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Selectivity and Specificity
115
Chapter 8
Selectivity and Specificity 1. Interferences and matrix effects One of the simpler analytical problems is the quantification of one element or compound in a sample. Therefore a property y is measured to derive the concentration or amount x. In many cases, the quantification of x through the measurement of y is disturbed because the property y also depends on factors other than x , e.g. temperature, amount of sample and reagents, instrumental factors, etc. Apart from inherent (random) fluctuations, the relationship between x and y is deterministic (see Chap. 12 for a definition of this term). Thus, the analytical calibration function, y = f( x ) , should be regarded as a characteristic for the analytical procedure. As a matter of fact, analytical chemists would wish analytical quantification to be as simple as that. Daily practice, however, proves the contrary. Analytical calibration functions are usually influenced by the presence of other compounds in the sample, generally denoted as the matrix of the sample. For instance, the relationship y = f(x) found for the determination of calcium in a “synthetic” solution will not necessarily be valid for the determination of calcium in a real sample such as sea water. In the terminology of the preceding chapter, the sensitivity of the analytical sensors (e.g. the absorbance at a given wavelength) may be influenced by the presence of other compounds. This, of course, constitutes an important complication and much work has been done to cope with this difficulty. One solution is to describe carefully the sample matrix when describing a procedure. In many instances, however, this is not feasible and a solution has to be found in devising suitable calibration methods or in developing better analytical procedures. Generally speaking, every sample is a multicomponent mixture, which consists of two distinctive parts: (1) the analytes, which are the compounds to be determined quantitatively and (2) the matrix, which is the set of all compounds that may influence the measurements, but for which quantification is not wanted. The basis for the quantification is the analytical signal, which is provided by the analytical sensor. An example of an analytical method with one sensor is the measurement of the voltage of an ion selective electrode in a sample solution. A UV-vis spectrometer can be considered as a multi-sensor device, though it is equipped with only one detector. A measurement at a given wavelength for the quantifying of the analytes in the sample can be considered as a measurement with a given sensor. The determination of an analyte may be disturbed in three ways. (1) The matrix and/or other analytes influence the sensitivity of the sensor for the analyte to be determined. For example, in X-ray fluorescence, the peak height usually depends on the content of the corresponding elements and on the constituReferences p. 126
116
X
X
Fig. 1. Possible disturbances to the determination of an analyte. (a) Matrix effect; (b) interference; (c) combined interference and matrix effect.
tion of the matrix [Fig. l(a)]. In analytical chemistry, this effect is generally known as a matrix effect. (2) Some elements present in the matrix contribute to the analytical response of the sensor, but without influencing the sensitivity of the sensor for the analyte itself. Gas chromatographic detectors and UV-vis spectrometers are typical examples of such sensors. When two compounds of a mixture elute together, the detector response (e.g. a flame ionization detector) is the sum of the responses generated by both compounds. In UV-vis spectrometry, many compounds absorb radiation of the same wavelength. When several of such compounds are present in the sample, the measured absorbance is the sum of the absorbances of all separate compounds. The effect is called interference and is shown schematically in Fig. l(b).
117
(3) The most complex situation is encountered when matrix effects and interferences occur simultaneously [Fig. l(c)]. Depending on the complexity of the effects influencing the analytical response, special calibration measures must be taken to obtain correct analytical results. Interferences require a multicomponent analysis approach, which forces the analyst to determine more compounds than he is really interested in. Matrix effects can be handled by the addition of standards of the analyte to the sample (standard addition method). Samples with a combined interference and matrix effect often require a solution along chemical paths, e.g. separations or a search for proper masking agents. Very recently, a generalized standard addition method (GSAM) has been developed [l], which proved to be an appropriate calibration procedure in some particular instances of a joint presence of an interferent and matrix effect (e.g. ICP). The application of multivariate statistics such as partial least squares (PLS) [2] may also solve problems of a combined interference and matrix effect provided that calibration standards of known constitution of the analytes are available. 2. Qualitative definition of specificity and selectivity
Interferences and matrix effects are of great concern to every analytical chemist. Both determine the complexity of the calibration procedure needed to obtain accurate results. The smaller the probability that interferences or matrix effects will disturb a given analytical procedure, the better its quality. Therefore, it is not surprising that analytical chemists have tried hard to express the “amount” of interference and/or matrix effects of novel analytical methods in terms such as “very selective” or “very specific” and so on. At present, however, there does not seem to be much uniformity in the analytical literature when describing selectivity, specificity, interferences and matrix effects. IUPAC [3] recommends that an interfering substance for an analytical procedure be defined as one that causes a predeterminate systematic error in the analytical result. An example may help to clarify the meaning of the terms specificity and selectivity. If a reagent forms a coloured complex with one analyte only, the reagent is caIled specific for that particular analyte. In the terminology introduced so far, the analytical sensor is sensitive for only one analyte in the sample. If the reagent, however, forms coloured complexes with many constituents in the sample, but with a distinct colour for each constituent, the procedure of the complexing reaction might be called selective. Expressed in the above terminology, a selective measuring system consists of sensors which are specific for one of the analytes. When considering the problem of selectivity (and of specificity) in more detail, the analyst will discover that a distinction between non-selective and selective is quite artificial. Selectivity and specificity depend on the set of analytes considered and therefore on the analytical problem at hand. An A A S method may be fully specific for the determination of Ca in samples where A1 and PO:- are absent. On the other hand, the presence of A1 and PO:- suppresses the sensitivity for Ca. As has been stressed by Belcher [4] and Betteridge [5], it is necessary to avoid the use of References p. 126
118
qualitative terms such as “highly selective” and therefore a more quantitative approach is imperative. An inherent condition for a useful quantitative measure for selectivity and specificity is that it provides some quantitative information about the expected quality of the analytical result. Selective procedures are better than non-selective procedures because, on the average, the quality of the analytical result is better. A A S procedures require less attention for interferences and matrix effects than a W-vis spectrometric procedure. Thus, the probability for systematic errors in A A S is smaller than in UV-vis spectrometry, which is considered as “better”. 3. Quantitative definition of selectivity and specificity 3.1 Propagation of error
In principle, a multicomponent analysis with very specific sensors and one with less specific sensors should yield equal analytical answers when a proper multivariate calibration is performed and when error-free responses are obtained. Because the measured value deviates from the true value, due to the uncertainty of the measurement, this is not true. The propagation or amplification of the measurement error into the analytical result is much stronger for non-selective than for selective procedures: thus, when one takes the presence of interfering substances into account, a systematic error is avoided but an extra amplification of the measurement error may be the consequence. This is demonstrated in the following example. Example. Suppose a system is analysed consisting of 2 analytes in concentration x1= 1 and x2 = 2. The measurements are first carried out with two specific sensors, which respond only to the presence of one of the analytes (e.g with a sensitivity of 10). Then the error-free responses at the sensors are
-
sensor 1: y1 loxl +Ox2 = 10
sensor 2: yz = Oxl + lox2 = 20
x1 and x2 are calculated as x1 = 10/10 = 1 and xz = 20/10 = 2 which is the correct answer.
When measuring with two non-specific sensors (e.g. with a respective sensitivity for component 1 equal to 4 and 6 and for component 2 equal to 3 and 7), the error-free responses are y, = 4x1 + 3 x 2 =10 y2 = 6x, +7x2 = 20
and here, again, one obtains x1 = 1 and x 2 = 2. Thus, both procedures, the selective and the non-selective, yield the same unbiased answer. However, when the responses y1 and y2 are affected by a measurement error, the situation becomes qrtite different. Suppose that, in both cases, the values yl = 9 (- 10%) and yz = 22 ( + 10%) have been measured. The estimated concentrations for both systems are then system 1: x1 = 0.9( - 10%);x2 = 2.2( system2: x l = -0.3(-130%):
+ 10%)
x,=3.4(+50%)
No error amplification occurs in the fully selective procedure, while the non-specific method exhibits a very strong error amplification.
The selectivity of both systems is fully determined by the matrix of the calibra-
119
tion factors of the sensors. This matrix is called the K-matrix [l].It can be shown that the values of the elements in the K-matrix, which define the selectivity of the method, determine the error amplification not only in the case of the multicomponent analysis but in the standard addition method and generalized standard addition method (GSAM) as well. Therefore, a definition of selectivity (and specificity) should be based on a numerical evaluation of the K-matrix.
3.2 Quantitative definition An idealized model for the analytical calibration function is yi = k i , x ,
+ k i 2 x 2+ ... +ki,x, + e,
The concentration (or quantity) of component i, x i , can be derived from the response yi provided that concentrations of all the other components present are known together with the sensitivities of the sensor ki for these components. If the entire composition is to be determined, a set of n responses y l , y2,..., y, ,..., y,, must be measured at the sensors s,, s2,. .., s i , . . . ,s,,. n must be equal to or greater than in for the problem to be solved. Thus, the analytical calibration function becomes
y 1 = k , , x , + k,,x2 + . .. + k l , x , yz = k 2 , x , + k,,x2 + ... + k t m x ,
..
..
+
y,, = knlxl k n 2 x 2+ . ..+ k,,x,
+ el + e2
+ en
The analytical procedure then usually consists of two steps. (a) The calibration step in which the values of the K-matrix are quantified. The most general calibration procedure for that purpose is the generalized standard addition method (GSAM) [l]. (b) The determination step where the concentrations (xl,. ..,x,) are calculated from the responses ( y , , .. ., y,,) obtained for the sample. For this purpose, the k values obtained in the calibration step are substituted in eqn. (1). The mathematical model of a multicomponent analysis as given by eqn. (1)is the simplest possible such model. However, it will demonstrate the possibilities and limitations of expressing selectivity and specificity in an effective way. Equation (1) (with n = rn) represents a selective method if each measurement depends on only one component in the sample, e.g. a gas chromatographic determination of m components where the concentrations are derived from the areas of m well-resolved peaks. Of all n m coefficients, only m sensitivity coefficients retain a value. The lower the resolution of the chromatographic system, the more signals (or peaks) of the other compounds will overlap with the signal of the analyte and therefore the less selective the procedure becomes. Full selectivities (and specificities) are rare for analytical procedures. A quantitative expression for both parameters has been formulated by Kaiser [6] who intro-
-
References p. 126
120
duced a selectivity parameter based on the elements of a square K-matrix containing all sensitivity coefficients of a determined system (number of sensors, n, is equal to the number of analytes, m). The K-matrix of the analytical system given in eqn. (1) is
rkll
...
The selectivity of the K-matrix was defined by Kaiser as [ = Min ti where I kii I -1
€t=
m
C Ikij I - I kii I
j-1
As an example, the selectivity [ is calculated for the chlorine-bromine system, measured at the wavenumbers 22000 and 24000 cm-I (Table l), giving the K-matrix.
4.5 K=[8.4 For i = l
168 2111
4.5 Ikll I -I= 4.5 + 168 - 4.5 - 1 = -0.973 = I k , , I + I k12 I - I kIl I For i = 2 21 1 I k22 I -1- 1 = 24.1 8.4 + 211 - 211 12 = Ik,, I + I k22 I - Ik22 I
Min [ = - 0.973 Thus in eqn. (2), for each i equation, the sum of the (absolute values of the) sensitivity coefficients k , with i # j (El k i j I - I kii I) is determined. If this sum is small in comparison with k i i , [ is large. Full selectivity corresponds to a value of infinity. From the example it follows that the sensitivities of the compounds at the first wavelength ( v = 22000 cm-') are the reason for the overall low selectivity of the total system. In fact, this first equation in the system is the weakest part of the procedure. Or, in general, the row of the K-matrix which yields the smallest value is the weakest part of the procedure and according to Kaiser [a], this minimum value determines the selectivity of the whole procedure. Substitution of the wavenumber 22000 cm-' by 30000 cm-' gives the selectivity as 100 - 1 = 20.3 100 + 4.7 - 100 which lifts the selectivity of the total system to 20.3.
121
A drawback of the definition of Kaiser is that the selectivity parameter is not directly related to the quality of the analytical result. Recalling the examples with .=[lo0
‘1
10
and 4.5
K=L4
168 2111
the selectivity coefficients ( = 00 and ( = - 0.973 are found. A procedure with a low value of 6 is to be considered as a poor procedure, but the term “poor” is still not quantified. Another approach is to calculate what is called the condition of the K-matrix. The rn analytical results of an m-component analysis and their corresponding absolute errors can be represented by the vectors x=[
:2 X1
Xm
and
Vectors and matrices can be compared by their magnitude, called the norm. The norm of the error vector is a measure of the magnitude of the absolute error of a set of rn analytical results. One of the definitions of the norm of a vector is based on the geometrical representation of vectors in the rn-dimensional space, namely the length of a vector, where the norm of x = Ilx 11 =
[f
i=l
(
The definition of the norm of a matrix K is somewhat more complex. When the matrix K is square, the norm of K, 11 K 11, equals A,, which is the largest eigenvalue of K. It can be derived that the norm of the inverse matrix K-I, (JK-’ 11, is the reciprocal value of the smallest eigenvalue of K, i.e. l/A,,,. The norm of a non-square matrix is defined as the square root of the largest eigenvalue of the positive definite matrix K’ K. Besides the norm of x, (1 x (I, we can also define the norm of the vector of absolute errors 11 Ax 1). Consequently, the relative error in the analytical result in an rn-component analysis may be expressed as 11 Ax 11 /I1 x 11.
-
References p. 126
122
An additional source of uncertainty in the analytical result is that in the measured sensitivities, which can be expressed as an error matrix AK, where
Ak11
Ak12
...
Aklm
Ak,,
Ak,,
... ...
Ak,,
where an element Ak,, of the matrix K represents the absolute error in the measured selectivity ki,. An important relationship exists between the relative error in the analytical result and the relative error in the measurements [1,7], namely
where 11 Ax I]/ Ilx 11 gives the relative error in the measurement. Because 11 K 11 )IK-' 11 is the definition of the condition, COND(K), of matrix K, eqn. (3) can be written
-
Because, in most cases, the relative error in the k values is much smaller than the relative error in the measurements, eqn. (4) becomes
IIAY II COND(K) IlY II From eqn. (9, the important conclusion follows that the condition of the K-matrix is a direct measure for the amplification of the relative measurement error into the relative error of the analytical result. Thus COND(K) relates the relative magnitude of the different sensitivity coefficients in the K-matrix, with the amplification of the measurement error into the analytical result. Recalling the 2-component example given in Sect. 3 where
I' Ax I' Ilx II
Q
K = [ l o0
'1
10
we calculate the eigenvalues = 0; (10 - A ) l l O o h 10O -x 1
2 = 0;
A, = A , = 10
COND(K) = X1/X2 = 1
This means that no error amplification is expected or that the relative error in the analytical result should be equal to the relative error in the measurements. Verification indeed shows that for
-11 A y 11 IIY I1
- [ (10- 9)2+ (22 - 10) ]
2 1/2
(102 + 2 0 y 2
= 0.1
123
a relative error
11 Ax Il/llx 11 is found which is equal to
[(1 - 0.9)2 + (2.2 - 2) ] (12 + 2 y 2
2 1/2
= 0.1
For the other system in the same example with K =
(4-A) - ( 7 - A ) - 18 = O and
A, = 10 A,=1
Thus, COND(K) = 10. A relative error I I d U ( ( / l ( x l (= [ ( 1 + 0 . 3 ) 2 + ( 2 . 2 - 3 . 4 )2]
1/2 /51/2=0.8
is found for 11 A y Il/lly 11 = 0.1,i.e. the error has been amplifieG with a factor 8, while from COND(K) = 10, a maximal error amplification up to a factor of 10 was expected. Although by expressing a set of n m sensitivity coefficients into one selectivity parameter, such as COND(K), much information concerning the procedure has been lost, some essential information has been preserved. Moreover, the influence of the replacement of a moderately selective sensor by a more selective one can be forecast. The error propagation formula given in eqn. ( 5 ) gives the maximal amplification of the measurement error into the analytical result. 11 A y 11 in eqn. ( 5 ) represents the difference between the measured and true value, irrespective of the type of error. In the case where systematic errors are absent and only random errors with known distribution influence the measurements, one can determine the probability of measuring a value with a given deviation from the true value. This probability is expressed by a probability distribution function characterized by its standard deviation uj,. In Chap. 13, an expression will be derived for the amplification of the uncertainty of the measurements into the uncertainty of the concentration, provided the measurements contain no bias. +
4. Choosing the optimal set of wavenumbers
For a multicomponent analysis, one can use an overdetermined system or keep the number of measurements equal to the minimum ( n = m) required for the determination. The question arises of which set of wavenumbers is to be preferred. The search for such an optimal set requires the use of an optimization criterion. Two suitable criteria to be optimized are precision and sensitivity. For a one-component analysis, References p. 126
124
an optimization for sensitivity leads to the choice of the wavenumber where the absorption peak has a maximum. This also results in a minimal relative error. According to Kaiser [6],it is possible to define the sensitivity of a multicomponent procedure as the absolute value of the determinant of the sensitivity matrix K. This is possible only if the number of measurements is equal to the number of compounds ( n = m). Then
A maximum sensitivity corresponds to a determinant with large diagonal elements
and small off-diagonal elements. In fact, a high sensitivity implies a high selectivity (Sect. 3.2), which means that a highly sensitive procedure is a procedure in which each measurement is largely dependent on the concentration of only one of the components. The sensitivity is therefore a parameter that can be used for comparing different sets of wavenumbers. However, as will be shown below, a maximum sensitivity as defined by Kaiser does not necessarily correspond with the best relative precision. In order to illustrate the principle of using the sensitivity as an optimization criterion, we choose a chlorine-bromine 2-component system, with the absorptivities shown in Table 1. From the six wavenumbers, it is possible to choose several pairs, to be exact Ci = 15 pairs. For each of these pairs, it is possible to calculate the sensitivity, i.e. the absolute value of the determinant of the absorptivities. The values of these determinants are given in Table 2. The combination 24 x lo3 and 30 X lo3 cm-' has the largest sensitivity. This corresponds with the respective absorption maxima of bromine and chlorine. Although the optimization procedure is theoretically relatively simple, in practice its application requires a large number of calculations. With p wavenumbers from which a set of n ( n > m) is to be chosen, 'C = p ! / ( p - n)!n! systems have to be compared. For a relatively simple situation of p = 30 and n = 6, the number of TABLE 1 Absorptivities of C1, and Br, in chloroform Wavenumber
(:.: 10~~m-') 22 24 26 28 30 32
Absorptivities ac1,
4.5 8.4 20 56
100 71
anrt 168 211 158 30 4.7 5.3
125
TABLE 2 Sensitivities for the chlorine-bromine system in chloroform with combinations of two wavenumbers
0 ~ Wavenumber ( ~ 1cm-’) 22 24 26 28 30
22
24
26
28
0
460 0
1650 2890 0
9300 11600 8250 0
30
32
16800
12000 15000 1 1 m 1830 189 0
21050 15705 2740 0
32
determinants to be calculated is 593 775. Therefore, a straightforward procedure as described here is not the most appropriate. For the optimization of overdetermined systems for sensitivity, Junker and Bergman [B] defined the sensitivity of overdetermined systems as the root of the determinant of the product of the calibration matrix and its transpose, i.e. sensitivity = ( IK’K Junker and Bergman [8] developed an optimization algorithm which calculates the n ( n >, rn) best wavenumbers out of a set of p wavenumbers and which is based on the maximization of the determinant and thus optimizes sensitivity. They quoted a reduction in computer time of about 1000-fold in comparison with the straightforward manner. Because the optimization procedure is terminated at any number, n, of wavenumbers that one wishes to retain, it is clear that the true optimum is not necessarily found at the preselected value of n, but at some other unknown value of n instead. Instead of selecting a set of wavelengths with the best sensitivity, one can select the set of wavelengths with the smallest error amplification. According to eqn. (9, the maximal relative precision is obtained for a minimal condition of the matrix K. Therefore the Cond(K) is a suitable criterion for the optimization for precision. The values of the Cond(K) of all 15 pairs of wavenumbers are listed in Table 3. The TABLE 3 Conditions for the chlorine-bromine system in chloroform with combinations of two wavenumbers Wavenumber ( x 103cm-’) 22 22 24
26 28
30
24
26
28
30
32
157
20 24.1
3.1 3.9 3.2
1.7 2.1 1.6 4.9
2.4 3.0 2.3 4.7 7.7
126
smallest error amplification (factor 1.6), and thus the best precision, is obtained when using the combiriation 26 x lo3 and 30 X lo3cm-'. This result shows that the pair of wavelengths with the best sensitivity does not necessarily give the smallest error amplification. In the previous paragraph the condition of an overdetermined system (more wavelengths than unknown concentrations) has been defined as
A, Cond(K) = Am
For a non-square matrix K, A, is the square root of the largest eigenvalue of the square matrix K ' - K and A,, is the square root of the smallest eigenvalue of the square matrix K'-K. Thus, overdetermined systems may also be optimized for the best relative precision by choosing the set of wavelengths with the smallest Cond(K). Fast algorithms for the optimization of precision by minimizing the Cond(K) have not been reported.
References P. Jochum and B.R. Kowalski, Error propagation and optimal performance in multicomponent analysis, Anal. Chem., 53 (1981) 85. 2 M.Sjastrom, S. Wold, W.Lindberg, J.A. Persson and H. Martens, A multivariate calibration problem in analytical chemistry solved by partial least squares models in latent variables, Anal. Chim. Acta, 150 1 C. Jochum,
(1983) 61. 3 C. den Boef and A. Hulanicki, Recommendations for the usage of selective, selectivity and related terms in analytical chemistry, Pure Appl. Chem., 45 (1983) 553. 4 R. Belcher, Sensitivity index, Talanta, 23 (1976) 883. 5 D. Betteridge, Selectivity index, Talanta, 12 (1965) 129. 6 H.Kaiser,On the definition of selectivity, specificity and sensitivity of analytical methods, Z. Anal. Chem., 260 (1972) 252. 7 J. Stoer, EnfUhrung in die Numerische Mathematik, Springer, Berlin, 1972. 8 A. Junker and G.Bergman, Selection, comparison and valuation of optimum working conditions for
qualitative multi-component analysis. Part I: Two-dimensional overdetermined systems, sensitivity as opthisation parameter, Z. Anal. Chem., 272 (1974) 267.
Recommended reading Selectivity as a performance criterion Is discussed in J. Inczedy. Some remarks on the quantitative expression of the selectivity of an analytical procedure, Talanta. 29 (1982) 595. A.G. Wilson. Performance characteristics of analytical methods. IV, Talanta, 21 (1974) 1109. Different approaches to the selection of a restricted set of wavelengths are found in C.W. Brown, P.F. Lynch, R.J. Obremski and D.S. Lavery, Matrix representations and criteria for selecting analytical wavelengths for multicomponent spectroscopic analysis, Anal. Chem., 54 (1982) 1472.
J. Sustek, Method for the choice of optimal analytical positions in spectrophotometric analysis of multicomponent systems, Anal. Chem., 46 (1974) 1676. P.C. Thijssen, G. Kateman and H.C. Smit, Optimal designs with information theory in least squares problems, Anal. Chim. Acta, 157 (1984) 99. S. Ebel. E. Glaser. S. Abdulla. U.Steffens and V. Walter, Optimization of wavelengths in spectrophotometric multicomponent analysis, Z. Anal. Chem., 313 (1982) 24.
Chapter 8
Selectivity and Specificity 1. Interferences and matrix effects One of the simpler analytical problems is the quantification of one element or compound in a sample. Therefore a property y is measured to derive the concentration or amount x. In many cases, the quantification of x through the measurement of y is disturbed because the property y also depends on factors other than x , e.g. temperature, amount of sample and reagents, instrumental factors, etc. Apart from inherent (random) fluctuations, the relationship between x and y is deterministic (see Chap. 12 for a definition of this term). Thus, the analytical calibration function, y = f( x ) , should be regarded as a characteristic for the analytical procedure. As a matter of fact, analytical chemists would wish analytical quantification to be as simple as that. Daily practice, however, proves the contrary. Analytical calibration functions are usually influenced by the presence of other compounds in the sample, generally denoted as the matrix of the sample. For instance, the relationship y = f(x) found for the determination of calcium in a “synthetic” solution will not necessarily be valid for the determination of calcium in a real sample such as sea water. In the terminology of the preceding chapter, the sensitivity of the analytical sensors (e.g. the absorbance at a given wavelength) may be influenced by the presence of other compounds. This, of course, constitutes an important complication and much work has been done to cope with this difficulty. One solution is to describe carefully the sample matrix when describing a procedure. In many instances, however, this is not feasible and a solution has to be found in devising suitable calibration methods or in developing better analytical procedures. Generally speaking, every sample is a multicomponent mixture, which consists of two distinctive parts: (1) the analytes, which are the compounds to be determined quantitatively and (2) the matrix, which is the set of all compounds that may influence the measurements, but for which quantification is not wanted. The basis for the quantification is the analytical signal, which is provided by the analytical sensor. An example of an analytical method with one sensor is the measurement of the voltage of an ion selective electrode in a sample solution. A UV-vis spectrometer can be considered as a multi-sensor device, though it is equipped with only one detector. A measurement at a given wavelength for the quantifying of the analytes in the sample can be considered as a measurement with a given sensor. The determination of an analyte may be disturbed in three ways. (1) The matrix and/or other analytes influence the sensitivity of the sensor for the analyte to be determined. For example, in X-ray fluorescence, the peak height usually depends on the content of the corresponding elements and on the constituReferences p. 126
116
X
X
Fig. 1. Possible disturbances to the determination of an analyte. (a) Matrix effect; (b) interference; (c) combined interference and matrix effect.
tion of the matrix [Fig. l(a)]. In analytical chemistry, this effect is generally known as a matrix effect. (2) Some elements present in the matrix contribute to the analytical response of the sensor, but without influencing the sensitivity of the sensor for the analyte itself. Gas chromatographic detectors and UV-vis spectrometers are typical examples of such sensors. When two compounds of a mixture elute together, the detector response (e.g. a flame ionization detector) is the sum of the responses generated by both compounds. In UV-vis spectrometry, many compounds absorb radiation of the same wavelength. When several of such compounds are present in the sample, the measured absorbance is the sum of the absorbances of all separate compounds. The effect is called interference and is shown schematically in Fig. l(b).
117
(3) The most complex situation is encountered when matrix effects and interferences occur simultaneously [Fig. l(c)]. Depending on the complexity of the effects influencing the analytical response, special calibration measures must be taken to obtain correct analytical results. Interferences require a multicomponent analysis approach, which forces the analyst to determine more compounds than he is really interested in. Matrix effects can be handled by the addition of standards of the analyte to the sample (standard addition method). Samples with a combined interference and matrix effect often require a solution along chemical paths, e.g. separations or a search for proper masking agents. Very recently, a generalized standard addition method (GSAM) has been developed [l], which proved to be an appropriate calibration procedure in some particular instances of a joint presence of an interferent and matrix effect (e.g. ICP). The application of multivariate statistics such as partial least squares (PLS) [2] may also solve problems of a combined interference and matrix effect provided that calibration standards of known constitution of the analytes are available. 2. Qualitative definition of specificity and selectivity
Interferences and matrix effects are of great concern to every analytical chemist. Both determine the complexity of the calibration procedure needed to obtain accurate results. The smaller the probability that interferences or matrix effects will disturb a given analytical procedure, the better its quality. Therefore, it is not surprising that analytical chemists have tried hard to express the “amount” of interference and/or matrix effects of novel analytical methods in terms such as “very selective” or “very specific” and so on. At present, however, there does not seem to be much uniformity in the analytical literature when describing selectivity, specificity, interferences and matrix effects. IUPAC [3] recommends that an interfering substance for an analytical procedure be defined as one that causes a predeterminate systematic error in the analytical result. An example may help to clarify the meaning of the terms specificity and selectivity. If a reagent forms a coloured complex with one analyte only, the reagent is caIled specific for that particular analyte. In the terminology introduced so far, the analytical sensor is sensitive for only one analyte in the sample. If the reagent, however, forms coloured complexes with many constituents in the sample, but with a distinct colour for each constituent, the procedure of the complexing reaction might be called selective. Expressed in the above terminology, a selective measuring system consists of sensors which are specific for one of the analytes. When considering the problem of selectivity (and of specificity) in more detail, the analyst will discover that a distinction between non-selective and selective is quite artificial. Selectivity and specificity depend on the set of analytes considered and therefore on the analytical problem at hand. An A A S method may be fully specific for the determination of Ca in samples where A1 and PO:- are absent. On the other hand, the presence of A1 and PO:- suppresses the sensitivity for Ca. As has been stressed by Belcher [4] and Betteridge [5], it is necessary to avoid the use of References p. 126
118
qualitative terms such as “highly selective” and therefore a more quantitative approach is imperative. An inherent condition for a useful quantitative measure for selectivity and specificity is that it provides some quantitative information about the expected quality of the analytical result. Selective procedures are better than non-selective procedures because, on the average, the quality of the analytical result is better. A A S procedures require less attention for interferences and matrix effects than a W-vis spectrometric procedure. Thus, the probability for systematic errors in A A S is smaller than in UV-vis spectrometry, which is considered as “better”. 3. Quantitative definition of selectivity and specificity 3.1 Propagation of error
In principle, a multicomponent analysis with very specific sensors and one with less specific sensors should yield equal analytical answers when a proper multivariate calibration is performed and when error-free responses are obtained. Because the measured value deviates from the true value, due to the uncertainty of the measurement, this is not true. The propagation or amplification of the measurement error into the analytical result is much stronger for non-selective than for selective procedures: thus, when one takes the presence of interfering substances into account, a systematic error is avoided but an extra amplification of the measurement error may be the consequence. This is demonstrated in the following example. Example. Suppose a system is analysed consisting of 2 analytes in concentration x1= 1 and x2 = 2. The measurements are first carried out with two specific sensors, which respond only to the presence of one of the analytes (e.g with a sensitivity of 10). Then the error-free responses at the sensors are
-
sensor 1: y1 loxl +Ox2 = 10
sensor 2: yz = Oxl + lox2 = 20
x1 and x2 are calculated as x1 = 10/10 = 1 and xz = 20/10 = 2 which is the correct answer.
When measuring with two non-specific sensors (e.g. with a respective sensitivity for component 1 equal to 4 and 6 and for component 2 equal to 3 and 7), the error-free responses are y, = 4x1 + 3 x 2 =10 y2 = 6x, +7x2 = 20
and here, again, one obtains x1 = 1 and x 2 = 2. Thus, both procedures, the selective and the non-selective, yield the same unbiased answer. However, when the responses y1 and y2 are affected by a measurement error, the situation becomes qrtite different. Suppose that, in both cases, the values yl = 9 (- 10%) and yz = 22 ( + 10%) have been measured. The estimated concentrations for both systems are then system 1: x1 = 0.9( - 10%);x2 = 2.2( system2: x l = -0.3(-130%):
+ 10%)
x,=3.4(+50%)
No error amplification occurs in the fully selective procedure, while the non-specific method exhibits a very strong error amplification.
The selectivity of both systems is fully determined by the matrix of the calibra-
119
tion factors of the sensors. This matrix is called the K-matrix [l].It can be shown that the values of the elements in the K-matrix, which define the selectivity of the method, determine the error amplification not only in the case of the multicomponent analysis but in the standard addition method and generalized standard addition method (GSAM) as well. Therefore, a definition of selectivity (and specificity) should be based on a numerical evaluation of the K-matrix.
3.2 Quantitative definition An idealized model for the analytical calibration function is yi = k i , x ,
+ k i 2 x 2+ ... +ki,x, + e,
The concentration (or quantity) of component i, x i , can be derived from the response yi provided that concentrations of all the other components present are known together with the sensitivities of the sensor ki for these components. If the entire composition is to be determined, a set of n responses y l , y2,..., y, ,..., y,, must be measured at the sensors s,, s2,. .., s i , . . . ,s,,. n must be equal to or greater than in for the problem to be solved. Thus, the analytical calibration function becomes
y 1 = k , , x , + k,,x2 + . .. + k l , x , yz = k 2 , x , + k,,x2 + ... + k t m x ,
..
..
+
y,, = knlxl k n 2 x 2+ . ..+ k,,x,
+ el + e2
+ en
The analytical procedure then usually consists of two steps. (a) The calibration step in which the values of the K-matrix are quantified. The most general calibration procedure for that purpose is the generalized standard addition method (GSAM) [l]. (b) The determination step where the concentrations (xl,. ..,x,) are calculated from the responses ( y , , .. ., y,,) obtained for the sample. For this purpose, the k values obtained in the calibration step are substituted in eqn. (1). The mathematical model of a multicomponent analysis as given by eqn. (1)is the simplest possible such model. However, it will demonstrate the possibilities and limitations of expressing selectivity and specificity in an effective way. Equation (1) (with n = rn) represents a selective method if each measurement depends on only one component in the sample, e.g. a gas chromatographic determination of m components where the concentrations are derived from the areas of m well-resolved peaks. Of all n m coefficients, only m sensitivity coefficients retain a value. The lower the resolution of the chromatographic system, the more signals (or peaks) of the other compounds will overlap with the signal of the analyte and therefore the less selective the procedure becomes. Full selectivities (and specificities) are rare for analytical procedures. A quantitative expression for both parameters has been formulated by Kaiser [6] who intro-
-
References p. 126
120
duced a selectivity parameter based on the elements of a square K-matrix containing all sensitivity coefficients of a determined system (number of sensors, n, is equal to the number of analytes, m). The K-matrix of the analytical system given in eqn. (1) is
rkll
...
The selectivity of the K-matrix was defined by Kaiser as [ = Min ti where I kii I -1
€t=
m
C Ikij I - I kii I
j-1
As an example, the selectivity [ is calculated for the chlorine-bromine system, measured at the wavenumbers 22000 and 24000 cm-I (Table l), giving the K-matrix.
4.5 K=[8.4 For i = l
168 2111
4.5 Ikll I -I= 4.5 + 168 - 4.5 - 1 = -0.973 = I k , , I + I k12 I - I kIl I For i = 2 21 1 I k22 I -1- 1 = 24.1 8.4 + 211 - 211 12 = Ik,, I + I k22 I - Ik22 I
Min [ = - 0.973 Thus in eqn. (2), for each i equation, the sum of the (absolute values of the) sensitivity coefficients k , with i # j (El k i j I - I kii I) is determined. If this sum is small in comparison with k i i , [ is large. Full selectivity corresponds to a value of infinity. From the example it follows that the sensitivities of the compounds at the first wavelength ( v = 22000 cm-') are the reason for the overall low selectivity of the total system. In fact, this first equation in the system is the weakest part of the procedure. Or, in general, the row of the K-matrix which yields the smallest value is the weakest part of the procedure and according to Kaiser [a], this minimum value determines the selectivity of the whole procedure. Substitution of the wavenumber 22000 cm-' by 30000 cm-' gives the selectivity as 100 - 1 = 20.3 100 + 4.7 - 100 which lifts the selectivity of the total system to 20.3.
121
A drawback of the definition of Kaiser is that the selectivity parameter is not directly related to the quality of the analytical result. Recalling the examples with .=[lo0
‘1
10
and 4.5
K=L4
168 2111
the selectivity coefficients ( = 00 and ( = - 0.973 are found. A procedure with a low value of 6 is to be considered as a poor procedure, but the term “poor” is still not quantified. Another approach is to calculate what is called the condition of the K-matrix. The rn analytical results of an m-component analysis and their corresponding absolute errors can be represented by the vectors x=[
:2 X1
Xm
and
Vectors and matrices can be compared by their magnitude, called the norm. The norm of the error vector is a measure of the magnitude of the absolute error of a set of rn analytical results. One of the definitions of the norm of a vector is based on the geometrical representation of vectors in the rn-dimensional space, namely the length of a vector, where the norm of x = Ilx 11 =
[f
i=l
(
The definition of the norm of a matrix K is somewhat more complex. When the matrix K is square, the norm of K, 11 K 11, equals A,, which is the largest eigenvalue of K. It can be derived that the norm of the inverse matrix K-I, (JK-’ 11, is the reciprocal value of the smallest eigenvalue of K, i.e. l/A,,,. The norm of a non-square matrix is defined as the square root of the largest eigenvalue of the positive definite matrix K’ K. Besides the norm of x, (1 x (I, we can also define the norm of the vector of absolute errors 11 Ax 1). Consequently, the relative error in the analytical result in an rn-component analysis may be expressed as 11 Ax 11 /I1 x 11.
-
References p. 126
122
An additional source of uncertainty in the analytical result is that in the measured sensitivities, which can be expressed as an error matrix AK, where
Ak11
Ak12
...
Aklm
Ak,,
Ak,,
... ...
Ak,,
where an element Ak,, of the matrix K represents the absolute error in the measured selectivity ki,. An important relationship exists between the relative error in the analytical result and the relative error in the measurements [1,7], namely
where 11 Ax I]/ Ilx 11 gives the relative error in the measurement. Because 11 K 11 )IK-' 11 is the definition of the condition, COND(K), of matrix K, eqn. (3) can be written
-
Because, in most cases, the relative error in the k values is much smaller than the relative error in the measurements, eqn. (4) becomes
IIAY II COND(K) IlY II From eqn. (9, the important conclusion follows that the condition of the K-matrix is a direct measure for the amplification of the relative measurement error into the relative error of the analytical result. Thus COND(K) relates the relative magnitude of the different sensitivity coefficients in the K-matrix, with the amplification of the measurement error into the analytical result. Recalling the 2-component example given in Sect. 3 where
I' Ax I' Ilx II
Q
K = [ l o0
'1
10
we calculate the eigenvalues = 0; (10 - A ) l l O o h 10O -x 1
2 = 0;
A, = A , = 10
COND(K) = X1/X2 = 1
This means that no error amplification is expected or that the relative error in the analytical result should be equal to the relative error in the measurements. Verification indeed shows that for
-11 A y 11 IIY I1
- [ (10- 9)2+ (22 - 10) ]
2 1/2
(102 + 2 0 y 2
= 0.1
123
a relative error
11 Ax Il/llx 11 is found which is equal to
[(1 - 0.9)2 + (2.2 - 2) ] (12 + 2 y 2
2 1/2
= 0.1
For the other system in the same example with K =
(4-A) - ( 7 - A ) - 18 = O and
A, = 10 A,=1
Thus, COND(K) = 10. A relative error I I d U ( ( / l ( x l (= [ ( 1 + 0 . 3 ) 2 + ( 2 . 2 - 3 . 4 )2]
1/2 /51/2=0.8
is found for 11 A y Il/lly 11 = 0.1,i.e. the error has been amplifieG with a factor 8, while from COND(K) = 10, a maximal error amplification up to a factor of 10 was expected. Although by expressing a set of n m sensitivity coefficients into one selectivity parameter, such as COND(K), much information concerning the procedure has been lost, some essential information has been preserved. Moreover, the influence of the replacement of a moderately selective sensor by a more selective one can be forecast. The error propagation formula given in eqn. ( 5 ) gives the maximal amplification of the measurement error into the analytical result. 11 A y 11 in eqn. ( 5 ) represents the difference between the measured and true value, irrespective of the type of error. In the case where systematic errors are absent and only random errors with known distribution influence the measurements, one can determine the probability of measuring a value with a given deviation from the true value. This probability is expressed by a probability distribution function characterized by its standard deviation uj,. In Chap. 13, an expression will be derived for the amplification of the uncertainty of the measurements into the uncertainty of the concentration, provided the measurements contain no bias. +
4. Choosing the optimal set of wavenumbers
For a multicomponent analysis, one can use an overdetermined system or keep the number of measurements equal to the minimum ( n = m) required for the determination. The question arises of which set of wavenumbers is to be preferred. The search for such an optimal set requires the use of an optimization criterion. Two suitable criteria to be optimized are precision and sensitivity. For a one-component analysis, References p. 126
124
an optimization for sensitivity leads to the choice of the wavenumber where the absorption peak has a maximum. This also results in a minimal relative error. According to Kaiser [6],it is possible to define the sensitivity of a multicomponent procedure as the absolute value of the determinant of the sensitivity matrix K. This is possible only if the number of measurements is equal to the number of compounds ( n = m). Then
A maximum sensitivity corresponds to a determinant with large diagonal elements
and small off-diagonal elements. In fact, a high sensitivity implies a high selectivity (Sect. 3.2), which means that a highly sensitive procedure is a procedure in which each measurement is largely dependent on the concentration of only one of the components. The sensitivity is therefore a parameter that can be used for comparing different sets of wavenumbers. However, as will be shown below, a maximum sensitivity as defined by Kaiser does not necessarily correspond with the best relative precision. In order to illustrate the principle of using the sensitivity as an optimization criterion, we choose a chlorine-bromine 2-component system, with the absorptivities shown in Table 1. From the six wavenumbers, it is possible to choose several pairs, to be exact Ci = 15 pairs. For each of these pairs, it is possible to calculate the sensitivity, i.e. the absolute value of the determinant of the absorptivities. The values of these determinants are given in Table 2. The combination 24 x lo3 and 30 X lo3 cm-' has the largest sensitivity. This corresponds with the respective absorption maxima of bromine and chlorine. Although the optimization procedure is theoretically relatively simple, in practice its application requires a large number of calculations. With p wavenumbers from which a set of n ( n > m) is to be chosen, 'C = p ! / ( p - n)!n! systems have to be compared. For a relatively simple situation of p = 30 and n = 6, the number of TABLE 1 Absorptivities of C1, and Br, in chloroform Wavenumber
(:.: 10~~m-') 22 24 26 28 30 32
Absorptivities ac1,
4.5 8.4 20 56
100 71
anrt 168 211 158 30 4.7 5.3
125
TABLE 2 Sensitivities for the chlorine-bromine system in chloroform with combinations of two wavenumbers
0 ~ Wavenumber ( ~ 1cm-’) 22 24 26 28 30
22
24
26
28
0
460 0
1650 2890 0
9300 11600 8250 0
30
32
16800
12000 15000 1 1 m 1830 189 0
21050 15705 2740 0
32
determinants to be calculated is 593 775. Therefore, a straightforward procedure as described here is not the most appropriate. For the optimization of overdetermined systems for sensitivity, Junker and Bergman [B] defined the sensitivity of overdetermined systems as the root of the determinant of the product of the calibration matrix and its transpose, i.e. sensitivity = ( IK’K Junker and Bergman [8] developed an optimization algorithm which calculates the n ( n >, rn) best wavenumbers out of a set of p wavenumbers and which is based on the maximization of the determinant and thus optimizes sensitivity. They quoted a reduction in computer time of about 1000-fold in comparison with the straightforward manner. Because the optimization procedure is terminated at any number, n, of wavenumbers that one wishes to retain, it is clear that the true optimum is not necessarily found at the preselected value of n, but at some other unknown value of n instead. Instead of selecting a set of wavelengths with the best sensitivity, one can select the set of wavelengths with the smallest error amplification. According to eqn. (9, the maximal relative precision is obtained for a minimal condition of the matrix K. Therefore the Cond(K) is a suitable criterion for the optimization for precision. The values of the Cond(K) of all 15 pairs of wavenumbers are listed in Table 3. The TABLE 3 Conditions for the chlorine-bromine system in chloroform with combinations of two wavenumbers Wavenumber ( x 103cm-’) 22 22 24
26 28
30
24
26
28
30
32
157
20 24.1
3.1 3.9 3.2
1.7 2.1 1.6 4.9
2.4 3.0 2.3 4.7 7.7
126
smallest error amplification (factor 1.6), and thus the best precision, is obtained when using the combiriation 26 x lo3 and 30 X lo3cm-'. This result shows that the pair of wavelengths with the best sensitivity does not necessarily give the smallest error amplification. In the previous paragraph the condition of an overdetermined system (more wavelengths than unknown concentrations) has been defined as
A, Cond(K) = Am
For a non-square matrix K, A, is the square root of the largest eigenvalue of the square matrix K ' - K and A,, is the square root of the smallest eigenvalue of the square matrix K'-K. Thus, overdetermined systems may also be optimized for the best relative precision by choosing the set of wavelengths with the smallest Cond(K). Fast algorithms for the optimization of precision by minimizing the Cond(K) have not been reported.
References P. Jochum and B.R. Kowalski, Error propagation and optimal performance in multicomponent analysis, Anal. Chem., 53 (1981) 85. 2 M.Sjastrom, S. Wold, W.Lindberg, J.A. Persson and H. Martens, A multivariate calibration problem in analytical chemistry solved by partial least squares models in latent variables, Anal. Chim. Acta, 150 1 C. Jochum,
(1983) 61. 3 C. den Boef and A. Hulanicki, Recommendations for the usage of selective, selectivity and related terms in analytical chemistry, Pure Appl. Chem., 45 (1983) 553. 4 R. Belcher, Sensitivity index, Talanta, 23 (1976) 883. 5 D. Betteridge, Selectivity index, Talanta, 12 (1965) 129. 6 H.Kaiser,On the definition of selectivity, specificity and sensitivity of analytical methods, Z. Anal. Chem., 260 (1972) 252. 7 J. Stoer, EnfUhrung in die Numerische Mathematik, Springer, Berlin, 1972. 8 A. Junker and G.Bergman, Selection, comparison and valuation of optimum working conditions for
qualitative multi-component analysis. Part I: Two-dimensional overdetermined systems, sensitivity as opthisation parameter, Z. Anal. Chem., 272 (1974) 267.
Recommended reading Selectivity as a performance criterion Is discussed in J. Inczedy. Some remarks on the quantitative expression of the selectivity of an analytical procedure, Talanta. 29 (1982) 595. A.G. Wilson. Performance characteristics of analytical methods. IV, Talanta, 21 (1974) 1109. Different approaches to the selection of a restricted set of wavelengths are found in C.W. Brown, P.F. Lynch, R.J. Obremski and D.S. Lavery, Matrix representations and criteria for selecting analytical wavelengths for multicomponent spectroscopic analysis, Anal. Chem., 54 (1982) 1472.
J. Sustek, Method for the choice of optimal analytical positions in spectrophotometric analysis of multicomponent systems, Anal. Chem., 46 (1974) 1676. P.C. Thijssen, G. Kateman and H.C. Smit, Optimal designs with information theory in least squares problems, Anal. Chim. Acta, 157 (1984) 99. S. Ebel. E. Glaser. S. Abdulla. U.Steffens and V. Walter, Optimization of wavelengths in spectrophotometric multicomponent analysis, Z. Anal. Chem., 313 (1982) 24.