Anal. Chem. 1983, 55,565-567
565
CORRESPONDENCE Precision and Stability for the Generalized Standard Addition Method Sir: Recently, an alternate experimental design and a mathematical data analysis procedure were introduced (1) which simultaneously corrects for matrix effects and interferences in multicomponent analysis. The method is called the generalized standard addition method (GSAM) and is based on the method of standard additions. The GSAM has successfully been applied to inductively coupled plasma atomic emission spectrometry ( 2 ) ,anodic stripping voltametry (3), ion-selective electrode potentiometry (4),and spectrophotometry (5). I t has been shown to possess the ability to characterize multicomponent instruments with minimal effort (2)thereby providing a tool for analyte interferences detection. The theory of the GSAM has additionally been extended to allow for the detection and correction of drift in an analytical signal occurring during the course of an analysis ( 6 ) . Also, since the GSAM requires that standard additions of all analytes be made, a laborious task in complex analysis, a completely automated instrument has been designed (7) to implement the GSAM under computer control. However, most of this earlier work has only made use of the case where the number of sensors (p) is equal to the number of analytes ( r ) (p = r). Reference 6 contains a study where p > r was used solely for the detection of potential interferences. Several workers have shown that by using p > r for spectrophotometry, the highest precision for multicomponent analysis is obtainable (8-13). With p > r, the system of linear absorption equations is overdetermined which permits the most problable composition to be estimated as compared to the exact solution of the linear equations for p = r. Another advantage of p > r is that improved signal-tonoise ratios are obtainable which result from the overdetermined system using all spectral information available throughout the absorption bands. This is similar to the mathematical representations for the integration of signals over the spectrum preriented by Hirschfeld (14).However, this reasonable compai*isonhas not been proven. A logical conclusion is then that the GSAM should also benefit from using p :> r for spectrophotometry or other multicomponent instrumentation. T o demonstrate this extension of the GSAM, the experimentally obtained data of Zscheile and co-workers was used (9). In their work i t was shown that a more stable system is acquired if 36 wavelengths are used for a four-component mixture analysis as opposed to using only four wavelengths, the minimum necessary for solution of the linear equations. After a brief review of the theory of the GSAM, this paper will deal with precision improvement and enhanced stability for a four-component mixture analysis by incorporating p > r in the GSAM procedure.
THEORY The background and theory of the GSAM have been discussed in numerous previous papers (1-3,5, 6). Therefore, only key equations are presented here. The GSAM requires that when there are r analytes to be determined, the responses from p sensors (p 2 r) be recorded before and after n standard additions are made ( n 2 r ) . The model in matrix notation is
R = CK
(1)
0003-2700/83/0355-0565$01.50/0
where R is the n X: p matrix of measured responses, C is the n X r concentration matrix, and K is the r X p matrix of linear responses constants showing the contribution of each of‘ the r analytes to each d the p sensors for a given sample matrix. Since concentrations are not always additive, using volumecorrected changes for the responses (6), eq 1 becomes AQ = m K (2) where AQ is now the n X p volume-corrected response-change matrix and LWis the n x r absolute quantity standard addition matrix (e.g., moles, micrograms). By use of the method of multiple linear least squares to solve for K in the presence of an overdetermined system ( n > r), the generalized inverse of AN (15)is needed resulting in
Ir: = ( W A N Y ’ W A Q
(3)
where the generalized inverse of the matrix AN is defineld as (@AiV)-l@. After calculation of the K matrix, the vector of initial analyte quantities no, is recovered by solving
qo = P n o
(4)
where qo is the volume corrected initial response vector. Normally, when p = r, solution for no entails multiplying eq 4 through by the inverse of p.On the other hand, for the case when p > r, K is not a square matrix and consequently the inverse of IP‘no longer exists. Therefore, a generalized inverse of K is needed which will allow the estimation of no as
no = (KIcr)-lKq0 (5) where the generalized inverse of K is defined as ( W ) - I K . This is the favored1 solution for no since in the presence of experimental error for both no and K , it is not possible to solve for no exactly. EXPIERIMENTAL SECTION As noted in the introduction, the experimental data of Zsheile et al. was used in this study. The apparatus and experimental procedure have been previously described in detail (9). The molar concentrations of the four components used are listed in Table I. Also shown in Table I are the wavelengths used and the experimentally determined molar absorptivities which are synonymous with the entries in the GSAM K. Procedure. Three standard additions of one unit each per analyte were made. Volumes were kept constant. The molar concentrations and molar absorptivities presented in Table I were used to calculate initial responses (qo,before any standard ,additions are made) and to calculate the corresponding responses after standard additions are made (AQ as expressed in eq 2). Random noise was introduced to qo and A Q by using a Monte Carlo method (16).A normal distribution was assumed witlh a mean zero and a standard deviation of 2% relative standard deviation (RSD) of the noise-free synthetic data responses. A 2% RSD was assumed by Zscheile et al. also. One hundred Monte Carlo perturbations were performed to compute 100 sets of random responses (100 sets of qo and A&). These 100 sets of Monte Carlo responses were used in the GSAM to calculate 100 sets of no. The 100 3ets of no were used to estimate standard deviations for n,. This is very similar to the procedure of response perturbation used by Zscheile et al. All calculations were done by using “GSAM”a Fortran IV program available from Infometrix, Inc., Seattle, WA. All computations were performed on the Montana State University VAX 11/780 computer. 0 1983 American Chemical Society
566
ANALYTICAL CHEMISTRY, VOL. 55, NO. 3, MARCH 1983
Table I. Experimental Design no* Of Wavemolar absorptivities (X10-3)a wave- length, length nm adenylic cytidylic guanylic uridylic
1 2 3 4 5 6 7 8 9 10 11
12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
2 20 222 2 24 226 228 230 232 234 236 238 240 242 244 246 248 250 252 254 256 258 260 262 264 266 268 270 272 274 276 278 280 282 284 286 288 290
7.00 5.50 4.40 3.70 3.22 3.30 3.60 3.88 4.45 5.10 6.00 7.08 8.10 9.26 10.45 11.40 12.35 13.30 13.68 13.70 13.40 12.95 12.38 11.68 10.47 9.10 7.60 6.20 4.90 3.84 2.95 2.25 1.70 1.20 0.80 0.50
8.70 7.94 7.00 5.95 4.85 3.96 3.26 2.65 2.30 2.00 1.90 1.90 2.02 2.30 2.70 3.20 3.80 4.45 5.15 5.98 6.90 7.90 8.65 9.50 10.36 11.12 11.80 12.30 12.55 12.73 12.59 12.20 11.60 10.90 9.88 8.66
4.50 3.57 2.90 2.60 2.50 2.65 3.06 3.65 4.45 5.18 6.05 7.00 7.90 8.90 9.75 10.45 11.00 11.36 11.42 11.35 11.00
10.45 9.90 9.33 8.79 8.40 8.15 8.00 7.92 7.79 .7.60 7.30 7.00 6.57 6.03 5.38
4.30 3.55 2.92 2.50 2.20 2.10 2.25 2.50 3.00 3.45 4.00 4.62 5.30 6.07 6.85 7.60 8.30 8.83 9.25 9.55 9.70 9.62 9.37 8.97 8.27 7.58 6.70 5.80 4.80 3.80 2.85 2.03 1.30 0.80 0.40 0.18
a Molar concentrations: adenvlic. 1.44 X 10.’ M ; cytidylic, 1.55 X lo-’ M ; guanylic, 1:51 X lo-’ M; uridylic, 1.54 x 10-5M.
RESULTS AND DISCUSSION Table I1 presents the results of the GSAM experiment. In the previow study (9) various seta of wavelengths were chosen. In this study, two of those sets were duplicated. Namely, wavelengths 10,25, 28, and 34 and the set containing all 36 wavelengths. The 10, 25, 28, 34 set was chosen because Zscheile et al. found the results to be better than any other
combination of four wavelengths. One additional study was performed making use of only the odd wavelengths 1,3, 5,7, ...,35 totaling 18 wavelengths. With a 2% RSD, there is seen a substantial improvement in the accuracy and precision for the 18 and 36 wavelength sets as opposed to the four wavelength set. This is analogous to the findings of Zscheile et al. as shown in Table 11. Nevertheless, the GSAM is observed to have better accuracy and precision for both sets of wavelengths listed. This can be reasoned as being due to the statistical equilibration obtained by multiple measurements at each sensor for the GSAM compared to only one measurement for the normal multicomponent procedure used in the previous study. Additionally, using the GSAM would also allow for matrix effect corrections while the procedure of Zscheile only corrects for spectral interference. The data in Table I1 show that there is little difference in the accuracy whether wing 18 sensors or 36 sensors. However, there exists an improvement to the precision of the analysis by wing 36 wavelengths as expressed by the RSD’s for no.The lower RSD values for 36 sensors suggest that a greater stability to noise can be obtained. This results from the signal-to-noise enhancement by using 36 sensors over 18 sensors. With the 36 wavelengths there is more complete utilization of the spectrum allowing a greater tolerance to noise in the data. In some instances, no chemical information is available to the additional sensors and unwanted noise may be added to the responses. This can be minimized by weighting the various responses to their noise content (13, 14). This was not the case for the present study as verified from Table I or the absorption spectra presented in ref 9. The time required for the GSAM calculation when increasing the number of sensors from 4 to 18 to 36 was 11 s, 40 s, and 90 s, respectively. The 90 6 required for a GSAM calculation represents a negligible loss as compared to the improvement of precision and accuracy obtained. When interferences are present, as is the case here, for a multicomponent analysis procedure, theory states that measurement errors can actually be amplified to produce larger uncertainties in the estimated K matrix and initial analyte quantities, no (5). This error amplification can be expressed by the condition number of the matrix K (cond (K)) (5, 17). Cond (K)represents an upper bound to this error magnification and can also be used to characterize the stability of the linear system. Listed in Table I11 are cond ( K ) for the three sets of wavelengths. For the set containing four wavelengths cond ( K ) is 65.6, meaning that the sum of the relative error in measuring
Table 11. Results of GSAM and Zscheile and Co-workers with a 2% RSD Zscheile GSAM calcd calcd compo- molarity X molarity x nenta lo-’ relerror,b % RSDC 10‘ re1 error, % A C G U A C G U A C G U
2.08 1.78 1.04 0.805 1.67 1.56 1.49 1.20 1.71 1.56 1.50 1.23
44.4 14.8 31.1 47.7 16.0 0.65 1.32 22.1
136.7 92.4 89.3 646.0 13.7 2.79 6.13 28.7 9.35 2.16 4.93 19.3
2.24 1.90 0.89 0.72
55.5 22.6 41 .O 53.2
RSD
wavelength combinationd
93.0 211.0 77.9 2331.0
10, 25, 28, 34
1 , 3 , 5 , 7,..., 35
0.95 34.0 26.2 1-36 0.64 2.29 1.56 5.60 1.49 1.32 37.7 15.2 2.12 Relative Relative error = 100 X (ltrue - calcdl)/true. a A = adenylic; C = cytidylic; G = guanylic; U = uricyclic. Refer to Table I for numerical correspondence to wavelength. standard deviation = 100 x (standard deviation/calcd). 18.8
0.64 0.66 20.1
Anal. Chem. 1983, 55,567-570
Table 111. Condition K and the (Determinant)'" of K K T for Experimentally Obtained K Matrix wavelength corn hination 10, 25, 28, 34 1 , 3 , 5 , 7)...,55
1-36
cond K
(det. KKT)"'
65.6 20.3 12.1
2.21 x 10' 1.28 X 10' 3.22 X 10"'
the responses and the relative errors in the estimated calibration constants can be multiplied by 65.6 to yield the relative error in no,The worst accuracy and precision were found for the four sensor system. This is reflected both in cond ( K )and the relative errors and the RSDs listed in Table 11. The best accuracy and precision were found when all 36 sensors are used for analysis of the four-component system. This is also expressed in the cond ( K ) shown in Table I11 and the relative errors and the RSD's listed in Table 11. It appears that cond ( K ) can be used as an indication for the stability of the system to noise perturbations while at the same time it can be used for a measure of accuracy and precision. That is, a minimum cond (ac)means minimum relative errors and RSD for no. Kaiser has shown (28) that it is possible to define the total sensitivity of a multicomponent procedure as the absolute value of the determinant of the calibration matrix K. Maximum sensitivity will correspond to a K matrix with large diagonal elements andl low off-diagonal elements. These same circumstances will minimize cond ( K ) . Hence a minimum cond ( K ) should be reflected as a maximum determinant. "he method of Kaiser is limited to the case of p = r. Junker and Bergmann (19)have generalized Kaiser's definition of sensitivity for p 2 r as the square root of the determinant of the product of the K matrix and its transpose. Again, we would expect that for p > r, maximum sensitivity, greatest determinant would also be represented as a minimum in cond ( K ) . This was found to be true for the present study. Shown in Table I11 are the determinants of K p for the three sets of sensors. The sensor set containing all 36 wavelengths is seen to have the minimum cond ( K ) and the maximum sensitivity. Consequently, it appears that the sensitivity of the calibration matrix K compliments the information contained in cond ( K ) . Jochum et al. (5) have also used the cond ( K ) to determine the optimal sensors for multicomponent analysis. This is accomplished by obtaining the minimum cond ( K ) for various combinations of the available sensors. Similarly, Junker and Bergmann (19)have used the sensitivity number as a method of optimization of a multicomponent system. Other methods for the choice of optimal analytical sensors in multicomponent systems exist (8-10); these involve tedious mathematical calculations and necessitate the direct comparison of several numbers rather than the comparison of single numbers as in the methods of the GSAM and Junker and Bergmann. With the cond ( K ) or determinant of K p as criteria for optimal
567
selection of sensors, all 36 sensors should be used for optimal performance. This agrees with the experimental findings presented in Table 11. These results suggest digitization at a higher resolution for optimal performance. Namely, measurement at 1.0 nm intervals should increase the precision and accuracy accordingly. Other studies have been performed to show the optimal number of standard additions that should be made (20, 21). This paper describes a new approach for using the GSAM to perform analytical multicomponent analysis and ai new interpretation of the condition number of matrix K. It was demonstrated th,at, whenever possible, the best results are obtained when p > r , as expressed by cond (K). This paper is in agreement with a previous study (22)where similar results were found for a series of computer-simulated experiments and real data. The real data were those obtained by Jochum et al. ( 5 ) ,but only consisted of selected wavelength positions in the absorption curves rather than the complete absorption spectra as done here.
LITERATURE CITED (1) Saxberg, B. E.; Kowalski, B. R. Anal. Chem. 1979, 57, 1031--1038. (2) Kaiivas, J.; Kowalski, B. R. Anal. Chem. 1981, 53, 2207-221 2. (3) Gerlach, R. W.; Kowalski, B. R. Anal. Chem. Acta 1982, 734, 119. (4) Moran, M.; Kowalski, B. R., Laboratory for Chemometrlcs, Department of Chemistry, Unlversity of washington, unpublished work, June 1981. (5) Jochum, C.; Jochum, P.; Kowalski, B. R. Anal. Chem. 1961, 53, 85-92. (6) Kalivas, J.; Kowaiski, B. R. Anal. Chem. 1982, 54,560-565. (7) Kalivas, J.; Jochum, C.; Kowalski, B. R. Presented as paper NO. 440 at the Pittsburgh Conference, Atlantic Clty, NJ, March 1982. (8) Sustek, J. Anal. Chem. 1974, 46, 1676-1679. (9) Zscheile, F. P.; Murray, H. C.; Baker, G. A.; Peddicord, R. 0. Anal. Chem. 1962, 3 4 , 1776-1780. (10) Sustek, J.; Llvar, M.; Schiessl, 0. Chem. Listy 1972, 66, 168. (11) Certontaln, H.; Duin, H. 0.J.; Vollbracht, L. Anal. Chem. 196:3, 35, 1005-1007. (12) Milano, M. J.; Kim, K. Anal. Chem. 1977, 49,555-559. (13) Ratzlaff, K. L. Anal. Chem. 1980, 52, 1415-1420. (14) Hirschfeld, T. A@. Spectrosc. 1976, 30, 67-69 (15) Neter, J.; Wassorman, W. "Applied Linear Statistical Models"; Rlchard D. Irwln, Inc.: Homewood, IL, 1974;Chapter 6. (16) Naylor, T. H.; Baiintfy, J. L.; Burdlck, D. S.; Chu, K. "Computer Simulatlon Techniques"; Wiley: New York, 1966;Chapter 4. (17) Dahlquist, G.;Bjorck, A.; Anderson, N. "Numerical Methods", Pretntica Hall: Englewood Cliffs, NJ, 1974;Chapter 5. (18) Kaiser, H. Pure Appl. Chem. 1973, 34, 35-61. (19) Junker, A.; Berpmann, G. 2.Anal. Chem. 1974, 272, 267. (20) Franke, J. P.; de Zeeuw, R. A. Anal. Chem. 1978, 50, 1374-1380. (21) Ratziaff, K. L. Anal. Chem. 1979, 57, 232-235. (22) Kalivas, J.; Kowaiski, B. R., Laboratory for Chemometrics, Department of Chemistry, University of Washington, unpublished work, June 1982.
'
Present address: Laboratory for Chemometrics, Department of Chemistry, BElO, University of Washington, Seattle, WA 98195.
J. H.Kalivas' Department of Chemistry Montana State University Bozeman, Montana 59717 RECEIVED for review August 19, 1982. Accepted November 12, 1982. This work was supported by the Department of Chemistry, Montana State University, Bozeman, MT.
Temperature! Dependence of the Modified Pararosaniline Method for the Determination of Formaldehyde in Air Sir: There is currently considerable interest in methods for the measurement of formaldehyde in nonindustrial indoor environments, in particular, buildings insulated with urea formaldehyde foam insulation. The N.I.O.S.H. recommended method ( I , 2 ) employri chromotropic acid and sulfuric acid
for the analysis of the sampled air. The chromotropic ,acid method is relativelly insensitive and is potentially subject to interferences by both organic and inorganic compounds (3). Miksch et al. have recently published ( 3 ) a modified pararosaniline method. which has superior sensitivity for the
0003-2700/83/0355-0567$01.50/0 0 1983 American Chemical Society
