mass

Anal. Chem. 1083, 55, 835-841

Modern capillary GC/MS techniques, such as on-column injection and insertion of the column into the MS source, can guarantee the introduction of a known amount of compound into the ionization region much more reliably than earlier GC/MS methodologies. These experimental techniques offer potential as a method of measuring relative electron impact cross sections which is compatible with modern instrumentation. The covergence of improved GC/MS instrumentation with a theoretical basis for predicting GC/MS response should be of significant import to the field of quantitative GC/MS.

LITERATURE CITED (1) Sauter, A. D.; Betowski, L. D.; Smith, T. R.; Strlckier, V. A,; Beimer, R. G.;Colby. B. N.; Wilklnson, J. E. HRC CC, J . Hlgh Res. Chromatogr. Chromatogr. Commun. 1961, 4 366. (2) Buriinaame. A.; Deli, A.; Russell, D. H. S. Anal. Chem. 1962, 54, R363h409. Sauter, A. D.; Mills, P. E.; Fitch, W. L.; Dyer, R. HRC CC. J . Hlgh Res. Chromatogr. Chromatogr. Commun. 1982, 5 , 27-30. Sauter, A. 01.; Fltch, W. L.; Lopez-Avila, V. Presented at the I982 Amerlcan Chemical Society Meetlng, Las Vegas, NV, March 1982. Otvos, J. W.; Stevenson D. P. J . Am. Chem. SOC. 1956, 78, 546. Lampe, F. W.; Franklin. S. L.; Fleld, F. H. J . Am. Chem. SOC. 1957, 79, 6129. Rapp, D.; Enlplander-Golden, P. J . Chem. Phys. 1965, 43, 1464.

835

(8) . . Harrison. A. G.; Jones, E. G.;Gupta, S. K.; Nagy, G. P. Can. J . Chem. 1966, 4 4 , 1967. (9) Beran, J. A.; Kevan, L. J . Chem. Phys. 1969, 73, 3866. (10) Albertl, R.; Genoni, M. M.; Pascuai, C.: Vogt, J. Int. J . Mess Spectrom. Ion Phys. 1974, 14, 89. (11) Kleffer, L. J.; Dunn, G. H. Rev. M o d . Phys. 1966. 38, 1. (12) Fleld, F. H.;Franklin, J. L. “Electron Impact Phenomena and the Properties of Gaseous Ions”, Revised Edtlon; Academic Press: New Yo&, 1970:

181).

(13) Crowe,rA.; McConkey, J. W. Int. J . Mass Spectrom. Ion Phys. 1077, 24, 181. (14) Stevie, F. A.; Vasile, M. J. J . Chem. Phys. 1961, 7 4 , 5106. (15) Crable, G. F.; Coggeshall. N. D. Anal. Chem. 1956, 30, 310. (16) Reeher, J. H.; Flesch, G. D.; Svec, H. J. Org. Mass Spectrom. 1076, 1 1 , 154. (17) Stevenson, D. P.: Schlssler, D. 0. “The Chemlcal And Biological Actions of Radlation”; Haissinsky, M., Ed.; Academic Books Ltd.: London, 1961; Vol. 5. (18) Oono, Y.; Nlshimura, Y. Bull. Chem. SOC.Jpn. 1977, 50, 1379. (19) Bevlngton, P. R. “Data Reduction and Error Analysis for the Phycilcal Sciences”; McGraw-HIII: New York, 1969. (20) Mann, J. B. J . Phys. Chem. 1964, 68, 441. (21) Bondi, A. J . Phys. Chem. 1964, 88, 441. (22) Haberditzl, W. Angew. Chem., Int. Ed. Engl. 1966, 5 , 288. (23) Beran, J. A.; Kevan, L. J. Chem. Phys. 1969, 73, 3866.

RECEIVED for review December 2, 1982. Accepted January 24, 1983.

Confidence Limits in Isotope Dilution Gas ChromatographyIMass Spectrometry G. F. Moler,” R. R. Delongchamp, W. A. Korfmacher, B. A. Pearce, and R. K. Mltchum DepaHment of Health and Human Services, Food and Drug Administratlon, National Center for Toxicologlcal Research, Jefferson, Arkansas 72079

Sources of varlance In quanttlatlve analysls by Isotope dllutlon gas chromatogvaphy/mass spectrometry (ID-GC/MS) are Identlfled. Four ID-GC/MS llnear regresslon methods are examlned. A procedure for lncorporatlng all slgnlflcant varlances to produce slatlstlcally valld confldence llmlts Is described; the procedure has been Implemented wlth on-line GC/MS computcsr programs.

obtaining least-squares ID-GC/MS linear regressions.

SOURCES OF VARIANCE An ID-GC/MS standard curve is the line

Y=aX+b where X is the weight ratio

X= In determination of the concentration of the toxic pollutant 2,3,7,8-tetrachlorodibenzo-p-dioxin (TCDD) in environmental samples by isotope dilution gas chromatography/mass spectrometry (ID-GC/MS) (1,2), it became apparent that only calculating the cioncentration found was insufficient. In order for the quantitative results to have significant meaning in intralaboratory rind interlaboratory statistical analyses, it was necessary to obtain accurate confidence limits for these concentrations. No published comprehensive discussion of methods for deriving such confidence limits was found, therefore an investigation into this quantitative analysis problem was conducted, the results of which are presented here. In this paper, sources of variance which can affect IDGC/MS confidence limits are identified, the relative magnitudes of the variances are examined, and methods to reduce the magnitudes of some of these variances are explored. The most significant contributor to the widening of the confidence interval, the ID-GC/MS standard curve variance, is examined in detail. The performance of a new method developed during this study is coinpared to three conventional methods for

analyte weight isotope diluent weight

and Y is the area ratio

y=

analyte GC MS peak area -

isotope diluent GC MS peak area

( 3)

After the constants a and b are obtained from linear regression calculations, a portion of an isotope diluent-spiked extract of a sample S is injected into the GC/MS and the resulting sample area ratio, Ys, is used to calculate the sample weight ratio, Xs

x, = (Ys - b ) / a

(4) If the measured weight of the sample is M and the sample is spiked with a measured volume J of an isotope diluent solution of concentration Q, then the analyte concentration, C, in the sample is

C = JQXs/M

(5)

After estimates of the variances of components J,Xs,Q, and M a r e obtained, the variance, uc2,of C can be estimated by using the formula for the variance,,:,a of a product of two

This article not subject to U.S. Copyrlght. Publlshed 1983 by the American Chemlcal Soclety

838

ANALYTICAL CHEMISTRY, VOL. 55, NO. 6, MAY

random variables r and s, which have means spectively

+

urs2 = u;s2

U82F

+ ur2us

F

1983

and

g,

x2(N)

(6)

and the variance, VrIs, of a first-order Taylor's expansion of r / s about the means F and s

VrIs=

(s2a,2

+ Pu,2 + u;u;)/s4

+(Elm2

+ (E,Xd2)1+ XS2UJ2WI (9)

Confidence limits of C are estimated as conf. limits (C) = C f t N - z ~ c

UM/O.l

(11)

and Jmin

=

(13)

i=l

N

w, = icwi =l

(14)

The sum of weights, Wz, for a sample S which has been measured K times is K

w2

=

cwsj

(15)

]=1

It is usually desirable to construct upper and lower confidence intervals around X s of equal probability, and when one Student's t value is so chosen for both confidence intervals, the resulting confidence limits around X s are given by conf. limits =

L)'

(x + -) x s -a * ( 1- g

1-g

x

a

( X , - X)2 C ( X i - x ) 2 / u y ; (16)

However, when g is small (e.g., less than 0.1 ( 3 ) ) then , it can be ignored ( 3 , 4 )without significantly altering the computed confidence limits; when g is thus =O, the resulting limits are symmetric about X s conf. limits =

(10)

where t is Student's t (for some specified confidence interval), and N is the number of points in the ID-GC/MS standard curve. Equation 10 is a valid estimator if the coefficients of variation (cv = standard deviation/mean) of M and of J I 0.1. If these cv's were larger, then the probability density function of C would deviate significantly from being symmetric and one-sided confidence limits would be required. Under these cv restrictions, the minimum weight, M-, and minimum volume, Jmin, required to use eq 10 as a valid approximation are therefore

Mmin=

c (Yi - UXi - b y / uy;

(8)

Using eq 5-8, and noting that Q can now be considered a constant as its standard deviation has canceled, uc2 may be calculated with

(or,;

=

where uy,2 is the variance of Yi. A regression weight, Wi, is defined as 1/uy,2,and the regression sum of weights, W1, is defined as

(7)

There are two components of the variance, ux:, of Xs. One is the component from the ID-GC/MS linear regression, ureg2, and the other is due to errors made in preparing standards for this regression. are: will be presented in the following section on weighted least-squares linear regression. As far as errors in preparing standards are concerned, it was found, in the experiments conducted in this study, that variances from pipetting, diluting, and combining solutions with class A glassware were not significant compared to variances from weighing the standards. If the measured weights of analyte standard and isotope diluent standard are G1 and Gz, respectively, and the standard deviations of these weights are SIand Sz, respectively, then the relative errors, El = S1/G1 and E2 = S 2 / G 2 ,will be propagated throughout the concentrations CAN, and CID,of analyte and isotope diluent, respectively, in prepared standard solution i in such a manner that CAN,and CID, will have standard deviations CANzEland CID,E2, respectively. Further, if the isotope diluent spiking solution is prepared from the same stock as the standard m e standards, then the standard deviation component of Q due to the isotope diluent will cancel the standard deviation component of X s due to the isotope diluent. The remaining component of the standard deviation of X s due to errors in making the standards is E,Xs and the variance of X s may therefore be written

ax: = %,g2

N

re-

UJ/O.l

respectively. In practice, the largest component of cC2is usually ure:. This component is presented in the following section. WEIGHTED LEAST-SQUARES LINEAR REGRESSION Given the ordered pairs, ( X i , Y i ) ,i = 1 to N, a and b are obtained (3-6) by minimizing x 2 ( N ) :

ur,g2, for use in eq 9, is given by

ID-GC/MS LINEAR REGRESSION METHODS Conventional methods used to calculate ID-GC/MS standard curves include unweighted, Poisson weighted, and replicate measurement weighted linear regressions. The methods differ in their estimators of uy; used in the regression (eq 13). The unweighted regression is used under an implicit assumption of uniform variance and can be computed by setting cry:, i = 1 to N, equal to 1 which produce an unweighted estimator, Vycuw,of uy,2

V,"" = 1

(21)

In the Poisson regression method, the mass spectrometer is considered to be a simple pulse counting device, with the counts being Poisson-distributed. In this distribution, the variance of the counts is equal to the counts. The GC/MS peak areas are then treated simply as counts. Let N,be the

ANALYTICAL CHEMISTRY, VOL. 55, NO. 6, MAY 1983

analyte GC/MS peak area and let Di be the isotope diluent GC/MS peak area. Then

Yi = Ni/Di (22) If allowance is made for a constant hardware or computer then the Poisson variance estimator, V N ~ , software gain, of Ni and the Poisson variance estimator, VD:, of Di will be the gain squared times Niand Di, respectively V,P = GN; (23) and

VD; = GDi

(24)

By combining eq 7 , 22, 23, and 24, one obtains the Poisson estimator, Vyy,of uy;2

V,P = (.DiNi+ NF

+ GNi)G/D:

125)

In the third conventional regression procedure, L replicate measurements of Yiare obtained. The replicate measurement estimator, VyiRm,of uy;2 is

Replicate measurements must also be obtained for Ys,in order to estimate oys12with Vyspm. A new ID-GC/MS linear regression method, introduced in this paper, uses the recently developed (7)squared successive differences (SSD) estimator, VASSD, of the variance of the area, A , of a single chromatography peak. This SSD estimator of area variance for a peak consisting of P intensity values I distributed over K background intensity values B is given by (7) P-1

P-1

2=l

i=l

+

V p D = 1/2c(li - li+1)2 - YZC(SFi- SFi+1)2

where SF is a simoothing function (8) estimator of a theoretical smooth peak a bout which the chromatography peak is randomly distributed. In the SSD regression procedure, VN:sD and VDISSD are computed by using eq 27 and then are combined with eq ’7 and 22 to yield the SSD estimator, VyrSD,of CY:

Vy,SSD

= (D:V

SSD

Nc

+ N2V D,SSD + VNt s s D VD,S S D ) / D t 2

(28)

EiXPERIMENTAL SECTION GC/MS. The GC/MS used in this study has been previously described (1,2). It consisted of an atmosphericpressure ionization mass spectromieter (API-MS) (Extranuclear Laboratories, Pittsburgh, PA) interfaced to a Hewlett-Packard 5710 gas chromatograph. The chromatograph used a 50-m SP 2100 fused silica capillary column connected to a falling needle injector (Chrompack, Bridgewater, NJ). The API-MS was monitored with a Model 2400 INCOS Data System (Finnigan Corp., Sunnyvale, CA) which included the MSDS (revision 3) data acquisition software package. The GC oven was operated isothermally at 220 “C. Helium was the CC carrier gas and the linear flow rate was 25 cm/s. An produced 4 5 ion-molecule reaction between TCDD and 02dichlorc-1,2-benzoquinoneions ( I , 2 ) . These ions were monitored at mlz 176 and mlz 178 for W-TCDD and at m / z 182 for lacTCDD. Other Equipment. The balance used to weigh the standards was a Cahn Electrobalance G-2 (Ventron Instruments, Paramount, CA). The sample balance was a Model 2610 Dial-0-Gram (Ohaus, Florham Park, NJ). The spiking syringe was a 100-pLGC syringe. TCDD Standards. Uniformly ring-labeled 13C-TCDDwas synthesized under contract ( I ) , and l2C-TCDDwas purchased from Eco-Control (Cambridge, MA) and used as received (reported

41

837

isotopic purity was 98%). The 383 pg W-TCDD and 180 pg 13C-TCDDwere weighed out on the Electrobalance and pllaced into separate volumetric flasks,each containing 100 mL of toluene. These were the stock solutions from which 1 2 standards were prepared. The weight ratios (g of lZC-TCDD)/(gof I3C-TC!DD) of the 12 standards ranged from 0.0 to 4.79. All dilutions, and volumetric transfers were conducted at 20 “C. Estimates of uMz,uJ2,E l , and E,. The standard deviations, S1 and S2, of the weighings of 12C-TCDDand 13C-TCDD,respectively,were calculated from repetitively weighing 1 mg and 5 mg Class M calibration weights on the Electrobalance. Then El = S1/383 and E , = S2/160. An unweighted linear regression was performed with the sample balance (scale reading vs. weight), using temperature controlled H20 for standard weights. A total of 15 points were used and weights ranged from 3.0 to 18.0 g. An unweighted linear regreiasion was performed with the spiking syringe (syringe reading vs. volume), using temperature controlled HzOfor standard volumes. A total of 20 points were used and volumes ranged from 1.0 to 100 pL. uMzand oJ! were then estimated for any specified weight or volume, reaipectively, from eq 20. ID-GC/MS Standard Curves. Each of the 1 2 TCDD standards was injected three times into the GC/MS for a total of 36 injections. The ordered pairs, ( X I ,Y,),i = 1-36, where X was the standard weight ratio and Y was the ratio of the GC/MS peak area at m / z 176 to the GC/MS peak area at m/z 182, ‘were used to compute least-squares linear regression lines by the four discussed methods. In the SSD method, a nine-point quaclratic-cubic smoothing function (8)was used t o compute SF for eq 27. Relative Contributions to Error. After the TCDD SSD ID-GC/MS standard curve was computed, the standard curve point with weight ratio = 4.79 was sent back through the standard curve as 19 “samples”,with these simulated samples having sample weights (M) ranging from 0.5 to 10.0 g. Each of these were simulated to have been spiked at 100 parts-per-trillion (pptr), so that the spike volume (J)changed with sample size. Confidence intervalwidths (2tN-2~7,)for each of these samples were calculated. For each of the 19 simulated samples, relative contributions to error (RCE) of uJ, uM,ow, and EIXS were calculated as follows. The relative increase in uc due to uJ was calculated by computing uc (eq 9) normally with oJ included and then with uJ = 0. Then oc - UC(UJ = 0)

relative increase (uJ) =

UC

(29)

Relative increases were calculated for the other three components. Then the normalized RCE of component 2 (2 = oJ,oM, ore,c,or EIXs) was computed as relative increase (Z) RCE (2)= (30) E relative increases Poisson Hypothesis Experiments. Two experiments testing the Poisson hypothesis (eq 23 and 24) were conducted with the API-MS operating in the positive ion mode and the GC disconnected from the API source. In the first experiment, using single ion monitoring at m/z 36.0, a mass peak width of 0.01 amu, and an integration time of 0.2 s, 100 consecutive counts were recorded and the mean and variance of these 100 counts were computed. This process was repeated, incrementing the m / z by 0.1 amu up to m / z 37.9. The second experiment was conducted similarly to the first, except mlz ranged from m / z 27.0 to m / z 32.8 in increments of.O.2 amu, the integration times were 0.2 s, and the mass peak widths were 0.2 amu. Standardized Variable Cumulative Probability Disitributions. Simulation experiments were conducted with an I13M 4341 computer using FORTRAN IV and Statistical Analysis System (SAS Institute, Raleigh, NC) software packages. Each experiment consisted of the simulation and statistical analysis of 500 ID-GC/MS standard curves based on the line Y = X (a = 1, b = 0). The simulated standard weight ratios were 0.0, 0.1, 0.5,O 75, 1.0, and 1.5. At each standard weight ratio X,, one or more area ratios YI = N,/D, were generated as follows. D, was randomly selected from the uniform distribution, 20 000 5 D,5 80 000. D, so selected was then the area of a Weibull function (9) with peak

838

ANALYTICAL CHEMISTRY, VOL. 55, NO. 6, MAY 1983

Table I. ID-GC/MS Linear Regression Parameters parameter

rep1 measmt regression

SSD regression

a

b

x

P

X(Xi - x ) / u y ; X(Yi - P)/oy; X ( X i - X )(Yi- P ) / u y ; wi

I

Poisson regression

unweighted regression

1.379 -0.005945 0.1012 0.1335 4744

1.349 -0.0008127 0.1208 0.1621 9695

1.397 -0.01546 0.04926 0.05335 68910

1.420 -0.01997 0.9474 1.325 65.10

9076 6541 10900 7.36 x 10-4

17960 13080 194600 3.24 x 10-4

136000 96260 5710000 1.37 x 10-3

131.80 92.46 36 4.93 x 10-4

i

0.03;

6. 2 0 0

10

20

30

40

50

60

70

80

Q0

100

1

2

3 4 5 6 7 8 Sample Size (9)

9 10

TIME

Flgure 1. Simulated chromatography peak (dashed line) wlth a nolse level equal to 40 of a Welbull function (solld line) height. Weibull peak shape parameter = 2.0.

shape parameter = 2.0 about which a particular simulated chromatography peak of area Diwas generated. This peak consisted of 240 points randomly normally distributed about the Weibull function with standard deviations (noise levels) equal to 40% of the Weibull function value (see Figure 1). Every third point of the peak was sampled (to simulate ID-GC/MS peak switching (10))to yield a net of 80 points per peak. Di was the was not sum of 80 intensity values of this random peak. (Di corrected for peak switching (Le., multiplied by 3) as this factor would simply cancel in all pertinent equations.) Then Ni = Xi& and Niwas the area of a peak whish was similarly distributed about a Weibull function of area Ni. After all Y for a standard curve were generated,linear regreasion parameters were calculated. Then, for each of the 500 standard was curves in an experiment, a “true” sample weight ratio, Xt, randomly selected from either of the uniform distributions, 0 5 XtI 1.5 or 0 5 XtI 0.5. A sample area ratio, YS = Ns/Ds, was by the method described randomly generated about Yt ( =Xt) above. Then the estimated sample weight ratio, X,,was computed as XE = (Ys - b ) / a (31) The standardized variable, U , for each of the 500 standard curves and XE values was

XE - xt u =-

(32)

ureg

The result of each experiment was reported with a Kolmorgorov-Smirnov (K-S) (11)one sample goodness-of-fitstatistic. The K-S test result was expressed as a probability (p) that the cumulative probability distribution (CPD) of U could have been a CPD of Student’s t . Experiments were conducted with the unweighted,SSD, and replicate measurement regression methods.

RESULTS E l , E z , Jmin, and Mmin.The standard deviation of the Electrobalance weighing5 was 4 pg. Since the 12C-TCDD

L

SYRINGE REGRESSION

1 2 3 4 5 6 7 6 9 1 0 Sample Size (9)

Figure 2. (a) Influence of sample size on SSD ID-GC/MS linear regression confidence interval widths. (b) Influence of sample slze on relatlve contrlbutlons to error of four contributors to total error.

standard weighed 383 pg, El = 4/383 = 0.0104. 13C-TCDD weighed 160 pg, so Ez = 4/160 = 0.025. Mminwas calculated by using eq 20 of the sample balance’s linear regression using u y = M,i,,/O.l and was 0.95 g. Jmin was calculated similarly and was 3.1 pL. ID-GC/MS Linear Regression Parameters. Table I lists linear regression parameters for the ID-GC/MS standard curves computed by each of the four described methods. Each of the ID-GC/MS regression methods used the same 36 points generated from 12 standards with weight ratios from 0.0 to 4.79, using three injections per standard. Relative Contributions to Error (RCE). Figure 2a shows the influence of sample size on the 95% confidence interval width produced by the SSD ID-GC/MS linear regression for a simulated sample containing 479 pptr I2C-TCDD. Figure 2b shows the corresponding RCE’s for each of the four contributors to total error (uc). ID-GC/MS Confidence Limits. Table I1 lists 95% confidence limits computed by each of the four ID-GC/MS linear regression methods (the confidence limits were computed with eq 19, as g (eq 18) was negligible (see Table I) in all cases).

ANALYTICAL

CHEMISTRY, VOL. 55, NO. 6,

MAY 1983

839

Table 11. Coiparison of 95% Confidence Limits (Parts Per Trillion) from Four ID-GC/MSLinear Regression Methods repl Poisson SSD measmt unweighted standards regression regression regression regression 1 + 5 2 + 18 1+2 2+2 2+1 5+1 6 + 18 5i3 5i 1 6 +2 9 +1 I O + 18 101 3 lor3 10+1 17+3 P8+18 17i3 19+1 18+4 25i 2 2 5 + 18 24+4 25+ 5 24t 1 5 1 + 22 5 0 i 18 48+ 3 5 0 + 11 5 0 k 8 7 0 i 12 67 + 18 68i 6 72i: 4 6 8 + 10 l o o + 26 9 6 i 18 98+9 96i: 5 9 8 i 14 144r 8 1381.21 1 3 9 r 1 4 1 4 2 + 12 1 3 6 r 18 239 % 1 3 226 + 32 228 + 23 233 i 11 223 + 18 479 t 26 494 i 68 500 i 56 510 i 5 3 486 + 20 The data were simulated by sending 11of the standard curve points back through the standard curves as "samples" spiked a t 100 pptr. The spikes were defined to have zero variance (Le., u d = 0, uJ" = 0, and El = 0) so that variances due solely to the ID-GC/MS standard curves could be compared. The 95% confidence limits for the standards (column 1 of Table 11) were calculated by using the determined values of El,E2, and the estimator (eq 7) of the variance of a ratio. Poisson Hypothesis Experiments. Figure 3a shows a reconstructed positive ion mass spectrum between m/z 36.0 and m/z 37.9. The counts are the means of 100 consecutive measurements. Figure 3b shows G (the variance of the intensity divided by the mean of the intensity, eq 23 and 24) vs. m / z for the data of Figure 3a). Figure 4a shows a reconstructed positive ion mass spectrum between m / z 27.0 and m / z 32.8. The counts are the means of 100 consecutive measurements. Figure 4b shows G vs. m / z for the data of Figure 4a. Cumulative Probability Distributions (CPD's). Table I11 reports results of the CPD simulation experiments. Probabilities (p) were calculated from Table X.7 of Beyer (11, p 426). DISCUSSION RCE's. The results presented in Figure 2 illustrate effects of changing sample size on confidence limits and on RCE's. One would usually expect that the largest RCE would be that of the ID-GC/MS standard curve, with the other RCE's being significantly smaller, as is the case with the 10.0-g sample. As the sample size is reduced, however, the RCE's of the syringe regression and the sample balance regression might no longer be of small significance. One should, therefore, avoid using small samples or use more precise instruments than were used in the present study. For example, the Ohaus Model 2610 balance is an inexpensive general purpose instrument not designed for precisely weighing out small samples. Also, a spike level of 100 pptr for a 1.0-g sample required a spike solution volume of only 6.3 p L ; clearly, the coefficient of variation at such a volume for a 100-pLsyringe is high so one should, for such a volume level, use a 10-pL syringe instead. It should not be inferred from the foregoing discussion, however, that the Ohaus Model 2610 balance and the 100-pL syringe were unacceptable; indeed, a5 long as the sample

I , . . ,. ..,...,.., 36

38

37

m/z

Figure 3. (a)Reconstructed positive ion mass spectrum from m l z 36.0 to m l z 37.9. Counts are means of 100 consecutive measuremlents. (b) G (variancelmeanof counts) vs. m l z .C

20 15(1

10-

27

31

29

33

m/r

Figure 4. (a)Reconstructed positive ion mass spectrum from mlz 27.0 to m l z 32.8. Counts are means of 100 consecutive measurements. (b) G (variancelmeanof counts) vs. m l z .

weight was about 10 g or greater their RCE's were less than 20%. The RCE's of EIXs could easily have been less than they were. The standard deviation of weighings on the Electrobalance in the range from 1to 5 mg was essentially const,mt. If the 12C-TCI>Dstandard had weighed 5.0 mg instead of 383 pg, for example, then El would have been 4/5000 = 0.0008 instead of 0.0104 and the RCE's of E,Xs would have been reduced to insignificance.

--___

Table IIX. Kolmorgorov-Smirnov Goodness-of-Fit Probabilities

X, range 0-1.5

0-0.5

___-

0-1.5 0-1.5

points/ wt ratio 1 1

2 5

unweighted regression

(p) for

Cumulative Probability Distribution Experiments repl measmt regression

0.15 < p < 0.20

p 0.20 > 0.20 > 0.20

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ANALYTICAL CHEMISTRY, VOL. 55, NO. 6, MAY 1983

ID-GC/MS Linear Regression Methods. Table I1 shows a comparison of 95% confidence limits produced by each of the four linear regression methods, with each method using the same 36 points. Because of the limited number of points involved and the fact that the “sample” points were in fact taken from the standard curve points, this was by no means a statistically rigorous experiment, but it did expose some obvious trends. One conclusion from the data in Table I1 is that the means of the confidence limits for all methods were very similar. This indicates that the slopes and y intercepts calculated by the four methods were in close agreement, and this is confirmed by data of Table I. Another conclusion from Table I1 is that confidence limits calculated by the three weighted regression methods are remarkably similar, despite totally different formulas used to estimate weights. This may be due to the fact that for weighted regression accuracy, it is necessary only for the weight to be proportional to the inverse of the variance, because any proportionality constant will drop out of the relevant regression equations. Weighted regressions, therefore, tolerate a certain amount of inaccuracy in their methods of estimating weights. Table I shows that the sums of weights, W,, and therefore the individual weights, calculated by the three weighted regression methods are indeed widely different. Information concerning accuracies of the weighted regressions’ confidence limits can be deduced from Table 111. If points about a regression line are independent and normally distributed and estimated weights are proportional to the theoretical true weights, then the standardized variable U will be distributed as a Student’s t (3). Independence and normalcy were included in the specifications of the CPD simulations performed in this study, so major differences between the CPDs of U and Student’s t were due to inaccurate weight estimates (minor differences were expected due to the finite number of standard curves simulated). Conditions in the CPD simulations were severe. For example, the GC/MS peak areas of the isotope diluent were randomly uniformly distributed between 20 000 and 80 000. This was to simulate variances in the amounts of standards injected and in recoveries of spiked samples. Further, the noise levels of the simulated chromatography peaks were very high (40% of the Weibull function height, Figure 1). These conditions were deliberately inserted into the simulations in order to test the statistical robustness of each regression method tested, and also in some trace level ID-GC/MS work such conditions are not unusual. 1. Unweighted Method. It is not difficult to conclude from the data presented in Table I1 that the unweighted regression produced inaccurate confidence limits. First, the widths of the confidence bands for all but one concentration were essentially the same (after rounding off to integer values), contrary to known behavior of ID-GC/MS data. Second, confidence bands around the four lowest concentration levels include zero, which could allow for possible erroneous classification of %-TCDD determinations in these samples as being “not detected”. Third, the confidence band around the highest computed concentration (486 pptr) was narrower than the confidence band around the standard, which would imply that the linear regression contributed less variance at this point than was contained in the standard itself. It is seen from Table I11 that when X,was chosen from across the full range of standard curve X values (Le., 0-1.5) the unweighted regression’s confidence limits do not appear exceedingly inaccurate. But when Xt was chosen from only the lower third of standard curve X values (analogous to the lower concentration values in Table 11) accuracy diminished dramatically. Clearly, sampling from only the middle standard

curve X values tends to mask inaccurate confidence limits at each end of the standard curve. 2. Poisson Method. It can be seen from Figures 3 and 4 that the simple Poisson hypothesis was not supported on the AJ?I/IMS. Not only was G always found to be greater than the means of the counts, but G was not a constant. On the basis of these data, and other experiments conducted in this study (the data from which are not summarized in the figures), some trends could be seen for G. G tended to rise as the counts rose. G increased as the mass peak width decreased and decreased slightly as integration times decreased. From this behavior, one can conclude that the variance of the counts is in part a function of the counts, but it is not a simple relationship. The Poisson model has been used in certain isotope dilution mass spectrometry applications, such as the measurement of 13C02/12C02 ratios from gases bled into a mass spectrometer at a constant rate (12).Integration times in that application were long and instruments were stable. Schoeller (13) suggested that the Poisson model might apply to isotope dilution mass spectrometry standard curves in general, though he cautioned that for the model to be validly applied instrumental stability was required. It is clear that in the API-MS there were sources of significant variance other than Poisson processes. Analog-todigital converter noise (14), ac line noise, instability in the quadrupole, etc., might all have contributed. In addition, the chemical ionization process may have added variance due to reactant and product concentration fluxes. Although the simple Poisson model was not supported on the API-MS, the Poisson regression produced confidence limits which were similar to those calculated by the other weighted regressions (Table 11). As discussed earlier, if Poisson estimates of weights are only proportional to the true weights, the regression will be accurate. Thus, the Poisson method may perform adequately even when its model hypothesis is not strictly supported. 3. Replicate Measurement Method. As Table 111 illustrates, the replicate measurement method produces more accurate confidence limits when the number of points (L, eq 26) per weight ratio is increased. This is as expected, as the estimator V,Rm improves as L increases. There are theoretical problems and a practical limitation associated with the use of the replicate measurement method for ID-GC/MS standard curves. The theoretical problems stem from the assumption the VyRmis an accurate estimator of uy: for all points within a group, Le., that all points have the same variance which can be accurately estimated with VYRm.But in ID-GC/MS, the variances of individual points are in fact independent and are functions of the absolute amounts of material injected and of the GC-MS noise level at the time of injection, as much as being a function simply of weight ratios or peak area ratios. The practical limitation of using the replicate measurement method for ID-GC/MS standard curves is, of course, due to the requirement for a relatively high number of injections at each weight ratio (at least five injections, according to Kleijnen et al. (15)). Obeying this requirement may prove to be excessively laborious if the GC retention times are high, as is the case with some pesticides. Replicate injections must also be made for the sample in order to compute Ws. This latter requirement poses a problem with the determination of environmental pollutants at very low concentration levels (e.g., parts per trillion), because in this case it is advantageous, for sensitivity considerations, for each injection to use as large a fraction of the sample extract as possible. Despite these limitations, the replicate measurement method has two strong points. One, it allows for relatively easy

ANALYTICAL CHEMISTRY, VOL. 55, NO. 6, MAY 1983

computation OF an ID-GC/MS weighted linear regression and can be implemented with a calculator. Two, it provides more accurate confidence limits than does the unweighted method. 4. S S D Me!thod. Based on the data presented in Table I11 and on previously determined (7) characteristics, the SSD method for IDl-GC/MS has the following advantages. One, it produces confidence limits which are consistently more accurate than either the replicate measurement method or the unweighted regression method. Two, the number of standard curve points per weight ratio has no bearing on the accuracy of its calculated confidence limits, so replicate injections of standards and samples are not necessary. Three, variances of Y , are estimated independently, so the method imposes no theoretical requirement for identical injection sizes or for day-to-clay instrumental noise stability. And four, in the SSI)method it is unnecessary (7) to presuppose a particular distrikiution of the underlying noise of the mass spectrometer-a presupposition, for example, required of the Poisson method. Indeed, a Poisson noise distribution is merely a subset of distributions accommodated by the SSD method. The disadvantage of the SSD method is that it requires relatively complex calculations (eq 27), and therefore for its implementation an on-line computer is a practical necessity. 5. Other Methods. Other linear regression techniques exist which were not examined in this study. For example, one method (16) involves computing several regression lines from one set of' points by selectively deleting subsets of points and choosing the line which produces the narrowest confidence band around the unknown sample concentration. In another technique (17)the replicate measurement method is extended by performing a nonlinear regression of V$"' on X this tends to smooth out erratic values of VyFmwhich were computed from few points. These and other methods were not explored in the present study because either they assume a variance structure not applicable to ID-GC/MS data or they appeared to offer negligible improvement over the performances of the three conventional regression methods under normal experimental conditions. Recommendations. The SSD method should be implemented if the necessary computer facilities are available. Computer software implementing the SSD method has been prepared. This software consists of two programs. One program, called STANDARD,produces an SSD weighted linear regression ID-GC/MS standard curve. The other program, called SAMPLE,used parameters stored on a computer disk file by STANDARD to produce 95% confidence limits for a sample concentration. SAMPLE also performs qualitative analysis by analyzing molecular fragment ions; the programs can analyze up to four confirmation ions for both the isotope diluent and analyte, at the operator's discretion. (Contact the authors for

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software availability information.) If adequate computer facilities for implementing the SSD method are not available, then either the Poisson method or the replicate measurement method should be implemented. However, the Poisson method's hypothesis (eq 23 and 24) may require confirmation on the particular GC/MS in use if very accurate confidence limits are needed. If the replicate measurement method is used, then the amounts of standards and samples injectkd into the GC/MS should be tightly controlled to reduce intragroup variances. The unweighted ID-GC/MS regression will usually produce fair estimates of slope and y intercept but inaccurate confidence limits. This method should therefore be used only when accurate confidence limits are not required. ACKNOWLEDGMENT The authors acknowledge statistical assistance of D. W. Gaylor and R. L. Kodell. Registry No. TCDD, 1746-01-6. LITERATURE CITED Mltchum, H. K.; Moler, 0. F.; Korfmacher, W. A. Anal. Chem. 1980, 5 2 , 2278-2282. Mltchum, 1% K.; Korfmacher, W. A.; Moler, G. F.; Stalling, D. L. Anal. Chem. 1992, 5 4 , 719-722. Brownlee, K. A. "Statistical Theory and Methodology In Science and Engineering", 2nd ed.; Wlley: New York, 1965. Snedecor, G. W.; Cochran, W. 0. "Statlstlcal Methods", 7th ed.; Iowa State Unlversity Press: Ames, IA. 1980. Bevlngton, P. R. "Data Reduction and Error Analysis for the Physical Sciences"; McGraw-Hill: New York, 1969. Draper, N. R.; Smith, H. "Applied Regression Analysis"; Wlley: New York, 1966. Moler, G. F.; Delongchamp, R. R.; Mltchum, R. K. Anal. Chem. 1983, 5 5 , 842-847. Savltzky, A.; Golay, M. J. E. Anal. Chem. 1084, 36, 1627-1639. Hastings, N. A. J.; Peacock, J. B. "Statlstlcal Distributions"; \Niley: New York, 1975. Matthews, D. E.; Hayes, J. M. Anal. Chem. 1976, 4 8 , 1375-1382. Beyer, W. H. "Handbook of Tables for Probability and Statistics", 2nd ed., CRC Press: Cleveland, OH, 1968. Schoeller, D. A.; Hayes, J. M. Anal. Chem. 1975, 47, 408-414. Schoeller, D. A. Biomed. Mass. Spectrom. 1976, 3 , 265-271. Cheder, S. N.; Cram, S. P. Anal. Chem. 1972, 4 4 , 2240-2249. Kleijnen, J.; Brent, R.; Brouwers. R. Kenmerk 1980, 2 8 , 355. Mitchell, D. G.; Mills, W. N.; Garden, J. S.; Zdeb, M. Anal. Chem. 1977, 49, 1655-1660. Garden, J. S.; Mitchell, D. G.; Mllis, W. N. Anal. Chem. 1980, 5 2 , 2310-231 5.

RECEIVED for review June 28, 1982. Accepted January 21, 1983. Presented in part at the 37th Southweat Regisonal Meeting of the American Chemical Society, San Antonio, TX, 1981, at the 30th Annual Conference on Mass Spectrometry and Allied Topics, Honolulu, HI, 1982, and at the 184th Annual Meeting of the American Chemical Society, Kansas City, MO, 1982.