633
Anal. Chem. 1983, 55, 633-638
)0.367/0.033 0.627/0.668 2.027/-0.227 0.227/0.1221 = 111.121 0.939 -8.930
1.8611
Cross product of abstract 1 X abstract 2
[-x:
10.367*0.033 0.627*0.668 2.027*-0.227 0.227*0.1221 = 10.012 0.419 -0,460 0,0281 Line passing through (0.745, -0.400), (0.596, 0.039), (0.300, 0.916): E2 = -2.95631 1.802. In this example both the ratio and cross product indicate that the ratios corresponding to the second and third elements should be used to bound the concentration line. The line segment defining all possible concentrations of the two pure components is defined by that portion of the line E2 = -2.956E1 1.802 which is bounded by E2I8.930E1 and E2 1-0.939E1. The transformation matrix [TI is solved for by simultaneous solution of E2= -2.956E1 + 1.802 with E2= -0.939E1 and E2 = 8.93OE1
+
+
[PI 0.10 0.30
0.367
=
[AI[Tl 0.033
10.00 0.20 . 0 0 0.0.3 0.10 2.00 = 10.227 2.027 . 6 2 7 -0.227 0.122 0.668l
0.596 0.300 0.039 0.9161
LITERATURE CITED (1) Gillette, Paul C.; Koenig, Jack L. Appl. Specfrosc. 1982, 3 6 , 535-539. (2) Gillette, Paul C.; Koenig, Jack L. Appl. Specfrosc. 1082, 36, 661-665. (3) Koenla. Jack L.; DESDOSito, L.;Antoon, M. K. ADP/.SDeCtfOSC. 1977, 37,232-295. (4) Knorr, F. J.; Futreil J. H. Anal. Chem. 1070, 51, 1236-1241. (5) Ohta, N. Anal. Chem. 1973, 45, 553-557. (6) Sylvester, E. A.; Lawton, W. H.; Maggio, M. S. Technomet 1974, 16, 353-368. ... ... (7) Lawton. W. H.; Sylvester, E. A. Technomet 1971, 73, 617-633. (8) Macnaughtan, D.; Rogers, L. B.; Wernimont, G. Anal. Chem. 1972, 4 4 , 1421-1427. (9) Malinowski, E. R.; Howery, D. G. “Factor Analysis in Chemistry”; WIley: New York, 1960. (IO) Sharaf, Muhammad A.; Kowalski, Bruce R. Anal. Chem. 1982, 54, 1291-1 296. (11) Malinowski, Edmund R. Anal. Chlm. Acta 1082, 134, 129-137. (12) Malinowski, Edmund R. Anal. Chem. 1977, 49, 612-617. (13) Mallnowski, Edmund R. Anal. Chem. 1977, 49, 606-612. (14) Malinowski, Edmund R. Anal. Chim. Acta 1978, 703, 339-354.
[,.I51 1.354
The concentrations of these pure components in the starting mixtures can be directly computed as
RECEIVED for review July 19, 1982. Accepted December 20, 1982. The authors express their gratitude to the National Science Foundation for support of this research under Grant DMR80-20245.
Abstract Factor Analysis of Solid-state Nuclear Magnetic Resonance Spectra D. W. Kormos and J. S. Waugh” Department of Chemistty, Massachusetts Instlyute of Technology, Cambridge, Massachusetts 02 139
A method for component analysis of Solid-state NMR spectra Is described. By use of abstract factor analysis, AFA, slmuiated 13C powder and magic angle spinning (MAS) spectra were utilized to establish S / N levels necessary for successful delineation of factors. Chemical shift tensor information for p-dlmethoxybenzene, PMB, was used In the study. The use of dtfferentiai crosspolarization (CP) rates was suggested and demonstrated as an experimental means to produce factor anaiyzabie “spin mixture” spectra. 31P CP-MAS spectra of octacalcium phosphate, OCP, were analyzed to reveal three components. Additional areas of application are noted.
Early work applying factor analysis to liquid-state NMR data focused on elucidating the number of physically significant factors which produce solvent shifts of solutes in solution. The work of Buckingham, Schaefer, and Schneider showed these shifts to be a linear sum of various contributions ( I ) . Weiner, Malinowski, and Levinstone studied proton chemical shifts of substituted methanes in various media using factor analysis and concluded that three factors span the solventeffect space (2). Using the published data of Abraham, Wileman, and Bedford, Malinowski found similar evidence for three factors for 19Fshifts of organofluorine compounds
in nonpolar isolvents (3). Factor analysis has also been used to study solvent shifts for 13C and 29Siresonances of Me4% (4) as well as I6N resonances in amides (5). Two or three factors are generally found and are broadly ascribed to gasphase chemical shifts, anisotropic solvent susceptibilities, and van der Waals effects. Factor analysis has also been used in attempts to elucidate the variables which intrinsically control chemical shif‘ts. Studies of 13C chemical shifts of aliphatic and aromatic halides suggest two principal factors and one smaller factor are needed to correlate collected data (6, 7). In this pa.per, we investigate the use of AFA to extract component information present in solid-state NMR spectra which exhibit overlapping powder patterns or, in the case of magic angle sample spinning, overlapping sets of spinning sidebands. Results of AFA applied to model and experimental solid-state NMR spectra are presented. The model simulations which were employed to probe the practical limitations of the technique include powder and magic angle sample spinning 13Cspectra for p-dimethoxybenzene, PMB. The four independent 13C chemical shift tensors of PMJ3 were combined in varying proportionsto create chemical shift “spin mixtures”. Random noise was systematically added to the spectra. In this way, the ability of AFA to predict the number of spectral components. or chemical shifts, underlying the spectra was assessed empirically as a function of signal to noise. We then
0003-2700/83/0355-0633$01.50/00 1983 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 55, NO. 4, APRIL 1983
report on the use of AFA in investigating experimental magic angle sample spinning, proton enhanced 31Pspectra of octacalcium phosphate or OCP. OCP was chosen not only because of its relevance in the study of calcified tissues but also because of its complex phosphate structure. Rothwell, Waugh, and Yesinowski have previously found evidence for two and perhaps three phosphate components each with high sensitivity to cross-polarization (CP) contact times (8). The ability to use distinct cross-polarization contact times of a single sample to create suitable factor analyzable chemical shift spin mixtures seemed an enticing possibility.
THEORY Abstract Factor Analysis. Factor analysis is a multivariate statistical technique which consists of a series of operations. These operations are generally applied to data in matrix form which makes the method amenable to computerization. A prerequisite is that data must be expressible as a linear sum of product terms that relate, for example, to measurable quantities a and b
or in matrix form
[Dl = [AI[Bl Equation 2 may be rewritten with abstract row and column cofactor matrices, which reproduce the original data but are devoid of physical meaning [Dl = [Rlabstract[Clabstract (3) Beyond AFA, one may seek an appropriate transformation matrix, [TI, such that
tDI = [RIabstract[TIN [TI-'[cI
abstract1
[RIreal[Clreal (4) The [R],& matrix contains information about the factors (e.g., component spectra) and the [C],,, matrix the contribution of the factors in the data (e.g., concentrations). Methods to find the [TI matrix go beyond AFA and will not be discussed here. In the AFA of spectral mixture data, data matrices will generally be nonsquare and of the following form:
where dlt through dXlare the x sampled data points in each spectrum for each of the y mixture spectra. Data matrices can be pretreated in the first step of AFA according to the following linear equation: In this work [J]and [K] were chosen to be the unit and null matrices, respectively, for covariance about the origin of factor space. This preserves the origin of the factor space and relative error and statistically weights the data points in proportion to their absolute value. Construction of the covariance matrix, [Z], constitutes the second step of AFA. The [D] matrix is premultiplied by its transpose to obtain a square [Z] matrix
121 = [DI+[Dl
(7)
Standard eigenanalysis of the covariance matrix yields the eigenvalues and eigenvectors representing the data. Decision on the number of nonzero or insignificant eigenvalues composes the final step of AFA. The number of nonzero eigenvalues is equivalent to the rank of the covariance matrix. This
is also the rank of the original data matrix. Determining the rank is the pinnacle of AFA since this is the determination of the number of factors or components, n. Knowledge of the eigenvalues and eigenvectors leads to the determination of [Rlabstract and [Clabstract of eq 3. A number of approaches have been suggested to find the least number of eigenvalues needed to represent the data being analyzed. A plot of the common logarithm of the eigenvalues vs. a component index as suggested by Ohta (9) is frequently used. A major break in the initial slope of such a curve signals the transition from significant to noise eigenvalues. Noise eigenvalues will be present in all real data. An empirical IND or factor indicator function has been shown by Malinowski (3) to reach a minimum at the correct number of factors. It has the following form for covariance about the origin:
INDi =
RE (y - i)2
Y
2 Ej/x(y j=i+l
RE = [
The IND value for each eigenvalue Ei (i = 1 to y - 1) is calculated as a function of the remaining smaller eigenvalues, E,, using the residual error or RE. It is not defined for the smallest eigenvalue. Both of these methods will be utilized in the analyses presented here Suitability of Solid-state NMR Spectra. In cross-polarization NMR, decoupled spectra can to a first approximation be written as a linear sum of product terms n
&&,a$)
= Cg(ij)v(or)cO',cr)eO',~,~) (10) j=l
where the summation is over n spin components. DCp(i,a,t) is the signal intensity of spin mixture a at frequency i with cross-polarization contact time t;g ( i j ) is a line shape function for the NMR signal of type j spins at frequency i; u(a)is the volume of spin mixture or; c(j,a) is the concentration of type j spins in mixture CY, and e(j,a,t) is the cross-polarization efficiency for type j spins in mixture a with contact time t. For dilute, decoupled nonquadrupolar spins, g(i,j) will be determined by the chemical shift Hamiltonian. The chemical shift parameters a, 6, and q will dictate the shape of g ( i j ) for static powder patterns. Magic angle sample spinning renders the Hamiltonian time dependent and introduces the rotor spinning speed, wR,as another parameter to the line shape. Maricq and Waugh have given the explicit form of g(i,j) for static and spinning cases as a function of i i , 6 , q r and up, (IO). A single spinning speed for all spectra is necessary in order that CP-MAS data matrices be factor analyzable. It should be noted that the line shape function has in fact some dependence on contact time which will couple this term to the eQ,a,t) efficiency term. Consider the general case of an internuclear vector between an isolated I and S spin pair. When the angle between this vector and the applied static field approach the magic angle (54.7O), the cross-polarization efficiency is significantly diminished. A singularity would therefore be predicted to occur in the line shape function. Experimentally, anomalies of this type are rarely observed and for the purposes described here, we will consider eq 10 to be valid for C P data. This effect does not pertain to standard pulse experiments. Spectra obtained with these sequences should adhere to eq 10 with the deletion of the CP efficiency term. The spectra of spin mixtures resulting from such experiments should therefore also be factor analyzable. Equation 10 reveals two approaches for preparing a set of spin mixture spectra. If the CP contact time is kept constant, varying the relative concentrations of each spin species is acceptable. If the concentration is fixed, varying the contact
ANALYTICAL CHEMISTRY, VOL. 55, NO. 4, APRIL 1983
635
Table I. PMB I3CShielding Tensors H
U
nucleus a
b
C
d
011(ppm)
-80.4
-193
-198
-230
o22(PPm)
-71.5
-134
-136
-162
ogj(ppm)
-16.1
-12.0
-23
-74
;(0
-56
-113
-119
-155.3
ppm = TMS)
time will produce a set of spin mixture spectra. In this case eq 10 reduces to n
DCP(i,t)= Cg(ij)c'Ci)eCi,t)
(11)
j=l
where c'6) is a constant spin concentration for each spin species. If a single sample is used, u(a)is a constant and is accounted for in the ~'6)'s. Equation 11 predicts that if different spin species have sufficiently distinct cross-polarization characteristics, CP-MAS intensities can be influenced to create chemical shift spin mixture spectra from a single sample. Spin mixtures can be produced from a single molecular species as well as from physical mixtures of distinct molecules in this way. An analysis of polarization transfer and CP efficiencies for CP-MAS NMR is given by Stejskal, Schaefer, and Waugh (11). EXPERIMENTAL SECTION Instrumentation. A Digital Equipment Corp. LSI-11 computer was used for all calculations including spectral stimulation and factor analyses. All programs were written in FORTRAN. The computer, an integral part of a home-built 250-MHz proton spectrometer,was interfaced to an Analogic AP400 array processor for Fourier transformation and other manipulations of data sets. An Oxford Instruments widebore superconductingmagnet (5.872 T) was used with a double resonance single coil probe for 31P operation at 101.2 MHz with lH decoupling. OCP Spectra. 31PNMR spectra of OCP were obtained by using the technique of cross-polarization combined with magic angle sample spinning and high power 'H decoupling (CP-MAS). The decoupling field was 8 G. Principles of CP-MAS NMR and applications to solids are given by Pines, Gibby, and Waugh (12) and by Schaefer and Stejskal (13).Cross-polarizationwas done for signal enhancement and to decrease signal averaging recycle times. A Delrin sample rotor of Beams design was spun at 1201 Hz. The 'H spin temperature was cyclically alternated during half of the cross-polarizationcontads to minimize zero frequency spectral artifacts (14).Zero-filling of the 1K sampled points to 2K was done. With a spectral window of 40 kHz, an appropriate line-broadeningfunction was used to apodize the data. All OCP spectra were the result of 100 accumulationsat room temperature. Nine contact times incremented by 0.25 ms from 0.5 ms to 2.5 ms were used to create the OCP spin mixture spectra used for the factor analysis experiment. Computer PMB Simulations. The simulated 13CPMB static powder pattern spectra were computed by use of the tabulated PAS frame tensor elements for the four chemical shifts observed in the solid. The shieldingtensors values are summarized in Table I. A prescription detailed by Mehriig (15)with an approximation to the needed elliptic integrals given by Abramowitz and Stegun (16)was employed to do the calculations. The chemical shift line shapes obtained were weighted, suitably broadened, and added to obtain a powder pattern for the molecule. Various weighting factors corresponding to CP efficiencies were employed to simulate 10 cross-polarization spin mixtures. This modeled the effect of distinct CP efficiencies. The ratio of b to c efficiencies was kept
Flgure 1. Model pdlmethoxybenzene statlc I3C spectrum.
constant partly to simulate the fact that their true CP efficiencies should be quite similar and also as an added test for AFA to detect only three chemical shift components in spin mixtures displaying shielding information for four tensors. The b and c spins should therefore be detected as only a single component. Magic angle sample spinning simulations for I3C spectra of PMB were computed with the tensor information contained also in Table I for a spinning speed of loo0 Hz. A procedure described by Maricq and Waugh (10)was used to calculate the spinning sideband line shape for each of the four shielding tensors. For each of the Euler angles (a,8, y), 5O steps were used to obtain an orientation-averaged complex signal for one rotor period ( 2 ? r / w ~ ) . This time domain signal was replicated to fill a 1024point complex buffer to form a train of rotational spin echoes, suitably line broadened,zero-fiied to 4096 data points and Fourier transformed to give magic angle spinning line shapes. The line shapes were again weighted in the manner used for the preceding static case and combined to form 10 simulated CP-MAS spin mixtures. For both the spinning and static CP model mixtures, no line shape anomalies due to contact time dependencies were simulated. To further simulate real spectra, artificial random noise was added to each calculated spin mixture spectrum. Sets of spectra with SIN ratios from approximately 30 to 300 were created. The S I N was defined as S/N = 2(
5 (I; -
(12) f)2/"'2
i=A
where B
f = ( C I i ) / ( N+ 1 ) i=A
I,, is the difference between the-peak maximum intensity and the average base line intensity, I , defined for N points from spectral points A to B where no significant spectral features are seen. RESULTS AND DISCUSSION Simulated Spectra. Typical static and magic angle spinning 13Cmodel spectra for PMB are shown in Figures 1 and 2. The spectra shown are composed of equally weighted contributions of each of the four chemical shift tensors of the molecule. They represent spin mixture 1 of Table 11. For the static powder case, the number of chemical shift components is not obvious from the spectrum itself. The previously defined SIN ratios are also given. The region of each spectrum encompassed for factor analysis is indicated. For the static case, 520 data points were used in the analysis. For the magic angle spinning case, 600 data points were used. The results of AFA applied to 8 sets of model PMB I3C static spin mixture spectra are summarized in Figure 3.
636
ANALYTICAL CHEMISTRY, VOL. 55, NO. 4, APRIL 1983 9.0
8.0
-
7.0
W
3 J
-
s 6.0
2
-$
S/N 57
W
s 5.0
-1
lo
95 114 140 I87
280
4.0 KHZ
Model pdimethoxybenzene magic angle spinning I3C
Figure 2.
72
spectrum.
3.0 I
2
3
4
5
6
7
COMPONENT
8
9
1
0
INDEX
Figure 4. AFA eigenvalue results for model I3C PMB magic angle spinning spin mixture spectra. Three components are indicated.
Table 111. AFA Results of OCP Spin Mixture Spectra component log IND index (eigenvalue)
-I
9
8-
6.0-
W W
I
f)
5.0-
-
.
S/N
- -. 5973
4.0-
98
147
I
I
33
2
3
4
5
6
COMPONENT
7
8
9
I
IO
INDEX
Flgure 3. AFA eigenvalue results for model I3CPMB static spin mixture
spectra. Three components are indicated.
Table 11. CP Efficiency Factors for Spin Mixture Calculations nucleus spin mixture a b C 1 2 3 4 5 6 7 8 9 10
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
1.0 0.9 0.9 0.9 0.8 0.8 0.7 0.7 0.5 0.4
1.o 0.9 0.9 0.9 0.8 0.8 0.7 0.7 0.5 0.4
d 1.0 0.8 0.7 0.6 0.6 0.6 0.6 0.5 0.4 0.2
Plotted as the common logarithm of the determined eigenvalues vs. component index, a number of features appear. SIN values simulated range from 33/1 to 147/1 and differentiate the data sets. The highest S I N data consists of eigenvalues which level off at a component index of 4, indicating the clear presence of three factors or component spectra. These eigenvalues are distinguished in the figure from the remaining seven noise on zero eigenvalues. This is the expected result since the ratio of b to c spin species was kept constant in the spin mixture simulation (see Table 11). As the SIN decreases, a steady, uniform increase in eigenvalue magnitudes is noted
1 2 3 4 5 6 7 8 9
12.95 11.58 10.08 9.56 9.33 9.06 8.97 8.69 8.60
250.21 79.56 75.79 91.31 121.44 195.38 376.27 1430.9
for the noise eigenvalues. A smaller, consistent increase is observed for the smallest real eigenvalue. The result is a small dispersion in this eigenvalue for the data sets. The third eigenvalue is seemingly lost in the noise eigenvalues at a SIN value of 3311. Similar eigenvalue results for AFA of model 13C PMB MAS spin mixture spectra are shown in Figure 4. As for the static case, SIN values differentiate the seven data sets and range from 57/1 to 280/1. Three factors are again shown to characterize the data sets. A similar trend is noted with decreasing SIN values. The third factor’s eigenvalue is still perceptible at a SIN of 5711 but with lower S I N its recognition would be doubtful. These studies for PMB spin mixtures indicate a minimum S I N of approximately 7511 is necessary for a successful application of AFA. This requirement is rigorously applicable only to the mixture compositions modeled here and to the AFA of whole spectral regions. We suggest that AFA of MAS spectra will in general be more successful than for powder spectra. OCP Spectra. Representative cross-polarization, magic angle spinning 31PNMR spectra of octacalcium phosphate are presented in Figure 5. Spectra taken with four contact times are shown with the spectral region used for AFA indicated. The chemical shifts of the central peaks are 3.2 f 0.2, 1.8 f 0.2, and -0.7 f 0.2 ppm relative to an 85% HsP04 reference. Differing cross-polarization efficiencies are responsible for the intensity changes in the central and sideband peaks. The average SIN of the nine spectra used in the AFA was 9511. Tabulated eigenvalue and factor indicator function values in an AFA of nine OCP CP-MAS spectra are presented in
ANALYTICAL CHEMISTRY, VOL. 55, NO. 4, APRIL 1983
-1
IO KHZ
1-
Figure 5. Experimental 31P CP-MAS spectra of OCP. Contact times used were as follows: (A) 0.5 ms, (B) 0.75 ms, (C) 1.75 ms, and (D) 2.0 ms. 1
I
8.0 1
2
3
4 5 6 7 8 COMPONENT INDEX
9
Figure 6. AFA eigenvalue results for nine OCP spln mixture spectra. Three components are indicated with arrows. Table 111. Nine contact times ranging from 0.5 to 2.5 ms were used to collect the CP-MAS spin mixture spectra used in the analysis. A spectral region containingthe zero- and first-order sidebands spanning 195data points was d. The IND values reach a minimum at three components before turning up. This result is corroborated by an eigenvalue plot shown in Figure 6. A straight line can be drawn through the seven noise eigenvalues (n = 0.9894). It is useful in distinguishing the nearly constant difference in noise eigenvalues from the substantially larger difference between the third, smallest eigenvalue and the noise eigenvalues. In comparison to the model studies, the noise eigenvalues appear to fall on a line with a much larger negative slope. This may be attributed to small errors in peak phasing or base line artifacts not present in the simulated spin mixture spectra. A previous 13PNMR study of solid calcium phosphates by Rothwell, Waugh, and Yesinowski has indicated a tricomponent nature for octacalcium phosphate (8). This finding is
637
substantiated here both by the observed CP-MAS spectra (at higher resolution than the previous report) and by the use of AFA. The observed chemical shifts here are in basic agreement with their values. We accept the assignment of the high field peak (-0.7 ppm) to the acidic HPOZ- groups, the low field peak (3.2 ppm) to the “apatitic layer” PO-: groups, and the middle peak (1.8 ppm) to PO4“ groups near hydrated sites or adjacent HPOd2-groups. Applications. Abstract factor analysis has been successfully applied to model and experimental solid-state NMR spectra. The method offers an unbiased procedure for extracting additional information useful in characterizing complex overlapping powder or magic angle sample spinning sideband spectra. The number of isotropic chemical shifts is in many cases apparent in spectra. In cases where it is not, AFA may provide the information. In cases of coincidental near overlap of chemical shifts,detection of hidden resonances may be achieved. Entire spectral regions may be quickly analyzed subject only to data processing limitations. Observations using model data, although an exhaustive study was not attempted, are the following: Reasonably high SIN spectra are necessary in order that eigenvalues, small in magnitude, are not lost in noise eigenvalue background. This establishes a practical limitation especially for broad powder pattern spectra of isotopically dilute spin species. Similar reasoning suggests very small but real eigenvalues will commonly fall into the noise background. Such conditions imply that AFA will reveal only the minimum number of factors needed to reproduce the experimental data. Cross-polarization efficiencies, through the use of varied contact times, have been shown to be a convenient and powerful method to produce chemical shift spin mixture spectra which are factor analyzable. The spin mixture spectra need not result from molecular mixtures but also from single compounds containing different chemical shift species. A limitation is inherent for spin species with similar cross-polarization characteristics. These species will most likely be seen as a single component. The PMB model simulation has shown this clearly. We believe that there is substantial potential for factor analysis in solid-state NMR. Additional methods to clarify and extract new information from many varieties of solid-state NMR spectra are still needed. The complexity of coal, wood, and many biological solid sample spectra are but a few areas where AFA may be applied for component analysis. AFA might also supplement information in phase transition studies. The method as outlined here is not limited to chemical shift spectra and could easily be expanded to include other inhomogeneous interactions in NMR such as weak quadrupole or heteronuclear dipolar coupled spectra. In addition to AFA, the future application of spectral isolation procedures using factor analysis described by Malinowski and others (17-19) holds much promise in solid-state NMR and in other spectroscopies as well. ACKNOWLEDGMENT The authors wish to acknowledge W. E. Brown of the National Bureau of Standards, Washington, DC, for the octacalcium phosphate sample used in this investigation. Registry No. PMB,150-78-7;OCP, 13767-12-9. LITERATURE CITED (1) Bucklngham, A. D.; Schaefer, T.; Schnelder, W. 0. J . Chem. fhys. 1880, 32, 1227-1233. (2) Welner, P. H.; Mallnowski, E. R.; Levinstone, A. R. J . Phys. Chem. 1870. 74. 4537-4542. (3) Mailnowski, E. R. Anal. Chem. 1077, 49, 612-617. (4) Bacon, M. R.; Maclel, G. J . Am. Chem. SOC.1073, 95, 2413-2426. (5) Martin, G. J.; Bertrand, T.; Le Botlan, D.; Letourneux, J. J . Chem. Res., Synop. 1870, 12, 408-409. (6) Wlberg, K. B.; Pratt, W. E.; Bailey, W. F. J . Org. Chem. 1880, 45, 4936-4947.
638
Anal. Chem.
1983,55,638-643
(7) Bailey. W. F.; Cloffl, E. A.; Wiberg, K. B. J . Org. Chem. 1981, 46, 42 19-4225. (8) Rothwell, W. P.; Waugh, J. S.; Yesinowski, J. P. J . Am. Chem. SOC. 1980. 702. 2637-2643. (9) Ohta,’N. Anal. Chem. 1973, 45, 553-557. (IO) Marlcq, M. M.; Waugh, J. S. J . Chem. Phys. 1979, 70, 3300-3316. (11) Stejskal, E. 0.; Schaefer, J.; Waugh, J. S. J. M a p . Reson. 1977, 28, 105-112. (12) Pines, A,; Glbby, M. G.; Waugh, J. S. J. Chem. Phys. 1973, 59, 569-590. (13) Schaefer, J.; Stejskal, E. 0. J . Am. Chem. SOC. 1976, 98, 1031-1032. Schaefer, J. J . Magn. Reson. 1975, 78, 560-563. (14) Stejskal, E. 0.; (15) Mehrlng, M. “Hlgh Resolution NMR Spectroscopy In Solids”; SpringerVerlag: New York, 1978; Chapter 2.
Abramowitz, M.; Stegen, 1. ”Handbook of Mathematical Functions”; Dover: New York. 1965; p 591. Maiinowskl, E. R. Anal. Chim. Acta 1982, 734, 129-137. Howery, D. G. I n “Chemometrlcs: Theory and Appllcatlon”; Kowalskl, 8. R., Ed.; American Chemlcal Society: Washington, DC, 1977; ACS Symp. Ser. No. 52, pp 73-79. (19) Mallnowski, E. R.; Howery, D. G. “Factor Analysls In Chemistry”; WIley-Interscience: New York, 1960; Chapter 3.
RECEIVED for review October 12, 1982. Accepted December 22, 1982. This work was supported in part by the National Science Foundation [Grant No. 8111180-CHE]and in part by the National Institutes of Health [Grant No. GM 165521.
Determining the Lowest Limit of Reliable Assay Measurement Leonard Oppenhelmer, * Thomas P. Caplzzl, Roger M. Weppelman, and Hlna Mehta Merck Sharp & Dohme Research Laboratories, P.O. Box 2000, Rahway, New Jersey 07065
The use of external standards to determine the lowest iimlt of reliable assay measurement is commonly used to evaluate the ihnttations of a particular analytical technique and to make comparisons between dlff erent procedures. Ail previously devekqmi criteria have assumed a constant variance over the entire concentration range of interest. Assay limit criteria formulas have been developed for the heterogeneous varlance situation. Widely disparate results can be obtained dependlng on the choice of crlterla, whether a uniform variance or a nonuniform variance analysis is employed, the method used to select the weights, the models and/or transformations utilized, and experlmental deslgn conslderations. These Issues are examined by using data from an analytical procedure developed to monttor tissue residues for Ivermectln, an antiparasitic agent.
The “lowest limit of reliable assay measurement” (LLO-
RAM)is often used in analytical work since it makes a succinct statement about the limitations of an assay and allows for meaningful comparisons between different assays. It expresses how effectively one can distinguish between a measurement made when true activity is zero and one made when true activity is greater than zero. All too often the LLORAM is determined intuitively which compromises its utility. However, even an objective, well-defined criterion may be deficient if the statistical methodology used in its determination is inappropriate. Previous work on LLORAM determination (1-6) has assumed that the response variance was constant and thus independent of the response’s magnitude. In this case (the unweighted case) all values used for calculating the standard curve can be treated equally and results can be obtained by using ordinary least squares. However constant variance is probably not attained by most assays. For example, many assays involve transferring constant volumes of solutions containing variable quantities of the material to be assayed. The variance of the volumes transferred can reasonably be expected to be constant, but clearly the variance in the amount of material transferred will depend on its concentration in the solution, Le., the percent error will be constant and thus the variance will be proportional to the predicted response squared. This general point has been recognized by Smith and Mathews (7) who stated that for “typical chemical ap0003-2700/83/0355-0638$0 I b O / O
plication, the experimental conditions are controlled so as to make the percentage error a constant.” This structure with variance proportional to the square of the predicted value may also have application in atomic absorption spectrometry and anodic stripping voltammetry (8). Other structures are also possible, for assays based on Poisson distributed counting measurements (either photon or radioactive) the variance may be proportional to its predicted value (9,10) rather than its square. Garden, Mithcell, and Mills (11)have a less sanguine view: “It is reasonable to expect that much analytical data will not show constant variance nor would we expect the variance to be a simple function of Concentration.” The issue of nonconstant variance and the weighted least-squares approach in obtaining standard curves has been previously addressed (7-13),but this work has not been extended to determining assay measurement limits. This report describes the derivation of appropriate expressions for obtaining various types of assay limits under the more general conditions of heterogeneous variances. The results have been illustrated by using data from an assay developed to monitor tissue residues of Ivermectin, an antiparasitic agent, potent at very low doses. Rather dramatic differences have been observed depending on whether a valid weighted or an inappropriate unweighted analysis is used. In addition, related issues which also affect assay limit determinations are discussed and illustrated using the tissue residue data. These issues include the method used to select the variance weights, alternative models which transform the variance structure of the responses, and experimental design considerations used in obtaining the standard curve.
EXPERIMENTAL SECTION Data Description. A newly developed analytical procedure (14)was employed to analyze spiked tissue samples (fat, kidney, liver, muscle) at various concentrations (9.7 to 100 ppb) in four
species (cattle,swine, horses, sheep). The tissues were spiked with Ivermectin, a broad spectrum antiparasitic agent. The analytical procedure was based upon the detection of a fluorescent derivative of Ivermectin following high-performanceliquid chromatography. For each combinationof animal species and target tissue, multiple samples were assayed for at least four concentration residue levels which were believed to be in the linear range (observed response vs. concentration). Table I indicates the distribution of independently performed assays among species-tissue-concentration combinations. An assay which will be acceptable to governmental regulatory agencies must be able to reliably discriminate the marker residue response from the target tissue background at very 0 l9S3 American Chemical Soclety
