the graphical process of locating a point within an area defined by three experimental observations (analogously to graphical interpolation between two points on a line). The more seriously a system deviates from simple additive behavior, the more finely the surface must be defined, by local solution, in the region of the sample point. If a system is found to be highly reproducible in the sense that the surfaces a t t l and t 2 can be closely reproduced from day to day, the routine use of this method is possible simply by measuring extent of reaction of a sample under the standard defining conditions, and then graphically solving for the sample composition from the pre-determined triangular diagram. For non-ideal systems, local solutions can be incorporated into the calibration diagram. I n t e r p r e t a t i o n s of the G r a p h i c a l Method. This graphical interpolation method can be viewed as a geometric version of the method of proportional equations, as a matching strategy for the concentration-time matching method, or as an extension to three-component mixtures of the Lee-Kolthoff method. This multiplicity of viewpoints simply reflects the basic similarity of most kinetic methods for the analysis of mixtures. Aside from the unusual graphical treatment introduced here, the principal features to be noted in the concentration-time matching procedure are the independence of assumptions about the form of rate equations or additivity relationships, and the elimination of calculations in reaching a n analytical result.
H. Y. Malmstadt, C. J. Delaney, and E. A. Cordos, Anal. Chem., 44, (12), 79A (1972). H. Y. Maimstadt, C. J. Deianey, and E. A. Ccfdos, Crit. Rev. Anal. Chem., 2, 559 (1972). H. 9. Mark, Jr., Taianta, 20, 257 (1973). H. B. Mark, Jr., Taianta, 19, 717 (1972). H. A. Mottola, Crit. Rev. Anal. Chem., 4, 229 (1975). T. I. Munnelly, Anal. Chem., 40, 1494 (1968). J. B. Pausch and D. W. Margerum, Anal. Chem., 41, 226 (1969). D. W. Margerum, J. E. Pausch, G. A. Nyssen, and G. F. Smith, Anal. Chem., 41, 233 (1969). F. Willeboordse, J . Phys. Chem.. 74, 601 (1970). 9 . G. Willis, W. H. Woodruff, J. R. Frysinger, D. W. Margerum, and H. L. Pardue, Anal. Chem., 42, 1350 (1970). M. Nakanishi, Talanta, 19, 285 (1972). J. B. Worthington and H. L. Pardue, Anal. Chem., 44, 767 (1972). J. P. Hawk, E. L. McDaniel, T. D. Parish, a d K. E. Simmons, Anal. Chem., 44, 1315 (1972). L. C. Coombes. J. Vasiliades and D. W. Margerum. Anal. Chem.. 44. 2325 (1972). W. A. deDliviera and L. Meites, Anal. Chim. Acta, 70, 383 (1974). J. G. Kloosterboer, Anal. Chem., 46, ‘I 143 (1974). K. A. Connors, Anal. Chem., 47, 2066 (1975). K. A. Connors, Anal. Chem., 48, 87 (1976). H. C. Brown and R. S. Fletcher, J . Am. Chem. SOC.,71, 1845 (1949). S. Siggia. J. G. Hanna, and N. M. Serencha, Anal. Chem., 35, 362 (1963). S. Siggia. J. G. Hanna, and N. M. Serencha, Anal. Chem.,35, 365 (1963). J. Block, E. Morgan, and S. Siggia, Anal. Chem., 35, 573 (1963). S. Siggia, J. G. Hanna, and N. M. Serencha, Anal. Chem.,35, 575 (1963). J. G. Hanna and S. Siggia, Anal. Chem., 36, 2022 (1964). S. Siggia and J. G. Hanna, Anal. Chem., 33, 896 (1961). J. G. Hanna and S. Siggia, J . Polym. Sci., 56, 297 (1962). J. G. Hanna and S. Siggia, Anal. Chem., 34, 547 (1962). R. A. Greinke and H. 9. Mark, Jr., Anal. Chem., 39, 1572 (1967). R. G. Garmon and C. N. Reilley, Anal. Chem., 34, 600 (1962). I. M. Kolthoff and T. S.Lee, J . Polym. Sci., 2, 206 (1947). T. S.Lee and I. M. Kolthoff, A y . N . Y Acad. Sci., 53, 1093 (1951). S. H. Maron and C. F. Prutton, Principles of Physical Chemistry”, 4th ed., Macmillan, New York, N.Y. 1965, p 377. R. J. Washkuhn, Y. K. Patei, and J. R. Robinson, J . Pharm. Sci., 60, 736 (1971).
L I T E R A T U R E CITED (1) H. 9. Mark, Jr., and G. A. Rechnitz. “Kinetics in Analytical Chemistry”, Wiley-Interscience, New York, N.Y., 1968. (2) H. V. Malmstadt, E. A. Cordos. and C. J. Deianey, Anal. Chem., 44 (12), 26A (1972).
RECEIVED for review April 18, 1977. Accepted July 11, 1977.
MuItipIe-Curve Procedure for Improving Precision with Calibration-Curve-Based Analyses Douglas G. Mitchell,’ Wayne N. Mills, and John S. Garden Division of Laboratories and Research, New York State Department of Health, Albany, New York
12201
Michael Zdeb Office of Biostatistics, New York State Department of Health, Albany, New York
A procedure for improving precision and reducing the rlsk of gross error with calibrationcurve-based anatyses is descrlbed. Standards are selected and analyzed, the regression order is selected, and outlying standard measurements are rejected. Calibration data are then used to compute a serles of least-squares regression equations with confidence bands around predicted concentrations. For each sample analyzed, the regression equation based on contiguous standards enclosing the sample and yielding the narrowest confidence band is used to calculate sample concentration. I n an evaluatlon using atomic absorption data, this procedure produced dlscernible improvements for hlgh-quallty data and for analyses over narrow dynamic ranges. Large improvements in confldence bands (up to a factor of three) and In minimum reportable concentrations (up to a factor of seven) were obtained for poor-quallty data and for many nonlinear curves.
Many chemical analyses are based on the comparison of
12201
signals from samples with signals from standards. For maximum precision, the method of differential analysis should be used, with signals from the sample closely bracketed by signals from standards of slightly lower and slightly higher concentrations. In practice this procedure is time-consuming and expensive, so calibration curves are usually used. A series of standard solutions is analyzed, a regression equation is calculated, samples are analyzed, and sample concentrations are calculated using the regression equation. This equation is assumed to be valid for all analyses, with the assumption checked by analyzing control standards at regular intervals. In general, standards are chosen to bracket expected sample concentrations. Also, when possible, measurements are carried out a t signal levels which minimize relative concentration errors (e.g., in absorption spectrometry a t absorbance values of 0.25 to 0.6). Calibration-curve-based procedures yield their maximum precision over very limited concentration ranges. This can be deduced from the equation ( 1 ) : ANALYTICAL CHEMISTRY, VOL. 49, NO. 12, OCTOBER 1977
1655
h
est. s.e. ( Y o )=
P -
+
where cst. s.e. (Po)= estimated standard error for predicted signal Y ocorresponding to a sample of concentration X o and calculated from the regression line, s = standard error of estimate for the regression equation, n = number of calibration measurements, and X = mean concentration of the calibration standards, XI, X,. . . X,. T h e error in Y ois smallest at X o = X and increases with increasing difference between them. For example, if standards containing 0 , 2 , 4 , and 10 units are analyzed in duplicate, the est. s.e. ( Y o )is 0.353s a t X = 4; 0.518s a t X = 0, and 0.668s at X = 10. Samples a t a concentration of 4 could thus be analyzed with nearly twice the precision of samples at a concentration of 10. T h e fact that a given regression equation yields maximum precision over only a limited concentration range is not important when analyzing samples of similar concentration. For example, sodium is present in human serum a t between 120 and 160 mmol/L, and a single calibration equation will yield about the same precision over the entire concentration range. However, if near-maximum precision is required over a wide concentration range, a single equation with the conventional least-squares procedure has serious limitations. (a) Precision is best a t X o = X but increasingly degraded toward each end of the curve. (b) Constant variance is assumed a t all concentrations, but operations such as sample digestion and spectrometric measurement are carried out most precisely over a reasonably limited concentration range. The assumption of constant variance should be approximately true over a limited concentration range, but is not necessarily true over a wide range. (c) T h e chosen model (first-order, second-order, etc.) is assumed to be valid over the entire concentration range. This is less likely to be true with larger ranges. T h e net effect is that the assumptions upon which conventional least-squares procedures are based become decreasingly valid with increasing dynamic range. For example, an analyst using plasma emission spectrometry to determine sodium in natural waters should not use only one calibration curve, since levels can vary over three orders of magnitude. Samples could be sorted into groups with low, medium, and high sodium concentrations, and a calibration curve prepared for each group. This is practical only if approximate concentrations are known. Alternatively, a curve can be prepared for the low concentration range, and the more concentrated samples can be diluted as necessary. However, this entails considerable extra work, and additional error may occur from sample contamination and degradation in handling unnecessarily dilute samples. Another approach is to develop a calibration procedure that is valid over a wide dynamic range. If the curve is nonlinear, this requires the use of equations which fit the data more exactly than the commonly used second-order equations. If the variance is not constant, it is necessary to measure its value a t each standard concentration and compute a weighted least-squares curve of best fit. This is probably the most precise approach, b u t it requires considerable investment of time and effort. In this paper we describe a multiple-curve procedure for obtaining precise analytical data over wide concentration ranges. Its results are often significantly more precise than those of the conventional least-squares method. Additional computation is required but no additional experimental work. A single set of standard measurements is used for all sample analyses. Outlying standard measurements are rejected, and a series of regression equations are computed. For any given sample there will be several possible regression equations 1656
ANALYTICAL CHEMISTRY, VOL. 49, NO. 12, OCTOBER 1977
/
CE BAND AROUND
ONFIDENCE BAND ROUND REGRESSION
XI
x2
x3
xo
x,
x5
CONCENTRATION, X
Figure 1. Firstarder calibration curve showing confidence bands around signal Y o , the calibration curve, and the predicted concentration X ,
based on contiguous standards enclosing the sample. T h e equation which gives the narrowest confidence band around the predicted concentration is used to calculate the sample concentration. The procedure utilizes standard statistical techniques to calculate confidence bands, but their use for this application has not been described in the chemical literature. (The reader is referred to Natrella (2) for a discussion of confidence bands around calibration curves and to Miller (3)for a more detailed discussion, including reports on confidence bands around signals from analytical samples). Such multiple confidence-band calculations are feasible only with computerized data handling, a limitation which will become less important as an increasing number of routine analytical laboratories adopt computerization.
THEORY Basic Statistics. After choosing the appropriate regression order and calculating a least-squares curve of best fit, confidence bands around both (a) the sample signal and (b) the regression equation are calculated. The confidence band around the predicted sample concentration is then obtained by combining the two bands. The procedure is illustrated in Figure 1. ( a ) Signal. T h e confidence band, A Y , around Yo,a signal obtained from a sample measurement, includes the mean of the population of which Yois a member with a given level of probability, a. If u, the true value of t h e standard error of estimate for the regression equation, were known, Y owould be bounded by i=Zu, where Z is the percentile point from the normal distribution. T h e true value of u is not known, but a conservative estimate can be obtained from the calibration data using the inequality (3): where n is the number of calibration measurements, p is the number of predictors in the regression equation (first order, p = 2; second order, p = 3), s is the estimated standard error of estimate for the regression equation, x 2 is the lower percentile point from the chi-squared distribution for ( n - p ) degrees of freedom a t the a / 2 level of significance, and m is the number of replicate sample measurements. Hence a n estimate of the confidence band around Yo is given ( 3 ) by:
(3) This band is affected both by the number of standard and sample measurements and by day-to-day variations in precision. For example, for 01 = 0.1, p = 2, m = 1, the equation yields A Y = 3.9s for 6 standards and AY = 2.62s for 12 standards. Day-to-day variations in precision are reflected in varying values of s. ( b ) Regression Equation. The confidence band about the regression line for a first-order equation is given (3) by:
Y = b , + b , X * (pF;::,)”*s
1
1 (X,-X)* -n+ c ( X i - x ) ’
[-
l’*
(4)
where F is the percentile point from the F distribution for p , n p degrees of freedom and significance level cy. For the , expression second-order equation, Y = bo + b l X + b 2 X 2 this becomes more complex, with the expression in square brackets replaced by the matrix term [X’,(X’X)-lXo]where X is a matrix of n rows of the form [ l X X’], with one row corresponding to the concentration X I of each standard measurement. X’,, is a one-row matrix of the form [ l Xo Xo2]where X o is the concentration at which the confidence band is to be determined. T h e predicted concentration is calculated by solving the regression equation (Equation 4 without the confidence bands) for X , using the average signal Yofor Y. The upper confidence band for the predicted concentration (for a regression with positive slope) is calculated by solving Equation 4 with negative sign for X = X o , using the upper limit of the sample band and ( y o- AY). The lower confidence band is obtained using Equation 4 with positive sign and Yo A Y . This is shown diagrammatically in Figure 1. This statistical approach assumes that the data show constant variance, and that residuals, t (the increment by which any individual signal, Y , falls off the regression line) are normally distributed with a mean value of zero (1). ~
+
CHOICE OF CALIBRATION CONDITIONS These equations also provide useful guidelines for selecting calibration conditions. Choice of Standards. Standards should be chosen so that their mean concentration, X , is approximately equal to the most important sample concentration. From Equation 1, if Xo = X , est. s.e. (Po)= s/& and is thus a minimum. Theoretically, standard concentrations of 0 and m should be included in order to maximize B ( X , - X)*(Equation 1). However, this equation is based on the assumption that the chosen regression equation exactly fits the experimental data, an assumption which will be valid only over limited concentration ranges. A reasonable compromise is to use standards a t well-spaced concentrations that enclose the expected sample concentrations. Number of Standard Measurements. An increase in n will reduce A Y by increasing x2 and will narrow the band around the regression equation by reducing F. For example, for a first-order equation, ( p = 2 ) and cy = 0.1, m = 1,the use of 4 calibration standards (n = 4) yields values of 7.26s for A Y and 2.15s for the band around the regression at X = X o . Use of 8 standards greatly reduces the band widths, to 3.14s and 0.86s respectively. A further increase to 20 calibration standards produces smaller improvements, to 2.27s and 0.43s, respectively. Method Precision. Choice of an inherently precise method will result in low values for s and hence in narrower confidence bands. Note t h a t with only a few standards, choice of a high-order regression equation (e.g., a third-order equation
with four standards) allows s to be reduced to zero, since the curve can pass exactly through all points. However, thirdand higher-order regressions, with points of inflection, are not appropriate for most analytical measurements. Number of Sample Replicates. Using triplicate sample measurements (rn = 3) instead of a single one narrows the confidence band around the signal by 42% (Equation 3), and an additional three measurements reduces it by a further 17 % . Level of Significance, (Y, will affect band widths by influencing F , x2,and 2 values. It should be chosen to give realistic confidence bands and kept constant to yield consistent data. For the examples reported in this paper, LY = 0.1 was used for F and 2 and LY = 0.2 for x2.
PRINCIPLE OF CALIBRATION TECHNIQUE Theoretically the optimum calibration curve for each sample and for a given set of standard measurements can be obtained by choosing the combination of standard measurements and regression equation which yields the narrowest confidence band around the predicted sample Concentration. Unfortunately, this approach cannot be used in practice, since values of s vary unpredictably as different standard measurements are used. Accordingly, a partly empirical computational procedure was developed. After outlier rejection, a series of regression equations is calculated. For each sample measured, all regression equations based on standard measurements at three or more contiguous concentrations (four if second-order) which bracket the sample are inspected. The equation yielding the narrowest confidence band at the predicted concentration is used to calculate the sample concentration. These restrictions on choice of regression equation prevent the use of an inappropriate set of standards which would fortuitously yield a narrow confidence band. For example, it would eliminate use of a curve based on only XI, X 2 , and X 3 to predict Xo (Figure 1). This calibration procedure improves precision even though we often choose to not use all available measurements. This improvement occurs because: (a) The reduced number of standard measurements is almost compensated for by a more appropriate set of standards. Reducing the number of standard measurements reduces n ,thus increasing AY (Equation 3) and F (Equation 4). However this is almost compensated for by the change in X ,which can now more closely approximate Xo. For example, suppose standards of concentrations 0, 2, 4, 10, 20, and 40 are analyzed in triplicate. For a first-order equation ( p = 2 ) where m = 1and cy = 0.1, AY = 2.32s, 8 = 12.67, and the regression equation has a band width of 0.63s a t X o = 4. If we choose to use only standards of concentrations 0, 2 , 4 , and 1 0 , X = 4; AY = 2.63s and the regression has a band width of 0.70s. If the regression equation is Y = X, the confidence band around the predicted concentration is 4 f 2.96s using all six standard measurements and 4 f 3.37s using standards 0, 2, 4, and 10 only. (b) Errors due to model inadequacy can be reduced. For example, a calibration curve may be linear from 0 to 10 units but curve toward the concentration axis at higher concentrations. With a conventional least-squares procedure, a second-order equation would be fitted to the entire data set. This equation may not show significant lack of fit, but the standard error of estimate will be larger, and hence confidence bands will be wider than with a more appropriate model. The procedure described in this paper allows use of a first-order equation over the linear concentration range 0-10 units and of a second-order equation at higher, nonlinear concentrations. The multiple-curve procedure can be regarded as a mathematical simulation of manual plotting, which for many nonlinear calibration curves yields a better fitting line than a least-squares procedure with a second-order equation, since ANALYTICAL CHEMISTRY, VOL. 49, NO. 12, OCTOBER 1977
1657
the manual plotter is not limited to first- or second-order equations. (c) T h e multiple-curve procedure is less prone to errors resulting from nonconstant variance. T h e least-squaJes procedure chooses an equation which minimizFs X(Yt- Y)’, where Y , is an individual standard signal and Y is the signal predicted from the calibration curve. The least-squares procedure assumes constant variance. If this assumption is not justified, inappropriate equations may result. For example, a calibration curve over the range 0-100 units may have a variance of 1 a t concentration 1 and 10 at concentration 100. Although replicate measurements a t 100 would be much more widely dispersed than replicates a t 1, the least-squares procedure would give all measurements equal weight. This could result in an equation which fits the data a t high concentrations but is grossly erroneous at low concentrations. At low concentrations, the multiple-curve procedure would choose a calibration curve based on low concentration standards and with a low s. At high concentrations the multiple-curve procedure could inappropriately choose a curve based on low concentration standards if it were not restricted to using standards which bracket the sample. Confidence band statistics also provide meaningful minimum reportable concentrations. Assuming no systematic error, the confidence band around the predicted sample concentration includes the true sample concentration with a probability of 1 - a. If this band just includes zero, we conclude that, with 1 - cy + a / 2 probability, the analyte has a concentration greater than zero, Le., it is present in the sample. For the data user this is a much more useful definition than a detection limit, which is basically a signa1:noise ratio and does not directly inform the data user about the presence or absence of analyte. With the multiple-curve procedure, the minimum reportable concentration is obtained by computing confidence bands for the most appropriate regression equation a t successively lower concentrations until the band just includes zero.
DETAILED CALIBRATION PROCEDURE Figure 2 shows a flowchart for the overall procedure. All calculations were carried out using a FORTRAK program written for a Burroughs B 6700 computer. A FORTRAN program for the multiple-curve technique is available on request from the corresponding author. The six steps are as follows: 1. Select, Prepare, and Analyze Standards. 2. Select Regression Order. An appropriate order for a least-squares line of best fit can be chosen several ways-for example, by plotting calibration data and “eyeballing” the line or by selecting an equation that gives the maximum correlation coefficient, the minimum standard error of estimate, or no significant lack of fit. A lack-of-fit test was selected, since it yields a quantitative statement of goodness of fit. The test compares the variations within sets of replicate measurements with variations between each measurement and the calibration curve. A significant F value (mean square lack of fit/mean square pure error) is strong evidence that an inappropriate regression equation has been chosen. If there are too few replicates for the lack-of-fit test, the regression order can be selected using the extra-sum-of-squares test ( I ) . T h e lack-of-fit test is more useful than the extra-sum-ofsquares test because a showing of no significant lack of fit indicates that the model is not inadequate to fit the data (It does not show that the chosen model is optimum). The extra-sum-of-squares test merely shows that one model is or is not significantly better than the other, where both may be inadequate. Care should be taken in interpreting the lack-of-fit test. Imprecise data will rarely show significant lack of fit, since the test becomes less stringent with increasing standard error 1658
ANALYTICAL CHEMISTRY, VOL. 49, NO. 12, OCTOBER 1977
f
START
7
5-
L I i
I
Select and analyze standards.
2
Select regression order.
7 3
Reject outliers.
4
Analyze next sample. Reject
5
Select appropriate standard measurements. Calculate regression equations, confidence band, and sample concentrations. Select optimum data.
A Yes
1
I
Concentrat ion minimum detectable.
1
No
I
P
A. = VO
I
j
Figure 2. Abbreviated computer program for conventional and multiple-curve calibration procedures
of estimate. With very precise data, even a small bias may result in significant lack of fit. This may not be an important practical problem, and an advantage is that the test can detect errors such as incorrectly prepared standards, e.g., closely spaced replicate measurements which are “too far away” from the line. It can also detect reading errors, e.g., a technician reading replicate measurements from a stripchart recording and rounding off values to the same level. Such rounding would result in replicate measurements which are “too good for the curve.” During initial method development, the regres+on order should be further tested by plotting residuals, ( Y ,- Y), against X. Visual examination of residuals will detect errors in the model such as use of an inappropriate regression order or nonconstant variance. If variance is reasonably constant, with residuals approximately fitting a normal distribution, one can proceed with the analysis. If not, it will be necessary to use a weighted least-squares procedure. 3. Reject Outlying Standard Measurements. There are a number of procedures for rejecting outliers ( 2 ) . We chose to flag standard measurements as potential outliers when their residual exceeded twice the standard deviation of all the residuals. They can then be rejected if there is reasonable cause. For example, we can reject duplicate measurements which are both significantly above or below the curve, suggesting an incorrectly prepared standard. 4. Analyze Sample and Reject Outlying Measurements. If three or more sample measurements are taken,
-
Table I. Effect of Multiple Curve Calibration on Analytical Precision for Selected Calibration Curves Determination Highest standard (pg/mL) Calibration mode Minimum reportable concentration : No. measurementsa Order Value (pg/mL) Low samples (116) No. measurements Order Band % b Medium samples (112) No. measurements Order Band 7c High samples (516) No. measurements Order Band %
Chromium 1.5
Chromium 25.0
Iron 25.0
Copper 2.5
Conv
Mult
Conv
Mult
Conv
Mult
6.4 2 0.020
4.3
4.8 1
2
0.009
0.73
4.4 1 0.10
4.8
1
4.8
4.6
4.8
1
1
6.7
4.3 1 3.0
9.8
3.3
2 11.5
6.4 2 3.0
4.4 2 1.6
6.4 2 1.9
4.4 2
6.4 2
C
C
0.44
4.8 2 8.1
Conv
Mult
Conv
Mult
4.3
(4.8)-2
(4.3)-2
1
1
1
0.07
0.017
0.013
4.8 2 0.031
4.3 1 0.013
4.6 2 3.3
(4.8)-2
(4.6)-2 1 3.1
4.8 2 4.6
4.6
4.8
4.7
2 2.0
1.5
1
3.7
4.4 1
C
4.0 C
Copper 2.5
C
1
3.2 1
C
1.8
a Number of measurements used in calculating the calibration curve = (No. of concentration levels X No. of standards) (No. of outliers rejected). Relative band width around predicted concentration = [(Upper confidence band - Lower conData are not shown when the multiple curvc! procedure does not fidence band) X 1001/ ( 2 X Predicted concentration). improve data quality.
measurements can be flagged as potential outliers if they lie outside P f A Y . 5. Select A p p r o p r i a t e S t a n d a r d M e a s u r e m e n t s a n d C a l c u l a t e Regression Equations a n d Confidence Bands. All calibration curves based upon contiguous standards which enclose the sample are considered. In each case, first- and second-order least-squares equations are calculated and used to predict sample concentration and confidence bands around this concentration. (Second-order equations are considered only if a second-order equation is most appropriate using all calibration points.) For each sample, the expected concentration, X o , and a confidence band are calculated from the mean signal, 7 (Equation 4),using appropriate standards and regression orders. Consider, for example, a calibration curve based on standards XI, X p ,X s , X4,and X 5 . For a sample with a mean signal roughly corresponding to X1,three first-order equations, using (a) all five standard measurements; (b) X1,X 2 ,X 3 ,and X,only; (c) X1,Xz, and X 3 only, are calculated. If replicate standard measurements are made, all replicates a t each selected concentration (after outlier rejection) are used. The equation yielding the narrowest confidence band is used to predict sample concentration. 6. Test for Minimum Reportable Concentration. If the confidence band includes zero, the analyte is a t or below the minimum reportable concentration. EVALUATION T h e multiple-curve procedure always gives equal or narrower confidence bands than conventional calibration procedures. Hence the range of allowable concentrations is equal or narrower, and, assuming no bias, the predicted concentration should on the average be closer to the true value. T h e procedure was evaluated using calibration data obtained for metal analyses by conventional atomic absorption spectrometry. In each case, standards of a t least five concentrations (including zero) were prepared and analyzed. Because the effectiveness of the multiple-curve procedure increases with increasing number of standard measurements, each standard was measured in quadruplicate for a total of a t least 1 2 standard measurements for each calibration, (When n is low, choosing to use fewer than all available measurements results in large increases in band widths, and t h e multiple-curve technique is not likely to improve data quality.)
RESULTS Table I shows the results obtained from some selected calibration curves. Conventional and multiple-curve procedures are compared with and without outlier rejection. Minimum reportable concentrations were calculated along with relative confidence bands at concentrations corresponding and 5/s of the most concentrated standard. Each to ' I 6 ,'I2, was measured in quadruplicate, and relative confidence bands were calculated. Several important conclusions can be drawn from these data. (a) For many analyses, use of the multiple-curve procedure produced only trivial improvements in confidence bands. In general this occurred with calibration curves covering a limited concentration range or with linear curves based on high quality data (typically with correlation coefficients of 20.9997). (b) The multiple-curve procedure significantly narrowed the confidence bands for most second-order curves and for curves based on poor quality data. For example, one of the copper curves was linear up to 1.0 pg/'mL, with a gentle curve toward the concentration axis a t higher concentrations. Yet the conventional procedure without outlier rejection requires use of a second-order equation over the entire range, while the multiple-curve procedure allowed use of a first-order equation for the two lowest concentrations. Similarly, the chromium data included a number of potential outliers at high concentrations. With these retained, the multiple-curve procedure produced much better results than conventional calibration for the low-concentration sample. (c) The greatest improvements were obtained at low concentrations, with improvements in minimum reportable concentrations of up to a factor of seven. This occurred because the program could choose not to use high concentration standards, often allowing use of a first-order equation and avoiding the effects of an inadequate mathematical model. (d) As expected, outlier rejection reduced confidence band widths. In principle, data should be rejected only if the analyst has a valid reason for doing so. In general, if potential outliers are retained, the multiple-curve procedure handles the more widely dispersed data better than the conventional calibration procedure. DISCUSSION The multiple-curve procedure described in this paper is in effect a systematic mathematical technique for plotting a ANALYTICAL CHEMISTRY, VOL. 49, NO. 12, OCTOBER 1977
1659
family of least-square curves. T o compute the concentration of each sample, a curve from this family is chosen which gives equal or better precision than a conventional least-squares curve using all calibration data. The procedure is not useful (but not damaging) with small numbers of calibration standards. Its major application is for highly automated, high-throughput analyses, in which multiple standard measurements can be made at little extra cost. For that purpose, it is relatively simple and effective. More rigorous mathematical techniques, such as using a weighted least-squares approach to compensate for variable variance data or identifying more appropriate equations than first- and second-order linear models, require considerably more development. T h e multi-curve procedure has several important advantages. (a) Calibration curve calculations can be automated with less risk of gross error than in conventional procedures. (b) Data quality is improved, particularly for calibration over wide dynamic ranges and with nonlinear curves. Alternatively, with initially adequate data quality, the multicurve procedure allows use of wider dynamic ranges, often with increasing analytical efficiency.
(c) If data quality is inadequate for a particular application, confidence band statistics will suggest ways t o improve precision. (d) Confidence bands and minimum reportable concentrations are meaningful and should be reported to the data user-along with a warning that they do not account for less easily quantified errors such as systematic analytical error and unrepresentative sampling. (e) The procedure requires additional computation but no additional experimental work. As with all statistical techniques, the final result is only as good as the experimental data. The analyst should not allow the use of this sophisticated mathematical technique to induce overconfidence in the quality of the data.
LITERATURE CITED ( 1 ) N. R . Draper and H. Smith, “Applied Regression Analysis”, J . Wiiey and Sons, Inc., New York, N.Y., 1966. (2) M. G. Natrella, Section 17.3, “Experimental Statistics”, National Bureau of Standards Handbook 91, (1963). (3) R. G. Miller, “Simuhneous Statistical Inference”, McGraw-Hill, New York, N.Y., 1966.
RECEIVED for review November 18, 1976. Accepted July 13, 1977.
Solution of General Equilibrium Systems by the System Equation Jan Blaha National Textile Research Institute, Centre for Research and Application of Ionizing Radiation, 6647 1 Vev. 132~%ka,Czechoslovakia
The general qualitative and quantitative computational simulation for solution of equillbrium systems is described. The qualitative testing simulation set of equations consists of a quantitative simulation equations set extended by experimentally measurable criterion quantities. The quantitative simulation equations set consists of a set of experimentally nonmeasurable quantlties. A general system is not solvable by a mathematical simulation equation set directly. The solvability is conditioned by forming a sufficient number of partial simulation equations set transformable into a single algebraic equation of the nth degree. This transformability of the simdation equations set was formalized by the general system equation. This equatlon determlnes a general law of qualitative and quantitative relationships of indivldual subsystems of the general equilibrium system. The system equation was applied to a system of cyanide-heavy metals in aqueous solution.
T h e arrangement of systems based on system theory was first demonstrated in Biology ( I ) and later applied to other variable natural systems (2). Solutions of compounds involving complex ion equilibria have until now been studied from a quantitative point of view using a non-systems approach. The qualitative composition of the system (specification of the components formed in the system) was determined either by estimation (3) or by an iterative calculation ( 4 ) . The most probable estimate ( 3 )of the system composition was determined by the concentration 1660
ANALYTICAL CHEMISTRY, VOL. 49, NO. 12, OCTOBER 1977
of separated components and pH of the solution. The iterative calculation ( 4 ) presupposes systematic exhaustion of all possibilities of solution composition followed by completing the necessary calculations to find the single available combination which reproduces the actual qualitative composition. These methods ( 3 , 4 )are either of low accuracy ( 3 )or are at the expense of a fairly long and elaborate program which is difficult to apply ( 4 ) . The number of possible combinations of the qualitative composition of the system increases in a power series with an increasing number of complex forming system members. Therefore, it is not possible to feasibly apply these methods to more complicated equilibrium systems in a reasonable calculation time. The system scheme proposed here uses a simple method, from the point of view of formal mathematical formulation, enabling the calculation to be simplified. With increasing numbers of system members, the increase in the number of necessary calculations is additive. Thus the computation time for more complicated systems does not become burdensome. This study presents a general analysis of the principles of formulation of the system equations used in the solution of general equilibrium systems. The basic viewpoint in the classification of the systems is their separation into qualitative and quantitative characteristics. The analysis of preconditions enables the formulation of both partial qualitative methods of solution and a single quantitative one. The precondition for the formation of a solvable system is the ability to transform the system into a single algebraic equation of the nth degree. This is formalized as the general system equation which arranges the general equilibrium system according to
