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Anal. Chem. 1988, 58, 1167-1172
spectrum increases, it is probably easier than estimating all of the parameters of each of the bands as many algorithms require. It is worth noting that both area renormalization and walls should be applicable to other methods of resolving spectra, including the gradient techniques. Area renormalization could serve to add additional constraints to the parameter estimates, and the judicious placement of walls with the error space minimization methods should make convergence more likely and quicker. The final iterations could still be performed without the walls to ensure that best estimates are obtained. Some references to walls used for similar purposes are present in the literature (9, 11). We are currently working to adapt this method to other band types. An implementation that is able to resolve Lorentzian as well as Gaussian bands has already been developed. We are also attempting to better characterize the accuracy of this method in the presence of noise and model failure. The results of this work will be presented in a future paper.
ACKNOWLEDGMENT We thank Dennis L. Tebbe for early conversations that helped crystallize some of the ideas presented here, Pickering
Laboqatories of Mountain View, CA, for their generous assistance and patience in the acquistion of the chromatogram presented in Figure 4, and Spectra-Physics of San Jose, CA, for aid in interfacing their integrator to our computer.
LITERATURE CITED (1) Allen, Geofrey, C.;McMeeking, Robert F. Anal. Chim. Acta 1978. 103, 73-108. (2) Pithia, J.; Jones, Norman R. Can J . Chem. 1966, 4 4 , 3031-3050. (3) Fraser, R. D. B.; Suzuki, Eikichi Anal. Chem. 1966, 3 8 , 1770-1773. (4) Bartecki, Adam; Soltowski, Jahusz; Kurzak, Krzysztof Comput. Enhanced Spectrosc. 1983, 1 , 31-38. (5) Lam, C. F.; Forst, Art; Bank, Harvey Appi. Spectrosc. 1979, 3 3 , 273-278. ( 6 ) Binsch, Gerhard Comput. Methods Chem. R o c . I n t . Symp . 1980, 15-36. (7) Slavic, Ilfan A. Nucl. Instrum. Methods 1976, 134, 285-289. (8) Abramowitz, Stegun “Handbook of Mathematical Functions”; Natlonal Bureau of Standards: Washington, DC, 1970; p 932. (9) Roberts, S . M.; Wilkinson, D. H.; Walker L. R. Anal. Chem. 1970, 4 2 , 886-893. (10) Barker, B. E.; Fox, M. F.; Hayon, E.; Ross, E. W. Anal. Chem. 1974, 46, 1785-1789. (11) Anderson, Andrew H.; Gibb, Terence C.; Littlewood, Anthony B. J . Chromatogr. Sci. 1970, 8 , 640-646.
RECEIVED for review July 23, 1984. Resubmitted January 9, 1986. Accepted January 9, 1986.
Error Propagation and Figures of Merit for Quantification by Solving Matrix Equations Avraham Lorber Nuclear Research Centre-Negev, P.O. Box 9001, Beer-Sheva 84190, Israel
Quantitatlon from one-dimensional data enables the slmultaneous determination of all components contributing to the spectrum. However, the appllcablllty of the procedure is llmked because of the figures of merk error propagation, signal to noise, llmlt of detection, precision, accuracy, sensttlvtty, and selectivity are not determined for each component. I t Is suggested that by considering the “net analyte signal”, error propagation and other figures of merit are defined for each component. Net analyte slgnal Is defined as the part of the signal that Is orthogonal to the spectra of the other components. The mathematical results were applied to absorbance data of a four RNA nucleotides mixture, and lt was found that It succeeds well in predicting both preclslon and accuracy.
Determination of a multicomponent mixture by using the full spectrum of the sample can be accomplished by using linear algebra methods. The benefits of using several data points are resolution of overlapping spectrum (1)and accurate background subtraction (2). However, until now, it was impossible to estimate both the precision and accuracy in determining each component when matrix manipulations are involved. This shortcoming imposes a severe restriction on the applicability of matrix computation methods for quantitative analysis. One can hardly speak about quantitative analysis without knowledge of the amount of signal, error, and related quantities for each component. The signal-to-noise ratio (S/N) of an analytical system is the most useful figure of merit that can be used to characterize
an analytical technique for a specific application. The S/N is related to three other features of merit: (a) the precision expressed as percent relative standard deviation (RSD) of the concentration measurement; (b) the limit of detection (LOD), which is the concentration of analyte that corresponds to a SIN = 3 (3);and the sensitivity of the analytical method, which is the instrumental response to a certain concentration of analyte and is measured as the slope of the analytical calibration curve. Quantitation from an overlapped spectrum necessitates the definition of two additional figures of merit: the error propagation, which is the ratio of the precision in the determined concentration to the precision of the instrumental response; and the selectivity (4-61, which measures the possibility of deconvoluting the overlapped spectrum. In fact, a boundary for the error in solving matrix equations is given by the condition number, K . K is usually computed as the ratio of the largest to the smallest eigenvalues of the matrix. The product of K by the error in the measured values gives an upper bound on the error in the determined concentrations vector. This approach of estimating error propagation was introduced to analytical chemistry problems by Jochum et al. (7). The condition number tool was introduced in numerical analysis to estimate the possibility of inverting a matrix and usually results in over estimation by up to 1order of magnitude (8). Thus, this approach may be regarded as a qualitative tool for error estimation. A more severe restriction that prevents its practical application to analytical quantification is that the estimate is for the sum of the errors. To demonstrate this problem, we consider a three-component mixture. The spectra of two of
0003-2700/66/0358-1167$01.50/00 1986 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 58, NO. 6, MAY 1986
them are similar and that of the third is distinct. Because two spectra are similar, the third eigenvalue will be small and the condition number will be large. Nevertheless, the component with the distinct spectrum may be determined without error propagation. Thus, it is clear that error estimate for each component is needed for analytical applications. An alternative approach for estimating both dependence between a pair of components and error propagation is to examine the correlation matrix and its inverse. The off diagonal values of the correlation matrix represent the dependence between a pair of components (9). The diagonal elements of the inverted correlation matrix represent information on error propagation for the respective components. These diagonal elements are the so-called “variance inflationary factors” ( I O ) . However, these factors are merely used as qualitative tools to indicate which component is responsible for the ill-conditioned situation. Marquardt ( I O ) suggested that a factor value larger than 10 indicates an ill-conditioned case. Brown (11) presented formulas to compute the confidence region for each estimated variable. The covariance matrix is used to calculate the confidence region. The number of equations to be solved is equal to the number of data points, and the equations are quadratic. Solution of such a set of equations is time-consuming and should be carried out for each sample. Moran and Kowalski (12)represented a similar approach for estimating errors in the generalized standard addition method. In this paper it is presented that individual error estimate and analyte signal determination for each component separately can be defined for the one-dimensional case. In order to demonstrate the usefulness of the derived mathematical method, we selected to present here the application to spectrometric analysis of a mixture of four RNA nucleotides.
THEORY Problem Formulation and Notation. Boldface capital letters are used for matrices, e.g., X, superscript T for transposed matrices, e.g., XT, boldface small characters for vectors, e.g. x and x,,and small characters for scalers, e.g., x . Superscript + denotes the pseudoinverse, e.g., X+. The pseudoinverse (13)is used to solve a system of linear equations and differs from the inverse by that nonsquare, and singular matrices may be solved. llxll designates the Euclidian norm of the vector, defined by the square root of the sum of the squared elements of the vector, llXll is the matrix norm usually computed as the square root of the largest eigenvalue of the matrix. The unity matrix is designated by I. There are m (i = 1, ..., m ) data points for each sample (sensors, wavelengths, or time intervals) and n 0‘ = 1, ..., n ) analytes. The model considered here is the linear additive model d = Sc = ACo-lc Here S is an m x n matrix of sensitivities. d is the vector of instrumental responses with m elements. c is the vector of determined concentrations for n analytes. The matrix of sensitivities is the product ACo-I, where A is the matrix of responses and Cois a diagonal matrix whose elements, c;, are the concentrations that correspond to the responses in the calibration matrix A. The formal solution of this model is c = S+d = Co A+d
(2)
We use here the pseudoinverse and not the usual matrix, (ATA)-’AT,because transposing and inverting a matrix is both a time-consuming operation and a source for numerical instabilities. Powerful methods that use transformations in order to solve eq 1 are commonly used. The pseudoinverse is a general expression to all methods of solution.
Because each measurement is subjected to some kind of uncertainty, the model in eq 1 should be written as
(d + Ad) = (S + AS)(c + Ac)
(3) The problem is how the errors in the instrumental response Ad and in the sensitivities A S affect the determined concentrations. When eq 3 is solved as (c
+ Ac) = (S+ AS)+(d + Ad)
(2a) the error in the concentration A c is a nonlinear function of the errors in the matrix of sensitivities and the errors in the response data. Thus, calculation of errors Ac of the estimated concentration is not at all straightforward. It is important to note that the errors in the measured response Ad do determine the precision, and the errors in the matrix AS are the cause of systematic deviation from the true value of concentrations. The combination of both sources of errors does determine the accuracy. In the condition number approach, the estimation of errors in the determined concentrations is given as
IlAcll llcll =
IlASll
IlAdll
.[mi-+m]
(4)
The error is the sum of contributions of two terms. The term IlASll/llSll is the error in determining the sensitivity matrix S and the term IlAdll/lldll is the precision in the measured response. This approach was developed mainly for estimating the effect of the computer round-off and truncation errors. However, as previously discussed, this is not satisfactory for analytical application. The approach for error estimation presented here accounts for the contribution of each component to the measured instrumental response. The error is related to the computed net contribution instead of for the total measured data. The point of this paper is well-illustrated by referring to the quantitation of an unknown, using single point measurement. The analytical model is given by d = sc b t (5)
+ +
Here d is the instrumental response signal whose measurement is accompanied with error, t. The signal is the sum of contributions of the analyte signal, a, and the background signal b. s is the sensitivity of the instrument. It is obvious that the relative precision of the determined concentration, c, is equal to
Ac = - - -t c
d-b
-
d
t
d - b d
(6)
The term t / d is identified as the relative precision in the measurement of the instrumental response signal. The error propagation, which is the “ratio of the precision in the determined concentration to the precision of the instrumental response”, is now simply identified to equal the “ratio of the gross to net analyte signal” d / ( d - b). Also S / N is identified as (d - b ) / t ;thus, relative precision in concentration is also equal to the inverse of S/N. These considerations suggest that if the net analayte signal is determined, all important figures of merit may be evaluated. Net Analyte Signal Determination for One-Dimensional Data. The main postulate of this paper is that “net analyte signal for a component is equal to the part of its spectrum which is orthogonal to the spectra of the other components”. The reasoning for this definition is not arbitrary, and stems from the properties of solving a set of linear equations. The part of the spectra that is not orthogonal to the data of the other component is a linear combination of the spectrum of the others. Only the orthogonal part is unique to the sought-for component. Therefore, this part is the data
ANALYTICAL CHEMISTRY, VOL. 58, NO. 6, MAY 1986
that is useful for quantitation. It is possible to find the part of a vector u that is orthogonal to a matrix X by v = (I - XX+)u
(7) Here, v is the orthogonal part. The matrix I - XX+ is a projection matrix (14) that has two properties. It is a symmetric matrix (i.e., (I - XX+)T= I - XX+, and idempotent (i.e., (I - XX+)2= I - XX+). The property of eq 7 is a result of the Moore-Penrose relations (13) for the generalized inverse
x = xx+x;x+ = X+XX+
(8)
I t is clear that if the vector u is a linear combination of the vectors in the matrix X, then multiplication by the matrix XX+ results in the vector itself. Multiplication by I - XX+ will result in a vector of zeros. Therefore, multiplication of a vector by the matrix I - XX+ cuts off the part that is orthogonal to the matrix X. The part a,* of the vector a, (which corresponds to the j t h component in the calibration matrix) that is orthogonal to the other components is given by
aJ*= (I - AJAJ+)a,
(9)
Here A, is the m X (n - 1)matrix of spectrum of the components except the j t h component. The part d,* of the response vector d for the unknown sample that is due to the j t h component is also found by multiplying it by the projection matrix
d,* = (I - A,A,+)d
(10)
According to the above identification of the net analyte signal, the vector dJ*found by this equation is the net analyte signal of the j t h component. Now the j t h component may be quantified by combining eq 9 and 10
cj0(I - A,A,+)d = c,(I - A,A,+)a,
cjOlldj*112= cjdTaj* and
(djnet)z= lld,*112 = dTa,*c,/cJO
c,"d,* = cI a, *
It is observed that by multiplying the vectors by the projection matrix, we arrived at an equation that enables quantitation of a single component in the presence of spectral interferences. Solution of eq 11for the concentration of the j t h Component requires multiplying both sides by the transpose of (I A,A,*)a, to give
dTa;* Now eq 13 may be reorganized as
Ilaj*ll cj-
cjO
dTaj* =-
Ilaj*ll
The term 11a,*ll/c: is recognized as the sensitivity. It is also observed that eq 18 has the form a = d - b = sc. Thus, the problem of quantitation from one-dimensional data has been brought to a form of scalar data, which enables simple extraction of the figure of merit. Relation between the Net Signal and the Pseudoinverse. Computation of all a,* requires computing n projection matrices, which is a tedious computation work. When the interest is merely in one component, the QR decomposition (13) may be used to find a,*. However, this possibility will not be presented here because the value of a]* may be readily computed from the pseudoinverse. The solution by the pseudoinverse was presented in eq 2. The j t h row of the pseudoinverse will be designated as the column vector y,. With this designation the solution for the j t h component is given as
c, = c?yjTd
(19)
When this is compared to eq 14, it is clear that
yjTa,* = 1
(20)
In order for this relation to hold, the vector y, should be a scalar multiple of the vector a,*. Thus, we arrive at the following relations: (a) yJ = llyj112a,*
,
(b) a,* = IIa * II Y, (c) 1 1 3 1 1= IIy,II-l
(21)
cPajT(I- A,A,+)T(I- AJA,+)d= cjajT(I- A,A,+)T(I - A,A,+)a, (12)
Now, the net j t h analyte signal is given by
Using the symmetric and idempotent properties of the projection matrix this equation is simplified to
and the equation which is analogous to eq 18 is
cjOajT(I - A,A,+)d = c,a?(I
- A,A,+)a,
(16)
Inserting the value of c, found in eq 14 into this equation and taking the square root, results in
(11)
or
1169
d,"et = YJTd/llY,II
(22)
(13)
or
cI O a, *Td = c,11aj*112 and the concentration is given by
a,*Td The recognition of the net analyte signal to equal d,* is not enough for the purposes of this paper because the noise is expressed as a scalar and therefore the net signal should also be expressed as a scalar. The transformation from vector to scalar is simply accomplished by multiplying the vector by its transpose and taking the square root of the inner product. The scalar value of the net analyte signal of the j t h component, dFt, is found by applying this procedure to eq 11
The sensitivity is now recognized as l / ( c , " ~ ~ y , ~ ~ ) . By these relations, it is clear that calculation of the net signal requires no additional computation work beyond what is necessary to solve the matrix equation. Precision. The error Ed in the data may be computed in the case that m > n (14) as €d=[
]
dT(I - AA+)d ' I 2 m-n
(24)
This equation can be also used to estimate the errors in the calibration data. When m = n error estimation cannot be derived from the data of the sample and should be estimated by other means. The S / N for the j t h component, (S/N)j, is found by
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ANALYTICAL CHEMISTRY, VOL. 58, NO. 6, MAY 1986
’
It is observed that because d contains information from other components, (S/N), and all other figures of merit that are derived from it are dependent on the concentration of other components in the sample. The relative precision in the instrumental response data for the sample is the ratio of the error cd to the total measurement: ed/lldll. The relative precision, Ac/c,, in the concentration of the j t h component is given as the inverse of (S/N),
-AC-- IIa,*IkIIdII
Ed I
dTa,*
Cj
(26)
Thus, the error propagation for the j t h component, Kd, which is the ratio of the precision in concentration to the precision in the instrumental response, is given by
When only the j t h component is present, according to eq 13 Kd is equal to ~ ~ a , ~ ~ / This ~ ~ avalue , * ~ is~ an . additional figure of merit that is concentration-independent and accounts for the possibility of dismissing this component with others because of two reasons: (a) it has the same spectrum as another component; (b) the spectrum of the j t h component is a linear combination of the spectra of the others. This figure of merit will be identified as the inverse of selectivity. Selectivity. Obviously, an analytical procedure would be called “fully selective” (in the colloquial language of analytical chemists) for the j t h component in the sample, if there is a row, i, in the matrix of sensitivities for which all the sensitivities are zero except the sfJterm. The fulfillment of this condition permits quantitation of this component regardless of others. When this condition holds for all n components, the matrix can be arranged as an upper n X n diagonal matrix and the analytical procedure will be fully selective for all components. Attempts to define selectivity for non “fully selective” procedures were proposed on the basis of the requirement that the matrix shall be solvable. Kaiser in his last paper (4)considered the solution of the matrix equation by an iterative process. He considered a square matrix which is arranged in the form that the value of the diagonal elements are the largest in the row. The first approximation is obtained by taking into consideration only the matrix elements in the diagonal. In this step one proceeds as if the analytical procedure were fully selective. Convergence will occur only when in each row of the sensitivity matrix the element in the diagonal is larger than the sum of all other elements in the same row. For each ith row the following inequality should be valid lsjjl
>1
The larger these quotients are, the better the iteration procedure converges. If, therefore, we take the smallest value of these quotients, a quantitative measure C; of selectivity is given by C; = min
lsjjl
For a “fully selective” procedure, 4 becomes very great. When the value of C; is only little above zero, one can hardly speak of selectivity. Fujiwara et al. (6) presented a criterion to assess the acceptability of the selectivity as
4 Ildll/cd (30) This criterion is essential in evaluating whether more separation steps or more instrumental resolution is needed. They also extended the concept of selectivity to treat intercomponent effects. With the condition number approach, the solvability of a matrix equation is determined by the magnitude of the condition number. Therefore, the selectivity may be recognized as the condition number. In this definition of selectivity, a value of one means a fully selective procedure. The criterion for the acceptability of the selectivity is given by IldAll (31) This condition follows from the restriction on the applicability of the condition number for estimating error propagation (9). When the value in the criterion is greater than one, rank degeneracy occurs and formally the matrix may not be inverted. A new definition of selectivity, which is based on the results of this paper, is proposed. In the proposed approach, the selectivity measures the degree of overlap. Several choices to measure the degree of overlap may be used. The following possibility is used because it is identified as the inverse of error propagation for the calibration data, and it is useful in determining the accuracy. The part of the vector a,* that is not overlapped with the other components is the orthogonal part. Division of the norm of this vector by the norm of the total data measures the degree of overlap of the j t h component
4; = ll~,*ll/ll~Jll
(32)
With this definition, Kaiser’s observation on the connection between selectivity and solvability is valid. However, three improvements are gained: (a) the selectivity is connected to the actual computation method; (b) in Kaiser’s approach the multiplex advantage of using more data points than analytes was not taken into account; and (c) selectivity is assigned to each component. A “fully selective” procedure for the j t h component has a value of unity as in the definition of selectivity for the definition by the condition number. It is also clear that for a diagonal matrix the selectivity equals unity. The criterion for accepting the selectivity will be presented below along with the accuracy estimation. Sensitivity. If two components have the same spectrum, although each spectrum may be sensitive, there is no possibility to assign it to an individual component. Thus, in this case, the high sensitivity is not useful for chemical quantification. Kaiser has shown (5) that it is possible to define the total sensitivity for a multicomponent procedure as the absolute value of the determinant of the matrix of sensitivities S. Maximum sensitivity will correspond to a matrix with large diagonal elements and low off-diagonal elements. Kalivas (15) suggested using the condition number for the same purpose. The condition number value is interpreted as described for the selectivity. The sensitivity was identified in eq 18 to equal Ila,*ll/c,O, and there is no need to add an additional measure of deconvolutability other than the selectivity. Accuracy. The accuracy of an analytical quantitation is the outcome of several factors that cause deviation from the true chemical quantity: (a) the precision in the measured response of the unknown sample, (b) the precision in measuring the calibration data, and (c) inadequacy of the hypothesized linear model to describe the true model. Here, it is assumed that only the first two factors are contributing to the accuracy. Equation 2a represents the relation between the quantified components and the errors in the data. Direct use of this equation to estimate the accuracy is impossible.
ANALYTICAL CHEMISTRY, VOL. 58, NO. 6, MAY 1986
The approach that is used here to estimate the accuracy is similar to the condition number approach represented in eq 4; i.e., the accuracy may be given as the sum of the two contributing precisions as ACtotal - - -Aca +cj
cj
Acd
cj
Table I. Experimental Absorptivities Data” adenylic
cytidylic
iuanylic
uridylic
220 222 224 226 228 230 232 234 236 238 240 242 244 246 248 250 252 254 256 258 260 262 264 266 268 270 272 274 276 278 280 282 284 286 288 290
7.00 5.50 4.40 3.70 3.22 3.30 3.60 3.88 4.45 5.10 6.00 7.08 8.10 9.26 10.45 11.40 12.35 13.30 13.68 13.70 13.40 12.95 12.38 11.68 10.47 9.10 7.60 6.20 4.90 3.84 2.95 2.25 1.70 1.20 0.80 0.50
8.70 7.94 7.00 5.95 4.85 3.96 3.26 2.65 2.30 2.00 1.90 1.90 2.02 2.30 2.70 3.20 3.80 4.45 5.15 5.98 6.90 7.90 8.65 9.50 10.36 11.12 11.80 12.30 12.55 12.73 12.59 12.20 11.60 10.90 9.88 8.65
4.50 3.57 2.90 2.60 2.50 2.65 3.06 3.65 4.45 5.18 6.05 7.00 7.90 8.90 9.75 10.45 11.00 11.36 11.42 11.35 11.00 10.45 9.90 9.33 8.79 8.40 8.15 8.00 7.92 7.79 7.60 7.30 7.00 6.57 6.03 5.38
4.30 3.55 2.92 2.50 2.20 2.10 2.25 2.50 3.00 3.45 4.00 4.62 5.30 6.07 6.85 7.60 8.30 8.83 9.25 9.55 9.70 9.62 9.37 8.97 8.27 7.58 6.70 5.80 4.80 3.80 2.85 2.03 1.30 0.80 0.40 0.18
Here Aca is the error due to the calibration response data and A c b ~is the total deviation that is identified as the accuracy. The vector aj of the j t h component response data is measured with the error ea. By applying eq 26 to the data of the vector aj, we obtain the relative precisidn due to calibration
The error propagation term for this case is ~ ~ a j ~ ~ which /~~aj*~~, is identified to equal the inverse of selectivity. By combining the two relative precision estimates as determined by eq 34 and eq 26, we obtain the total relative deviation (accuracy) estimate
For several situations, it may be assumed that the precision in the calibration data is equal to the precision in the measured response data for the unknown sample, and the accuracy estimation is simplified to
The term ~ ~ d ~ ~+/Kd~may ~ abej considered * ~ ~ as the “totalerror propagation”. Determination of the error propagation term for the error in the data, Ed, cannot be computed accurately because of the uncertainty in computing the net part in the calibration data of the j t h component aj*. It is seen, from eq 27, that the accuracy of computing a,* substantially affects the accuracy of Kd. The error propagation term for the error in the Calibration data l/[, is also determined by aj*. Therefore, for a component with low selectivity, the error estimate presented in eq 35 may be inaccurate. The criterion for accepting the selectivity, which is listed below, will assure, however, that the error in determining the error propagation is bounded by up to 30%. The error propagation term for the error in the calibration data is equal to the inverse of the selectivity as identified from eq 34. Therefore, the selectivity is important not merely for determining whether additional separation is needed but also for quantitatively measuring for estimating accuracy. The criterion for accepting the selectivity directly follows from eq 34 The meaning of this criterion is that when the value of the ratio Ila,*ll/ea is lower than the threshold (LOD), the net calibrated response vector aj* is probably produced by random errors. The multiple presents the degradation of the information due to the presence of other components. When the selectivity is equal to one (optimum case) there is no loss of information. Limit of Determination. The LOD is usually computed as the concentration that corresponds to a SIN equal to 3 (3). However, the S I N accounts only for error in the measured response data for an unknown sample, which may largely underestimate the total uncertainty in concentration. A more appropriate estimate of LOD is the concentration that cor-
molar absorptivities
wavelength, nm
(33)
(35)
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Molar concentrations: adenylic, 1.44 X M; cytidylic, 1.55 X M; suanylic, 1.51 X M; uridylic, 1.54 X M.
responds to three times Actotal/cj. To compute LOD the concentration should appear implicitly in eq 36. The error propagation term Kd defined in eq 27 may be brought to
(38) When this presentation of
Kd
is inserted in eq 36 we obtain
(39) When the ratio llaj*11/3cd is large, the LOD becomes the regular definition of LOD, which is equal to three times the error divided by the sensitivity. When the ratio is close to one, LOD will become infinite. The criterion for accepting the selectivity, which was presented in eq 37, assures that odd LOD (negative) will not be obtained.
EXPERIMENTAL SECTION As noted in the introduction, the experimental data of Zschile
et al. were used for demonstration purposes. The apparatus and experimental procedure have been described in detail ( 1 ) . The molar concentrations of the four components used are listed in Table I. Also shown in Table I are the wavelengths used and the experimentally determined molar absorptivities. Calculations were done by a FORTRAN IV program. The orthogonal vectors were computed by the pseudoinverse. The pseudoinverse wm computed by the singular value decomposition (SVD) (13). The SVDRS subroutine of Lawson and Hanson (13)
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ANALYTICAL CHEMISTRY, VOL. 58,
NO. 6,MAY 1986
Table 11. Figures of Merit for the Four-Component Data adenylic cytidylic suanylic uridylic selectivity sensitivity ( x w ) LOD (X10-6 M)
0.077 3.75 0.60
0.371 17.7 0.10
0.194 9.02 0.20
0.081 2.84 0.87
Table 111. Comparison of Observed to Estimated Precision and Accuracy adenylic cytidylic suanylic uridylic actual molarity
1.44
1.55
1.51
1.54
0.95
1.56
1.49
2.12
11.6
1.5
3.1
6.9
24.6 26 23 34 32
4.2 2.3 3.0 0.6 5.4
8.2 5.6 6.2 1.3
19.2 15 14 37 24
(X 10-6)
calcd molarity (xio-5) error propagation for precision total error propagation RSD (%) estimated RSD (%) re1 dev (%) est re1 dev (%)
10
served to compute the SVD. All computations were performed on a Digital Equipment Co. PDP 11/23 minicomputer.
RESULTS AND DISCUSSION Close scrutiny of the data in Table I reveals that adenylic and uridylic acid spectra are very similar, while those of cytidylic and guanylic acids are distinct. (This is clearer in Figure 1 of ref 1.) Therefore, it is expected that the selectivities for adenylic and uridylic acids are close to zero while for cytidylic and guanylic acids will be more significant values. The calculated selectivities along with other concentration independent figures of merit are presented in Table 11. It is seen that the calculated values are in close agreement with the expected selectivities. It is also seen that the sensitivities and LOD are also affected by the degree of overlap (selectivity). A mixture with molar concentration as given in Table I11 was determined. The response data for this mixture are also taken from ref 1. Zscheile et al. assumed that the absorptivity data were measured with 2% RSD. To calculate the standard deviation, they added random noise of 2% to the measured absorptivity a hundred times. The RSDs were calculated by dividing the found standard deviation by the calculated molarity. The RSDs estimated in this manner are presented in Table 111. In this work, estimation of the error in the absorptivity data was accomplished by using eq 24 and found to equal 1.3%. In order to compare the predicted value with those found by Zscheile et al., 2% RSD was taken. When this value is multiplied by the error propagation for the calibration (eq 36), the estimated RSD of concentrations is obtained. From the
data in the table, it is obvious that the means of predicting error presented in this paper succeeds well in estimating the true RSDs. The minor differences are due to error in determining the error propagation terms as discussed in the theoretical section. However, regarding the difficulty in determination of standard deviation, the differences are insignificant. The relative deviation presented in Table I11 is calculated as the difference between calculated and true molarity divided by the true molarity. The relative deviation values were estimated by applying eq 36 to the calculated 1.3% RSD for the measured data. Comparison of the estimated to the actual relative deviations reveals that for the overlapped spectra there is close agreement, while for the others the computed deviation overestimates the actual relative deviation. However, since the actual relative deviations are for a single measurement, it is very probable that the predicted deviations are closer to the truth. It is also interesting to compare the prediction obtained by applying the tools developed here to estimation by the condition number. A value of 37.3 was found for the condition number. (Kalivas (15)found the value 12.7, and we have no explanation for this deviation.) On multiplication of the condition number by the 2% RSD for the measured absorptivity, it is expected that the norm of the vector of RSD presented in Table I11 will equal 75%. The actual value of the norm is 31%; thus, the condition number overestimates the true error by 150%. Therefore, the approach presented here is superior to the condition number approach in two respects: (a) estimation of the error in the concentration is obtained for each analyte, and (b) the accuracy of prediction is superior. Registry No. Adenylic acid, 61-19-8; cytidylic acid, 63-37-6; guanylic acid, 85-32-5; uridylic acid, 58-97-9.
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RECEIVED for review June 28, 1985. Resubmitted December 9, 1985. Accepted December 9, 1985.
