Anal. Chem. 1996, 68, 982-990
Standardization of Near-Infrared Spectrometric Instruments E. Bouveresse, C. Hartmann, and D. L. Massart*
ChemoAC, Pharmaceutical Institute, Vrije Universiteit Brussel, Laarbeeklaan 103, B-1090 Brussels, Belgium I. R. Last and K. A. Prebble
Analytical Development Laboratories, The Wellcome Foundation Ltd., Temple Hill, Dartford, Kent DA1 5AH, U.K.
In this paper, two different approaches are studied for the standardization of near-infrared spectrometric instruments. The first is a simple method based on a univariate slope/bias correction of the predicted values. The second approach proposed is the much more sophisticated piecewise direct standardization (PDS) based on a multivariate correction of spectra. Both standardization methods are applied to three different data sets, and the results obtained are compared. In certain cases, the simple slope/bias correction approach yields results at least as good as those obtained by the PDS procedure. In other cases, the complex PDS procedure is required to obtain acceptable results. A diagnostic tool is developed in order to decide whether the simple slope/bias correction approach can be applied instead of PDS. Calibration Transfer. Near-infrared (near-IR) spectroscopy has become a rapid and powerful analytical technique and is now used in many industrial applications.1-4 However, calibration models developed on one near-IR instrument can only be used for the prediction of new samples, if the instrumental response is identical to the one given by the instrument when the calibration samples were measured. If the measurements of new samples are performed on another near-IR instrument or on the same nearIR instrument after a certain time, the instrumental response is usually different from the one obtained when the calibration samples were measured, and this leads to erroneous predictions. In order to correct those differences in the instrumental responses, standardization of near-IR instruments has become a necessary step, and different standardization procedures have been proposed.5 Univariate Correction of the Predicted Values. One of the first standardization approaches was proposed by Osborne and Fearn6 in order to transfer the calibration equations built for the determination of protein and moisture in flour. This approach is based on a set of 20 samples that are measured on nine different (1) Hildrum, K. I.; Isaksson, T.; Naes, T., Tandberg, A. Near infra-red spectroscopy: bridging the gap between data analysis and near-IR applications; Ellis Horwood: Chichester, England, 1992. (2) Osborne, B. G.; Fearn, T.; Hindle, P. H. Practical near-IR spectroscopy, 2nd ed.; Longman Scientific and Technical: Essex, England, 1993. (3) MacDonald, B. F.; Prebble, K. A. J. Pharm. Biomed. Anal. 1993, 11, 10771085. (4) Jones, J. A.; Last, I. R.; MacDonald, B. F.; Prebble, K. A. J. Pharm. Biomed. Anal. 1993, 11, 1227-1231. (5) De Noord, O. E. Chemom. Intell. Lab. Syst. 1994, 25, 85-97. (6) Osborne, B. G.; Fearn, T. J. Food Technol. 1983, 18, 453-460.
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instruments, and a bias correction factor (bias referring to a constant systematic error) was computed in order to correct the prediction obtained when spectra obtained on the “slave” instruments were used with univariate or simple multilinear regression (MLR) models built on the “master” instrument. A similar approach was used by Jones et al.4 in order to transfer the calibration equations developed for the moisture determination in freeze-dried injection products. Two independent data sets were measured on two different near-IR instruments. Each data set was split into calibration and test sets, and independent calibration models were built on each instrument. Spectra of the calibration set measured on the first instrument were predicted with the calibration model developed on the second one, and a univariate linear model was computed in order to correct those predicted values. This correction method is called the slope/bias correction. Spectra from the test set measured on the first instrument were then predicted with the calibration model developed on the second instrument and the values obtained were corrected by the linear model. The same slope/bias correction of the predicted values was performed from the second instrument to the first one: good predictions of the test set samples were reported in both cases. However, those simple univariate corrections of the predicted values were successfully performed only with identical instruments. If more dissimilar instruments are used, the differences between their instrumental responses will be more complex, and it will usually not be possible to obtain satisfactory corrections of the erroneous concentration values predicted with nonstandardized spectra by simply fitting a straight line. Moreover, De Noord5 does not recommend this approach when the calibration models used are more complex than univariate or simple MLR models. Multivariate Calibration Standardization. To cope with those complex situations, different standardization methods have been described in the literature.5 Among those standardization methods, Wang et al.7 proposed a multivariate approach called “piecewise direct standardization” (PDS). This multivariate approach transfers the data obtained from the slave instrument to the master one, the transfer parameters being computed in a multivariate way. This method has been compared with three other multivariate standardization methods and with the univariate full-spectrum correction method proposed by Shenk and Westerhaus.8,9 The best results were obtained with PDS. Moreover, this method has already been applied to different standardization (7) Wang, Y.; Veltkamp, D. J.; Kowalski, B. R. Anal. Chem. 1991, 63, 27502756. (8) Shenk, J. S.; Westerhaus, M. O. Crop Sci. 1991, 31, 1694-1696. 0003-2700/96/0368-0982$12.00/0
© 1996 American Chemical Society
problems, such as calibration transfer across near-IR instruments of similar7 and different10 qualities in terms of resolution and signal to noise ratio, and to temperature-compensating calibration transfer,11 and good results were reported. However, one important step of PDS, namely, the determination of the local optimal number of principal components, can yield artifacts in the transferred spectra, if this step is not carefully performed.12,13 Another important point for standardization methods that correct spectral differences is the number and the representativity of the standardization samples used to estimate the transfer parameters. Strategy for the Selection of Standardization Method. If the two near-IR instruments involved are very different (e.g., different optical systems, different signal to noise ratios) or used in different conditions (e.g., different temperatures), the differences between the spectra of the same sample measured on both instruments will be large and complex (e.g., nonconstant wavelength shift, large differences in spectral intensities, different types and levels of noise). In this case, PDS or other sophisticated methods have to be applied to compute the standardization parameters, because no satisfactory standardization will be obtained by applying the slope/bias correction approach. However, if the two near-IR instruments are rather similar, and used in identical conditions, the differences between the spectral responses given by the two instruments can be much simpler (e.g., no significant wavelength shift, constant offset in spectral intensities, similar noise levels). In this case, satisfactory results can be obtained by the simple and rapid slope/bias correction approach, and it may be superfluous to apply the more sophisticated PDS approach which requires the computation of many multivariate regression models. An advantage of the slope/bias correction method is that it requires only basic statitical knowledge which can therefore be applied by nonspecialized people. In this work, both PDS and slope/bias correction approaches are applied to three different data sets, in order to verify whether the slope/bias correction approach can be considered as a much simpler method to be tested before the more sophisticated PDS is applied. A diagnostic is proposed, which helps the user to decide whether the slope/bias correction approach can be applied or not. THEORY Notations. For each spectral matrix A, Aji corresponds to the spectral intensity of the jth sample at the ith wavelength. Aj. represents the jth row of the A matrix and corresponds to the spectrum of the jth sample. Ai represents the ith column of the A matrix and corresponds to the spectral intensities of all samples at the ith wavelength. In order to indicate on which instrument the spectra were collected, a superscript letter is added in front of the matrix: mA contains the spectra collected on the master instrument, and sA contains the spectra of the same samples collected on the slave instrument. Calibration models are built on the master instrument with Nc calibration samples. The spectra of those Nc calibration (9) Bouveresse, E.; Massart, D. L.; Dardenne, P. Anal. Chim. Acta 1994, 297, 405-416. (10) Wang, Y.; Lysaght, M. J.; Kowalski, B. R. Anal. Chem. 1992, 64, 562-564. (11) Wang, Y.; Kowalski, B. R. Anal. Chem. 1993, 65, 1301-1303. (12) De Noord, O. E. Poster presented at the International Chemometrics Research Meeting, Veldhoven, The Netherlands, July 3-7, 1994. (13) Bouveresse, E.; Massart, D. L. Chemom. Intell. Lab. Syst., in press.
samples are collected on the master instrument, yielding mC. For each calibration sample, y-values are determined by a reference method yielding y. Regression coefficients relating mC and y are then computed yielding b. Ns new samples are measured on the slave instrument yielding sX. In order to determine the transfer parameters, Nt standardization samples are measured on both instruments, yielding mT and sT. Predictions of y-values can be obtained by multiplying any spectral matrix by the regression coefficients b. For instance, the predicted y-values obtained with the spectra collected on the slave instrument are given by s
y( X) ) sX × b
(1)
Subset Selection. Since no independent standardization samples were available, the standardization samples were selected among samples already measured. The subset selection approach usually presented in the literature consists of selecting a subset among the calibration samples collected on the master instrument. m
C f subset selection f mT
(2)
Those samples are then remeasured on the slave instrument, yielding sT. However, it is sometimes also possible to follow a second approach, which consists of selecting a few representative samples among those collected on the slave instrument. s
X f subset selection f sT
(3)
Those samples are then remeasured on the master instrument, yielding mT. The choice between those two approaches is usually based on practical considerations.13 Subset Selection Method. The original selection method proposed by Wang et al.7 consists of choosing the samples with high leverages. However, the Kennard and Stone14 algorithm will be used as subset selection method in this study, because it allows the selection of a more representative subset.12-13 To allow a better understanding, Figure 1 gives a graphical description of this algorithm applied to a simulated data set containing 27 objects and 2 variables. This algorithm starts by selecting the two spectra that are the farthest from each other (see Figure 1a). Then, steps 1-3 are repeated until the desired number of spectra has been included in the subset. Panels b-d of Figure 1 give graphical descriptions of steps 1-3 for the first iteration. Step 1: For each spectrum i not selected in the subset, the Euclidean distances d(k,i) between the considered spectrum and each spectrum k already selected in the subset are computed (see Figure 1b). It should be noted that those distances are computed from the raw spectra. Step 2: For each spectrum i not selected in the subset, the smallest Euclidean distance computed between the considered spectrum and the spectra already selected in the subset is found (see Figure 1c). This smallest distance is referred to as δ(i). Step 3: The nonselected spectrum i that has the highest δ(i) distance is found and selected in the subset. (14) Kennard, R. W.; Stone, L. A. Technometrics 1969, 11, 137-149.
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suggest computing this model by orthogonal least squares.15-18 Under the assumption of equal measurement variances on master and slave instruments, the bias and the slope of this linear model are given by
slope )
Syy - Sxx + x(Syy - Sxx)2 + 4(Sxy)2 2Sxy
bias ) jy(
mT)
s
- slope × jy( T)
(6) (7)
with
Figure 1. Graphical description of the Kennard and Stone algorithm: (a) selection of the first two points; (b-d) steps 1-3 (first iteration).
Since the PDS method corrects differences between spectra (i.e., in the X-space), the Kennard and Stone algorithm provides a good and representative subset for the PDS procedure. However, this algorithm probably does not provide the best subset for the slope/bias correction approach, which performs a correction of the predicted values (i.e., in the Y-space). It is theoretically possible to use another subset selection method based on the concentration values, but the subset obtained will be different from the one selected by the Kennard and Stone algorithm for the PDS approach. The application of a different subset selection method for each standardization method increases the total number of samples to be remeasured on the master instrument, which makes the standardization less attractive. For this practical reason, it is preferable to apply a single subset selection method, and we decided to apply the original Kennard and Stone algorithm. Slope/Bias Correction of the Predicted Values. The spectra of the subset samples collected on both instruments (master and slave) given in mT and sT are multiplied by the regression coefficients b obtained on the master instrument. (mT)
m
) T×b
(4)
y( T) ) sT × b
(5)
y
s
where y(mT) are the concentration values of the Nt subset samples determined by using both spectra and calibration model built on the master instrument, and y(sT) are the concentration values of the same subset samples determined by using spectra obtained on the slave instrument, without taking the different instrumental responses of the two instruments into account. The predicted values of y(mT) are plotted against the ones of s y( T), and a univariate linear model fitting those Nt points is computed. Osborne et al.6 and Jones et al.4 computed this univariate model by ordinary least squares (OLS). However, we 984
Analytical Chemistry, Vol. 68, No. 6, March 15, 1996
Sxy )
Sxx )
∑(y
Syy )
∑(y
∑(y
(sT)
(sT)
- jy( T))2
j
- jy(
(mT)
s
j
s
j
- jy( T))(y(
(8)
mT)
)2
mT)
j
- jy(
(9)
mT)
)
(10)
jy(sT) and jy(mT) being the means of the Nt predicted values obtained by sT and mT. It is important to note that the regression line obtained by OLS is based on the assumption that one variable is measured or known without error, which here of course is not correct. Ignoring the error in y(sT), which is as y(mT) subject to measurement and prediction errors, can seriously affect the regression coefficients as demonstrated in a Monte Carlo study,17 the slope being underestimated and the intercept being overestimated. It is more sensible to assume that the measurements on both instruments, y(mT) and y(sT), are affected by the same magnitude of measurement error. Then, a simplified version of Mandel’s method18 can be calculated. This orthogonal regression line of formulas 6 and 7 corresponds to the line of the first principal component after column mean-centering the data. The combination of the regression coefficients b and slope/ bias correction terms enables us to compute directly the corrected concentration values y(sX)std from the spectra collected on the slave instrument. s
y( X)std ) bias + slope × (sX × b)
(11)
Piecewise Direct Standardization. The PDS method was presented as an improved version of the direct standardization method.7 The PDS method relates the spectral intensities obtained at the ith wavelength on the master instrument to those contained in a moving spectral window containing a few neighboring wavelengths on the slave instrument. For each wavelength of the master instrument, a multivariate model is then built between the spectral intensity obtained at the ith wavelength on the master instrument and the corresponding spectral window, (15) Adcock, R. J. Analyst 1878, 5, 53-54. (16) Fuller, W. A. Measurement Error Models; John Wiley and Sons: New York, 1987. (17) Hartmann, C.; Smeyers-Verbeke, J.; Massart, D. L. Analusis 1993, 21, 125132. (18) Mandel, J. J. Qual. Technol. 1984, 1, 1-14.
and the regression coefficients are placed in a banded diagonal transfer matrix F. In order to transfer the spectra from the complete data set, the matrix sX is multiplied by the transfer matrix F, yielding sXstd. s
Xstd ) sX × F
(12)
Prediction can then be carried out by multiplying those transferred spectra by the regression coefficients b built on the master instrument. The combination of the transfer matrix F and the regression coefficients b enables us to compute directly the corrected concentration values ystd from the spectra collected on the slave instrument. s
y( X)std ) (sX × F) × b
s2(master), this means that the differences between instrumental responses can be corrected by a simple slope/bias correction. However, if the residual variance s2(slave) is much larger than the residual variance s2(master), this means that the simple slope/bias correction method is not sufficient to correct the complex differences between instrumental responses. In this case, the more sophisticated PDS has to be applied. In order to decide whether the slope/bias correction can be applied, those two variances have to be compared by means of an F-test.
F)
s2(slave) s2(master)
∑(y ) ∑(y
(sT)
(mT)
- [s0 + s1 × ysub])2
- [m0 + m1 × ysub])2
(16)
(13) Since the variance s2(slave) can not be significantly smaller than s2(master), the hypothesis test is formulated as follows:
Recently, it was shown that the results obtained with the PDS algorithm were improved by performing an additive background correction of the spectra.19 This additive background correction consists of removing baseline differences by column meancentering the spectra of the standardization subset before computing the transfer matrix. This elimination of baseline differences between the instruments is similar to a bias correction. Details concerning the PDS algorithm and the additive background correction can be found respectively in refs 7 and 17. It has to be noted that one of the main drawbacks of the PDS algorithm is that there is no guidance for the determination of the required number of samples to be included in the standardization subset. The determination of this number is rather important for the PDS algorithm, because the selected samples have to be representative of the calibration set for all local spectral windows. However, this parameter is not so crucial for the slope/bias correction method, because this method is based on a simple univariate linear model. If two samples are theoretically enough to estimate this univariate linear model, we advise one to use more standardization samples in order to get a more stable estimation of the univariate parameters. F-Test To Decide Whether the Slope/Bias Correction Should Be Applied. After having collected the spectra of the Nt standardization samples on both instruments, predictions are computed with eqs 4 and 5. The predicted values y(sT) obtained with the Nt spectra collected on the slave instrument are plotted against the corresponding y-reference values ysub, and a line is fitted to those Nt points by OLS, yielding intercept s0 and slope s1. The predicted values y(mT) are also plotted against the corresponding y-reference values ysub, and a line is fitted to those Nt points by OLS, yielding intercept m0 and slope m1. Then, the residual variances s2(master) and s2(slave) are computed. 2
s
s
(slave)
2 (master)
∑(y )
∑(y )
(sT)
(mT)
- [s0 + s1 × ysub])2 Nt-2
(14)
- [m0 + m1 × ysub])2 Nt-2
(15)
If the residual variance s2(slave) is similar to the residual variance (19) Wang, Z.; Dean, T.; Kowalski, B. R. Anal. Chem. 1995, 67, 2379-2385.
H0: s2(slave) e s2(master) (the slope/bias correction is adequate) f the slope/bias correction should be applied (17) H1: s2(slave) > s2(master) (the slope/bias correction is not adequate) f the slope/bias correction should not be applied (18) The choice of the level of significance R is based on practical considerations. A level of significance corresponds to the probability of incorrectly rejecting the null hypothesis when it is true. Usually, R is set equal to 1% or 5%. Since the PDS is a complex procedure, the simple and fast slope/bias correction method should be applied when possible. If the results obtained with the slope/bias correction are considered good enough, time and effort are saved, because there is no need to apply the sophisticated PDS. If the results are not considered good enough, very little time is lost, and the PDS can then be applied to the data. Therefore, a small level of significance has to be chosen: R is then set equal to 1%. Nt-2 being the degree of freedom of both residual variances, computed F-values will then be compared to the critical values of a one-sided F(0.99,Nt-2,Nt-2)-test in order to decide whether the slope/ bias correction can be applied or not. EXPERIMENTAL SECTION Near-IR Data Sets. Three different data sets were used for this study: The first data set contains 60 spectra of soy seed samples collected on two different near-IR instruments. The samples were scanned in the reflectance mode from 1100 to 2500 nm with a step of 2 nm for the first instrument and 4 nm for the second one. In order to have the same number of wavelengths, only one of every two wavelengths was used for the first instrument. Spectra were the average of five replicates. The near-IR spectra of those 60 samples collected on both instruments are given in Figure 2a, and the differences between the spectra collected on the two instruments are given in Figure 2b. For all samples, three variables (moisture, proteins, and oil) were determined by reference methods from wet chemistry. Details about the reference methods used can be found in the literature.20 Analytical Chemistry, Vol. 68, No. 6, March 15, 1996
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Figure 2. (a) Spectra of 60 soy samples scanned on two near-IR instruments and (b) spectral differences between the two instruments.
Figure 3. (a) Spectra of 30 pseudo-gasoline samples scanned on two near-IR instruments and (b) spectral differences between the two instruments.
The second data set was taken from Wise’s PLS Toolbox21 for MATLAB, where it is used for the demonstration of the PDS routine. This data set contains 30 pseudo-gasoline samples that were scanned in the transmission mode from 800 to 1600 nm with a step of 2 nm on two different near-IR instruments. The near-IR spectra of those 30 samples collected on both instruments are given in Figure 3a, and the differences between the spectra collected on the two instruments are given in Figure 3b. Five variables corresponding to the concentration of five different compounds present in the samples were used to determine the predictive ability of each standardization approach. No additional information about the handling of samples is available. The third data set contains spectra of 39 batches of a tablet product which were scanned in reflectance mode from 1100 to (20) Forina, M.; Drava, G.; Armanino, C.; Boggia, R.; Lanteri, S.; Leardi, R.; Corti, P.; Conti, P.; Giangiacomo, R.; Galliena, C.; Bigoni, R.; Quartari, I.; Serra, C.; Ferri, D.; Leoni, O.; Lazzeri, L. Chemom. Intell. Lab. Syst. 1995, 27, 189-203. (21) Wise, B. M. PLS Toolbox for Use with MATLAB, version 1.4, Eigenvector Technologies, West Richland, WA.
986 Analytical Chemistry, Vol. 68, No. 6, March 15, 1996
Figure 4. (a) Spectra of 38 pharmaceutical tablets scanned on two near-IR instruments and (b) spectral differences between the two instruments.
2500 nm with a step of 2 nm on two identical near-IR instruments. This data set contains 26 samples from production batches and 13 synthetic samples added to extend the calibration range. The samples were handled in the same way on both instruments. Duplicate spectra of each sample were collected in a randomly determined order, each spectrum being the average of 32 scans taken automatically by the instrument. The Cochran test22 was used for the detection of outliers in replicate measurements, and one sample was identified as outlier. After having visually checked that the corresponding duplicate spectra were indeed abnormal, this sample was removed from both data sets. The two spectra of each of the remaining 38 samples were then averaged. The two data matrices containing the near-IR average spectra of the 38 samples scanned on each instrument were used in this study. The near-IR spectra of those 38 samples collected on both instruments are given in Figure 4a, and the differences between the spectra collected on the two instruments are given in Figure 4b. For all samples, the concentration of the active compound was determined by a well-established UV method as reference method. This concentration was used in order to determine the predictive ability of each standardization approach. Software. All computations were performed in MATLAB (The MathWorks, Natick, MA) with programs developed by us and with programs from Wise’s PLS Toolbox.21 Calibration Models. In practice, calibrations are developed on the master instrument with a high number of samples. Since measurements of additional samples were not available on the master instrument in our case, the near-IR spectra of the whole data set obtained on the master instrument and the corresponding concentration reference values were used to compute the regression coefficients. The different calibration models developed use the whole wavelength range, and the calibration coefficients were determined by partial least squares (PLS). The optimal number of latent factors was determined in all cases by the repeated evaluation set (RES)23 method. Statistics about the different calibration models developed are given in Table 1. (22) ISO Standards Handbook, 3rd ed.; (Statistical Methods) International Organization for Standardization: Geneva, 1989. (23) Forina, M.; Drava, G.; Boggia, R.; Lanteri, S.; Conti, P. Anal. Chim. Acta 1994, 295, 109-118.
Table 1. Calibration Model Built on the Master Instrument, for the Determination of the Active Compound (AC) in the Tablets, the Three Variables (V1-V3) in the Soy Seeds, and the Five Compounds (C1-C5) in the Pseudo-Gasoline Samples
AC V1 V2 V3 C1 C2 C3 C4 C5 a
concn range (%)
RMSEP (CV)a
no. of PCs used
34.7-42.5 5.9-18.4 29.0-43.4 14.7-22.9 9.8-29.9 9.9-29.9 29.9-50.0 3.9-15.3 0.9-32.9
0.61 1.05 1.36 1.24 0.63 0.15 0.08 0.15 0.56
6 7 7 5 6 8 8 8 6
CV, cross-validation.
Standardization. The two approaches proposed are applied on both data sets. Selection of subsets containing different numbers of selected samples is performed with the Kennard and Stone algorithm.14 Concerning the PDS method, each local regression is performed by PCR with or without additive background correction for all data sets. The positions of the spectra from the subset collected on both instruments are carefully compared with the wavelength index correction step of the algorithm proposed by Shenk and Westerhaus8,9 in order to detect wavelength shifts. Since no significant wavelength shift was detected, the width of the spectral window was then set equal to three channels. This corresponds to a 8 nm window for the first data set and to a 4 nm window for the second and third data sets. For both approaches, the transfer parameters are computed, and the root mean square errors of prediction (RMSEP) are computed with the calibration models built on the master instrument and with the spectra collected on the slave before and after standardization.
Figure 5. Percentage of the concentration range covered by the subset samples as a function of the number of samples included in the subset. Those percentages are computed for the active compound (+) of the third data set, for the three y-variables of the first data set (O), and for the five compounds of the second data set (f).
x∑(y - y( X))2 )
(19)
x∑(y - y( X)std)2
Figure 6. Score plot PC1-PC2 of the third data set (tablets). Production samples are represented by circles and synthetic samples by crosses. The standardization samples selected by the Kennard and Stone algorithm are numbered in the order they were selected.
(20)
data set, the concentration range of the active compound (AC) is well described by the subset samples if at least four samples are selected in the subset. This is due to the heterogeneity of the third data set, which can clearly be seen on the PC1-PC2 score plot shown in Figure 6. From the first three standardization samples selected by the Kennard and Stone algorithm, only one is a synthetic sample with a high concentration. In order to cover the concentration range well, it is necessary to select another synthetic sample with a low concentration. This is done by selecting a fourth standardization sample (see Figure 6). The fifth and sixth standardization samples belonging to the production samples, the percentage of the concentration range covered by the subset samples remains constant. Comparison of the Two Standardization Approaches. The predicted concentration values were computed with spectra collected on the master instrument and on the slave instrument before and after standardization with the slope/bias correction approach. Figures 7-9 show the predicted concentration values
s
RMSEP(before)
xNs
s
RMSEP(after) )
xNs
RESULTS AND DISCUSSION Subset Selection. For all data sets, the location of the selected subset samples in both spectral and concentration spaces is studied. As expected, the points selected by the Kennard and Stone algorithm cover the whole spectral domain well. Concerning the location of the selected samples in the concentration space, Figure 5 gives the percentage of the concentration range covered by the subset samples, when different numbers of samples were selected in the standardization subset. For the first data set, the range of the three studied variables (V1-V3) is well described by the subset samples if at least five samples are selected in the subset. For the second data set, Figure 5 indicates that the subset samples cover well the concentration ranges of all compounds (C1-C5), except perhaps for the fifth compound. For the third
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Figure 7. Predicted concentration values versus reference concentration values plotted for the first variable V1 of the first data set. The prediction were computed with the 60 spectra collected on the master and with the 60 spectra collected on the slave before and after standardization computed with six subset samples and with the slope/bias correction approach.
Figure 9. Predicted concentration values versus reference concentration values plotted for the third data set. The prediction were computed with the 38 spectra collected on the master and with the 38 spectra collected on the slave before and after standardization computed with six subset samples and with the slope/bias correction approach. Table 2. RMSEP (in %) for the Concentration of the Active Compound (AC) in the Tablets, the Three Variables (V1-V3) in the Soy Seeds, and the Five Compounds (C1-C5) in the Pseudo-Gasoline Samplesa variables
RMSEP (CV)
before
3
4
5
6
AC V1 V2 V3 C1 C2 C3 C4 C5
0.61 1.05 1.36 1.24 0.63 0.15 0.08 0.15 0.56
2.05 11.3 15.4 3.07 6.19 2.63 2.02 4.00 3.48
0.73 1.25 1.42 1.41 3.17 0.79 1.44 1.80 2.53
0.67 1.33 1.37 1.38 2.90 0.52 1.26 1.78 2.85
0.60 1.32 1.33 1.38 2.99 0.52 1.20 1.80 2.92
0.61 1.32 1.27 1.37 2.61 0.48 1.20 1.82 3.14
a RMSEP are computed with spectra collected on the slave instrument before standardization and after standardization. Standardizations using different numbers of subset samples are computed with the slope/bias correction approach. The RMSEP of cross-validation (CV) obtained on the master instrument is given for comparison.
Figure 8. Predicted concentration values versus reference concentration values plotted for the first variable C1 of the second data set. The prediction were computed with the 30 spectra collected on the master and with the 30 spectra collected on the slave before and after standardization computed with six subset samples and with the slope/bias correction approach.
vs the reference concentration values. Concerning the first and third data sets, it can be observed on Figures 7 and 9 that the predictions obtained with the spectra collected on the slave seem to be simply shifted when they are compared to those obtained with the spectra collected on the master instrument. By using the slope/bias correction approach, it is possible to eliminate this shift and to obtain predictions after standardization that seem similar to those obtained with the spectra collected on the “master”. However, for the first compound (C1) of the second data set, Figure 8 clearly shows that the predictions obtained with the spectra collected on the slave after standardization with the slope/bias correction approach are not as good as those obtained with the spectra collected on the master instrument. 988 Analytical Chemistry, Vol. 68, No. 6, March 15, 1996
The RMSEP values obtained by the slope/bias correction approach for different numbers of standardization samples are summarized in Table 2, and those obtained by the PDS approach with or without additive background correction are summarized in Table 3. For the first data set, the slope/bias correction method and the PDS method with the additive background correction lead to similar RMSEP values. Whatever the subset size, the results obtained with PDS without additive background correction are much worse. This confirms the information given by Figure 7, and this clearly indicates that the slope and bias correction method can lead to results as good as those obtained with PDS with additive background correction. For the second data set, the best RMSEP values are obtained with PDS with additive background correction, whatever the subset size. Those results are better than those obtained without additive background correction, which is in accordance with the results obtained by Wang et al.19 Moreover, they are better than those obtained by the slope/bias correction approach, as shown in Figure 8. For the third data
Table 3. RMSEP (in %) for the Concentration of the Active Compound (AC) in the Tablets, the Three Variables (V1-V3) in the Soy Seeds, and Five Compounds (C1-C5) in the Pseudo-Gasoline Samplesa variables
RMSEP (CV)
AC V1 V2 V3 C1 C2 C3 C4 C5
0.61 1.05 1.36 1.24 0.63 0.15 0.08 0.15 0.56
AC V1 V2 V3 C1 C2 C3 C4 C5
0.61 1.05 1.36 1.24 0.63 0.15 0.08 0.15 0.56
before
3
4
5
6
With ABC 2.05 0.76 11.3 1.17 15.4 1.56 3.07 1.38 6.19 1.17 2.63 0.75 2.02 0.16 4.00 0.37 3.48 0.78
0.72 1.16 1.51 1.35 1.08 0.38 0.17 0.25 0.85
0.75 1.13 1.43 1.35 0.98 0.35 0.16 0.25 0.74
0.86 1.13 1.45 1.34 0.81 0.33 0.12 0.21 0.64
No ABC 2.05 0.80 11.3 1.71 15.4 2.81 3.07 1.59 6.19 1.72 2.63 0.90 2.02 0.64 4.00 1.40 3.48 2.04
0.66 1.48 2.70 1.64 1.96 1.03 0.62 1.43 2.17
0.66 1.46 2.64 1.61 1.88 1.09 0.54 1.27 1.96
0.65 1.45 2.65 1.59 1.20 0.71 0.45 1.04 1.26
a RMSEP are computed with spectra collected on the slave instrument before standardization and after standardization. PDS with or without additive background correction (ABC) was used as standardization method. Standardizations using different numbers of subset samples are computed with PDS. The RMSEP of cross-validation (CV) obtained on the master instrument is given for comparison.
set, the results obtained by the three standardization methods are very similar to each other. However, the results obtained by the slope/bias correction are slightly better than the results obtained by PDS with additive background correction. Since only one variable is studied, it is difficult to know whether this difference is significant or not. However, it is possible that the results obtained by PDS could have been influenced by the strong heterogeneity of this data set due to the presence of synthetic and production samples. The PDS algorithm corrects the differences between two subsets of standardization samples. If the data are heterogeneous in the X-space, problems can occur with the PDS method, but not with the slope/bias correction method, which is based on the correction of the predicted values (Y-space), and which is therefore not influenced by heterogeneity in the X-space. This can perhaps be an additional advantage of the slope/bias correction approach.
F-Test. The experimental F-values are computed for each variable of each data set and are compared to the critical F(0.99,Nt-2,Nt-2)-values. The results are presented in Table 4. For the first data set, the computed F-values are systematically smaller than the critical F(0.99,Nt-2,Nt-2)-values, whatever the subset size. According to the F-test, satisfactory results should be obtained with the slope/bias correction. Indeed, the results obtained in Tables 2 and 3 confirm that the decision was right. For the second data set, the computed F-values are much larger than the critical F(0.99,Nt-2,Nt-2)-values. According to the F-test, the application of the slope/bias correction method to this standardization problem will not yield acceptable results. This decision is clearly confirmed by the results presented in tables 2 and 3. F-Values smaller than the critical values are only obtained for low numbers of standardization samples. Those anormal values can be explained by the fact that regression lines are determined with three or four points only. Including more standardization samples in the subset leads to more reliable results. For the third data set, the computed F-values are systematically smaller than the critical F(0.99,Nt-2,Nt-2)-values. As for the first data set, the slope/bias correction method seems to be the most suitable for this standardization problem. However, the computed F-value is higher than the critical F(0.99,Nt-2,Nt-2)-value when five standardization samples are included in the subset. This is because the addition of this fifth point does not modify the regression coefficients m0 and m1 given by the first four points but modifies quite strongly the regression coefficients s0 and s1 given by the first four points. The value of s2(master) remaining unchanged, the increase of s2(slave) leads to an anormal increase of the F-value. However, the results obtained with more standardization samples selected in the subset indicate that this is not due to a real difference between the variances but only to a particular case. CONCLUSION The slope/bias correction method is a simpler standardization procedure than the piecewise direct standardization. Since this slope/bias correction is a univariate approach, the results cannot be damaged by artifacts such as those coming from bad local rank determination. Moreover, this approach is not influenced by problems occuring in the spectral space, such as data heterogeneity, because it is based on an univariate linear correction of predicted values. However, this simple approach can only be used for instrument standardization when the differences between the instrumental
Table 4. Comparison of the Experimental F-Values Obtained with the Critical Values Given in Statistical Tables for a One-Sided F-Test with r ) 1%a std samples (Nt)
3
4
5
6
7
8
9
10
F(0.99,Nt-2,Nt-2) AC V1 V2 V3 C1 C2 C3 C4 C5
4052 1.58 0.72 7.89 0.28 55.7 88500 4500 14500 44.8
99.0 19.9 11.89 7.57 1.75 138 59200 1270 685 44.7
29.5 45.6 1.42 6.03 0.90 145 3830 1000 563 44.1
16.0 6.81 1.15 7.42 0.82 135 115 976 487 23.9
11.0 5.23 0.53 3.63 0.84 130 139 560 829 32.0
8.47 4.18 0.63 3.53 0.86 101 153 539 648 27.3
6.99 3.36 0.64 3.45 0.78 100 138 450 673 35.5
6.03 2.48 0.66 2.04 0.78 80.4 127 472 676 38.1
a
Boldface values indicate cases where the experimental F-value is smaller than the critical value.
Analytical Chemistry, Vol. 68, No. 6, March 15, 1996
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responses are rather simple. When more complex differences occur, this simple slope/bias correction cannot yield acceptable results anymore. Therefore, it is important to be able to decide whether the slope/bias correction method can be used or not. This can be achieved by applying the diagnostic tool proposed. However, this diagnostic can sometimes yield erroneous conclusions, when very few standardization samples are included in the subset. Therefore, we recommend use of at least five standardization samples, in order to obtain more reliable results. The proposed procedure allows a fast and simple selection of the most appropriate standardization approach and, therefore, the most reliable results in cases where instrument standardization is necessary.
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Analytical Chemistry, Vol. 68, No. 6, March 15, 1996
ACKNOWLEDGMENT The authors acknowledge Prof. M. Forina (University of Genoa, Italy) for providing the soy seed data set, and Sheelagh Halsey (Perstorp Analytical/NIRS System) for near-IR measurements.
Received for review October 24, 1995. Accepted January 3, 1996.X AC9510595
X
Abstract published in Advance ACS Abstracts, February 1, 1996.
