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Anal. Chem. 1984, 56, 462-466
from its normal and derivative chromatograms. Few sets, usually 3 to 5, of five peak heights of the normal and its derivative chromatograms with fixed intervals of time points, i.e., At, are required for this purpose. These peak heights can be measured manually or electronically. From these peak heights, all necessary peak parameters are easily calculable by a small hand calculator with reasonable accuracy.
LITERATURE CITED (1) Yamaoka, K.; Nakagawa, T. Anal. Chem. 1975, 4 7 , 2050. (2) Pauls, R. E.; Rogers, L. 6. Anal. Chem. 1977, 49, 625. (3) Littlewood, A. E. “Gas Chromatography”, 2nd ed.; Academic Press: New York, 1970; p 169. (4) McWilliam, I . G.; Bolton, H. C. Anal. Chem. Ig89, 4 7 , 1755. (5) McWilliam. I . G.; Bolton. H. C. Anal. Chem. 1989, 4 1 , 1762. (6) Anderson, A. H.; Gibb, T. C.; Littlewood, A. E. J. Chromafogr. Sci. 1970, 8, 640. (7) Gladney, H. M.; Dowden, E. F.; Swalen, J. D.Anal. Chem. 1989, 4 1 , 883.
(8) Chesler, S. N.; Cram, S. P. Anal. Chem. 1975, 45, 1354. (9) Dondi, F.; Bettl, A.; Blo. G.; Blghl, C. Anal. Chem. 1981, 5 3 , 496. (10) Dondi, F. Anal. Chem. 1982, 5 4 , 473, and earlier references cited therein. (11) Grushka, E. Anal. Chem. 1972, 4 4 , 1733. (12) Grushka, E.; Myers, M. N.; Giddings, J. C. Anal. Chem. 1970, 42, 21. (13) Sternberg, J. C. “Advances In Chromatography”; Glddlngs, J. C., Keller, R. A., Eds.; Marcel Dekker: New York, 1966; Vol. 2, pp 205-270. (14) Yau, W. W. Anal. Chem. 1977, 49, 395. (15) Barber, W. E.; Carr, P. W. Anal. Chem. 1981, 53, 1939. (16) Savltzky, A.; Golay, M. J. E. Anal. Chem. W64, 36, 1627. (17) Kalvoda, R. “Operational Amplifiers in Chemical Instrumentation”; Wiley: New York, 1975; p 49. (18) Chesler, S. N.; Cram, S. P. Anal. Chem. 1971, 43, 1922. (19) Beyer, W. H. “Handbook and Mathematical Science”, 5th ed.; CRC Press: Cleveland, OH, 1975, pp 40-43. (20) O’Haver, T. C.; Green, G. L. Anal. Chem. 1976, 4 8 , 313.
RECEIVED for review March 29, 1983. Accepted November 4,1983. We are grateful to the Korean Traders Scholarship Foundation for support of this work.
Alternate Indexes of Variation for the Analysis of Experimental Data Jimmy W. Worley,* James A. Morrell, David L. Duewer, and Laurie A. Peterfreund Monsanto Agricultural Products Company, Research Department, 800 North Lindbergh Blvd., St. Louis, Missouri 63167
Three alternate Indexes of varlatlon, In addltlon to the standard devlatlon and the coefflclent of varlatlon, have been examlned for the evaluatlon of analytical method valldatlon data. One of the Indexes, the “coefflclent of root varlatlon” or CRV, Is Intermediate between the standard devlatlon and the coefflclent of varlatlon and mathematlcally models the OSHA slldlng scale requlrements of lndustrlal hygiene methods used In the prlvate sector. The utlllty of the alternate Indexes Is Illustrated for the determlnatlon of anlllnes In alr by a modlfled NIOSH procedure.
The analytical chemist working on the development of a new industrial hygiene method t y p i d y treats the uncertainty in experimental data primarily from a perspective which dictates that it not be ”too big”. For example, the NIOSH criterion of accuracy in the development of analytical methods for 385 OSHA regulated materials, the Standards Completion Project ( I ) , states that a method should give results within f 2 5 % (95% confidence limits) of the true value at 0.5,1, and 2 times the permissible exposure limit (PEL). This criterion, which is now used routinely when developing new industrial hygiene methods, is a special example of the general case where demonstration of homogeneity of the statistical index of variation known as coefficient of variation (CV) is required. Alternatively, OSHA (2) allows a “sliding scale” accuracy requirement for methods used to meet the exposure limits of several health standards. Here, the method is required to give results within only 3~50%of the true value at half the PEL, &35% at the PEL, and &25% at two times the PEL. A problem with the OSHA requirement is that it fails to specify limits on a method‘s variance at other than these three specific levels. In this paper, we introduce a concept which generalizes the OSHA requirement to all levels within the vicinity of the PEL.
There are, of course, other reasons for a chemist to be concerned with an analysis of the variance in experimental data. In the first place, such an analysis can be quite useful in the search for the causes of the variance. For example, if the greatest source of error in the analyst’s work is the so-called measurement error, or errors proportional to the value of the measurement being made, it is expected that the coefficient of variance will be nearly constant over all the levels being examined. On the other hand, if the greatest source of error is due to instrumental error, then it is expected that the standard deviation will be constant over all the levels. Another type of error, sampling error, is usually manifested by the situation in which the variance is very large at the lower levels being examined and is much smaller at the higher levels. The major cause of such error is often lack of sample homogeneity. Obviously, there may not be just one principal source of error; multiple sources may be in existence simultaneously. Finally, another reason why the analyst should pay attention to the nature of variance in his data is that usually a linear model will be applied to the data using standard linear regression techniques in order to generate a calibration curve. Such techniques, to be valid, require the variance to be constant a t all levels in the model. This requirement is usually at odds with the analyst’s attempts to eliminate both sampling and instrumental errors, and thereby achieve a constant CV. Methods used to circumvent this apparent dilemma are generally of two types. First, a transformation can be applied to the data which would make the variance of the transformed data constant. With constant CV data, for example, the log transformation is often appropriate (3, 4 ) . It is the transformed data which is then evaluated by the linear regression. An alternative method is to perform a weighted regression. The question of what weights should be applied has been discussed elsewhere (5), but generally focuses on an analysis of the variance in the original data. Either of these methods can be acceptable. A discussion of which method should be
0003-2700/84/0356-0462$01.50/0 0 1984 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 56, NO. 3, MARCH 1984
used in which situation is not relevant to the scope of this paper. The point being made here is that a regression on the original data exhibiting a constant CV without prior data “manipulation” or weighting is incorrect. We now report three alternate indexes of variation (other than the standard deviation and the CV) which have been found useful in expanding the analytical chemist’s ability to provide a better mathematical interpretation of the variance or dispersion of his data. These indexes can be used directly to respond to the issues raised above. One of these indexes, termed the “coefficient of root variation”, or CRV, is shown to be a general case of the OSHA “sliding scale” requirements. We also report on the utility of the indexes in the validation of two methods for anilines in air by sampling on silica gel and analysis by gas chromatography. In one method, thermionic detection is used; the other method uses infrared detection. By calculation of the five indexes of variation and use of the concepts presented here, it is shown that, although both methods satisfy the NIOSH requirements, they are not equally “good”. By examination of trends in the indexes of variation, it is possible to identify possible sources of error present in each method.
EXPERIMENTAL SECTION Materials. Samples of aniline, N-isopropylaniline (NIPA), 2-ethyl-6-methylanhe (EMA),and 2,6-diethylaniline (DEA) were available from other projects in these laboratories. All were greater than 99% purity by GC. 3,5-Dichloroanhe was Aldrich technical grade. Methanol was Burdick and Jackson distilled in glass. Silica gel sampling tubes containing 65 mg in the front section and 35 mg in the back section were purchased from Zink Safety Co., St. Louis, MO (catalog no. 226-10-100);the contents were removed and packed in new tubes as described below. Sample Generation-GC/Thermionic Detection Procedure. The NIOSH method for “aromatic amines” consists of sampling on silica gel, desorbing with methanol, and analyzing by gas chromatography with flame ionization detection (6). Desorption efficiency is determined by the phase equilibrium technique, and GC calibration is by the external standard method. This general procedure was applied to the determination of aniline, N-isopropylaniline, 2-ethyl-6-methylaniline, and 2,6-diethylaniline in air in the 2.5-10.0 ppm range (PEL for aniline itself, established by OSHA, is 5.0 ppm), with a few modifications for improved accuracy and specificity. An internal standard, 3,5dichloroaniline, was added to the methanol used for desorption, thermionic GC detection was used to maximize selectivity for nitrogen-containing materials, and a “flash loading” dynamic standard system, described below, was developed. One valuable consequence of these changes when the method was used at a manufacturing site was the elimination of an interfering GC response for an aromatic hydrocarbon in some of the samples. Becher (7)recently has described air sampling for similar anilines using an internal standard, capillary gas chromatography, and nitrogen-selective detection. A “flash loading” technique was used to prepare samples for the GC nitrogen selective detection work. The contents of a commercial silica gel sampling tube were removed and repacked into a 15 cm x 1/4 in. 0.d. glass tube containing a dimple, for holding the packing in place, about 4 cm from one end. The other end was inserted into the injection block of a Hewlett-Packard 5840A GC and secured in place with a 1/4 in. Vespel ferrule and stainless steel nut. The dimensions of the tube were such that the column of silica gel just came up to, but did not enter, the stainless steel nut (and therefore the metal surfaces heated by the injection block). For sample generation, the injector temperature was 275 “C, and the oven was at ambient temperature. Carrier gas flow was 25 mL of helium/min. A 10-pL injection of a methanol solution containing the equivalent of -2.5, 5.0, or 10.0 ppm each in air, for a 20-L air sample, of the four anilines was made. After 10 min, the sampling tube was removed from the GC and connected to a portable sampling pump (Du Pont Model 4000). Twenty liters of ambient air was then pulled through the tube at a rate of 50-100 mL/min to simulate a real sample and thus check for possible
463
Table I. Spectral Regions Used for GC/IR Quantitation spectral band compound used, cm-’ aniline NIPA EMA DEA DCA
1664-1569 1643-1566 1689-1566 1664-1580 1666-1516
breakthrough of anilines into the backup section. The two sections of silica gel then were removed and placed in separate l-dram vials with Teflon-lined caps. A 2.0 mL volume of methanol containing 0.66 g/L 3,5-dichloroanilinewas pipetted into each vial as an internal standard for GC analysis. The vials were capped and sonicated for 1.0 h or allowed to stand overnight before analysis. Sample Generation-GC/FTIR Method. For validation, 10.0-pL manual injections were made of methanol solutions containing in 2.0 mL the equivalent of -2.5,5.0, or 10.0 ppm each of the four anilines and 10.0 ppm of 3,5-dichloroaniline, for an 80-L air sample, For example, the injection amounts for a 2.5 ppm equivalent sample were 3.8 pg of aniline, 5.5 pg of N-isopropylaniline, 5.5 pg of 2-ethyl-6-methylaniline, 6.1 pg of 2,6diethylaniline, and 6.6 Mg of 3,5-dichloroaniline. GC/Thermionic Detection. A Hewlett-Packard 5840A GC equipped with a nitrogen-phosphorus detector operating in the nitrogen mode was used. The injector temperature was 275 OC, and the detector temperature was 250 OC. On-column injection was done on a 1/4 in. 0.d. X 2 mm i.d. X 6 f t length glass column containing 10% OY-11 on 100-120 mesh Gas Chrom W-HP. Carrier gas was 25 mL of helium/min. Manual sample injections of 6.0 fiL were used. GC/FTIR. A Nicolet 7199 Fourier transform infrared spectrometer with a KBr-Ge beam splitter, gold-plated light pipe (400 mm X 2.5 mm i.d.), and liquid nitrogen cooled mercury-cadmium-telluride detector (MCT-A, 5000-850 cm-’) was used. Data were collected with a scanning velocity of 0.712 cm/s (2048 data points/scan) and were apodized with the Happ-Genzel function. A Varian 3700 gas chromatograph was used. The column was identical with the one used for the nitrogen-selectivework above. The end of the column was connected to a nominal 1003 splitter, with the smaller portion going to an FID and the larger portion to the light pipe via a 1/18in. 0.d. glass-linedstainless steel tubing transfer line. The light pipe and transfer line were held at 260 “C. The GC injector temperature was 250 “C. Carrier flow was 24 mL of nitrogen/min. After injection, the column was held at 140 “C for 1min and then programmed to 210 “C at 10 “C/min and held there for 5 min. For quantitation of each aniline, a strongly absorbing IR region was chosen to be used in the integration software supplied by Nicolet. The exact spectral regions used for each component are listed in Table I. The integrated values were then summed for all spectra collected during the elution of the material of interest. The resulting summation value, which is a function of time, intensity, and wavelength, was used as the component response. It was normalized by ratioing to the component response of the internal standard.
RESULTS AND DISCUSSION Statistical Analysis. The two most common statistical treatments of variance are the standard deviation (a) and the coefficient of variation ( u / X ) , or CV, where X is the mean of measurements a t a level of interest. This index is also known as the “index of dispersion”. The statistical validation protocol used by NIOSH in the Standards Completion Project ( 1 ) was to analyze six samples at each of three levels (0.5, 1, and 2 times the permissible exposure level), check the data for outliers by Grubbs’ test (8,9), calculate the coefficient of variation for each level, calculate the pooled coefficient of variation across the three levels (eq 1)and test the data for homogeneity of variance by Bartlett’s test (10) (eq 2)
(m)
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ANALYTICAL CHEMISTRY, VOL. 56, NO. 3, MARCH 1984
Requirement of the NIOSH SCP protocol Yt - 0.25Yt IY, IYt where is the pooled coefficient of variation, n is the number of levels, fi is the degrees of freedom at level i and is equal to the number of observations minus one, CVi is the coefficient of variation for level i, and f is the total degrees of freedom.
But since Yt = Y, 0.75(Y,
f tu
+ 0.25Yt
we have the further constraint
+ tu) I
Y, I 1.25(Ym- tu)
or, after further reduction tu I0.25Y, or u/Y,
I 0.25/t
or, after pooling the CV, I 0.25/t i=l
Bartlett’s test yields a x2 value which is compared against critical values in a standard x2 table a t the significance level of interest. NIOSH used a 1%significance level. The critical value of x2 at this level for two degrees of freedom (number of test levels less one) is 9.210. Therefore, a method whose validation data yields a x2 value less than 9.210 cannot be considered to have an inhomogeneous or nonuniform coefficient of variation throughout the range of 0.5 to 2 times the PEL. If it is not inhomogeneous, the best estimated value of that coefficient over the range is given by The magnitude of cV then dictates whether the NIOSH accuracy criterion of f25% is met. For three levels at the 95% confidence level, has to be less than 0.058 for an unbiased method. This number is derived simply by dividing the target percentage accuracy (0.25) by the appropriate Student t statistic (ti!?2= 4.303). (It should be noted that the NIOSH protocol stated that the estimated value of had to be less than 0.105; it is not clear how this number was derived, and we recommend the more restrictive value calculated above.) A poor alternative when constant coefficient of variation is not observed by the NIOSH protocol is to claim that the f25% requirement is met at the three measurement levels if each of the three individual CV’s are less than the target value. This, however, dilutes the statistical information of the already relatively small sample population of 18 data points, and, of greater concern, it allows no valid conclusions to be reached about the variance at points other than the three levels. Another, better, solution is to look for other statistically significant functions which model the variance as some function of X. Such functions would utilize all the available information and, more importantly, would allow a smooth mathematical interpretation of the variance throughout the measurement range. Knowledge of the functional form of the variance would also dictate the nature of an appropriate weighting function for a proper least-squares regression analysis of the data. One way to express the functional form of the variance is through the use of an appropriate index of variation. For example, when u is proportional to X (formally, when u, = kX,),one obtains a constant CV. Of course, the best estimate of k is ?%, the pooled coefficient of variation. Analogously, if u is proportional to the square root of X (or u, = kXL1/’) the index a/X1/2will be constant over the levels. A convenient name for this index is the “coefficient of root variation” or CRV. Similarly, the best estimate of k , in this case, is CRV, the pooled coefficient of root variation. That constancy of CRV encompasses the OSHA “sliding scale” requirements is established by the following lines of reasoning. Both the NIOSH and the OSHA protocols have the f 2 5 % requirement for measurements at the 2PEL level. The sliding scale case, however, allows the relative error to increase by a factor of 2lI2 (to about 35%) at 1/2 of 2PEL and by a factor of 4ll2 (to 50%) at 1/4 of BPEL, or, in general, by n1/2a t l / n of 2PEL. These relationships are shown below:
m.
cV
where Y, is the measured concentration, Yt is the true concentration, and t = t&-n, the Student t statistic for n degrees of freedom a t an appropriate significance level a. Similarly, for the OSHA sliding scale protocol
+
Yt - 0.25(2YPEL/Yt)l/2Yt IY, 5 Yt 0.25 (2 YPEL / Yt) ‘”Yt and since Yt = Y, f t u , we have (Ym
+ to)(l - 0.25(2YpEL/Y$/2)
5 Y, 5 (Y, - tu)(l 0.25(2YPEL/Yt)1/2)
+
or, after further reduction
t u I0.25(Y,(2YPEL)/Yt)1/2 and since Y, = E(Yt) this can, at a practical level, be reduced to U/Ymliz I0.25(2YpEL)1/2/t or, after pooling
-
CRV I 0.25(2YPEL)l12/t where YPELis the concentration at the permissible exposure level. Therefore, the critical value for a pooled coefficient of root variation (CRV) to pass the 50-35-25 OSHA sliding scale requirements is CRV 5 0.25(2YpEL)1/2/tfor an unbiased method, or for three levels (2 degrees of freedom) at the 95% confidence level CRV 5 0.082( YPEL)lI2. There are, of course, a number of ways to determine the best functional form for the variance in a set of experimental data. NIOSH suggests that Bartlett’s test, described earlier, be used to determine the constancy of CV. For consistency in the comparison of the different indexes of variation, we have also used Bartlett’s test in the evaluation of the data presented here. The absolute error becomes smaller for both CV and CRV a t lower levels of measurement. It is possible, however, for absolute error to increase at lower levels of measurement. This might result, for example, if one took increasingly smaller samples of a nonhomogeneous sample. Two analogous indexes which would describe this behavior of the variance would be the inverses of the CV and the CRV. These indexes, designated as the “coefficient of inverse variation (CIV)” and “coefficient of root inverse variation (CRIV)”, are defined mathematically as a X and uX1izrespectively. A summary of all five indexes of variation is given in Table 11. To test the potential utility of the new indexes of variation, the validation data for 20 industrial hygiene methods from the NIOSH Standards Completion Project ( I ) , chosen randomly from the total of 385, were evaluated. Although the majority of the 20 methods gave the lowest x2 from Bartlett’s test for constant CV, five of the 20 methods gave the lowest x2 for constant CRV and two gave the lowest x2 for constant SD. As explained previously, a lower x2 means a better “fit”
ANALYTICAL CHEMISTRY, VOL. 56, NO. 3, MARCH 1984
Table 11. Summary of Indexes of Variation index of variation
abbreviation
coefficient of variation coefficient of root variation standard deviation coefficient of root inverse variation coefficient of inverse variation
definition
confidence rangea
ax -- 1
t tcvY, t tCRVYm1l2 -
-
cv CRV SD CRIV CIV
ax-l/2 0
465
t tSD
-
-
tCRIV Ym- l I 2 i- tcIVY,-l
aX”2 -
?:
ax
----
t is the appropriate Student’s t statistic. Ym is the mean measured concentration. CV, CRV, SD, CRIV, and CIV are pooled (over all levels) values of the corresponding index of variation. a
Table 111. x2 Values Calculated by Bartlett’s Test for Four Anilines by Five Variation Models with GC/Thermionic Detection
x2 for index of variationa compound
CV
CRV
aniline NIPA EMA DEA
4.Sb
14.8
1.5b
1.6
2.0b 3.0
5.4 0.04*
SD
CRIV
CIV
27.6 8.2 11.3 2.6
41.5 18.6 18.9 9.7
55.9 30.9 27.5 19.8
a Critical value is 9.2 at the 1% significance level. Calculated values less than this indicate the corresponding index may not be rejected as being constant or homogeneous. Lower values indicate a better fit to the model. For each aniline, this is the lowest of the five calculated xz values. The corresponding index of variation describes the best model of the exDerimenta1 variance.
of the experimental data to the variation model being tested. Thermionic GC Results. The modified NIOSH method for aromatic amines described in the Experimental Section was used to analyze eight samples a t each of three levels of interest (2.5,5.0, and 10.0 ppm) for aniline, N-isopropylaniline, 2-ethyl-6-methylaniline,and 2,6-diethylaniline in air. The five indexes of variation described in Table I1 were calculated for each of the anilines and these were tested in each case for homogeneity by Bartlett’s test. The resulting x2 values are in Table 111. Aniline, N-isopropylaniline, and 2-ethyl-6methylaniline all fit the constant coefficient of variation model best. The results for 2,6-diethylaniline, however, fit the CRV model best, indicating that there may have been larger than “normal” errors in the sample generation for this material. It is the highest boiling of the four anilines, and its “flash vaporization” onto the sampling tube may not have been cleanly reproducible. GC/FTIR Results. An infrared spectrophotometer may be considered also as a potential “nitrogen-selective” GC detector ( I I ) , just as is the thermionic detector. GC/IR is beginning to be applied more frequently as a qualitative technique to trace environmental problems (12,13).We investigated it in the present case as a quantitative procedure. Our initial results are encouraging and indicate the desirability of further work in this area. The potential for qualitative confirmation of a GC peak as a specific material is indicated in Figure 1,which shows a series of consecutive IR spectra collected in a typical GC run. The region where each of the five anilines elutes is easily identified in the contour plot. For each material, the component response represents the area of a slice of one of the ”hills” of maximum intensity. With the IR instrumentation acting essentially as a multichannel analyzer, i t is clear that more than one spectral region could be quantified if desired for confirmation of the results. We are currently investigating the use of the statistical techniques presented here to aid in the window selection process. For validation of the IR procedure, six aliquots of each solution (2.5,5.0, and 10.0 ppm equivalent) were injected and the results were evaluated statistically by the same procedures
1
000
4690
1580
1470
060
I250
WAVENUMBERS
Flgure 1. Contour plot of a GClIR run for five anilines, at a level equivalent to 2.5 ppm each in air, for an 80-L air sample. The time coordinate increases from back to front in the figure. Injection amounts (in order of elution) were 3.8 pg of aniline, 5.5 pg of N-isopropyianiline (NIPA), 5.5 Mg of 2-ethyl-6-methyianiline (EMA), 6.1 pg of 2,6diethylaniline, and 6.6 Mg of 3,5-dichloroaniline.
Table 1V. xz Values Calculated by Bartlett’s Test for Four Anilines by Five Variation Models with GC/FTIR Detection xz for index of variationa compound CV CRV SD CRIV CIV 15.1 24.4 34.3 aniline l . S b 7.2 9.0 17.4 NIPA 6.9 2.2b 3.1 2.2b 6.8 14.5 3.3 EMA 10.0 4.0 10.1 DEA 4.0 0.5 0.45b a Critical value is 9.2 at the 1% significance level. Calculated values less than this indicate the corresponding index may not be rejected as being constant or homogeneous. Lower values indicate a better fit to the model. For each aniline, this is the lowest of the five calculated x2 values, The corresponding index of variation describes the best model of the experimental variance.
described for the thermionic detection case. In this instance, however, only the results for aniline fit the constant CV model best. N-Isopropylaniline fit the CRV model best while both 2-ethyl-6-methylaniline and 2,6-diethylaniline best fit the constant SD model. The x2 values for each of the indexes of variation are shown in Table IV. Although all the samples met the NIOSH f25% criteria, it is clear from the above results that the “fixed“ errors in this methodology are, in general, more predominant than those errors proportional to the measurement level, We speculate that a possible source of these errors is in the nature of the quantitation software supplied by Nicolet. Although the concept of integrating over a spectral window may be beneficial to increase the effective signal/noise ratio, there is no a priori reason to suppose that a (spectral) peak width or band shape is in any way proportional to the concentration of the material being studied. Beer’s law only relates the concentration of a sample to a peak absorbance value-not to the area of a peak. We are currently investi-
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Anal. Chem. 1904, 56, 466-470
gating whether the use of peak height rather than peak area significantly improves the quantitation capability of GC/FTIR techniques. Without the examination of the indexes of variation as shown in Tables I11 and IV, it is doubtful that a consideration of the sources of error would have occurred. It is in this manner that the use of the indexes of variation, as presented here, becomes most beneficial to the analytical chemist.
CONCLUSIONS The analytical chemist working on the development of a new industrial hygiene method must already go through a number of calculations to verify that the variance or dispersion in his data is not “too large”. We have demonstrated here that by expanding slightly on those calculations and calculating in a similar manner the coefficient of root variation, the standard deviation, and one or both of the “inverse” coefficients, the chemist can, after pooling and analyzing these coefficients, better make use of his data to shed light on the nature of trends in the variance of the data. Furthermore, one of the indexes described here, namely, the CRV, has been shown to encompass the OSHA sliding scale requirement. We believe that it expands on concepts originally developed as the motivation for OSHA’s implementation of the sliding scale requirement. Finally, an identification of the functional form of the dispersion or variance in a set of data can be used to properlv develop a weighted regression model of the data. In our laboratory, we have found that the use of these indexes significantly helps the chemist identify possible sources of error in a potential method and also aids in his or her selection when the choice of several similarly functioning
analytical methods is available. In this study an analysis of the indexes of variation led us to the conclusion that there may be a sampling problem for the higher boiling anilines in the method using “flash loading” and thermionic detection and also that the quantitation procedure for the GC/FTIR method may need improvement. Had only the coefficient of variation been calculated, as is usually done, it is doubtful that these potential problems would have been considered. Registry No. NIPA, 768-52-5; EMA, 24549-06-2; DEA, 57966-8; DCA, 626-43-7; aniline, 62-53-3.
LITERATURE CITED (1) Taylor, D. G.; Kupel, R. E.; Bryant, J. M. “Documentation of the NIOSH Validation Tests”; DHEW (NIOSH) Publicatlon No. 77-185, Clnclnnatl, OH, 1977. (2) Code of Federal Regulatlons; Title 29, Part 1910.17: Vinyl Chloride. (3) Bartlett, M. S. Biometrics 1947, 3 , 39. (4) Kurtz, D. A. A n d . Chim. Acta 1983, 150, 105. (5) Draper, Norman; Smith, Harry ”Applied Regresslon Analysls”; Wiley: New York, 1966; pp 77-81. (6) Crable, J. V.; Taylor, D. G. “NIOSH Manual of Analytical Methods”; DHEW (NIOSH) Publication No. 75-126, Cincinnatl, OH, 1974. (7) Becher, G. J . Chromtogr. 1981, 21 1 , 103. (8) Grubbs, F. E. Technometrics 1972, 14, 847. (9) Grubbs, F. E. TeChi7Of?7etf~CS1969, 1 1 , 1. (IO) Bethea, R.; Duran, B.; Bouillion, T. “Statistical Methods for Engineers and Scientists”; Marcel Dekker: New York, 1975; pp 247-251. (11) Hall, R. C. CRC C f k . Rev. A n d . Chem. 1978, 7 , 323. (12) Worley, J. W.; Rueppel, M. L.; Rupel, F. L. Anal. Chem. 1980, 52, 1845. (13) Erickson, M. D. Appl. Specfrosc. Rev. 1979, 15, 261.
RECEIVED for review October 22, 1981. Resubmitted October 20, 1983. Accepted November 18, 1983. Presented in part at the American Industrial Hygiene Conference, Portland, OR, May 25-29, 1981.
Reiterative Least-Squares Spectral Resolution of Organic Acid/Base Mixtures S. D. Frans and J. M. Harris* Department of Chemistry, University of Utah, Salt Lake City, Utah 84112
A method of data anaiysls for acid/base equllibria employing multiwavelength detectlon Is shown to recover both the dissociation constants and absorption spectra of organic acid/ base pairs in complex mixtures. The resolvlng power of the method is studied by using synthetic data and demonstrated experlmentaiiy on multicomponent mixtures. The multiple wavelength tltratlon curves are used to form a matrix, D, which Is then factored Into matrices A, containing the absorption spectra of the components, and C, contalnlng their pH-dependent concentrations. The correct pK,’s and absorption spectra are arrlved at by a leastgquares minimization in which the only parameters varied are the pK,’s.
The analysis of complex multicomponent samples has been greatly facilitated by the use of “hyphenated” analytical methods (1-3), which can facilitate the identification and resolution of individual components of a mixture (4-10). The informing power of a two-dimensional measurement lies in the large number of information channels which increase by the product of the number of channels in each dimension, 0003-2700/84/0356-0486$0 1.50/0
provided that the measurement dimensions are uncorrelated. When one measurement dimension, which has few degrees of freedom and can be modeled, is combined with a second measurement dimension of higher informing power (more independent information channels),one can utilize the benefits of each dimension for the resolution of mixtures (11,12). By fitting the data along the lower informing power dimension using a physical model to provide a functional form, one can extract the richer analytical information from the other dimension. Because of the independent nature of the two dimensions, considerably more overlap can be tolerated than with one-dimensional curve fitting (13). Example applications of this reiterative least-squares method have included timeresolved fluorescence spectrometry (11) and GC/MS (12). The reiterative least-squares technique could naturally be applied to data involving chemical equilibria vs. some higher informing power dimension. For the purposes of this study, a pH-dependent, acid/base equilibrium is chosen where the second dimension is based on UV/Vis spectrophotometry. The use of pH as a variable in spectrophotometric studies, especially for biochemical systems, appears to be universal (14). Numerous other techniques have been used to determine 0 1984 American Chemical Soclety
