS-CM procedure for assessing a screening qualitativeethod and its natural adaptation for this kind of task is described.

ince it is about detecting the possible presence of a mixture of sul-onamides in milk by recording the emission spectra, in Section 2.2nd Appendix A the concept of multivariate analytical sensitivity isetailed.

Next, in Section 4, a first analysis of the six sulfonamides withamples of a brand of milk, fortified independently with each sul-onamide, is carried out. A joint PLS calibration model is built since

ultivariate analytical sensitivity is equal for all of them. After that,nother PLS model is built from samples of two brands of milk for-ified with a mixture of the six sulfonamides, whose spectra areecorded in two days. And then, once the validity of the methodor quantifying the sum of concentrations is assessed, a study ofhe capability of the method for screening samples with mixtures

f the six sulfonamides is studied. In summary, a front-face fluo-escence spectrophotometric method is proposed as a rapid andimple method for the screening of six sulfonamides (SD, SMR,MT, SCP, STZ and SMO) in semi-skimmed milk samples. It con-ists of recording the emission fluorescence spectra of the sample

ica Acta 657 (2010) 136–146 137

after a very simple derivatization step with fluorescamine. Next,the screening of the milk samples is carried out by PLS-CM, theprocedure being validated according to Decision 2002/657/EC forscreening the six sulfonamides with front-face fluorescence at theMRL level.

2. Theory

2.1. Class-modelling via PLS regression (PLS-CM)

The PLS-CM procedure uses a PLS regression where the predictorvariables are the spectra and the response is a binary variable whichequals one if the sum of the found concentrations of sulfonamidesdoes not exceed the MRL (compliant samples), and two if the sumof the found concentrations is higher than the MRL (non-compliantsamples).

Given a new sample, its emission spectrum is recorded and, bymeans of Hotelling’s T2 and Q statistics, it is decided if it is sim-ilar to the samples of the training set of the PLS regression andtherefore if the built model can be applied to it. This allows one toidentify samples outside the two categories (compliant and non-compliant), for example because of a matrix effect. On the contrary,if the sample is similar to the training samples, it will be classifiedin one of the two categories depending on whether the responsecomputed by the PLS regression is closer to 1 or to 2. In order todecide if a unknown sample belongs to one or another class, athreshold value, tv, between 1 and 2 must be established. If thevalue estimated by PLS is higher than tv the sample is declarednon-compliant. For estimated values lower than tv it is consideredas compliant.

The assessment of a class model is given by its sensitivity, whichis defined as the proportion of samples of the class that are correctlyassigned, and its specificity, defined as the proportion of samplescorrectly rejected. These two parameters are related to the proba-bilities of false non-compliance, ˛, and false compliance, ˇ, whichare referred to in the European Decision [5] for screening meth-ods for which a MRL is established. Sensitivity is an estimation of(1 − ˛) × 100 and specificity estimates (1 − ˇ) × 100. The sensitivityand specificity are widely used to compare different class modelsas UNEQual class model, UNEQ [25], feedforward and radial basefunction neural networks [26], Genetic Inside Neural Networks,GINN [27] or Soft Independent Modelling Class Analogy, SIMCA[28].

In this way, each one of the class models can be consideredas a hypothesis test to decide if a PLS response value for a sam-ple belongs or not to the distribution of the compliant samples.Formally, the hypotheses of the test are:

- Null hypothesis, H0: the sample belongs to class 1 (its concentra-tion is equal or lower than MRL).

- Alternative hypothesis, H1: the sample belongs to class 2 (its con-centration is greater than MRL).

With the values computed by the PLS regression, a distributionof probability for each class is obtained, and the probabilities offalse non-compliance (˛ = pr {to reject H0|H0 is true}) and of falsecompliance (ˇ = pr {to accept H0|H0 is false}) are calculated for any

clear that the increase of one of the parameters can be assumed if itleads to a noticeable decrease of the other. Using the risk curve, theanalyst can select the classification model, that is the tv threshold,with the probabilities ˛ and ˇ suitable for using the fluorescenceas a screening method.

1 Chimica Acta 657 (2010) 136–146

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38 N. Rodríguez et al. / Analytica

.2. Analytical sensitivity

The possibility of developing a screening method for detectingimultaneously the total amount of several sulfonamides dependsn whether it has equal analytical sensitivity for all of them. Inther case, since in real samples it cannot be predicted which com-ination of substances will be present, a mixture of sulfonamidesor which the method is less sensitive could yield a signal equalo another mixture with less concentration of the substances butor which the method is more sensitive, i.e. it could not be distin-uished a sample that exceeds the MRL from another that does notxceed it. So, when proposing a method of multianalyte screeningor the total concentration of sulfonamides it is imperative to studyhe analytical sensitivity of the method for each of them and tostablish its equality.

.2.1. Univariate analytical sensitivityThe IUPAC ([30], chapter 18, p. 25) defines sensitivity as “the

lope of the calibration curve. If the curve is in fact a “curve”,ather than a straight line, then of course sensitivity will be a func-ion of analyte concentration or amount”. This definition is usuallymployed in many works.

The concept of sensitivity refers to the change in the responsesignal) of a system (measurement method) for a small change ofhe stimulus (analyte concentration) causing the response, i.e. theatio of signal change to change in concentration causing it. Thelope of the calibration line seems to reproduce this concept sincemethod with higher slope discriminates better small differencesf the analyte concentration. However, as the slope of the calibra-ion line is a parameter determined experimentally, it is affectedy experimental variability and this uncertainty has to be taken

nto account to decide whether a method is more sensitive thanther. In addition, the slope of a regression line is not invariant by ahange of scale and it has no sense that the sensitivity of a methodf analysis depends on the units in which the signal is expressed.

This has been pointed out many times, for example in ref. [31],y Massart et al. (p. 435 of ref. [32]) and by Commission SFSTP [33]hat proposes an approximate expression to estimate the differ-nce between two concentrations based on the ratio between thetandard deviation of the signal and the slope of the calibration line/b.

It seems adequate to define the analytical sensitivity such thathese problems do not exist but taking into account the idea under-ying the definition: a method with a high sensitivity will be ableo discriminate two concentrations closer than another with lowerensitivity. Because the word sensitivity has been defined by reg-latory bodies (IUPAC) as the slope of the calibration function, thexpression analytical sensitivity will be used instead of sensitiv-ty to describe the ability of a procedure to distinguish two closeoncentrations.

One approach to the problem consists of evaluating the smallerifference in concentration that can be distinguished between twoignals. Conceptually, this definition was proposed by Mandel andtiehler [34]; a wide discussion can be found in ref. [35]. But thetatistic justification is obtained in the way of a hypothesis test byeneralizing the method of Clayton et al. of ref. [36]. The methodf Clayton allows one to estimate the capability of detection of aethod by using a linear calibration, the probability of false positive

nd false negative being guaranteed. This procedure for calculatinghe detection limit, programmed in ref. [37] has been adopted bySO [38] and also by IUPAC ([33], Chapter 18, p. 36) which also

dvises to not use as quantification or detection limit a multiple ofhe standard deviation of a blank (Note 3 in page 37 of the samehapter 18).

Formal definitions of analytical sensitivity as a bilateral hypoth-sis test, as well as the estimator obtained from the calibration line

Fig. 1. Elements which define the multivariate analytical sensitivity, xd , and theirmutual relation as a function of �x (non-centrality parameter of a non-central Stu-dent’s t-distribution), wx0 (a factor which depends on values of abscissas xi), �

(residual standard deviation) and b (slope).

and the more relevant properties are included in the appendix ofthis manuscript to facilitate their consultation. In this formaliza-tion, X denotes the concentration variable and Y the correspondingsignal, the linear relationship between them is

Y = a + b X + ε (1)

where ε is a random variable, independent, normally distributedwith zero mean and variance, �2, constant for all x in the calibrationrange.

According to Proposition 1 of the Appendix A, if one has n exper-imental values (xi,yi), i = 1,. . ., n, the analytical sensitivity at x0, i.e.the minimum concentration, xd, that can be distinguished fromx0, with probability ˛ of stating that the concentration from x0 isgreater than xd when it is false and with probability ˇ of wronglystating that its distance from x0 is less than xd, is given by theequation

xd = |x − x0| = �x wx0 �

b(2)

In addition, the analytical sensitivity, xd, is the same when usingthe data (xi, xi) i = 1,. . ., n, being xi for i = 1,. . ., n the concentrationsestimated for the standards by means of the calibration line Y =a + b X , see Proposition 3 of the Appendix A.

2.2.2. Multivariate analytical sensitivityWhen using first or higher order signals, the impossibility of

obtaining the multivariate analytical sensitivity analogous to aslope is due to the fact that in a multivariate calibration (as PLSor PCR) an infinite number of signals x which give the same con-centration exists. An extensive revision of these concepts for theunivariate and multivariate case can be seen in Olivieri et al. [39].

A discussion about these questions and the proposal of definingthe multivariate analytical sensitivity by means of the regression“calculated concentration, X , versus true concentration, X”, can beseen in ref. [40]. This definition agrees with the analytical sensitiv-ity in the univariate case, and has the advantage of being fit intothe regulatory context of ISO and IUPAC for the capability of detec-tion and its multivariate generalization [41]. In addition, it doesnot depend on the kind of regression used for calibrating, beingalso applied to non-linear calibrations, e.g. multivariate calibrations

based on neural networks or support vector machines, since onealways has the values of concentration estimated for the standardsby means of calibration, i.e. the pairs (xi, xi), i = 1,. . ., n.

Fig. 1 graphically shows the elements which define the multi-variate analytical sensitivity and their mutual relationship.

N. Rodríguez et al. / Analytica Chimica Acta 657 (2010) 136–146 139

Table 1Characteristics of samples sets used in this work: milk, fortified, sulfonamides concentration levels and number of days where they have been measured.

Data set(dimensiona)

Milk Fortified Concentrationlevels (�g L−1)

Days

Case A (48 × 101) Brand of milk S Only onesulfonamide (SD orSMR or SMT or SCPor STZ or SMO)

0, 30, 60, 90, 120,150, 180, 210

1

Case B (28 × 101) Brand of milk S or T Mixture of 6sulfonamides(SD + SMR + SMT + SCP + STZ + SMO)

0, 50, 100, 150, 200 2

Case C (42 × 101) One mixture of 11brands of milk (P-Q-R-S-T-U-V-W-X-Y-Z)

Mixture of 6sulfonamides(SD + SMR + SMT + SCP + STZ + SMO)

0, 30, 60, 100, 140,170, 200

1

Case D (46 × 101) Brand of milk S or T or Mixture of 6sulf(SD

0, 125, 150, 200 4

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a Number of samples × number of emission wavelengths.

The aim of this work is to determine the analytical sensitivityt the MRL for the combined total residues of sulfonamides, so theRL is the nominal x0 of X, having to verify that the multivariate

nalytical sensitivity at the MRL is the same for each one of the sixnalysed sulfonamides. For that, it will be shown that slopes bk,= 1,. . ., 6, and standard deviations �k, k = 1,. . ., 6, of the six linear

egressions X versus X for each sulfonamide are significantly equal.s the same calibration standards will be used, the factor wx0 is theame and, as a consequence the analytical sensitivity estimated byeans of Eq. (2) is also the same for all of them at the MRL.

. Experimental

.1. Chemicals and reagents

The six sulfonamides used in this work, sulfadiazine (SD), sul-amerazine (SMR), sulfamethazine (SMT), sulfachloropyridazineSCP), sulfathiazole (STZ) and sulfamethoxazole (SMO), are of ana-ytical grade and were obtained from Sigma (Steinheim, Germany).

able 2ata set B: brand of milk (S or T), day of measurement, true and calculated sum of sulfonrediction (P) subsets.

Sample number Milk Day True sum of concentrations (�g L−1)

1 T 1 02 T 1 503 T 1 1004 T 1 1505 T 1 2006 T 1 07 S 2 508 T 1 1009 S 2 150

10 T 1 20011 S 1 012 S 1 5013 S 1 10014 S 1 15015 S 1 016 S 1 5017 S 1 10018 S 1 15019 T 2 020 T 2 5021 T 2 10022 S 2 15023 T 2 20024 S 2 025 T 2 5026 S 2 10027 S 2 15028 S 2 200

onamides+ SMR + SMT + SCP + STZ + SMO)

The stock (0.1 g L−1) and diluted standard solutions (12.5 mg L−1)for every sulfonamide were prepared in methanol (HPLC grade).Another solution with a total concentration of 12.5 mg L−1 of equalproportions of a mixture of the six sulfonamides was also preparedin the same solvent.

Fluorescamine was purchased from Sigma (Steinheim, Ger-many) and a solution 0.1% (w/w) was prepared in acetone (Merck,Darmstadt, Germany).

Semi-skimmed milk of eleven commercial brands was acquiredin a supermarket.

3.2. Experimental procedure

Firstly, milk samples were spiked with the suitable amount

of sulfonamide to obtain the desired antibiotic concentrations.Only the addition of 40 �L of fluorescamine was needed toobtain the fluorescence signal. Fluorescamine is widely used asa reagent for the fluorimetric determination of primary amines.This non-fluorescent compound reacts with primary amines to

amides concentrations, and absolute value of relative error for calibration (C) and

Calculated sum of concentrations (�g L−1) |Relative error| (%) Subset

28.31 – P53.67 7.34 P

109.51 9.51 C147.11 1.93 C197.92 1.04 C

3.28 – C58.97 17.93 P94.92 5.08 C

104.30 30.46 P193.08 3.46 P

11.73 – C48.48 3.04 C

100.61 0.61 P144.36 3.76 C

1.43 – C58.44 16.89 C

103.43 3.43 C155.90 3.93 C

-2.05 – C47.67 4.65 P91.00 8.99 C

139.63 6.91 C209.56 4.78 C

6.40 – C23.58 52.85 C

111.58 11.58 C158.41 5.61 P193.80 3.10 C

140 N. Rodríguez et al. / Analytica Chimica Acta 657 (2010) 136–146

Table 3Data set D: brand of milk, sum of sulfonamides concentration, and day of measurement.

Sample number Milk Sulfonamides concentration (�g L−1) Day Sample number Milk Sulfonamides concentration (�g L−1) Day

1 T 125 1 24 Z 0 12 T 125 1 25 Z 125 13 U 0 1 26 Z 125 14 U 0 1 27 U 150 25 U 125 1 28 U 200 26 U 125 1 29 U 150 27 V 0 1 30 U 150 28 V 0 1 31 U 150 29 V 125 1 32 U 200 2

10 V 125 1 33 Y 200 211 W 0 1 34 Y 200 212 W 0 1 35 U 150 313 W 125 1 36 U 200 314 W 125 1 37 U 150 315 X 0 1 38 U 200 316 X 0 1 39 Y 150 317 X 125 1 40 Y 200 318 X 125 1 41 T 150 419 Y 0 1 42 S 200 4

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nation, R , are shown in Table 4. It is clear that these regressionmodels, with R2 values between 0.04 and 0.94, have not enoughanalytical quality to perform the successful determination of thesesulfonamides and so they are not valid to estimate the analyticalsensitivity either.

Table 4Characteristics of univariate regression models for each sulfonamide obtained fromdata set A.

Sulfonamide Intercept Slope Residual standarddeviation

R2

SD 422.21 −0.13 48.13 0.04

20 Y 0 121 Y 125 122 Y 125 123 Z 0 1

ield derivatives, which upon excitation at 390 nm emit stronguorescence at 475–490 nm. This reaction occurs rapidly at roomemperature [42], nevertheless samples were measured 20 minfter their preparation in all cases.

It is possible that other substances, such as aminoacids orroteins, which are present in milk, were also labeled with fluo-escamine. Although commercial milks are very homogeneous, theistribution of the fat can be different in them. In order to take intoccount the effect that these two factors could make on the fluores-ence signal, several kinds of commercial milk have been includedn the analysis. The samples sets used are summarized in Table 1.ext, a more detailed description is shown.

Set A. For every of the six sulfonamides, milk samples spiked withincreasing amounts of antibiotic at eight levels of concentration(see Table 1) are prepared using a brand of milk, called S. All the48 samples are measured the same day.Set B. A total of 28 milk samples of two different brands (S and T)are fortified with different volumes of a solution of the mixtureof the six-sulfonamides, concentrations are shown in Table 1. Thespectra of these samples are measured in two different days. Thedata set is divided in two subsets: calibration (20 samples) andprediction (8 samples) sets. Characteristics of these samples areshown in Table 2.Set C. A solution with a total volume of 1.1 L of milk is preparedwith 0.1 L of each of the eleven different brands of semi-skimmedmilk. This solution is used to obtain a set of 42 milk samples spikedwith the mixture of the six sulfonamides at seven levels of sum ofsulfonamides concentrations (see Table 1), each sample being sixtimes replicated.Set D. Another set of 46 samples spiked with different amounts ofthe six sulfonamides mixture, prepared with different brands ofmilk and measured in different days, as can be seen in Table 3, isused.

.3. Instrumentation and software

Measurements were performed at room temperature on a

erkinElmer LS50B Luminescence Spectrometer equipped with aenon discharge lamp and a gated photomultiplier. Fluorescenceignals were recorded in front-face mode. Excitation and emissionlits were both set to 3 nm and the scan speed was 200 nm min−1.he incidence angle was set at 60◦. The emission spectra were

43 S 200 444 T 150 445 T 200 446 T 150 4

recorded at �excitation = 390 nm from 430 to 530 nm (intervals of1 nm).

The FL WinLab software (PerkinElmer) was used, data wereimported to Matlab using the INCA software [43]. The PLS Toolboxfor Matlab [44] was employed to perform the PLS models. Home-made programs NWAYDET and NDETBIL were used to estimate thecapability of detection, decision limit and multivariate analyticalsensitivity. STATGRAPHICS [45] was employed for building and val-idating the linear regressions and for performing the Cochran andBartlett tests.

4. Results and discussion

4.1. Individual sulfonamides calibration. Analysis of themultivariate sensitivity

It has already been pointed out in Section 2.2 the need to demon-strate that the analytical sensitivity of the method is the same forthe six sulfonamides. Firstly, a univariate approach is tested. Thus,fluorescence intensities at the maximum peak (�emission = 480 nmand �excitation = 390 nm) are recorded for the 48 samples of set A,in such a way that six univariate regression lines, “intensity ver-sus true sulfonamide concentration”, are separately built for everysulfonamide. The parameters of these regression models, intercept,slope, standard deviation of regression and coefficient of determi-

2

SMR 303.84 0.38 31.34 0.49SMT 301.05 0.78 28.49 0.82SCP 302.70 0.68 32.06 0.74STZ 363.86 0.44 21.30 0.73SMO 375.17 0.50 10.23 0.94

N. Rodríguez et al. / Analytica Chimica Acta 657 (2010) 136–146 141

Table 5Characteristics of regression models “concentration calculated with PLS calibrationversus true concentration” for each sulfonamide built with data set A.

Sulfonamide Intercept Slope Residual standarddeviation

R2

SD 0.08 0.99 2.19 0.99SMR 0.04 0.99 1.63 0.99SMT 0.07 0.99 1.97 0.99

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Table 6p-Values of the tests performed for validating the regression models “Ccalc. versusCtrue”: significance of the regression, normality of residuals, homoscedasticity ofvariances and lack of fit tests (see the text to know the equations fitted) for data setsA and B.

Joint calibration of sixsulfonamides (data set A)

PLS calibration(data set B)

Regression significancea 0.000 0.000Kolmogorovb 0.556 0.234Cochranc 0.088 0.054Bartlettc 0.406 0.335Lack of fitd 0.138 0.451

a Null hypothesis: the linear model is not significant.

SCP 0.04 0.99 1.64 0.99STZ 0.01 0.99 0.87 0.99SMO 0.01 0.99 0.81 0.99

Therefore, a first order regression technique, PLS, is secondlyried. To do it, emission spectra of samples of set A are recorded andrranged in six matrices (one for each sulfonamide) with dimen-ions 8 × 101, where 8 is the number of standard samples and 101he number of emission wavelengths. A PLS calibration model isuilt with each of the six matrices (raw data previously centred byolumn) taking as response variable the corresponding true antibi-tic concentration. The number of latent variables chosen was threeor all the models; which minimizes the root mean squares error inross-validation (RMSECV) with the leave-one-out procedure. Theetection of outliers is carried out by calculating Q residual andotelling’s T2 statistics and removing those objects whose valuesxceed the corresponding threshold at 95%. From this point of view,one of the objects was considered outlier in the six models.

Regression models “concentration computed by PLS versus trueoncentration” are built. The parameters of these regressionsintercept, slope, standard deviation and R2) can be seen in Table 5.he property of trueness is fulfilled in all cases since the regres-ion lines have slope and intercept statistically equal to 1 andrespectively at a confidence level of 95%. This means that theultivariate calibration technique used, PLS, is able of success-

ully extracting and modelling the contribution to the signal ofhe sulfonamides, avoiding the possible contribution of interfer-nts that could also been derivatized with fluorescamine in there-treatment step.

According to that set in Section 2.2, in order to decide if the mul-ivariate analytical sensitivity is equal for the six sulfonamides, its necessary to verify that the six slopes are statistically equal. Arocedure to carry out this consists of fitting a joint model with theata of the six calibrations, using indicator variables to identify thealibration of each sulfonamide, and testing that the coefficients ofhese variables are significantly null. This model contains an inter-ept, the true concentration, x, and an indicator variable di for eachulfonamide except the first. The model fitted can be seen in Eq.3):

= (0.08(0.94)

− 0.04d2(0.98)

− 0.015d3(0.99)

− 0.04d4(0.98)

− 0.07d5(0.96)

− 0.07d6(0.96)

) + (0.99(0.00)

+0.0003d2(0.98)

+ 0.0001d3(0.99)

+ 0.0003d4(0.98)

+ 0.0006d5(0.96)

+ 0.0007d6(0.96)

)x

(3)

ere y is the sulfonamide concentration calculated by PLS, x is therue sulfonamide concentration and each di, i = 2,. . ., 6 is a binaryariable that is equal to 1 if the standard belongs to group (sulfon-mide) i-th and zero in the contrary case. For example, for SD (group) all di are zero, being the resulting equation: y = 0.08 + 0.99x. ForMR data (group 2), d2 = 1 and d3 = d4 = d5 = d6 = 0, the resultingodel being y = (0.08 − 0.04) + (0.99 + 0.0003)x. A similar interpre-

ation can be made for the remaining sulfonamides in Eq. (3).

Coefficients of binary variables estimate the change in the inter-

ept and slope when going from one sulfonamide to another.-Value of each coefficient is written in brackets under it in Eq.3). Since all coefficients, di, are significantly equal to zero the indi-idual regression lines for every sulfonamide are statistically equal

b Null hypothesis: the residuals are normally distributed.c Null hypothesis: the variances are homoscedastic.d Null hypothesis: the regression model adequately fits the data.

at a significance level of 0.05. In particular, the six slopes bk aresignificantly equal.

Also the residual standard deviations, �k, corresponding to thesix regression models “concentration calculated by PLS versus trueconcentration” (Table 5) are significantly equal according to theCochran and Bartlett tests (p-values equal to 0.390 and 0.139respectively).

As the calibration standards for each sulfonamide have the sameconcentrations, xi, i = 1,. . ., 8, in the six regressions (see Table 1, dataset A), it is concluded that the multivariate analytical sensitivity isthe same, as it has already been explained in Section 2.2.

Therefore, as the multivariate sensitivity is statistically equal forall the six determinations carried out with PLS, instead of individualperformance characteristics, common values can be estimated forthe multivariate determination of these six sulfonamides in milkby working with a joint regression line “calculated versus true con-centration”, taking into account simultaneously the 48 samples tobuild it. The regression model built in this way is y = 0.043 + 0.999x,(syx = 1.42 and � = 0.999). In the second column of Table 6, the p-values of different hypothesis tests carried out to validate thismodel are shown. At the significance level of 0.05, it can be con-cluded that the regression is significant, the residuals are normallydistributed, the variances of the residuals are homoscedastic andthere is not lack of fit.

The mean of the relative errors in absolute value for the samplesis shown in Table 7. Next, it is checked that the proposed methodfulfils the property of trueness, showing that slope and interceptare statistically 1 and 0 respectively at a significance level 0.95 (seesecond column in Table 7).

Decision limit (CC˛) and capability of detection (CCˇ) of the pro-cedure are determined according to Commission Decision [5], ISO[38] requirements and IUPAC [30] recommendations. CC˛ meansthe limit, in concentration, at and above which it can be con-cluded with an error probability of ˛ that a sample has analyte,whereas CCˇ is the smallest content of the substance that may bedetected, identified and/or quantified in a sample with a false nega-tive probability equal to ˇ and a false positive probability equal to ˛.Both performance characteristics are estimated for this procedurefrom the regression model “calculated versus true concentration”,according to the implementation to the multivariate case (PLS mod-els) shown in ref. [41]. The CC˛ and CCˇ values are 2.5 and 4.9 �g L−1

respectively for ˛ = ˇ = 0.05.It is also possible to estimate the values of CC˛ and CCˇ but at

the MRL, which has analogous meaning: the concentration above

which it can be said with a probability of error ˛ that the concentra-tion of the sample is higher than the MRL and with a probability ˇof wrongly affirming that the sample does not exceed the MRL. Theestimators have an expression very similar to that of Eq. (2) as it isshown in ref. [40]. The values of CC˛ and CCˇ vary with the concen-

142 N. Rodríguez et al. / Analytica Chimica Acta 657 (2010) 136–146

Table 7Performance characteristics determined for the analytical method based on the joint calibration of the six sulfonamides: Mean of the absolute value of the relative errors(MAE), trueness, limit of decision (CC˛), capability of detection (CCˇ) and multivariate analytical sensitivity at 100 �g L−1 for ˛ = ˇ = 0.05.

Joint calibration of six sulfonamides (data set A) PLS (data set B)

Mean of absolute value of relative errors (%)Calibration 1.35 9.12Prediction – 10.01

TruenessIntercept (p-value of hypothesis test)a 0.043 (0.909) 1.562 (0.662)Slope (p-value of hypothesis test)b 0.999 (0.891) 0.983 (0.585)Residual standard deviation, � 1.420 9.427

CC˛ (�g L−1) at x0 = 0 �g L−1 (decision limit) 2.5 17.7CCˇ (�g L−1) at x0 = 0 �g L−1 (capability of detection) 4.9 35.0CC˛ (�g L−1) at x0 = 100 �g L−1 102.4 117.0CCˇ (�g L−1) at x = 100 �g L−1 104.8 133.6

5.3

txsaueTca

EvwTcIt

4

asf

FaAD

0

Multivariate analytical sensitivity (�g L−1) at x0 = 100 �g L−1 10

a Null hypothesis: the intercept is zero.b Null hypothesis: the slope is one.

ration x0, because the term wx0 varies depending on the position of0 in the range of the calibration standards. The Commission Deci-ion [5] requires the determination of CC˛ and CCˇ at the MRL aspart of the performance characteristics of the analytical methodssed in the determination of veterinary residues in foodstuffs. Thestimated values at the MRL (x0 = 100 �g L−1) are shown in Table 7.hey are rather less because the concentration 100 �g L−1 is moreentred in the calibration range than the null concentration; bothre really close to the MRL.

The multivariate analytical sensitivity is estimated according toq. (2). Curve (a) in Fig. 2 shows the probability of false complianceersus the multivariate analytical sensitivity at the MRL for ˛ = 0.05,hereas in Table 7 the estimated value for ˛ = ˇ = 0.05 can be found.

hat is, the minimum combined total residues of sulfonamides thatan be distinguished from 100 �g L−1 with ˛ = ˇ = 0.05 is 5.3 �g L−1.n this case, the multivariate analytical sensitivity is also very closeo the MRL.

.2. Mixture of six sulfonamides calibration

Once checked that it is possible to work with all the six sulfon-mides jointly because they have the same multivariate analyticalensitivity when they are individually determined in milk byront-face fluorescence spectroscopy, samples spiked with differ-

ig. 2. Probability of false compliance, ˇ, versus multivariate analytical sensitivityt the MRL for ˛ = 0.05 for: (a) the joint calibration for the six sulfonamides (data set) and (b) PLS model for the mixture of the six sulfonamides calibration (data set B).ashed lines represent the multivariate analytical sensitivity value at ˇ = 0.05.

137.5

ent amounts of a mixture of them, data set B, are analysed. As it isshown in Table 2, two brands of milk are used, the spectra beingrecorded in two different days. In this way, the variability that canbe expected in the routine use of the analytical method is includedin the analysis.

To achieve it, a PLS model with the emission spectra from the 20calibration samples in Table 2 is performed, that is with a matrixof dimensions 20 × 101. Firstly, data are centred by column, andnext the regression models are built in such a way that the numberof latent variables chosen is the one which minimizes the mean ofabsolute value of relative errors (MAE) calculated from the eightprediction samples in Table 2. Models are fitted with three, fourand five latent variables, reaching the minimum value of MAE inprediction (MAEP) with four latent variables when more than 98.3%of the response is explained (see bold values in Table 8). None ofthe samples in this model was considered outlier by calculating Qresidual and Hotelling’s T2 statistics at the corresponding thresholdof 95%.

The concentrations estimated with the PLS regression for thecalibration and prediction samples are shown in Table 2, togetherwith the absolute value of the relative errors. The regression model“concentration computed by PLS versus true concentration” forthe calibration samples is checked for significance of regression,normality of residuals (Kolmogorov test), homoscedasticity of vari-ances (Cochran and Bartlett tests) and lack of fit by means of thecorresponding hypothesis tests, fulfilling them at the significancelevel of 0.05, as it can be seen in the third column of Table 6.

Table 7 shows the values of some performance characteristics(MAE for calibration and prediction samples, trueness, CC˛, CCˇand multivariate analytical sensitivity) calculated for the analyt-ical method. By comparing these results with the ones obtained

in Section 4.1 (with milk of a single brand fortified with individ-ual sulfonamide amounts that was analysed the same day) it canbe concluded that the effect of having a mixture of six sulfon-amides, samples from two brands of milk (different compositionof proteins, aminoacids and fat) measured in two different days

Table 8Number of latent variables (L.V.), explained variance and mean of absolute value ofrelative errors in prediction (MAEP) for the PLS models. The chosen model is in bold.

L.V. Explained variance (%) MAEP (%)

X-block Y-block

1 99.51 9.922 99.55 71.323 99.59 93.83 14.254 99.63 98.31 10.015 99.66 99.73 14.63

N. Rodríguez et al. / Analytica Chimica Acta 657 (2010) 136–146 143

Table 9Means, x, and standard deviations, �, of the computed PLS values for the compliantand non-compliant samples and p-values for the four normality tests applied to bothdistributions.

Class 1 (compliantsamples)

Class 2 (non-compliantsamples)

n 18 11x 1.14 1.77� 0.24 0.25Chi-square (p-value) 0.74 0.32

idiuMx1ri

4s

ompatssb82s

tcvtoda

msndd

ftca1fibw

ecm

non-compliant samples.Since consumers’ health must be guaranteed by means of

current legislation, the presence of false compliant samples inscreening methods must be avoided, therefore the Decision

Shapiro–Wilks (p-value) 0.86 0.62Skewness (p-value) 0.90 0.92Kurtosis (p-value) 0.86 0.38

ncreases all performance characteristics of the method. This isue to the increase of the residual standard deviation, �, which

nfluences all of them. For example, CC˛ and CCˇ take the val-es 17.7 and 35.0 �g L−1 respectively, which are well below theRL for the determination of a mixture of sulfonamides in milk. At

0 = 100 �g L−1 CC˛ changes from 102.4 to 117.0, CCˇ from 104.8 to33.6 and multivariate analytical sensitivity from 5.3 to 37.7 �g L−1

espectively. The curve (b) in Fig. 2 shows the analytical sensitivityn these new conditions.

.3. Class modelling (PLS-CM) for screening of mixtures ofulfonamides

In Sections 4.1 and 4.2 the viability of using frontal-face flu-rescence emission spectra for quantifying the total amount of aixture of sulfonamides is shown. The aim of this section of the

aper is to build a classification model in order to be able to say ifmilk sample is compliant (has an amount of sulfonamides equal

o or lower than 100 �g L−1) or non-compliant (has an amount ofulfonamides higher than 100 �g L−1) according to European Deci-ion [5]. To achieve it, the procedure explained in Section 2.1 haseen followed. C and D data sets are used jointly and the total8 milk samples are divided into two groups: a training set with9 milk samples from data set C and a test set with the rest milkamples.

As in the previous multivariate analysis, the data are mean cen-red before building the PLS calibration models, however in thisase the response variable is not the concentration but a binaryariable which differentiates among those samples which exceedhe MRL and those that do not, as it is outlined above. The minimumf RMSECV (0.37) is reached with two latent variables. No outlierata are found according to Q residual and Hotelling’s T2 statisticst a threshold of 95%.

On the other hand, according to the p-values of the nor-ality tests (chi-square, Shapiro–Wilks, skewness and kurtosis)

hown in Table 9, the values predicted by PLS for compliant andon-compliant samples are normally distributed. Therefore bothistributions are characterised by their means and their standardeviations which are also shown in Table 9.

The graphical representation of the probabilities ˛ and ˇ as aunction of the value tv can be seen in Fig. 3. For a screening methodhe Decision 2002/657/CE [5] requires that the probability ˇ of falseompliance is less than 5%. From the curves in Fig. 3, ˇ = 0.05 ischieved with a probability ˛ equal to 0.17 and a critical value of.37. This means a sensitivity of the model equal to 83% and speci-city of 95%, i.e. the model accepts 83 out of 100 samples whichelong to the right class and rejects 95 out of every 100 sampleshich do not belong to it.

The risk curve corresponding to class modelling by the hypoth-sis test is shown in Fig. 4. With the previous results, it can beoncluded that the proposed procedure fulfils with the require-ents of the Decision 2002/657/CE.

Fig. 3. Probabilities of false non-compliance, ˛, and of false compliance, ˇ, as afunction of threshold value tv. Dashed lines represent ˇ = 0.05, which correspondsto tv = 1.37 and ˛ = 0.17.

In order to assess in prediction the class model previouslydescribed, it is applied to the 58 milk samples of test set. The sam-ples of this data set are shown in Table 10; the first 46 are those ofset D, whereas the rest 12 correspond to samples of set C (with 100and 170 �g L−1 of combined total sulfonamide concentration).

Those samples with values of Q residual and Hotelling’s T2 statis-tics higher than the critical values at a confidence level of 95% aremarked in Table 10. Samples 7, 9 and 10, are prepared from the Vmilk, whereas samples 23, 24, 25 and 26 are from Z milk, which indi-cates a matrix effect related to these brands of milk that would haveto be studied. In any case, PLS-CM avoids an inadequate applicationof the screening method to these samples.

Table 10 also shows the computed by PLS regression valuesfor the other 50 samples. By regarding these values and once thethreshold value of the class model hypothesis test is fixed at 1.37, itcan be concluded that samples 17 and 18 are considered false com-pliant samples whereas samples 51 and 55 are considered false

Fig. 4. Risk curve corresponding to the PLS-CM fitted to samples whose concentra-tion is equal or lower than MRL (100 �g L−1).

144 N. Rodríguez et al. / Analytica Chimica Acta 657 (2010) 136–146

Table 10True and calculated by PLS encoded response for the 58 test samples in the screening method.

Sample number True encoded response Calculated encoded response Sample number True encoded response Calculated encoded response

1 2 1.84 30 2 1.412 2 1.80 31 2 1.443 1 0.88 32 2 1.394 1 1.05 33 2 1.445 2 1.81 34 2 1.386 2 1.60 35 2 1.527 1a – 36 2 1.578 1 0.95 37 2 1.459 2a – 38 2 1.63

10 2a – 39 2 1.5011 1 1.29 40 2 1.4712 1 1.30 41 2 2.0813 2 1.53 42 2 1.8814 2 1.72 43 2 1.9915 1 0.74 44 2 1.6916 1 0.93 45 2 1.6917 2 1.25 46 2 1.7818 2 1.11 47 1 1.3019 1 0.83 48 2 1.5620 1 1.17 49 1 1.2221 2 1.51 50 2 1.6322 2 1.51 51 1 1.6623 1a – 52 2 1.5624 1a – 53 1 1.2225 2a – 54 2 1.7126 2a – 55 1 1.3927 2 1.61 56 2 1.85

2ta4e

5

mtt

fssooca

cstinf5wD

A

C

28 2 1.5929 2 1.57

a Sample with high values of Q index and Hotelling’s T2 at the 95% threshold.

002/657/CE [5] requires probabilities ˇ of false compliance lesshan 5%. By only paying attention to test samples encoded with 2s response in Table 10, it can be concluded that from a total of0 samples, 2 false compliant samples are found which means a ˇrror of 5% which is in accordance with European Decision.

. Conclusions

It has been checked that the amount of sulfonamide contained inilk samples measured by means of front-face fluorescence spec-

roscopy cannot be determined with an univariate approach andhat PLS models with emission spectra are needed.

It has been verified that the multivariate analytical sensitivityor the six sulfonamides, estimated from the PLS calibrations, isignificantly equal at a confidence level of 95%. Therefore it is pos-ible to handle their fluorescence signals jointly. Thus, the effectf working with a mixture of the six sulfonamides and samplesf two brands of milk measured in two days, makes increase thealculated performance characteristics, that for the multivariatenalytical sensitivity goes up from 5.3 to 37.5 �g L−1.

Finally, it is shown that front-face fluorescence spectroscopyombined with PLS-CM is an efficient screening method to clas-ify milk samples as compliant or non-compliant without the needo quantify the combined total sulfonamide concentration andndependently of the brand of milk analysed. Probability of falseon-compliance, ˛, equal to 0.17 is obtained in the proposed model

or ˇ = 0.05. Moreover, this PLS class model is applied to a set of8 test samples and two false compliant samples have been foundhich means a ˇ value equal to 0.05 required by the Europeanecision.

cknowledgments

The authors thank Ministerio de Ciencia e Innovación (ProjectTQ2008-02264), Junta de Castilla y León (Project BU024A07),

57 1 1.2858 2a –

FEDER funds, University of Burgos and Caja de Burgos for financialsupport. N. Rodríguez thanks the JCyL for her FPU grant.

Appendix A.

A.1. Univariate analytical sensitivity.

Consider an analytical method for which an univariate calibra-tion model may be established (signal Y versus concentration X ofan analyte) according to the following linear model

Y = a + b X + ε (A1)

The residuals ε in Eq. (A1) are supposed to be normallydistributed random variables, independent of the value of the con-centration, with null mean and common variance, �2.

Let (xi,yi), i = 1,. . ., n be the values of the concentration and thecorresponding recorded signals in n calibration standards.

Let a, b and �2 be the estimates, obtained by least squares, ofparameters a, b and �2 of the model in Eq. (A1).

Definition 1. The hypothesis test to decide whether or not theconcentration determined in a sample by using the analyticalmethod is equal to a nominal value x0 is as follows—Null hypoth-esis: “the concentration of the sample under consideration is notdifferent from the nominal value” against the alternative hypoth-esis: “the concentration under consideration is different from thenominal value”.

Formally X = x0 is the null hypothesis and X /= x0 the alterna-tive one. If the null hypothesis is true, the probability of wronglyrejecting it is ˛.

Definition 2. The critical value of the response variable, y , is the

c

value of the signal increment that corresponds to the critical valueof the hypothesis test in Definition 1.

Definition 3. The minimum discriminable value of the analyte con-centration specifies the true value of the analyte concentration for

Chim

wˇ

Pnse

x

Plw

E

a

V

w

w

a

�

It

nvf

C

a

˛

ˇ

nto

y

w

iii

wab

�

c

N. Rodríguez et al. / Analytica

hich the probability of wrongly accepting the null hypothesis is. The power of the test is 1 − ˇ.

roposition 1. The minimum discriminable concentration, xd, at theominal level x0 with previously fixed values ˛ and ˇ (analytical sen-itivity of the method) is obtained from the solutions of the followingquation:

d = |x − x0| = �xwx0 �

b(A2)

roof. The estimated signal y at a concentration x of the ana-yte follows a Student’s t-distribution with n − 2 degrees of freedom

hose mean is

(y) = a + b x (A3)

nd whose variance is

ar(y) = w2x �2 (A4)

here

2x = 1

k+ 1

n+ (x − x)2∑n

i=1(xi − x)2(A5)

nd

ˆ 2 =∑n

i=1(yi − yi)2

n − 2(A6)

n Eq. (A5) k denotes the number of replicated determinations onhe test sample.

For a given sample, the decision on accepting or rejecting theull hypothesis must be established on the basis of the responseariable Y. The critical region, CR, of the test can be written asollows:

R = {|Y − y0| > yc} (A7)

The false non-compliance and false compliance probabilities, ˛nd ˇ respectively, are the conditional ones

= pr{|Y − y0| > yc |X = x0} (A8)

= pr{|Y − y0| ≤ yc |X /= x0} (A9)

The false non-compliance, ˛, is a Student’s t-distribution with− 2 degrees of freedom, a mean y0 = a + bx0 and a variance equal

o w2x0

�2. Thus Eq. (A8) allows one to determine the critical valuef the signal, yc, for the significant level ˛

c = wx0 � t˛/2 (A10)

here t˛/2 is the critical value of a Student’s t with n − 2 d.f.Nevertheless, the false compliance, ˇ, is a function of the x, that

s, depends on how much the true concentration x differs from nom-nal x0 one. For each x, by substituting the value of yc in Eq. (A9) its:

ˇ(x) = pr{|Y − y0| ≤ wx0 � t˛/2|X = x}

= pr

{∣∣∣∣Y − y0

wx0 �

∣∣∣∣ ≤ t˛/2|X = x

}

= pr{t(�x) ≤ t˛/2} − pr{t(�x) ≤ −t˛/2}

(A11)

here t(�x) is a non-central Student’s t-distribution with n − 2 d.f.,mean equal to y − y0 and a parameter of non-centrality estimatedy means of Eq. (A12)

x = b|x − x0|wx0 �

(A12)

Eq. (A12) allows one to estimate the minimum discriminableoncentration xd at the nominal level x0 by means of Eq. (A2).�

ica Acta 657 (2010) 136–146 145

Definition 4. For each value of probability ˛, the operating charac-teristic curve of the hypothesis test of Definition 1 is built by meansof Eq. (A12) which allows one to calculate the value of ˇ for eachconcentration, x.

Once the least squares regression line is obtained, one has theestimated concentrations xi for the signals yi, i = 1,. . ., n, recordedfrom the calibration standards.

Proposition 2. If data (xi, yi) are transformed into (xi, y′i), i = 1,. . ., n,

by means of a linear transformation, that is y′i= p + q yi, the regression

of Y′ on the concentration X gives the same analytical sensitivity as theregression of Y on the concentration X stated in Proposition 1.

Proof. With the transformation proposed, the model of Y′ whenX = x is

Y ′ = p + q(a + b x + ε)= (p + qa) + qb x + qε= a′ + b′ x + ε′

(A13)

The estimates of the new residual standard deviation, � ′, andthe slope, b′ in relation to the earlier ones are

(� ′)2 =∑n

i=1(y′i− y′

i)2

n − 2

=∑n

i=1((p + q yi) − (p + q yi))2

n − 2= q2�2

(A14)

and

b′ = q b (A15)

The other two parameters of Eq. (A2) are: the non-centralityparameter �x, which depends on ˛ and ˇ. The factor wx0 is obtainedby substituting x by x0 in Eq. (A5) and only depends on abscissas xi,therefore it does not change when transforming the response. As aconsequence, if |x − x0|′ denotes the amount that can be discrimi-nated with the new regression Y′ versus X it is

|x − x0|′ = �xwx0 � ′

b′= �xwx0 q �

q b= |x − x0| � (A16)

So, it can be concluded that the analytical sensitivity is the same.

Proposition 3. The analytical sensitivity of a method obtained bymeans of the calibration line signal, Y, versus true concentration, X, isthe same that the one obtained when making the regression estimatedconcentration, X , versus true concentration, X.

Proof. It is enough to apply Proposition 2 to the case where q =1/b and p = −a/b. In this way, for each i = 1,. . ., n it is

y′i = yi

b− a

b= xi (A17)

In other words, the analytical sensitivity estimated through thevalues (xi,yi), i = 1,. . ., n, is the same that the one calculated with thevalues (xi, xi), i = 1,. . ., n.�

A.2. Multivariate analytical sensitivity

In the multivariate case, the calibration function has, in general,the form x = f (v), x being the concentration and v a vector (firstorder signal) or a tensor (second or higher order signal). In practice,

independently of the form of the function f one has the values ofconcentration of calibration standards, xi, and of the concentrationscomputed by means of the calibration function, xi. By applying Eq.(A2) to the linear regression X versus X, the analytical sensitivity ofthe method is obtained.

1 Chim

R

[[[

[

[[

[[

[[

[

[[

[

[

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[[

[

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46 N. Rodríguez et al. / Analytica

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