638
Anal. Chem. 1983,55,638-643
(7) Bailey. W. F.; Cloffl, E. A.; Wiberg, K. B. J . Org. Chem. 1981, 46, 42 19-4225. (8) Rothwell, W. P.; Waugh, J. S.; Yesinowski, J. P. J . Am. Chem. SOC. 1980. 702. 2637-2643. (9) Ohta,’N. Anal. Chem. 1973, 45, 553-557. (IO) Marlcq, M. M.; Waugh, J. S. J . Chem. Phys. 1979, 70, 3300-3316. (11) Stejskal, E. 0.; Schaefer, J.; Waugh, J. S. J. M a p . Reson. 1977, 28, 105-112. (12) Pines, A,; Glbby, M. G.; Waugh, J. S. J. Chem. Phys. 1973, 59, 569-590. (13) Schaefer, J.; Stejskal, E. 0. J . Am. Chem. SOC. 1976, 98, 1031-1032. Schaefer, J. J . Magn. Reson. 1975, 78, 560-563. (14) Stejskal, E. 0.; (15) Mehrlng, M. “Hlgh Resolution NMR Spectroscopy In Solids”; SpringerVerlag: New York, 1978; Chapter 2.
Abramowitz, M.; Stegen, 1. ”Handbook of Mathematical Functions”; Dover: New York. 1965; p 591. Maiinowskl, E. R. Anal. Chim. Acta 1982, 734, 129-137. Howery, D. G. I n “Chemometrlcs: Theory and Appllcatlon”; Kowalskl, 8. R., Ed.; American Chemlcal Society: Washington, DC, 1977; ACS Symp. Ser. No. 52, pp 73-79. (19) Mallnowski, E. R.; Howery, D. G. “Factor Analysls In Chemistry”; WIley-Interscience: New York, 1960; Chapter 3.
RECEIVED for review October 12, 1982. Accepted December 22, 1982. This work was supported in part by the National Science Foundation [Grant No. 8111180-CHE]and in part by the National Institutes of Health [Grant No. GM 165521.
Determining the Lowest Limit of Reliable Assay Measurement Leonard Oppenhelmer, * Thomas P. Caplzzl, Roger M. Weppelman, and Hlna Mehta Merck Sharp & Dohme Research Laboratories, P.O. Box 2000, Rahway, New Jersey 07065
The use of external standards to determine the lowest iimlt of reliable assay measurement is commonly used to evaluate the ihnttations of a particular analytical technique and to make comparisons between dlff erent procedures. Ail previously devekqmi criteria have assumed a constant variance over the entire concentration range of interest. Assay limit criteria formulas have been developed for the heterogeneous varlance situation. Widely disparate results can be obtained dependlng on the choice of crlterla, whether a uniform variance or a nonuniform variance analysis is employed, the method used to select the weights, the models and/or transformations utilized, and experlmental deslgn conslderations. These Issues are examined by using data from an analytical procedure developed to monttor tissue residues for Ivermectln, an antiparasitic agent.
The “lowest limit of reliable assay measurement” (LLO-
RAM)is often used in analytical work since it makes a succinct statement about the limitations of an assay and allows for meaningful comparisons between different assays. It expresses how effectively one can distinguish between a measurement made when true activity is zero and one made when true activity is greater than zero. All too often the LLORAM is determined intuitively which compromises its utility. However, even an objective, well-defined criterion may be deficient if the statistical methodology used in its determination is inappropriate. Previous work on LLORAM determination (1-6) has assumed that the response variance was constant and thus independent of the response’s magnitude. In this case (the unweighted case) all values used for calculating the standard curve can be treated equally and results can be obtained by using ordinary least squares. However constant variance is probably not attained by most assays. For example, many assays involve transferring constant volumes of solutions containing variable quantities of the material to be assayed. The variance of the volumes transferred can reasonably be expected to be constant, but clearly the variance in the amount of material transferred will depend on its concentration in the solution, Le., the percent error will be constant and thus the variance will be proportional to the predicted response squared. This general point has been recognized by Smith and Mathews (7) who stated that for “typical chemical ap0003-2700/83/0355-0638$0 I b O / O
plication, the experimental conditions are controlled so as to make the percentage error a constant.” This structure with variance proportional to the square of the predicted value may also have application in atomic absorption spectrometry and anodic stripping voltammetry (8). Other structures are also possible, for assays based on Poisson distributed counting measurements (either photon or radioactive) the variance may be proportional to its predicted value (9,10) rather than its square. Garden, Mithcell, and Mills (11)have a less sanguine view: “It is reasonable to expect that much analytical data will not show constant variance nor would we expect the variance to be a simple function of Concentration.” The issue of nonconstant variance and the weighted least-squares approach in obtaining standard curves has been previously addressed (7-13),but this work has not been extended to determining assay measurement limits. This report describes the derivation of appropriate expressions for obtaining various types of assay limits under the more general conditions of heterogeneous variances. The results have been illustrated by using data from an assay developed to monitor tissue residues of Ivermectin, an antiparasitic agent, potent at very low doses. Rather dramatic differences have been observed depending on whether a valid weighted or an inappropriate unweighted analysis is used. In addition, related issues which also affect assay limit determinations are discussed and illustrated using the tissue residue data. These issues include the method used to select the variance weights, alternative models which transform the variance structure of the responses, and experimental design considerations used in obtaining the standard curve.
EXPERIMENTAL SECTION Data Description. A newly developed analytical procedure (14)was employed to analyze spiked tissue samples (fat, kidney, liver, muscle) at various concentrations (9.7 to 100 ppb) in four
species (cattle,swine, horses, sheep). The tissues were spiked with Ivermectin, a broad spectrum antiparasitic agent. The analytical procedure was based upon the detection of a fluorescent derivative of Ivermectin following high-performanceliquid chromatography. For each combinationof animal species and target tissue, multiple samples were assayed for at least four concentration residue levels which were believed to be in the linear range (observed response vs. concentration). Table I indicates the distribution of independently performed assays among species-tissue-concentration combinations. An assay which will be acceptable to governmental regulatory agencies must be able to reliably discriminate the marker residue response from the target tissue background at very 0 l9S3 American Chemical Soclety
ANALYTICAL CHEMISTRY, VOL. 55, NO. 4, APRIL 1983
639
Table I. Available Data: Number of Observations concentration, ppb animal tissue 9.7 10 19 19.4 20 29 41 46 50 97 100 swine 3 3 muscle 3 3 liver 3 3 3 3 kidney 3 3 3 3 fat 3 3 3 3 horse muscle 3 3 2 3 3 2 3 3 liver 3 3 1 3 3 kidney 3 3 fat 3 3 1 3 3 sheep 3 3 3 3 muscle 3 3 3 3 liver kidney 3 3 3 3 3 3 3 3 fat cattle muscle 3 2 2 3 2 2 liver 3 2 3 3 1 3 3 3 kidney 3 3 4 3 fat 3 3 4 3 low concentrations,i.e., the lowest limit of reliable assay measurement must be below a designated safe tissue residue level. Computation, All the computations discussed below, both weighted and unweighted regression analysis, and graphics have been computer generated by wing the data management, graphics, and statistical package SAS (statistical analysis system), Helwig and Council (15). RESULTS AND DISCUSSION Assessment Criteria. Altshuler and Pasternack (1)initially used the statistical concepts of a type I and type I1 error to obtain more objective criteria to calculate the lowest limit of reliable measurement for an analytical procedure. A type I error is to falsely conclude there is sample activity when there is none (a false positive), while a type I1 error is to falsely conclude there is no activity in the sample when there is some (a false negative). Since then a plethora of criteria have been introduced, none of which are totally unique, inviolate, or universally appropriate. For example, some of the criteria are only concerned with type I errors and not type I1 ( 2 , 6 ) ,some criteria use two-sided test statistics rather than one sided (4), some criteria use linear calibration curves (5), some more complicated logistic curves (121,while others are model free (4). Whichever definition and method are employed, it is essential to specify the exact criteria, all required assumptions, and the method of analysis. This is especially important because of the often deceptive and conflicting terminology that has been employed. The three different assessment criteria introduced by Currie (3) are extended to the nonuniform variance situation, i.e., (i) the critical level, (ii) the detection level, and (iii) the determination level. The notation, assumptions, derivation, and results are given in the Appendix, Figures 1 and 2 for the weighted case. Currie’s (3) unweighted results are then a special case of this more general formulation and can be obtained by setting all the weights equal, Le., ki = 1 for all i. The critical level (Lc,), eq 2, is the assay response above which an observed response is reliably recognized as detected, i.e., the response at which one may decide whether or not the result of an analysis indicates the presence of a residue. The detection limit ( L h ) ,eq 3, is the actual net response which may be a priori expected to lead to detection. The determination limit (LQw), eq 4, is the level at which measurement precision will be satisfactory for quantitative determination, i.e., a result which is satisfactorily close to the limiting expected value. In general, we have that Lc, ILD, 5 L Q Note ~ that even for a specified criteria, results will vary depending on the prob-
0
* I
‘Cw
x ;Dw
Figure 1. The critical level (L&, x ~and) the detection level (Lb, x,) defined. Includes the predlcted calibration curve (-) with its upper (1 - a)(- -) and lower (1 - p) (- -) confidence intervals for a predlcted observation. The horizontal x axis refers to concentration and the vertical y axis to the assay response.
-
“P,
Figure 2. The determlnatlon level (L x,) defined,Le., the coefficient of varlatlon of the predicted value at xQwis sufficientlysmall, Le., equal to the quantity C; also includes the predicted calibration curve (-) and the predicted calibration curve plus one standard error for a predlcted observation (---) and (1 + C) times the predicted value (*). The horizontal x axis refers to concentration and the vertical y axis to the assay response. ability levels chosen for committing a type I (CY)or type I1 (0) error, whether a one-sided or two-sided test statistic is employed, and the measurement precision required. Unweighted Analysis. The LLORAM estimates for the various tissues and species were similar and hence only overall results (combining all the observations) have been reported. Table I1 presents summary statistics for the assay response by spiked tissue concentrations. Assay limits were computed by using eq 2, 3, and 4 in the Appendix for ordinary (unweighted, ki = 1 for all i) least squares both with and without the inclusion of the high concentration (97-100 ppb) and are reported in Table 111. Examination of Figure 3 and Table I1 indicates that the variability is not homogeneous but increases along with the spiked concentration levels. When the high concentration (97-100 ppb) results are omitted from the analysis the overall standard deviation drops by one-third (4.70 to 3.07) and so do the various LLORAM estimates, xc,, xDw, XQ, (respectively, 9.3 to 6.0, 18.7 to 12.0, 28.2 to 18.2). The
640
ANALYTICAL CHEMISTRY, VOL. 55, NO. 4, APRIL
I983 20.0-
Table 11. Summary Statistics by Concentration concn, no. of std coeff of observns mean dev variation ppb 9.7 10.0 19.0 19.4 20.0 29.0 41.0 46.0 50.0 97.0 100.0
36 16 24 3 23 1 30 6 25 35 12
7.68 8.59 16.33 16.67 17.35 25.00 32.93 37.33 43.68 80.71 83.83
0.93 1.21 1.57 1.53 1.82
12.0 14.1 9.7 9.2 10.5
2.89 5.47 5.70 8.27 8.28
8.8 14.6 13.0 10.2 9.9
* 17.5-
* 15.0 b
7. 5-
Table 111. Results for Different Situations situations considered high intercept concn
y=A+Bx y=A+Bx y=A+Bx y = A + Bx y=A+Bx y=A+Bx y=A+Bx Bx y = A iy=AxB y=AxB
*
5.0-
reason for the drop is that in the unweighted analysis all observations are weighted equally and thus observations at 100 ppb, far removed from the region of interest, have the same influence on the results as observationsmade at 10 ppb. Note the 3-fold spread for these different assay limit criteria, and also that xCw 5 xDW as required and that zDW5 zQwas expected. Weighted Analysis. Earlier work on assay detection limits, including Currie's results (3) have all assumed a constant variance around expected values, for the entire range of responses considered, Le., ki = 1 for all i. There are many situations when such an assumption is untenable. For example, the plot of residuals from fitting an unweighted least-squares line vs. predicted values (Figure 3) indicates a funnel shape, i.e., increasing variability with increasing magnitude of response. Ordinary (unweighted) least squares treats all observations equally but is not appropriate when heterogeneity exists. In that case, weighted least squares is used which gives observations with a large variance, less weight in fitting the model, than observations with smaller variance. Weighted regression equations, when the variance associated with the ith observation is proportional to the quantity ki and not all the kiare equal, are presented in the Appendix with corresponding LLORAM expressions denoted by x b , x k , X% given by eq 2,3, and 4, respectively. Weighted least squares would not be expected to alter the slope estimate, b, a great deal but it will have a large effect on the estimate of precision, s, especially at the lower concentrations (II), where more precise results are required. Thus a dramatic effect on the ~ readily resulting detection limit estimates xcW, XD,, X Q can occur. Empirical Weights. In order to apply a weighted leastsquares analysis one must assign weighting values, ki,to the various observations. In most applications the true ki values
R
E
2.5-
I 0
0.0-
L
none
none
*
-2. 5-
* * *
* *
* * * * *
**
(I
** * *
*
.*
*
* * *
-5.0-
*
* *
-7.5-
b
**
*.
-10.0-
* b
-12.5-
-15.0-
**
-
1
0
5
10
7
15
20
25
SO
.
35
40
45
50
5
55
80
*
1
75 80 E5
70
Figure 3. Residuals from the unwelghted least-squares analysis vs. predicted values indicating variance heterogeneity.
are not known. With sufficient replication at each concentration, a t least 10 replicates (II), an empirical estimate of ki from the observed data can be obtained, i.e. weight = l/(estimated kj) = l/(observed sample variance) This weighting scheme will give more variable observations lower weight than less variable observations. In order to have more precise variance estimates, with at least 9 degrees of freedom (111, nearby concentrations have been pooled whenever possible as indicated in Table 11. Table I11 indicates the substantial reduction in the assay limits by employing a weighted least-squares analysis. In this case, omitting the high concentration does not alter results much, since even when these observationswere included they had a small weight and little influence on the results. Although the slope and intercept estimates, (a, b), have not changed much from the unweighted analysis, the estimate of precision at the lower concentrations is dramatically affected (11) and this accounted for the change in assay limits. The
model fitting slope std b
deviation
0.099 -0.177 -0.067 -0.083 -0.195 -0.195 -0.312 -0.312 0.810 0.787
0.834 0.846 0.836 0.837 0.844 0.844 0.854 0.854 1.007 1.017
4.70 3.07 1.00 1.00 1.01 1.01 1.00 1.00 0.116 0.121
k
n/ad
nla
assay limits by criteria lack of detec- deterfit critical= tion* mination' statistic xc XD "Q 0.8ge 2.72f
9.3 6.0 2.1 2.1 2.1 1.2 2.1 1.2 1.2 1.2
18.7 12.0 4.2 4.2 4.2 2.4 4.2 2.5 2.4 2.5
nla nla modeled nla modeled nla modeled nla modeled nla 2.23 none nla 2.58f none nla C = 0.10. Not applicable. 01 = P = 0.05,one sided test statistic. a 01 = 0.05,one sided test statistic, statistically significant, P > 0.06. f Statistically significant lack of fit, P < 0.05. empirical empirical
65
PREDICTED YRLUE
a
weighting included
*
* * *
IO. 0-
~
models
*
12.5-
1.0 1.0 1.0 0.0109 1.0 0.0109
e
28.2 18.2 6.0 6.0 6.0 1.0 6.0 1.0 1.8 1.8
Not
ANALYTICAL CHEMISTRY, V M . 55, NO. 4, APRIL 1983
weighted analysis uses a more appropriate estimate of precision which reflects the actual situation a t the lower concentrations. Note that the overall standard deviation, s, for the weighted analysis is constrained to unity (within rounding error) by the choice of weights used. (This may not be true for other weighting schemes.) The expressions for determining the assay limits for a weighted analysis are a function of the variance at the assay limit. Thus the initial term under the radical sign, ko, kD,, and k,, in eq 2, 3, and 4 would generally not be known. In this case we have set k = 1since this makes the variance equal to that observed at the lowest experimental concentration, 10 ppb (refer to Table 11). This is probably a conservative approach since we would expect the variance to continue to decrease as the concentration went below 10 ppb. Note that a certain amount of restraint is required with these assay limits since the results involve an extrapolation below observed experimental results, i.e., below 9.7 ppb. Modeled Weights. An alternative to using the reciprocal of the computed variance as weights is to model the variance m a function of concentration. The model may be a theoretical construct, e.g., count response follows a Poisson distribution for which the variance is proportional to the mean, or it may be an empirical model which is fit to the data. Empirical models which have been considered for radioligand assays (12, 13) and may be appropriate in other general situations are as follows:
+ l)B(power function) uZ2= Co + C,x + C2x2(quadratic function) ax2=
A(x
The power function model has been fit to the tissue residue data by using ordinary least squares with resulting estimates: a = 0.0109, b = 1.9186. This is quite close to a quadratic fit (1.9186 vs. 2). To avoid the unrealistic assumption of zero variance at zero concentration, ( x 1) rather than the usual x was used in specifying the model. This model was then used to estimate the expected variance a t various concentrations, and the reciprocals of these estimated variances were used as weights. Once more the initial term under the radical sign, k , in eq 2,3, and 4 is required to calculate the assay limits. With a model for the variance structure we could now iterate to obtain an explicit solution. In Table I11 we have presented results using two values of k: (i) k = 1,and (ii) k = 0,0109. The first is chosen so that results can be compared to the conservative (upper bound) empirical weighting results. Not surprisingly these two results are very similar. Alternatively, the choice of k = 0.0109 is a lower bound and correspondsto the variance that would be estimated when the concentrationis zero. Using k = 0.0109 gives xc,, x D , estimates which are almost half k = 1.0 results; however, the XQ, result has been reduced even more, by a factor of 6. Our experience indicates that results from a single assay, without supplementary information, rarely contain enough information to adequately model the variance structure. In fact, more data may be required to model the variance function than is needed to establish the form of the calibration curve. Finney and Phillips (13) have obtained poor results in attempts using sophisticated statistical methodology to fit either the quadratic or power functions to data from a number of radioimmunoassay experiments from a single laboratory. Alternative Models and/or Scale Transformations. Statistical tests of model validity (lack of fit tests) can be performed since replicate results are available a t several concentrations (16). For the linear model, observed assay response vs. spiked concentration, lack of fit exists when the high concentration is excluded (Table 111). Adequate fit with all concentrations included is somewhat deceptive in that the
+
641
“pure error” term (16) used in the statistical test of fit is artificially inflated due to the inclusion of the high concentration. Alternative models might provide an improvement in fit. A linear model of the form log observed assay response vs. log spiked concentration was also examined (Table 111)which is equivalent to the following assay response = A(concentratiodB Note that the log transformation has the added advantage of stabilizing the variance, and thus unweighted least squares can be used to obtain estimates of A and B. The estimates obtained (a and b) are similar with or without the inclusion of the high concentration and indicate that the assay detects about 80% of the spiked residue concentration, Le., a = 0.80, b = 1.0, approximately. Since the model is now fit using log concentration rather than concentration, the intercept occurs when x = 1not x = 0. This may have an effect on LLORAM estimates when the assay limits turn out to be extremely low. A solution to this numerical anomaly would be to change scales to parts per trillion (pptr) from parta per billion in order to remove results from the region near the intercept. Design Considerations. The following experimental aspects are controllable: (a) the total number of standards (spiked tissue samples) to be used; (b) the range of the concentration of standards; (c) the spacing of the concentrations and the number of replicates at each concentration; and (d) the number of unknowns examined. The strategy is to manipulate these aspects so as to minimze the xc,, xDW, xQW estimates given in the Appendix. Hubaux and Vos (5) have examined selected design aspects for the homogeneous variance case while Franke et al. (8) considered improvements in the method of standard addition when a constant coefficient of variation exists. In addition to optimality considerations the experimental design should permit the assumptions required in the analysis to be checked and allow for remedial actions if necessary. For the tissue residue example the availability of multiple concentrations with replicate observations (at least eight to ten replicates) at each concentrationallowed for checking variance homogenity and the application of a weighted least-squares analysis as well as the ability to check model adequacy. However, it would have been beneficial to have additional data at lower concentrations so that less extrapolation would be required both for the variance structure and for the actual estimates obtained. The availability of enough concentrations to allow for the examination of several models, other than just linear relationships, is to be encouraged. CONCLUSIONS A cautious and conservative attitude toward assumptions, methodology, and extrapolation seems warranted. In our illustrative example, even using well-defined objective criteria we obtained a wide range of results for LLORAM estimates (1.0-28.2) and found that many basic assumptions were not satisfied. The experimental design should allow for the ability to check required assumptions, and low concentrationsshould be included so as to preclude the necessity of undue extrapolation. Whichever LLORAM definition is used, it is essential to specify the exact criteria, all required parameters (a= type I error rate, /3 = type I1 error rate, one or two sided test statistics, C = measurement precision required, etc.), and the estimation methodology employed. For the tissue residue data examined a LLORAM estimate of 4.2 seems appropriate. This is obtained by using the detection limits for a weighted least-squares analysis, using the empirical weights, and setting k = 1. Detection limits L D , and x h are recommended since they offer protection against
642
ANALYTICAL CHEMISTRY, VOL. 55, NO. 4, APRIL 1983
both types of error, a type I error which concludes there is a residue present when there is none and a type I1 error which concludes there is no residue present when there is some. Setting the probability of committing a type I error equal to that of committing a type I1 error (say at a = 0 = 0.05) and using one sided test statistics would be reasonable. The variance heterogeneity has to be addressed through either a weighted least-squares approach or by using an appropriate (e.g., log) transformation. Since a test of the log transformed model indicated lack of fit, the weighted leastsquares approach seemed preferable. The approach of using empirical weights and setting k = 1 (a conservative choice) was selected rather than a modeled variance structure with an alternative choice of k since there would be less chance of bias caused by a poor fitting variance structure model and less extrapolation beyond observed experimental results. APPENDIX Statistical Notation, Assumptions, and Formulas. Linear model: yi = A B x ~ ~i i = 1, 2, ...,N (1)
+
+
Assumptions: 6; values are independent, identically distributed normal random variables with variance ( ~ i )= ki$. Unweighted least squares applies when ki = 1 for all i. Weighted least squares applies when not all the ki's are equal. Estimated linear model: fwi = a, bGi
0confidence interval for the predicted value of an individual observation at xDW equals Lc, (see Figure l),i.e. Lcw =
,
implies LDw
=
Since
+
and also is much less than kD, l / x ( l / k i ) , then a computationally simpler but adequate approximation for LD, is as follows: LDw
=
+
with slope = b, =
C[(xi- aw)yi/ki]/C[(xi - f , ) 2 / k i ]
and intercept = a, = 9, - b,a, where jiw ~w
C[yi/kiI/C[l/kiI
= C [ ~ j / k i /IC [ l / k j I
sw2 = C[(yj- 9wJ2/kiI/(N- 2) is the weighted residual variance. t(l-a)is the upper a percentage point of the Student t distribution with ( N - 2) degrees of freedom. Estimated variance for a predicted value, fwi, is as follows:
where kD, = value of ki a t x = xDW. Note that the approximation will provide a conservative estimate, i.e., it will increase the assay limit. However, since the (kD, + l/x(l/ki)) term will tend to dominate the expression under the square root radical, the effect of the substitution should be minimal. The type I1 error rate = p, Le., will accept a blank when x = xD, only 0% of the time. Determination Limit: LQ,, xQ,. Definition: Find the value x = xQ, (corresponding value LQ, = ~ ( x Q , ) ) such that the coefficient of variation of f(xQw)is sufficiently small, e.g., equals the quantity C (see Figure 2). Thus d9ariance[9(xQw)l = C, say, 0.10 ~(XQW)
implies
Upper (1 - a)% confidence interval for f,j, i.e., predicted value a t a concentration xj is as follows:
and thus
yWj+ t(l-,)dvariance@,j) Refer to Draper and Smith (16)for additional clarification and details. Critical Level L m xcw. Definition: The upper (1- a)% confidence interval for a predicted value of an individual observation at a blank value, Le., x = 0 (see Figure 1). The type I error rate = a, Le., reject a blank when it is really a blank a% of the time. Lcw =
I
xcw = (Lcw - a,)/b,
As for the detection limits, a computationally simpler, but adequate approximation for LQ, is as follows:
. .
where kQ, = value of ki at x = XQ,. Registry No. Ivermectin, 70288-86-7. (2)
where lzo = value of ki when x = 0. Detection Level: LD,, XD,. Definition: The value x = xDw[corresponding value L D , = f(xDw)] such that the lower
LITERATURE CITED (1) Altshuler, B.;Pasternack, B. Health Phys. 1863, 9 , 293-298. (2) St. John, P. A,; McCarthy, W. J.; Winefordner, J. D. Anal. Chem. 1967, 39, 1495-1497. (3) Currle, L. A. Anal. Chem. 1866, 4 0 , 586-593. (4) Gabrlels, R. Anal. Chem. 1970, 42, 1439-1440.
Anal. Chem. 1983, 55, 643-648 Hubaux, A.; Vos, G. Anal. Chem. 1970, 42, 849-855. Ingle, J. D. J . Chem. Educ. 1974, 57, 100-105. Smith, E. D.; Mathews, D. M. J . Chem. Educ. 1967, 44, 757-759. Franke, J. P.; de Zeeuw, R. A.; Hakkert, R. Anal. Chem. 1978, 50, 1374-1 380. (9) Schwartz, L. M. Anal. Chem. 1977, 49, 2062-2068. (IO) Schwartz, L. M. Anal. Chem. 1979, 51, 723-727. (11) Graden, J. S.; Mitchell, D. G.; Mills, W. N. Anal. Chem. 1980, 52, 2310-23 15. (12) Rodbard, D. Anal. Biochem. 1978, 90, 1-12. (13) Flnney, D.J.; Phillips, P. Appl. StatisNcs 1977, 28, 312-320.
(5) (6) (7) (8)
643
(14) Tway, P. C.; Wood, J. S., Jr.; Downing, G. V . J . Agrlc. FoodChem. 1981, 29, 1059-1063. (15) Helwlg, J. T.; Council, K. A. “SAS Users Guide”; SAS Institute: Raleigh, NC, 1979. (16) Draper, N. R.; Smith, H. “Applled Regression Analysis”, 2nd ed.; Wiley: New York, 1981; pp 33-42.
RECEIVEDfor review September 20,1982. Accepted January 6, 1983.
Partial LeasbSquares Method for Spectrofluorimetric Analysis of Mixtures of Humic Acid and Ligninsulfonate Walter Llndberg* and Jan-Ake Persson Department of Analytical Chemistry, Unlversity of Urn&, S-901 87 Umea, Sweden
Svante Wold Research Group for Chemometrlcs, University of Umeh, 6-90 1 87 Ume.4, Sweden
Quantitative determinations of multicomponent fluorescent mixture have been made. The test substances used were humic acid, iigninsuifonate, and an optical whitener from a detergent. The fluorescence spectra from these substances have slmllar features with severe overlap In the whole wavelength reglon. For resolving these spectra and quantlfying the substances, we used a numerlcal method, the partial least squares in latent variables (PLS). The results of its application and the theory of the method are presented and a comparison Is made with other numerical methods.
Molecular fluorescence has attained popularity as an analytical technique due to its high sensitivity and relative selectivity. In complex samples, however, spectral overlap is often a serious limitation. In order to avoid time-consuming cleanup procedures, attempts to resolve complex spectra by using instrumental approaches (1) or various numerical methods have been made. The numerical methods used can be divided into two classes depending on whether all substances in the sample are known or not. If all substances in the sample are known and no interactions causing nonlinearities occur, then multiple regression (2, 3) is applicable. Individual contributions from each substances to a spectrum can be obtained and thereby their concentrations can be determined. However, usually these conditions are not fulfilled for unknown samples. In such cases, principal component analysis (4)has been employed for resolving the spectra obtained. With a related method called the partial leastsquares models in latent variables (PLS) developed by Wold and co-workers (5),a new approach to the multivariate calibration problem is demonstrated. This method is preferable to principal component analysis since it not only describes the emission matrix but also at the same time correlates the measured intensities with the concentrationsof each substance in the standard solutions. Ligninsulfonate, a compound released into water from sulfite pulp mills, contributes to the general pollution of seawaters and may be fatal to fish resources. This compound 0008-27n0/83/0351i-Q643$01.50/0
has earlier been determined with some success by using fluorescence spectrometry (6). With this method, possible interferences arise from humic acid and detergents containing optical whiteners. Emission spectra of these compounds are severely overlapped and no spectral region with a single emitting compound can be found. The aim of this work was to evaluate the prospectives of the PLS method for quantitative determinations in various mixtures of these compounds. MATHEMATICAL METHODS Problem Formulation and Notation. We use boldface capital letters for matrices, e.g., X and Y,primes for transposed matrices, e.g., X’, boldface small characters for vectors, x9q,and ym, and italic small characters for scalars, e.g., s, pa,and xcke The samples are divided into two matrices, the calibration matrix, the training set, and the test matrix. the validation set We use the first set containing the samples 1, 2 , ~ . .i,, ..., pt to establish a calibration model which relates the measured concentrations Y , k of the constituents 1,2,..., k ,..., m to the measured spectroscopic data xcj (wavelength j = 1, 2, ...,p ) . As a first step we subtract the means %, and Qk from the data xCjand Yckr respectively, and then scale each variable xi and Y k to unit variance. The thus centered and normalized data are collected in the n X p matrix X and the n X m matrix Y. The centering makes the following computations numerically well conditioned. The normalization gives each variable (Leo,emission wavelength) equal influence in the initial stage of the data analysis. If prior information about the information content of the variables is at hand, this can be utilized by weighting the variables proportionally to this information. We note that this is not directly related to the precision of the variables, but rather a function of the degree of “nonoverlap” between the chemical constituents in the variable in combination with its precision. A priori information about this information content was not at hand in the present example and hence the variables were normalized to unit variance. Assuming that the spectroscopic data, X,and the concentrations, Y,are linearly related, we have the calibration model (Figure 1) with the m X p coefficient matrix C still to be 0 1983 American Chemical Society
