Anal. Chem. 1004, 56, 457-462
due to contamination from acids and chemical reagents by the use of many blanks. These were found to contain Cr contents in the 5 to 10 ng range. PIXE Analysis. For elements such as V, Cr, Mn, Fe, Co, Ni, Cu, and Zn, the PIXE method can detect quantities of less than 1ng OII the target. The limit of detection for a given element depends on the X-ray emission (counting) rate of the next element on the spectrum. In the case of chromium, a high manganese level on the target is a serious problem. However, the main limitation of the present method for obtaining 1ng in a suitable form on the target is found in the preliminary sample preparation. With plasma volumes of 10 or 20 mL, we can easily measure 1 ng/mL, and even 0.3 ng/mL. Targets can be prepared and kept for 1or 2 months in a desiccator covered with aluminum foil to avoid damage to the polycarbonate from light.
ACKNOWLEDGMENT We gratefully acknowledge the cooperation of Magendie’s Internal Medicine Center. Special thanks are due to C. Conri’s department for the collection of many blood samples. The
457
authors also thank A. MacKenzie Peers for help with the manuscript. Registry No. Chromium, 7440-47-3.
LITERATURE CITED (1) Mertz, Walter I n “Chromium Nutrition and Metabolism”; Shapcott, D., Hubert, J., Eds.; Eisevier/North-Holland: Amsterdam, 1979: pp 1-14. (2) Slmonoff, M.; Llabador, Y.; Slmonoff, G. N.; Besse, P.; Conrl, C. Nucl. Instrum. Methods in press. (3) Speecke, A.; Hoste, J.; Versieck, J. “Sampling of Blologlcal Materials”; Bureau of Standards: washington, DC, 1976; NBS Special publication 422 (Proc. 7th IMR Symp., Gaithersburg, Oct 7-11, 1974), p 299. (4) Anand, V. D.; Ducharme, D. M. “Accuracy in Trace Analysis”; Bureau of Standards: Washington, DC, 1976; NBS Special Publicatlon 422, pp 611-619. (5) Chao, S. S.; Plckett, E. E. Anal. Chern. 1980, 52, 335. (6) Jones, G. B.; Buckley, R. A,; Chandler, C. S. Anal. Chlm. Acta 1975, 80, 369-392. ( 7 ) Mien, M. R.; Fishman, M. J. At. Absorpt. News/. 1967, 6 , 128-131. (8) Subramian. K. S.; MOranger, J. C. Int. J . Envlron. Anal. Chem. 1979, 7 , 25-40. (9) Versleck, J.; Hoste, J.; DeRudder, J.; Barbier, F.; Vanballenberghe, L. Anal. Left. 197% 12, 555-562.
RECEIVED for review July 11,1983. Accepted November 22, 1983.
Characterization of Peak Shape Parameters with Normal and Derivative Chromatograms Kyung-Hoon Jung,*Sun Jin Yun,and Sung Hoon Kang
Department of Chemistry, Korea Advanced Institute of Science and Technology, P.O.Box 150, Chongyangni, Seoul 131, Korea
The objective of the present study is to develop and assess a new method to extract peak shape parameters from the exponentially modified Gaussian peak model. This method requires both normal and derivative peak heights at four or five different time points with the same time Interval. The essential peak shape parameters can be readily evaluated by solving the cubic or quartic equation of T. Computer simulation studies showed that in the case of a peak-to-peak noise value of 1.0 %, the peak parameters were reproducible wtlhln 1.5% by the method developed In this study. The experimental works with the real chromatograms have shown that the parameters were recovered with the standard deviations no more than 1.57%.
The chromatographic peak shape is governed by several factors such as the characters of the column material and the external operation conditions of the column. The simplest input profile of 6 distribution is broadened asymmetrically and appears to be a skewed Gaussian form owing to the nonequilibrium mass transfer in column ( l ) dead , volume (2), nonhomogeneity of tube connection (3), and time lag of the detector-amplifier system ( 4 5 ) . The output peak profile has been well described, by adopting an exponentially modified Gaussian peak (EMGP) model, by several workers (4,6, 7). In the present study we report a new method to extract peak shape parameters from the EMGP model and the validity of the method by computer simulation study and experimental observations. 0003-2700/84/0356-0457$01.50/0
EMGP can be expressed as the convolution of a normal Gaussian with an exponential decay function
h(t) =
where A is the peak area, a is the standard deviation of the Gaussian component, t R is the retention time of the Gaussian component, and T is the time constant of the exponential modifier. In eq 1the EMGP model is valid over the whole range of time t and characterized fully by four peak parameters, Le., A, T , a, and t p These peak shape parameters reflect the process occurring in column and contain all the information necessary to optimize the column resolution (6, 7),the efficiency ( I ) , and the deconvolution of the overlapped peaks. A and tR determine the scale and the position of chromatographic peaks, respectively, while the asymmetricity of the peak depends upon the ratio of T to a. Some intensive studies have been made to find these parameters including the least-squares curve fitting technique (6,8),an alternative approach of the least-squares curve fitting technique (9, l o ) ,the moment analysis (2, 11-13), a modified moment analysis technique (14), and the graphical method (15). These techniques, however, suffer some degree of cumbersome extensive iterative search calculation and computerized data acquisition. In these regards our new technique has focused its attention on eliminating the cumbersome iterative procedure utilizing the normal and derivative chromatograms. The derivative chromatogram can be obtained 0 1984 American Chemlcal Society
458
ANALYTICAL CHEMISTRY, VOL. 56, NO. 3, MARCH 1984
+ h(tJ 7h’(t5)+ h(t5) =
Th’(t1)
T
f\
[
Th’(t2) Th’(t4)
+ h(t2) + h(t4)
1’
(6)
Rearranging eq 6 for 7 a simplified cubic equation is obtained in terms of 7 U3T3
+ U 2 T 2 + a17 + Uo = 0
(7)
where the coefficients consist of only normal and derivative peak heights at time points tl, t2, t 4 ,and t5. Upon determination of the T value from eq 7 one can then calculate u, tR, and A from eq 5 , 4 , and 2 subsequently. Derivation of the Quartic Equation of 7 . Using eq 2, one can derive eq 8 and 9 readily
TIME
Flgure 1. Typical normal and derlvatlve chromatograms. Five time points and their corresponding peak helghts are shown at r / u = 2.0 and zero noise level.
by electronic or numerical differentiation (16). The electronic differentiator can be built easily by use of a simple electronic circuit (17) or can be purchased commercially. The usage of the derivative chromatogram has the advantage of peak parameter calculations as well as the considerable improvement for the detection of minor peaks. Since noises on observed chromatographic peaks are unavoidable (18), we have also included the noise effect in our computer simulation study.
THEORY The method presented in this paper extracts peak shape parameters by solving either the cubic or quartic equation of 7, which are derived from the EMGP equation, and the coefficients of the corresponding polynomial equation are composed of only the peak heights of normal and its derivative chromatograms. h(t)in eq 1is the normal chromatographic peak height as a function of time in arbitrary units, and by differentiation, applying Leibniz‘s theorem, the corresponding derivative peak height, h’(t),is obtained. Combining eq 1 and its derivative, we obtain eq 2. The left-hand side of eq 2 is composed of
a normal peak height and its derivative peak height multiplied by the time constant which is equal to the Gaussian component a t time t. A typical EMGP and its derivative peak is displayed in Figure 1,where five time points are taken with same time interval At in order to derive the quartic and the cubic equations. Derivation of Cubic Equation of T. From eq 2, the ratio of the Gaussian components at times tl and t5 is given by
[
]
4Yd3 4At.t~ + h(ti) = e x p 2-Th’(t5) + M t 5 ) 82
Th‘(ti)
Similarly eq 4 and 5 are obtained
Since eq 3 and 4 are equivalent, by equating these two
(3)
An expression similar to eq 9 can be given for another set of times t z , t 3 , and t l , and eq 10 is obtained
[
7h’(tl)
+ h(t1)
+
7h’(t2) h ( t 2 )
[
I[
Th’(t3)
1-
+ h(t3) -
Th’(t2) + h(t2) + h(t2) T h ’ ( t 4 ) 7h’(t3) Th’(t3) + h ( t 3 ) Th’(t2)
I[
+ h(t4) + h(t3)
Rearranging eq 10 leads to a quartic equation of simplified form U4T4
+ U 3 T 3 + U2T2 + U17 + Uo = 0
T
]
(10)
in the (11)
Equation 11 also contains only peak heights at time points tl, t2,t3,and t 4 in its coefficients. The 7 value obtained from eq 11permits the evaluation of u, t R , and A values using eq 9, 8, and 2 in sequence. Execution of the Method. The roots of eq 7 and 11can be found conveniently by means of the absolute solutions (19) of cubic and quartic equations. Since three or four roots are generally obtained from the solutions, boundary conditions must be applied to remove physically meaningless roots of them. For these conditions the following criteria were applied. Firstly the 7 value must be real and greater than or equal to zero and zero is only for pure a Gaussian peak. The second condition presented by eq 2, Le., 7h’(tL)+ h(tJ > 0, must be satisfied for all time points used in derivation of eq 7 and 11, respectively. Finally, eq 5 and 9 imply another boundary condition that the left-hand side of the equations should be always greater than zero and less than one. In both cases of cubic and quartic equations, only one root remains through the boundary conditions when the noise-free chromatogram is analyzed, and even noisy a chromatogram mostly has one root to satisfy the conditions. Since the magnitude of At and the initial time value tl are chosen arbitrarily, it is possible to execute the process of extracting four essential parameters repeatedly by changing the values of At and tl. Consequently one can ascertain the validity of the calculated value and obtain the most probable value. Once four parameters are calculated, additional parameters such as skew, excess, and number of theoretical plates can be evaluated by simple relationships (11, 15).
COMPUTER SIMULATION Computer simulation was carried out by use of a FORTRAN IV program and IBM 370/145 computer. EMGP profiles of exactly known values of u, 7,tR,and A were gen-
ANALYTICAL CHEMISTRY, VOL. 56, NO. 3, MARCH 1984
Table I. Reproducibilities of
7
r / u = 0.5
h(tl)/h,
0.1 0.2 0.3 0.4 0.5 0.6 0.8 a
0
4.80 1.21
1.28 1.66 2.38 5.83 29.93
0.53 0.19 0.18 0.23 0.34 0.50 1.44
and
(I
with Variation of h(t,)/hpa reproducibility, % 7 / u = 1.0 710 = 2.0 T
1.44 0.86 0.74 0.78 0.96 1.34 4.37
0
0.88 0.31 0.23 0.24 0.31 0.44 1.42
T
46.82 1.07 0.83 0.77 0.74 0.83 2.08
0
5.05 1.85 1.28 0.80 0.59 0.60 1.39
i-/o
=
3.0
ria
7
0
32.13 1.53
21.72 14.83 4.42 2.02 1.93 1.18 1.55
1.18
0.87 0.83 0.85 1.54
=
3.5
7
0
4.80 2.15
38.23 25.08 6.13 2.94 1.90 1.78
1.10
0.88 0.88 1.41
459
Noise = 1.0%. Number of different initial times = 5.
erated with T / O ratio ranged from 0.1 to 3.0, and maintained a constant data density of 10 points/a. The random noise was superimposed on the peak profiles and was calculated by multiplying the normal random number by a given noise multiplier of the maximum peak height. The random number was generated by using the subroutine GGNML in IMSL, International Mathematical & Statistical Libraries, Inc., and employed noise levels were O.O%, 0.1%, 0.5%, and 1.0% of peak amplitude. Derivative chromatograms were obtained by means of both the electronic and the numerical differentiations, and the two cases were compared with each other. In the case of electronic differentiation, the simulated derivative peaks were generated by using eq 2, and then its noise level was determined to be two times as large as that of the normal peaks, since it was known that there is an increase by about a factor of 2 in the noise level of each successive derivative (20). In numerical differentiation, we employed Savitzky-Golay’s method (16) which performs differentiation simultaneously with smoothing. This least-squares procedure was executed by using seven data points, and the noisy normal chromatogram was also smoothed by the method. By means of the cubic and the quartic methods described in the preceding section, four peak parameters were extracted from the simulated chromatograms including a normal and its derivative peaks. The selection of time points was done in the following procedure. Firstly the ratio of the peak height at initial time point to maximum peak height, h(tl)/h,,was chosen, and then the magnitude of At was determined so that the peak height at initial time point was equal to that at the final time point. Since the number of different initial times was chosen, the number of time point sets was automatically selected with the same At value; the subsequent initial points of each set of data were chosen around the first set of time points. The number of different initial times is consequently the number of time point sets, i.e., the frequency of repeating a process to extract four peak parameters from a chromatogram. In all simulation procedures the statistical estimations such as mean values, confidence limit, reproducibilities, and absolute percent error of each peak parameter were calculated from 20 runs of each set of simulation study.
EXPERIMENTAL SECTION Reagents. Helium (99.9999% Matheson Co.) was used as the carrier gas. Ethylene and propane (99.5% and 99.6%, Matheson Co.) and n-hexane (min 95.0%, Wac0 Pure Chemical Industries, Go.) were used as solutes. The purifications of the solutes were performed by a low-temperature trap-to-trap distillation technique. Apparatus. A Hewlett-Packard 5880A gas chromatograph equipped with thermal conductivity detector was employed for the experimental observation. Samples were injected by use of a 1.0-mL internal loop gas sampling valve. Six in. 0.d. columns used in this study are as follows: 1.5 m stainless steel (ss) SE-30 20%/Chromosorb P SO/lOO mesh; 7.0 m ss SE-30 3O%/Chromosorb W 80/100 mesh; 4.0 m Ni OV-101 30%/Chromosorb P sO/lOO mesh; 4.0 m Cu OV-17 20%/ChromosorbP 1OO/120 mesh;
4.0 m Cu Apiezon L 20%/Chromosorb P SO/lOO mesh; and 1.1 m ss Spherocarb 100/120 mesh. Procedure. Several chromatogramsat various T/u ratios were generated by using these six columns and three solute materials. The estimated noise was 0.2% of its maximum peak height at our lowest solute concentration. The GC signals were digitized at the rate of 1data/0.6 s. The raw peak heights were smoothed and differentiated numerically by seven-point quadratic leastsquares fit. 7,U , tR, and A values were calculated by using the method described in the previous section in the h(t,)/h, range of 0.1-0.3. In order to evaluate the standard deviations, five sets of time points were selected for each chromatogram and the four parameters were calculated for each set.
RESULTS AND DISCUSSION Simulation Studies. Typical computer simulated normal and derivative chromatograms are displayed in Figure 1 together with their arbitrary time points and corresponding peak heights. In this display noise levels are not included for the simplicity. The statistical evaluations with 1.0% noise were performed by solving a quartic equation of 7 and are listed in Tables 1-111. A 1.0% noise level was chosen in this calculation as a possible worse case, though percentile noises are normally less than 0.5%. Since the reproducibility of parameters is significantly affected by the selection of initial time points and so h(t,)/h,, the optimization of parameters was performed by varying h(tl)/h,with respect to T / U values, and the results are tabulated in Table I. The optimal h(tl)/h,value in Table I has the tendency to move to higher value by increasing r/ u which is the measure of the peak broadening and the asymmetricity of the chromatogram. This observation indicates that the optimal h(tl)time point shows the same tendency along with the maximum peak height shift at a relatively larger value of 7 where the chromatogram is mainly governed by asymmetricity. At lower values of T where the major contribution comes from the Gaussian part, the reproducibility has shown the improvement at a relatively lower value of h(t,). However at a very low value of h(t,), for instance less than 0.1 of the h(t,)/h, ratio, a considerable increase of relative errors was observed for the calculation of parameters. The possible interpretation of this observation is that the reproducibility of h(tl)is more significantly affected by noise than h,, so that the small h(t,)/h,ratio can lead to the erroneous evaluation of the peak parameters particularly in the case of large T / U values. The effect of At values on the peak parameters cannot be investigated independently since as soon as the t , point is set from the optimization of h(tl)/h,to the t 5 point, the At value is automatically fixed. However, the measured distance from tl to t 5 at various T / U values was shown to be almost constant throughout the optimizations. Since At is one-fourth of the distance from t , to t6,it is a good indication that At has no significant effects on the reproducibilities of the parameters. This fact may be due to cancelling effects between the enhancement, by the larger At for increasing asymmetricity if there is no maximum peak height reduction
460
ANALYTICAL CHEMISTRY, VOL. 56, NO. 3, MARCH 1984
Table 11. Reproducibilities of Peak Area at Different Sets of Initial Timesa
h( t l )lhp
710
0.5
0.2
1.0
0.3
1.5
0.4
2.0
0.5
2.5
0.55
3.0
0.6
3.5
0.7
a 1.0% noise
reproducibility, %, with the following no. of different initial times
differentiation technique electronic numerical electronic numerical electronic numerical electronic numerical electronic numerical electronic numerical electronic numerical
1
3
5
1.07 1.39 0.80 0.29 0.56 0.46 0.64 0.56 0.74 0.74 0.86 0.95 1.11 2.02
0.08 0.18 0.13 0.25 0.17 0.42 0.23 0.50 0.30 0.54 0.46 0.62 0.80 0.80
0.08 0.13 0.13 0.16 0.18 0.27 0.29 0.31 0.51 0.37 0.90 0.46 1.39 0.92
added.
Table 111. Absolute % Error of Peak Parametersa 710
a
Noise
h(tl)lh,
0.5
0.2
1.0
0.3
1.5
0.4
2.0
0.5
2.5
0.55
3.0
0.6
=
% error
differentiation technique
7
0
tR
A
electronic numerical electronic numerical electronic numerical elecwonic numerical electronic numerical electronic numerical
-0.03 -2.19 -0.35 -0.83 -0.28 -0.13 -0.27 0.24 -0.21 0.50 -0.17 0.81
0.06 1.12 0.21 1.11 0.30 1.03 0.43 0.98 0.59 0.91 0.79 0.86
0.05 2.35 0.26 1.01 0.11 0.64 -0.08 0.52 -0.36 0.53 -0.78 0.54
0.14 0.33 0.05 0.35 0.25 0.44 0.61 0.39 1.20 0.19 2.16 0.07
1.0%;no. of different initial times = 5.
involved, and the reduction, by the lowering effect of the maximum peak height because of the r / u increment. In the later case, the large At will affect the wider spreading of time points and cause a left-hand side shift of the initial time. In turn the error will be increased due to lower h(tJ involvement where noise w ill play a significant role on the reproducibilities. The effect of the time points set or the number of initial times on the reproducibilities of the peak area is presented in Table I1 where the differentiations proceeded numerically with smoothing and electronically with no smoothing of the curves. In the numerical differentiation technique, the percentile relative error has been decreased tremendously by increasing the number of the time point sets. Since the smoothing procedure reduces high-frequency fluctuations and thus enhances the signal-to-noise ratio (SIN),better reproducibility is expected although the possibilities of peak distortion (16)cannot be totally removed by this method. On the other hand this trend is not observed in the case of electronic differentiation. Some improvement a t larger numbers of the initial time points should be regarded as simple removal of random errors. Attention should be called here on the later case. Since our major concern of the present study is to simplify the calculation of peak parameters and use a simple gadget which can be built in lab, we have not tried smoothing in electronic differentiation, because smoothing for electronic differentiation involves some expensive circuitries which may dilute the merit of this study. The effect of symmetrical peak broadening may be observed by changing the u value at r / u constant. With this observation we have found that the reproducibility does not show dependency on u but only on the r/ u value which may be a proof
L
0
6
'0: 2-
t
O
-
9 ' : =! n
Ea
o
o
A
A
. * . e o E *At * * * 0
Id'T *
0
0
0
A
A
A
A A
A
A A
A
W
LT
A
Id2
'
I
A
a
A
I
A
A
I
1
1
4
ANALYTICAL CHEMISTRY, VOL. 56, NO. 3, MARCH 1984
461
-
Table IV. Standard Deviations of Four Peak Parameters from Experimental Peaks standard deviation: % ?/a
conditions
A
tR
a
7
1.5 m ss SE-30 20%/Chromosorb P 80/100, n-hexane 7.0 m ss SE-30 30%/Chromosorb W 80/100, n-hexane 4.0 m N i OV-101 30%/Chromosorb P 80/100, n-hexane 0.19 4.0 m Cu OV-17 20%/Chromosorb P 100/120, n-hexane 0.10 4.0 m Cu Apiezon L 20%/Chromosorb P 80/100, n-hexane 0.14 1.1m ss Spherocarb; propane 0.03 0.01 1.1m ss Spherocarb? ethylene 0.59 0.02 4 m Cu OV-17 2O%/Chromosorb P 100/120 6.62 mL 0.31 0.01 post dead volume, n-hexane 4 m Cu OV-17 20%/Chromosorb P 100/120 6.62 mL 1.65 0.22 0.25 0.01 0.08 post dead volume and 6.62 mL predead volume, n-hexane Flow rate 13.6-20 mL/min; oven temperature, 50-60 "C. Flow rate, 30 mL/ a Number of different initial times = 5. min; oven temperature, 240 "C. Flow rate, 30 mL/min; oven temperature, 100 "C. 0.34 0.42 0.51 0.51 0.64 0.94 1.01 1.14
1.00
0.13
0.01
1.34 1.45 0.38 0.50 0.21 1.57 0.32
0.21
0.01 0.01 0.00 0.00
0.03 0.02 0.09 0.01 0.06 0.74 0.06 0.49
Table V. Standard Deviations of Four Peak I Parameters and Reduced Area Recovered by the Method pressure of propane: reduced torr area
1
1
0.5
A
I
1.0
I
I
I
I
1.5
2.0
2.5
3.0
7/U
Figure 3. Noise effect on the reproducibility of T . Time points sets are 5. Percentile noise levels are as follows: (0)1% : (A)0.5 % : (0) 0.1%; (A)0 % noise.
the rise of noises. The absolute percentile errors of each parameter are calculated by using the input values for the computer simulation study as the true value and are given in Table 111. From the foregoing discussion for the quartic method and the similar calculations for the cubic method where we have varied the T / u ratio from 0.5 to 3.0, h ( t J / h ,from 0.4 to 0.7, and the noise level L O % , we have found out that the reproducibilities of these methods are within the 1.0% range for the quartic and 1.5% for the cubic case, respectively. Experimental Studies. Experimental work was carried out to check unanticipated problems in performing the data reduction algorithm on real chromatograms. Peak parameters T , u, t R and A were calculated by solving cubic and quartic equations for T . Since the results of both calculations were almost same, we report only the resulb of the quartic equation of T in Tables IV and V and Figure 4. The standard deviations of peak parameters obtained from experimental peaks with r / u ratio in the range of 0.34-1.65 are represented in Table IV. The values have shown that the peak parameters are reproducible within 1% in most cases and 1.57% at the worst case. We have also investigated the linearity of peak area determinations, for example, to assess the availability of this method to quantitative analysis. The results of a typical case are presented in Table V as an example. The reduced areas are consistent with the expected values with a correlation factor of 0.9997. The superposition of an experimental chromatographic peak with the peak generated by using the values of T , u, tR,and A recovered from the chromatogram at the T / u ratio of 0.95
0.49 1.79 2.27 3.38 4.16 5.45 6.49 8.18
1.00
4.37 5.47 7.93 9.75 12.45 14.96 18.64
standard deviation,b % 7
a
0.48 0.21 0.36 0.96 0.21 1.70
0.18
1.08
1.52
0.57 0.91 0.70 0.01 0.57 1.53 1.79
tR
0.00 0.01 0.01
0.03 0.03 0.06 0.06 0.09
A
0.09 0.32 0.42 0.33 0.74 0.17 0.22 0.43
a
in. x 1.1m ss Spherocarb colum; temperature, 240 "C; flow rate, 30 mL/min. Number of different initial times = 5.
I
I
\
TIME Flgure 4. Comparison between the real chromatogram and the regenerated points from the calculatlon at r / u = 0.95: (solid line) the real chromatogram: (points)calculated values: column, ' I sin. X 1.1 m ss Spherocarb 100/120 mesh; carrier flow, 30 mL/min: oven temperature, 240 O C ; sample, propane.
is displayed in Figure 4. Excellent agreement is demonstrated between the peak profiles measured and calculated from the EMGP model. In conclusion, we assume that EMGP model is sufficient to describe the real observed chromatogram. The essential parameters of the chromatogram can be readily obtainable
462
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 “coefflclentof 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 f25% (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
