Multivariate decision and detection limits

Analytica Chimica Acta, 277 (1993) 205-214 Elsevier Science Publishers B.V., Amsterdam

205

Multivariate decision and detection limits Anita Singh Lockheed Environmental Systems & Technobgies Co., Environmental Sciences & Techmlogies Division, 980 Ke& Johnson Drive, Las Vegas, NV89119 (USA) (Received 3rd September 1992)

Abstract Principal component analysis (PCA) is used to develop an approach for estimating multivariate decision and detection limits (MDDLs) for gas chromatography-mass spectrometry studies where the instrument response is multivariate in nature. Many definitions and estimators have been published for univariate responses. In this article we extend these ideas to the multivariate case. When the first principal component explains most of the variation contained in the data, it may be used to express the multicomponent instrumental response as a univariate composite signal representing all of the monitored ions associated with the analyte of interest. The first principal component of these ions has been used to derive decision and detection limits through two approaches. Back-transformation of this composite instrument response to the original response variables will yield multivariate decision and detection limits for all of the ions considered shnultaneously. Keywords: Mass spectrometry; Principal component analysis; Calibration model; Detection limits

In environmental chemical measurement it is necessary to detect low-level concentrations of hazardous compounds in various media. The instrument response resulting from the analysis of samples or standards that have low concentrations may be difficult to distinguish from the inevitable background noise. Uncontrolled noise factors operate whether or not the analyte of interest is present. These uncertainties are addressed in this paper for multicomponent instrument responses that are characteristic of gas chromatography-mass spectrometry (GC-MS). The decision and detection limits (DDLs) are often expressed in terms of the analyte concentrations. In practice, however, a chemist observes the instrumental response first. Therefore, these limits are determined in the signal space first and then transformed to the concentration domain by Correspondence to: A. Singh, Lockheed Engineering and Sciences Co., Environmental Programs Office, 980 Kelly Johnson Drive, Las Vegas, NV 89119 (USA).

means of a suitable calibration model representing the relationship between the instrumental response and the analyte concentration. Thus, the problem is twofold: (1) obtain an appropriate calibration model for established standards using suitably designed experiments, and (2) develop appropriate procedures to estimate the DDLs using this calibration curve. Assessing the DDLs of an instrument is not an easy task. There is much controversy in the chemical literature regarding the definition of these limits. Because these limits indicate the performance of an instrument at detecting low-level concentrations, an analytical chemist should be clear about the method used in the derivation of these DDLs. In this paper, we will not try to distinguish between the instrument detection limits (based on standards) and the method detection limits (based on spiked samples for a particular matrix). It is assumed that the chemist using these procedures will differentiate between these limits accordingly.

0003-2670/93/$06.Otl 0 1993 - Elsevier Science Publishers B.V. All rights reserved

206

Currie [l] used the univariate instrumental response corresponding to blank analyses (in which the substance sought is absent) to establish univariate DDLs. He defined these limits in terms of the instrumental responses only. To define these limits in terms of analyte concentration of interest, it is necessary to rely on a calibration curve relating the instrument responses to concentration levels in established standards. Hubaux and Vos [2] used least squares regression of the univariate response onto the analyte concentration of a certain number of calibration standards. They used this regression function as an estimate of the true calibration curve. It is anticipated that any new standard will yield a signal falling in the neighborhood (regression band) of this regression line. Using this calibration curve, they defined the decision limit in the response domain and the detection limit in the concentration domain. Also Garner and Robertson [3] reviewed various methods of estimating these DDLs for the univariate case. In this article, we extend these ideas to the multivariate case, where the instrumental response is inherently multivariate such as in GCMS studies. Principal component analysis (PCA) [41 plays an increasingly important role in recent developments in chemometric literature. Due to the complex nature of the multicomponent analytical data obtained in GC-MS studies, in practice only one or two most abundant ions are used to quantitate the analyte concentration of interest. Information on the other ions is usually applied only for identification and confirmation. Due to the high precision in the measurement process in calibration experiments of the GC-MS studies, the first principal component of the relevant monitored ions typically accounts for most of the variation (2 99%) in the data. Without loss of significant amount of information, the first principal component can be used to represent the composite instrument response, which in turn can be used to quantitate the analyte of interest. Moreover, in calibration experiments, the magnitude of the last few principal components can be used to determine how well the first few principal components fit the observations. Using these first few principal components, one can back-trans-

A. Singh /Anal. Chim. Acta 277 (1993) 205-214

form the principal component scores to the original response variables (if desired). Delaney [5] used PC to obtain these limits using the Hubaux and Vos approach. Least squares regression has been used to obtain a calibration function for the composite instrument response onto the analyte concentration in established standards. Quantitation of the analyte concentration based on this calibration curve uses information on all of the relevant major ions and, in general, yields more precise estimates of the analyte concentration than the classical univariate techniques. Moreover, this calibration curve can be used to define the DDLs in terms of the analyte concentration. To obtain a calibration curve that can be used efficiently to estimate these limits, it is important to follow an appropriate design to carry out these experiments. Enough samples should be included in the design of experiments, so that one can obtain reasonably precise estimates of the unknown response parameters at low concentration levels. Moreover, because the principal components are linear functions of the original response variables, an appeal to the central limit theorem helps to justify the normality assumption for the principal components, even when the original response variables are not Gaussian. This justifies adaptations of the Currie and the Hubaux-Vos type DDLs, both of which assume Gaussian distribution for the instrumental response at all levels of the analyte concentration. In this article, we derive these limits by using two approaches, those of Currie and of Hubaux and Vos. Comparisons have been made between the results obtained by using (1) only the one or two most abundant ions (univariate approach) and (2) the composite response based on the first principal component (multivariate approach). The scores for the first principal component are transformed back to the original responses to give rise to MDDLs. It is important to transform the composite DDLs to MDDLs to ensure that these MDDLs lie within the predefined identification region for the target analyte to confirm its presence. Calibration data of 2,3,7,8-TCDD obtained according to the USEPA contract IFB WA 84-A002 have been used in this study. The pre- and post-transformed

A. Singh/Anal. China.Acta 277 (1993) 205-214

data are in close agreement in Table 3.

207

and are summarized

MATHEMATICAL FORMULATIONS

‘~1 be the multicompoLet R'=(rl, r2,..., nent response representmg the relative peak areas of p( 2 1) ions of interest included in a particular GC-MS study. Let xi, x2,. . . , x, be the n concentration levels in established standards included in the study. The observed response matrix thus obtained can be expressed as follows:

Ijj!

zz:

/_j

zz]=[z]

(I)

The dispersion matrix for the p-responses is given by

s=

~(R~-R)(R,-E)'/(w-~) i=l

where @ = f Ri/n i=l

Principal component analysis Let A,, AZ,..., A, be the p-eigenvalues of the dispersion matrix S and e,, e,, . . . , eP be the cor-

responding eigenvectors. The p-principal components are given by yi = ejR; i = 1, 2,. . . , p. The proportion of variation explained by the first k principal components is Av, = C~=ihi/CiP,iAi; k = 1, 2,. . . , p. In practical applications, when Av, is large enough ( 2 0.951, without loss of a significant amount of information contained in the data, one can use the first principal component y, as the composite instrument response representing all of the p ions included in the study. Estimates of the DDLs can be obtained in terms of this composite response, which in turn can be back-transformed to the p-dimensional original response vector L giving rise to the MDDLs. If these MDDL vectors fall

in the p-dimensional identification region I and meet alI of the identification criteria set by the researchers, then the presence of the analyte in a particular field-sample can be confirmed. The back-transformation can be performed by expressing the response vector Rj as a linear combination of the PCs as follows k

Rj=

Cyijei;

j=l,2

,..., n,and

k=1,2

,..., p

i=l (2)

magnitudes of the last few principal components determine how well the first few principal components fit the observations. That is Cf_ i yijei differs from Rj by cip,k+iyijei, when the first k principal components are used to fit the observations. In the dioxin study considered in this article, the first principal component alone was adequate to reproduce the original response matrix. The

Calibration model

A calibration curve is routinely used for quantitation of an analyte of interest. Moreover, the estimates of the DDI_s are first obtained in the response space and converted to the concentration domain through an appropriate calibration model. Thus, it is important to obtain a calibration model that is suitable for both purposes. The general calibration problem can be described by the following model Y(X) =Bo+&,

B) +E

(3)

where y(x) is the composite observed response at concentration level X, &, is the expected background noise, B is the vector of regression parameters, and E represents the random measurement error. Routinely, these measurement errors are assumed to be independently and identically distributed as N(0, a21 at all concentration levels. If the normality assumption is violated, a suitable Box-Cox type transformation given by Johnson and Wichem [6] can be used to transform the data to near-normality. Another assumption which is often violated is the homogeneity of variances. Variances are about the same at low concentrations, but are statistically significantly different at higher concentrations. Statistical tests

A. Singh /Anal. Chim. Acta 277 (1993) 205-214

208

should be performed to test the homogeneity of variances that in turn require sufficient number of observations at each concentration to be included in the calibration experiment. An estimate of the functional form h(x, B) of the model is obtained using the ordinary least squares (OLS) regression for homoscedastic responses or the weighted least squares GVLS) regression for heteroscedastic responses as described in Draper and Smith [71. In order to get the best possible calibration model, it is important to design the experiments with clear objectives about its usage in mind. If this model has to be used to estimate the DDLs, then enough experiments at low concentrations should be included in the study. If this curve is also to be used for quantitation and prediction purposes, then the researcher should include enough experiments covering the practical range in the concentration domain. With experiments that are appropriately designed, more meaningful and precise results can be extracted from the available data. The simplest calibration model is given by Y(X)

=Po+&x+E

(4)

where the estimates of &, and pi are obtained by using the OLS or the WLS regression on n calibration experiments. The estimates using the OLS regression are given by

by Eqn. 4 is also a maximum likelihood estimator (MIX) of x,,. This estimator, however, has infinite mean square error (see Shukla [S]). Because no other uniformly better (see Krutchkoff [9] and Wiliams [lo]) estimator is available in the literature, this classical MLE R, is widely used in practice. Moreover, the mean E[_&,] and variance V[&J of 4, are infinite. Therefore, the interval estimate (xL, x,) for x0 with confidence coefficient (1 -a+) is obtained by using the regression band around the model given by Eqn. 4. This (1 - a)lOO% confidence band is represented by the two curves given by Eqn. 6, for various values of the analyte concentration x within the region used in the calibration experiment.

*Bf[(x-~)2/s,,+l/r+l/n]1’2 (6) where jXx> is the average composite response based upon r replicates at concentration x. The corresponding confidence interval for a particular analyte concentration x0 within the experimental region is given by (xL, xJ, where XU

XL

&=

j~~(~~-x)(Yi-~)~j~~(xi-~)2

and

s2= i

(yi-Y"J2/(n-2)

i=l

with $i=&+$l~i;

i=l,

2,...,n

(5) Chemists measure the instrumental response first and then estimate the analyte concentration through a suitable calibration model. Thus, the inverse calibration problem is to estimate the analyte concentration x0 for an observed response y0 (based upon some r 2 1 replicates). The classical point estimator f, of x,,, obtained

I

=4,+

1 -x)&T+ (‘B/p^1)[&-q2/s,, (xl0

+(1-&(1/n

+

W)]1’2]/P-g) (7)

with g = t2G22/(&,,), s,, = c(xi -Zj2, and t is the student’s t-value with (n - 2) degrees of freedom, and is chosen such that P[ yL < y, < yu] = 1 - (Y,for a chosen level of significance (Y.The graphical representation of these expressions is given in Fig. 1. We now define some statistical terms. Suppose that a chemist has analyzed a sample and observed the (composite) response y which is assumed to be normally distributed with probability density function given by

f,(Y) =

-$yp[

-(Y

-cLx)2/2@Z] (8)

x

where x represents

the amount of the analyte

209

A. Singh/Anal. Chim. Acta 277 (1993) 205-214

in detection with probabilities (r and p respectively. Type I error. Reject HO when H,, is true, i.e, identify the background signal as the sample signal; this error constitutes a false positive. I)rpe 2 error. Reject Hi when Hi is true, i.e., identify the sample signal as the background signal; this error constitutes a false negative. (Y= P(Type

1 error) = P( y > yZ I H,,)

m

f(h)

=

/

/3 = P(me

Concontrrtion

Fig. 1. Definitions of (a) the limit of decision Lo; and (b) the lit of detection x:.

present in the sample, CL, and 0,’ represent the corresponding mean and variance of the instrument response y at concentration level of x. Unless otherwise necessary, we do not use the subscript x in the following discussions. In practice, the random background noise present during the analytical process operates, whether or not the analyte is present. This noise gives rise to the instrument response yr, distributed as A&.+ C& even when the analyte sought is absent from the sample. The parameters pg and ai are estimated from several replicates of blank samples. The central problem that the analyst now faces is to decide whether the instrumental response y is a result of the presence of this analyte or simply represents the background noise (or an analytical blank). In statistical terminology, an analyst wants to choose between the following two hypotheses:

dyB

2 error) = P( y < yi I HI)

where y, is the instrument response with mean CL, and variance u: corresponding to an amount c of the analyte found in the sample. As the amount c decreases, it becomes more difficult to distinguish between the true response y, and the background response yn. The detection problem that the chemist now faces is how to resolve the differences between yn and yC while keeping the error probability cy and /3 acceptably small. Throughout the rest of the paper, we will be making the assumption that the response standard deviations at low concentrations are equal, i.e., an = cL = a, d. These ideas are summarized in Fig. 2. ’

Ho: the analyte sought is absent, i.e., p= pa H,:

the analyte sought is present, i.e., p > pa

While resolving these uncertainties, the analyst is likely to commit the following two types of errors

(9

Yz

Inetrumontal Rooponse

Fig. 2. Error probabilities.

A. Singh /Anal. Chim. Acta 277 (1993) 205-214

210

Currie approach Assuming normality of the instrument responses, Currie [l] defined yz = pB + t,~n as the decision limit L, for (Y= 0.0013 and z, = 3.0. Statistically, L, represents the lowest meaningful (composite) response significantly different from the null response. Kaiser [ll], defined L, as the limit of detection. Also International Union for Pure and Applied Chemistry (IUPAC) [12] has recommended L, as the limit of detection. However, for samples with the instrumental responses centered around L,, the Type 1 error probability cy is small (= 0.0013), i.e., signals larger than L, represent the presence of the analyte with great precision, but the Type 2 error probability /3 is at least 0.50, i.e., signals smaller than L, can be interpreted as the absence (false negative) of the compound with poor precision, which is unacceptable. Thus L, cannot be used as a criterion of analyte detection for an analytical procedure. The response L, + z,a, becomes Currie’s detection limit L, for LY= 0.0013. Let yr = L, - zsaB. For instrumental responses centered around L,, a response smaller than y1 will be treated as the null response with /? as the probability of false negatives. For (Y= p = 0.0013, both yi and y, coincide at L,. These limits are expressed in Fig. 3. Thus a signal y greater than L, gets detected with high probability of at least 0.9987, whereas signals smaller than L, also get detected with

lesser accuracy. For example, a signal of magnitude L, + 1.960-, gets detected with probability 0.975. Thus far, L, and L, have been defined in terms of the composite instrument responses. In practice, the researcher may be interested in obtaining these limits in terms of the original p-responses, which are obtained by back-transforming the principal components to the original p-responses. These limiting response vectors are given as follows:

j=l

(10)

k

Lh = C Ldjej

=

(Lid, &j,. . . , Lpd)

j=l

where k is the number of PCs used to reproduce the observed response matrix given by Eqn. 1. The response vectors L, and L, should lie within the predefined p-dimensional identification region I to confirm the presence of the particular analyte of interest. These limits are defined in the response space, and by means of the calibration model, one can obtain the corresponding estimates of the DDLs in the concentration domain which we denote by X, and xd respectively. In practice X, and xd are estimated in more than one way. Different methods often lead to significantly different estimates of these limits. One commonly used approach (see Miller and Miller [13]) is to use the following equations to obtain the point estimates of x, and xd:

and

xdp=(Ld40)/~1

Inotrumontal Reapma Fig. 3. Definitions of (a) the limit of decision L,; and (b) the limit of detection Ld.

However, statistically the lowest analyte content that could be distinguished from being the null content is x,” (and not $1, where x,” and # are the corresponding upper fiducial limits given by Eqn. 7 for X, and xd respectively. We recommend the use of x,” and xy as the estimates of DDLs in the concentration space. Currie [l] used normal distribution in the derivation of these DDLs, assuming that pn and ai are known in advance. However, in practice

A. Singh/Anal. Chim. Acta 277 (1993) 205-214

these parameters have to be estimated from some b number of blank samples. Therefore, it is more appropriate to use the student’s t-distribution with (b - 1) degrees of freedom, rather than the conventional normal distribution in the derivation of these limits. Using student’s t-distribution, Currie-type DDLs in the response domain are given by the following equations:

and

where t,_, is the tabulated t-value with (b - 1) degrees of freedom for some significance level of (Y. Hubaux- Vos approach Hubaux and Vos [2] used the regression band around the calibration curve given by Eqn. 4 to derive the DDLs. They defined the decision limit in the response space and the detection limit in the concentration domain. The composite decision limit L, can be obtained from Eqn. 6 by lettmg r + 00 in equation L, = yJO), with jX0) = &,. The corresponding estimate of the detection limit x3 is, given by X: of Eqn. 7 with f, = (L, - &J/&. Thus, for appropriate values of (Y and /3, we have P(y,>L,Ip=Q=a and P[y, L, I H,) 2 1 - p. The decision limit vector L, in the original response space can be obtained by using Eqn. 10. In most applications, the instrumental responses are heteroscedastic, therefore, the WLS regression [7] should be used rather than the OLS regression to obtain the calibration model represented in Eqn. 4. But in practice, since enough experiments are not available at each of the concentration levels included in the calibration experiments, one is forced to use the OLS regression to obtain Eqn. 4. This overestimates the error variance (T’, which in turn yields inflated values of the DDLs when using the Hubaux and Vos approach.

211 EXPERIMENTAL

A GC-MS instrument was calibrated for 2,3,7,8-tetrachlorodibenzo-p-dioxin (TCDD) through three injections each of five standards. This experiment involved one instrument at a single laboratory. The calibration was performed as described in USEPA contract IFB WA 84-A002 for determining 2,3,7,&TCDD in soils and sedi. ments. The five standard solutions were provided by the USEPA. Each of these solutions contained the internal standards, r3C,,-labeled 2,3,7,8TCDD, at a concentration of 1 ng ~1~‘. These solutions contained unlabeled 2,3,7,8-TCDD at concentrations of 0.2, 1.0, 5.0, 20.0, and 40.0 ng are cI1-l, respectively. These concentrations equivalent to 50-~1 concentrated extracts of 10-g samples containing 1, 5, 25, 100, and 200 ppb, respectively. Through selected ion monitoring, responses were observed for the mass-to-charge (m/z> ratios 257,320,322,328,332, and 334. The 2,3,7,8-TCDD yielded responses at m/z 320 and 322, with a fragment at m/z 257, while the internal standard is observed at m/z 332 and 334. A surrogate compound, 2,3,7,8-TCDD labeled with 37C1 was present in the calibration also, with response at m/z 328, but surrogate information is used in quality control for sample analysis and is not part of our approach for estimating instrument detection limits. The least squares regression model (Eqn. 41, gives the most precise results when the measured. instrumental responses correspond to a point close to the centroid of this regression line. Because the main objective of this study is to estimate MDDLs, only solutions with low concentration levels of 0.2, 1.0, and 5.0 ng pl-’ of unlabeled 2,3,7,8-TCDD are used to develop the calibration model (Eqn. 4). The model thus obtained might not be ideal for general purpose calibration over the whole practical range of the analyte concentration. Six (b = 6) available reagent blanks are also included in the study. There are six observations at a concentration level of 0.2 ng pl-r, three from the initial calibration and three from the continuing calibration. In order to avoid the effect of the instrumental settings, the statistical

A. Singh /Anal. Chim. Acta 277 (1993) 205-214

212

analysis is carried out on the relative instrumental responses, rather than on the original mass counts. The relative responses for m/z 257, 320, and 322 are given as follows: ri = (peak area at m/r 257)/D, r2 = (peak area at m/z 320)/D, r3 = (peak area at m/z 322)/D, where D = (peak area at m/z 322 + peak area at m/z 334).

TABLE 2 Hubaux-Vos approach using 98% confidence band PC analysis Lo (decision limit) x: (detection limit) (ng ~I-‘) _rE (point estimate) (ng ~1~‘1

322 alone

320+322

0.03966 0.03371 0.05295 0.1157 0.1156 0.1115 0.0586 0.0586 0.0565

RESULTS AND DISCUSSION

The DDLs are obtained using the conventional univariate as well as the proposed multivariate approaches. The univariate DDLs have been obtained in two ways (1) using the most abundant ion 322 and (2) using the sum of the two most abundant ions 320 and 322. PC analysis has been carried out on relative response vectors R,; i = 1, 2,. .., 18. The three eigenvalues of the variance covariance matrix S are 1.51077,0.00015, and 0.0000285. The first PC yi alone explains 99.987% of the variation contained in the data. The calibration curve using this PC is given by yi = 0.0005761 + 0.6662~ with the correlation coefficient r = 0.9991. The calibration curve using response y at m/z 322 is given by y = 0.004643 +0.4959x with r = 0.9991; whereas the model based on the sum y of the responses at m/z 320 and 322 is given by y = 0.003078 + 0.8824x with r = 0.9991. Each of the models fits the data equally well. The DDLs using these models are summarized in Table 1 (Currie approach) and

Table 2 (Hubaux and Vos approach). The units used for all of the estimates in the concentration domain are in ng ~1~ ‘. Regression bands with a confidence coefficient of 0.98 and a t-value of 2.5835 (with 16 degrees of freedom) have been obtained for each of the three models. These confidence bands have been used in estimating the DDLs in the concentration domain for both of the approaches. In this study, from Tables 1 and 2, it is obvious that both of the multivariate as well as the univariate approaches produced similar results. However, one of the advantages of using the multivariate PCA approach is that, by means of back-transformation, the composite instrument response can be transformed to the original p monitored ions (here p = 3). The pre- and posttransformed data based upon the first principal component is given in Table 3. Using back-transformation, the following multivariate DDLs are obtained: L: = (0.00628, 0.011, 0.01412); L& = (0.00995, 0.01744, 0.02239) original Currie ap-

TABLE 1 Currie approach a

PB .

UB

L, XT (ng &l) x,” (ng ~1~l) Ld

x5 (ng @I-‘) xi (ng ~1~‘)

Original approach using normal distribution

Modified approach using Student’s t-distribution

PCA

322 alone

320 + 322

PCA

322 alone

320 + 322

0.00787 0.90370 0.01897 0.0276 0.0850 0.03007 0.0442 0.1016

0.00609 0.00316 0.01558 0.0220 0.0795 0.02507 0.0412 0.0984

0.00954 0.00447 0.02296 0.0226 0.0780 0.03637 0.0377 0.0930

0.02132 0.0312 0.0886 0.03477 0.0514 0.1085

0.01759 0.0260 0.0835 0.02908 0.0492 0.1064

0.02579 0.0258 0.0812 0.04205 0.0442 0.0994

a The r-value used in the blank estimates of the modified Currie approach is r5 = 3.3649.

213

A. Singh /Anal. Chim. Acta 277 (1993) 205-214 TABLE 3 Pre- and post-transformed relative responses Concentration (ng CLI-l) 0.0 0.0

0.0 0.0 0.0 0.0

0.2 0.2 0.2 0.2 0.2 0.2 1.0 1.0 1.0 5.0 5.0 5.0

Observed ion, 257

Back-trans. ion, 257

Observed ion, 320

Back-trans. ion, 320

Observed ion, 322

Back-trans. ion, 322

r1

11

r2

r2

r3

r3

0.006006 0.004429 0.002419 0.001233 0.005840 0.002276 0.034945 0.038642 0.040292 0.043042 0.030227 0.027873 0.184855 0.203213 0.198748 1.024497 1.128297 1.145603

0.003440 0.004270 0.001562 0.001532 0.003043 0.001556 0.042637 0.044128 0.043181 0.041539 0.041145 0.039634 0.224719 0.215732 0.219613 1.044511 1.123190 1.140042

0.005662 0.004712 0.002122 0.003536 0.003741 0.000893 0.070772 0.077429 0.072470 0.070000 0.069220 0.072144 0.399046 0.369567 0.375057 1.839075 1.960976 1.994428

0.006029 0.007485 0.002738 0.002685 0.005334 0.002726 0.074734 0.077348 0.075687 0.072809 0.072119 0.069472 0.393889 0.378136 0.384938 1.830825 1.968732 1.998272

0.006885 0.011697 0.003614 0.002917 0.006845 0.004608 0.102439 0.101663 0.100947 0.094983 0.099686 0.092323 0.519319 0.497636 0.511100 2.352620 2.530950 2.565619

0.007739 0.009608 0.003515 0.003447 0.006847 0.003500 0.095933 0.099288 0.097157 0.093462 0.092576 0.089178 0.505619 0.485397 0.494129 2.350152 2.527178 2.565096

preach - normal distribution; L: = (0.00705, 0.01236, 0.01587); L; = (0.01150, 0.02016, 0.02588) modified Currie approach - Student’s t-distribution (with t, = 3.3649); Lb = (0.0131, 0.0230,0.0295) Hubaux-Vos approach (using 98% confidence band). These multivariate limits L,,L, and L, should lie within the predefined identification region I (e.g., should satisfy the fingerprint criterion, etc.) to confirm the presence of the analyte sought. One of the criteria used in the identification of unlabeled 2,3,7,8-TCDD is that the ratio (m/z at 320)/(m/z at 322) should lie within the interval (0.67, 0.87). The three limits L,,L, and L,, given above, satisfy this criterion. In practical applications, there is a tendency to use the simple point estimates xi (Currie approach) or xi (Hubaux and Vos approach) as the limit of detection. But if one wants to associate some confidence level with these DDLs, it is recommended to use XI: (Currie approach) or xg (Hubaux and Vos approach) as the detection limit. From the above two tables, it is noticed that Hubaux-Vos approach generated slightly higher values of the DDLs. For hetroscedastic re-

sponses, it is recommended to use a separate regression model based upon data at low concentrations only for estimation of these DDLs. Finally, an instrumental response R' = r-J for which R belongs to the prede(q,Q,..., fined identification region I and ri 2 L, (Currie approach), for all i = 1, 2,. . . , p, indicates the presence of the analyte of interest, and a signal for which ri 2 L, (Currie approach), or rj r Li, (Hubaux-Vos approach) for all i = 1, 2,. . . , p, represents a signal that is statistically significantly different from the blank signal confirming the detection of the analyte with high probability. We acknowledge gratefully J. Armour, G.T. Flatman, N. Herron, D. Hewetson and J. Nocerino for their continuing help and valuable suggestions during the preparation of this manuscript. Although the research described in this article has been funded wholly by the U.S. Environmental Protection Agency through Contract 68X0-0049 with Lockheed Engineering and Sciences Co., it has not been subjected to Agency review. Therefore, it does not necessarily reflect the view of the Agency.

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A. S&h /Anal. Chim. Acta 277 (1993) 205-214

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