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Anal. Chem. 1086, 58,2814-2819
Comparative Study of Calibration Methods for Near-infrared Reflectance Analysis Using a Nested Experimental Design Howard Mark
Technicon Instrument Corporation, Industrial Systems Division, 511 Benedict Avenue, Tarrytown, New York 10591
Prlnclpal component callbration Is coming to be used as a callbratlon method for near-infrared reflectance anaiysls spectroscopy. We compared the performance of this anaanalysis on lytkal tedmlque w#h standard ntulUpb regrthe optlcal data, udng a multllevel nested statlstlcally designed experlment. Calibration uslng prlnclpal components has approximately the same analytical performance as the current CallbraUOn method, but neverthelees has advantages, whlch IncMe clrcumvenlng the need for complex variable search procedures w h k generatlng callbratkns that can be keyed directly Into commerclai near-Infrared Instruments In current use. The use of statWicaHy deslgned experiments permlis statlstlcally signlficant comparisons to be made, In contrast wlth comparisons made only on the bask of standard error of callbratlon or standard error of predlction, where differences are masked by the reference laboratory error.
Principal component analysis is a popular and widely used mathematical/statistical/chemometrictechnique for reducing the dimensionality of data (1,2). It has come to be used in several different areas of chemistry (3)and spectroscopy ( 4 4 , including the subset of analytical methods called near-infrared reflectance spectroscopy (7-1 1). The traditional method used for calibrating near-infrared reflectance spectrometers since the inception of the technique has been multiple regression analysis (12). Since that time there has been great interest in determining methods of improving the predictive capability of the technology. Comparisons of wavelength sets (13), data transformations (14), and different methods of calibration (15) have all been performed in order to determine the optimum conditions for performing near-infrared analysis. The usual method of performing such comparisons is to calculate the standard error of estimate (SEE) for the calibration and/or the standard error of prediction (SEP) from data not in the calibration set. The science of statistics allows a scientist to determine objectively whether a given result is sufficiently above the noise level to state that the observed result is, in fact, due to the phenomenon under study. However, statistical techniques contain many traps for the unwary. With near-infrared comparisons, one such trap is often encountered. The difficulty is associated with the fundamental assumption of regression analysis that all error is in the dependent (Y) variable. Indeed, it is usually found that the SEE and SEP from near-infrared measurements are close to the error of the reference laboratory measurments. Thus, this error is the dominating error source of the analysis, and when comparative studies are performed, all values of standard error are similar, since they are all dominated by this single error source. In many cases the differences found are not statistically significant. One example will suffice to demonstrate this. If we examine the sugar results in Table IV of McClure (14), we find that the value of F calculated between the largest SEP for sugar (1.047) and the smallest (0.936) is (1.047/0.936)2 = 1.251. However, since selection of the largest and smallest values is
a nonrandom method of comparison, the critical values selected in the usual manner from tables do not apply. Rather, since there are four values of SEP in McClure’s Table N,there are six possible pairwise comparisons, and we must use the considerations for multiple comparisons discussed previously (16). Thus, a proper F test must be made at a probability level of the sixth root of 0.95, which equals 0.991. The prediction set contained 100 samples, therefore we can find the critical F in standard tables: F(crit) is 1.53. Thus no significant differences exist for these four data transformations despite the use of an apparently large number of samples. Because of this aforementioned difficulty, in many comparative studies that have been performed the data were analyzed under the assumption that, since the variance due to the reference laboratory was constant, small differences in results represented real effects due to the phenomenon under study. Under less difficult conditions, ignoring the question of statistical significance of results in this manner would not be countenanced. Thus, as with the sugar results, lack of statistical significance is so common that scientists cease looking, which leads to errors in interpretation. This lack of statistical significance is a highly unsatisfactory state of affairs. Even though the reference laboratory results have a variance that is constant, in the absence of demonstrable statistically significant differences in the calculations, real differences due to the phenomenon under study cannot be distinguished from random variability of any other error sources affecting the data, such as noise or the effect of repack. To circumvent this difficulty it is necessary to make measurements of the effect of the phenomenon under study that exclude the reference laboratory error from the data for a sample, then statistical significance can be shown when real effects are present. In order to achieve this goal, a suitable experimental design must be used. Statistical experimental designs have recently been introduced into near-infrared calibration studies (10, 17). The experimental design used in the present study is suitable for comparing different data treatments because it allows calculation of several components of variance; then each component of variance can be compared between methods. Hrushka and Norris conducted a limited experiment of this nature by calculating the reproducibility of their readings (18). However, no other components were determined in that study. In the current study, we use a completely nested experimental design as described by Peng (19);the details will be discussed in the Experimental Section. Using this type of experimental design and analysis, we can determine four components of variance corresponding to instrument noise, sensitivity to orientation of the sample cup, repack variation, and sampling variation for both principal component regression and multiple linear regression on the original absorbance data. THEORY
,
The theories of multiple regression analysis (20) and of principal component analysis (5-11) have both been well described; therefore we need not repeat those derivations. However, it is possible to use the associative and commutative
0003-2700/86/0358-28 14$01.50/0 0 1986 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 58, NO. 13, NOVEMBER 1986
properties of vectors to convert the results of principal component regression into a particularly useful form as follows: Principal component scores are calculated from linear combinations of the optical data Si = GllXl GlzXz G13X3 + ... (14
+
+
... Where Si represents the ith principal component score, X j represents the optical data a t the j t h wavelength, and G , represents the principal component loading of the ith component for the data a t the j t h wavelength. The principal component regression relates the scores to the dependent (Y) variable thusly P = bo blSl b2Sz b3S3 + ...
+
+
+
Where P i s the predicted value of the dependent variable, and the bi terms are the regression coefficients for the scores. We replace the S, in eq 2 with their values from eq 1,expand each term, and collect the coefficients of the various X, P = bo + (blG11 + bzG21 + b3G31 + ...)XI + (b1Glz bzGz2 + b3G32 + ...)X, + ... (3)
+
Since all the expressions in parentheses are constants, this can now be written P = bo klXl kzXz k3X3 ... (4)
+
+
+
+
Thus we see that the results of a principal component regression can be converted into an equivalent set of coefficients of the original data. This form is particularly useful because it allows a principal component calibration to be directly compatible with current near-infrared instrumentation. It also allows a direct comparison of the coefficients obtained from a principal component regression with those obtained from a standard multiple linear regression. Another characteristic of eq 4 for a full spectrum is that a plot of the calibration coefficients vs. wavelength will be a reconstructed spectrum of the analyte, similar to that generated by a previous algorithm (21). The method using principal components reconstructs that spectrum via correlated structures rather than reconstructing the spectrum one wavelength a t a time. Previous use of principal components to reconstruct spectra has resulted in a nonunique reconstruction (6, 18). This approach results in a unique spectrum. On the other hand, the current approach produces spectra that are not corrected for interferences, unless extra steps are taken to make these corrections. Honigs et al. describe the necessary corrections (21). Calibration Statistics. Modern regression programs produce several statistics to help the analyst judge the quality of the regression. Global statistics include the standard error of estimate, the multiple correlation coefficient, and the F for the regression. Other statistics are the Student t values that are calculated for each of the regression coefficients. In addition to these standard regression statistics, other auxiliary statistics have been found useful for evaluating calibration equations generated from instrumental data. These statistics bridge the gap between the mathematical properties of the regression analysis and the physical properties of the instruments used for data collection. These statistics have been in use for many years, yet their derivations have not previously appeared in the open literature. These statistics are useful; therefore, we present the necessary derivations here: The first statistic of concern is the index of random error (IRE). The IRE is a measure of the sensitivity of the calibration equation to the random (noise) component of the absorbance data. We begin by noting that, for a linear com-
2815
bination of variables (e.g., eq 4),propogation of error considerations show that the variance of the result (Y) can be calculated from the variances of the data (X) thusly (22) var(Y) = k12 var(X,)
+ k$
var(X,)
+ ...
(5)
Under the assumption that all variables have the same amount of noise, then eq 5 can be factored var(Y) = var(X) C k i 2 Taking the square root of both sides
S.D.(Y) = S.D.(X) (Ch:)1/2
(7)
Thus the random component of the final reading can be related to the instrument noise through the quantity (Cki2)1/2, the sensitivity of the final result to the noise component of instrument variation, which is called the index of random variation. The second statistic found useful in evaluating empirical equations used for instrument calibration is the index of systematic error (ISE). This statistic is a measure of the sensitivity of a calibration equation to the systematic changes seen in reflectance spectra when samples are repacked (23, 24). The variations in the optical readings a t different wavelengths are proportional to each other; letting AX represent a measure of the average variation, the variation at each wavelength can be expressed as a different multiple of A X . Then the reading obtained after repacking a sample can be expressed Y AY = bo k l ( X I + a AX) + kZ(X2 + b AX) + ... (8) Expanding eq 8 and subtracting eq 4 results in the following expression: AY = kla AX k,b AX ... (9) A simplification can be achieved by making the assumption that a AX = b AX = ...; Le., the variations of the readings a t different wavelengths are equal. In some cases, reflectance data is found to approximate that property. In such a case, eq 9 can be rewritten:
+
+
+
+
A Y = AX C k j
(10) The term Ckj,the sensitivity to the common component of the change in optical readings, is the index of systematic error. EXPERIMENTAL SECTION
Thirty-one samples of hard red spring wheat were obtained and used for calibration. Since wheat is known to be a substance that is easily analyzed via near-infrared reflectance analysis, and has already been shown to be amenable to analysis via principal component regression (9),there seemed little point in reinventing the wheel, so to speak, by measuring another “prediction” set. Rather, in order to accomplish the goal of determining components of variance, a single large (approximately 5 lbs) sample of wheat, similar to the calibration samples, was also obtained. Each sample was mixed by passing it several times through a Boerner divider before proceeding further. Twenty-five aliquots of 18 f 0.5 g each were removed from the large sample. This amount of material was sufficient to afford two completely separate packs of each aliquot. Each aliquot was separately ground using a Udy Cyclotec grinder with a 1.0-mm screen installed. Each aliquot was packed twice into a closed sample cup; each pack was measured in two random orientations; for each orientation two measurements of the spectrum were obtained using a Technicon InfraAlyzer Model 500. This measurement process is shown schematically in Figure 1, and provides 24 degrees of freedom for measuring sampling error of the wheat; 25 degrees of freedom for measuring repack variation; 50 degrees of freedom for measuring cup orientation effects, and 100 degrees of freedom for measuring noise. Each sample of the calibration set was measured using the same protocol as for the aliquots of the prediction sample. This created
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ANALYTICAL CHEMISTRY, VOL. 58, NO. 13, NOVEMBER 1986
Table I. Results for Wheat Protein Calibrations from Absorbance Data
constant 1445 1680 1722 1734 1759 1778 1818 1940 1982 2100 2139 2180 2190 2208 2230 2270 2310 2336 2348
noncompressed data 3 wavelengths 19 wavelengths coeff t coeff t
compressed data 3 wavelengths 19 wavelengths coeff t coeff t
16.1
17.0
-165.3
9
-346.1
55
452.0
69
17.6 89.0 326.2 1141.2 -412.7 -422.2 -965.8 -65.8 89.7 -130.2 -13.6 182.9 -245.7 -46.1 634.5 182.2 -110.4 -42.1 -118.8 -92.2
2.2 2.3 3.3 0.9 0.9 2.5 0.3 1.1 1.2 0.0 1.2 1.1 0.1 3.3 1.5 0.9 0.3 1.0 0.9
-161.2
2.8
-348.7
17
450.9
23
9.1 218.5 187.6 847.7 1497.8 -988.2 -2400.9 335.2 -56.5 106.9 167.3 705.1 -2828.6 -73.7 3308.2 -583.8 -512.6 242.6 942.4 -1087.5
0.5 0.2 0.3 0.4 0.2 0.7 0.2 0.1 0.1 0.2 0.6 1.3 0.0 1.5 0.5 0.6 0.2 0.7 0.8
Calibration Statistics
SEE corr coeff F(regression1
ISE IRE
0.24 0.9876 3242 -59.4 592
0.19 0.9925 791 -19.8 1815
0.24 0.9890 404 -59.1 592
0.22 0.9962 77 27.6 5667
Calibration Components of Variation repack orientation noise
0.066
0.079 0.050 0.031
a
0.084 0.049 0.031
a
0.063
a
0.30
Prediction Components of Variation sampling repack orientation noise a
0.41 0.17 0.067 0.045
0.43 0.082
0.45 0.18 0.067 0.044
a
0.077
a
a a 0.44
ComDonent of variation not significant above lower levels of variation.
Table 11. Calibration Results Using All Principal Components component no. constant 1 2 3 4 0
6 r
1
8 9 10 11
12 13 14 15 16 17 18 19
eigenvalue 0.268 0.0105 0.00375 0.00102 1.42E-4 1.05E-4 1.26E-5 7.90E-6 3,343-6 2.153-6 1.91E-6 1.886-6 1.36E-6 1.033-6 7.163-7 4.593-7 3.63E-7 1.343-7 1.12E-7
noncompressed data coeff 17.6 -6.83 190.29 -205.56 11.65 -69.95 -13.58 233.74 -167.08 454.99 -181.57 -9.40 810.56 -182.05 -112.94 448.69 945.89 607.04 390.46 -751.26
t
eigenvalue
18 101 65 1.9 4.3 0.7 4.3 2.4 4.3
0.0318 0.00131 4.63E-4 1.01E-4 1.31E-5 7.37E-6 1.01E-6 5.54E-7 2.723-7 1.28E-7 8.993-8 3.57E-8 2.67E-8 2.09E-8 1.41E-8 1.08E-8 4.93E-9 2.78E-9 9.86E-10
1.4
0.1 5.7 1.1 0.6 1.9 3.3 1.9 0.7 1.3
compressed data coeff 9.17 -7.21 191.19 -206.83 13.90 -3.95 -6.43 241.85 61.59 589.89 -536.29 -1183.92 -815.39 1335.10 1731.71 4587.22 179.37 -979.84 1529.61 -350.79
Calibration Statistics
SEE corr coeff F
0.19 0.9925 791
0.22 0.9962 77
t 5 31 20 0.6 0.1 0.1 1.1 0.2 1.4 0.8 1.6 0.7 1.0 1.1
2.4 0.1 0.3 0.4 0.0
ANALYTICAL CHEMISTRY, VOL. 58, NO. 13, NOVEMBER 1986
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Table 111. Calibration Results Obtained from Principal Component Calibration, Converted to Equivalent Wavelength Coefficients for Noncompressed Data
Coefficients
constant 1445 1680 1722 1734 1759 1778 1818 1940 1982 2100 2139 2180 2190 2208 2230 2270 2310 2336 2348
17.6 89.0 326.2 1141.2 -412.7 -422.2 -965.8 -65.8 89.7 -130.2 -13.6 182.9 -245.7 -46.1 634.5 182.2 -110.4 -42.1 -118.8 -92.2
8.3 -123.75 -15.02 -10.33 -4.35 -38.29 -55.88 -41.48 -34.74 71.03 -116.36 -58.02 78.40 92.25 99.06 82.62 38.92 -18.29 -14.80 -3.42
5.8 -62.91 -1.98 8.51 10.88 -7.10 -17.83 -12.80 -120.75 -42.43 -42.48 -18.09 46.60 53.15 56.88 52.93 38.59 13.23 11.31 15.26
0.19 0.9925 791 -19.8 1815 0.066
0.23 0.9884 3451 -72.5 280
30.9 -62.18 -14.03 -19.93 -16.35 -32.34 -39.21 -29.76 -84.04 111.76 -75.72 -41.72 30.13 37.47 40.60 28.13 -1.46 -33.41 -28.01 -20.57
-5.5 -122.41 -14.03 -9.23 -3.24 -37.15 -54.72 -40.41 -32.76 72.74 -114.53 -56.23 80.06 93.89 100.65 84.18 40.70 -16.40 -12.91 -1.54
9.9 -80.47 -369.06 -60.37 229.77 -395.01 -156.72 461.61 -158.70 210.21 -205.03 252.57 -206.52 270.92 207.01 57.27 -124.20 -62.71 143.41 -102.47
13.6 -39.91 250.88 91.52 71.23 -132.98 -298.11 -310.40 -27.94 11.89 -172.25 236.81 36.27 -4.26 168.10 323.94 -334.21 -224.60 13.56 268.69
0.32 0.9771 2589 -43.3 280
0.27 0.9846 1929 -89.49 990
0.22 0.9895 2851 -71.8 858
Calibration Statistics SEE
corr coeff F(regression) ISE IRE
repack
a
0.84 0.8348 281 -19.0 190
1.27 0.5497 53 -82.5 205 a
a
a
a
a
0.088 0.029
0.100 0.042
0.086 0.063
0.445 0.205 0.095 0.036
0.345 0.278 0.107 0.059
0.457 0.182 0.108 0.079
Calibration Components of Variation orientation noise
a
0.063
0.096 0.029
0.097 0.049
0.071 0.028
Prediction Components of Variation sampling repack orientation
a
noise
0.077
0.43 0.082
0.398 0.194 0.102 0.038
0.549 0.279 0.101 0.069
1.34 0.304 0.078 0.042
Component of variation not significant above lower levels of variation. the capability of measuring the same components of variation (except for sampling variation) on the calibration set as on the prediction sample. For the purposes of this study, data corresponding to the 19 wavelengths found in interference-filter-based near-infrared instruments were transferred to an IBM Model 4381 mainframe computer and the calculations were performed using standard APL. The use of a limited number of wavelengths permitted us to perform multiple regression calculations using data at all available wavelengths, in order to compare those results with the principal component calibration results, which also use all available wavelengths.
RESULTS AND DISCUSSION Many factors affect the behavior of a calibration generated from a set of near-infrared data. In addition to the set of wavelengths used, or components retained in the case of principal component regression, transformations of the data can be expected to change the distribution of variance between the different sources of error. In addition, compression of the calibration data, or averaging together the several readings of a given sample, has been shown to change the distribution of both calibration and prediction variance, in different ways (17). Thus in the current work, all calibrations have been performed on both compressed and uncompressed data. For the multiple regression calibrations, two sets of wavelengths have been used: all available in the set of 19 and a
set of three. From long experience with calibration of nearinfrared reflectance analysis (NIRA) instruments for wheat protein, three wavelengths are known to be optimum for a multiple linear regression calibration. The results from multiple linear regression calibrations are shown in Table I. Comparison of the SEE for the three-wavelength calibration with that for the 19-wavelength case confirms this experience for this dataset; there is no significant difference between the two statistics (F = 1.59, F(crit) = 1.84). Principal component calibrations have been less extensively used. Table I1 presents the results of performing a principal component calibration on the calibration dataset, retaining all the principal component scores in the calibration. Due to the orthogonality of the scores, the coefficients for the scores do not change as scores are deleted from the set. However, different sets of coefficients for the optical data, as well as different calibration and prediction statistics result when different components are retained in the principal component calibration. Tables I11 and IV present, for calibrations on noncompressed and compressed data, respectively, the results obtained from various selections of scores retained. From these tables we conclude that retention of the three largest components generates the optimum principal component calibration. Several indicators lead to this conclusion. Compared to the use of all components, or compared to the use of four components, there is no statistically significant
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ANALYTICAL CHEMISTRY, VOL. 58, NO. 13, NOVEMBER 1986
Table IV. Calibration Results Obtained from Principal Component Calibration, Converted to Equivalent Wavelength Coefficients for Compressed Data (A) all
(E) 2, 3
(F) 1,2, 3, 4
Coefficients constant 1445 1680 1722 1734 1759 1778 1818 1940 1982 2100 2139 2180 2190 2208 2230 2270 2310 2336 2348
9.1 218.5 187.6 847.7 1497.8 -988.2 -2400.9 335.2 -56.5 106.9 167.3 705.1 -2828.6 -73.7 3308.2 -583.8 -512.6 242.6 942.4 -1087.5
6.2 -63.56 -1.94 8.55 10.94 -7.24 -18.06 -13.09 -121.16 -42.25 -42.91 -18.24 46.93 53.48 57.16 53.10 38.73 13.16 11.15 15.12
9.1 -122.10 -15.64 -10.84 -4.70 -38.80 -56.43 -41.56 -34.25 71.60 -117.77 -59.26 78.90 93.05 100.02 83.50 38.86 -19.16 -15.41 -3.91
32.1 -59.95 -14.75 -20.55 -16.82 -32.77 -39.58 -29.60 -84.83 112.05 -76.80 -42.89 30.21 37.85 41.19 28.75 -1.75 -34.32 -28.56 -21.02
-5.5 -120.68 -14.59 -9.68 -3.53 -37.58 -55.21 -40.42 -32.17 73.40 -115.84 -57.37 80.66 94.78 101.69 85.15 40.74 -17.17 -13.42 -1.92
10.4 -119.53 -11.07 -6.55 -0.45 -34.64 -52.31 -36.82 -36.01 73.31 -121.38 -61.96 78.51 93.14 100.98 85.17 36.78 -23.22 -18.90 -6.96
Calibration Statistics
SEE corr coeff F(regression)
ISE IRE
0.22 0.9962 77 27.6 5667
0.22 0.9907 476 -73.9 281
0.87 0.8377 32 -20.1 191
1.33 0.5504 6 -84.5 206
0.32 0.9788 320 -43.1 281
0.22 0.9908 349 -61.9 282
Calibration Components of Variation repack orientation noise
a
a
a
a
0.096 0.028
0.072 0.028
0.096 0.049
a 0.087 0.029
a
a 0.30
0.560 0.282 0.101 0.069
0.456 0.204 0.094 0.036
0.422 0.189 0.094 0.039
0.092 0.029
Prediction Components of Variation sampling repack orientation noise
a a a 0.44
0.431 0.196 0.102 0.038
1.34 0.304 0.078 0.042
Component of variation not significant above lower levels of variation.
the smallest increase, corresponding to deletion of component 1; F(crit) = 1.84). Multiple linear regression calibration on the full dataset is identical with principal component calibration on that dataset; this follows from theory. Thus the appropriate comparison is the comparison of the three-wavelength multiple regression calibration with the three-component principal component calibration. The calculated values of F for the comparison of corresponding components of variance are shown in Table V. We find that for the calibration data there is no statistically significant difference in noise level (F(crit) = 1.35), while the principal component calibration shows a significantly high sensitivity to sample cup orientation (F(crit) = 1.54).
Flgure 1. Protocol of experimental deslgn for measuring components of variation in wheat protein analysis via near-infrared reflectance analysis.
difference between the SEE’Sgenerated from the three sets of retained components for either the compressed or noncompressed data. Deleting any of three components does lead to a statistically significant increase in the SEE ( F = 1.93 for
On the prediction data, we find the multiple linear regression calibration more sensitive to noise (F(crit) = 1.35) while the principal component calibration is again more sensitive to cup orentation (F(crit) = 1.61). There is no difference between the two calibration methods as to the effect of repack (F(crit) = 1.96) or sampling. Since repack and sampling comprise the largest components of variance for both calibration methods and there is no difference between them, we conclude that for wheat protein, we expect the two calibration methods to have equivalent analytical performance.
ANALYTICAL CHEMISTRY, VOL. 58, NO. 13, NOVEMBER 1986
Table V. Values of F of Different Components of Variance for Comparison of Multiple Linear Regression and Principal Component Regression
component
using calibration f r o m compressed data
using calibration from noncompressed d a t a
A. F o r Caibration D a t a
b 2.32" 1.14
repack orientation noise
b 3.69" 1.14
B. F o r P r e d i c t i o n D a t a sampling repack orientation noise a
Statistically significant. I11 a n d IV.
1.1
1.28 1.96' 1.34'
1.02 1.30 2.32" 1.40a
N o t calculable due t o value missing
in Tables
Other Considerations. Precision and accuracy are only two (albeit the most important two) of the considerations that are encountered in the evaluation of an analytical method. Others include speed, ease of use, and other such auxiliary factors. one of the more difficult tasks of multiple linear regression calibration is the determination of the proper wavelength set to use to perform the analysis. To this end, many variable-selection methods have been devised, including stepwise selection (up and down) (251,row-reduction (26),and combinations searches (27). These search methods are very computation-intensive, since they involve calculation of many regression analyses, so that the best can be selected. In contrast, a single principal component regression using all the available components immediately reveals the correct ones to use. Examination of the results in Table 11shows that the t value for the coefficients of the principal component scores are significant for the first three components, immediately becoming nonsignificant for the fourth and beyond. As we have seen these three components are indeed the correct ones to use. Since wavelength selection is unnecessary, because all wavelengths are used, we find that this provides us with a much simplified variable selection method compared to multiple regression analysis on the optical data and, thus, a technique that is much simpler to use. Much more information is available by making comparisons within Tables I-IV; space limitations preclude considering them in detail here. However, the effect of compression, wavelength selection, and principal component selection on the distribution of variance among the different components of variance is an interesting study. It is worth noting, however, that large and statistically significant changes can be noted in the components of variance listed in these tables. This is directly pertinent to the discussion in the introduction. Small changes in the conditions of the calibration can introduce large changes in the amount of variance associated with the several components of variance. The increase in noise level for the different wavelength sets in Table I shows this clearly. The noise increased by a factor of 2 (for the calibration on noncompressed data) and by a factor of 10 (for the calibration from compressed data). Similar changes can be seen in the results for the principal component calibrations. These observations increase our misgivings concerning the interpretation of the small and statistically nonsignificant changes reported in some of the comparative studies in the near-infrared literature. CONCLUSIONS Use of components-of-variance analysis has shown statis-
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tically significant differences between principal component calibration and multiple linear regression on the optical data, when used for analyzing ground wheat. The differences were found in the noise level and sensitivity to sample cup orientation. Repack and sampling effects, however, were shown to be not different between the two methods. Since repack and sampling are the two largest sources of variance, unless other error sources can be shown in the future to be affected differently by the two calibration methods, we conclude that these two techniques are equivalent. Nevertheless, principal component calibration appears to be a desirable approach to the calibration problem, because it offers a much simplified method of variable selection, compared to the standard multiple regression approach. In addition, there is no need for special computer programs to do analysis with, because the principal component calibrations can be put into a form that is compatible with and that can be keyed directly into current near-infrared reflectance instrumentation. The changes seen in the components of variance as calibrations are performed under different conditions indicate that authors of comparative studies already in the near-infrared literature might reevaluate their work with attention to the question of statistical significance of their data and with a view toward determining whether their original conclusions are still valid under the objective criteria that the science of statistics provides. Use of experimental designs that avoid the inclusion of reference laboratory error in the results will help alleviate the problem. LITERATURE CITED Gnanadesikan, R. Method..for Statistical Data Analysis of MuMvariate Observations;Wiley: New York. 1977; Chapter 2. Morrlson, D. F. Munivariete Statistical Methods, 2nd ed.;McGraw-Hill: New York, 1976; Chapter 8. Martens, H.; Russwurm, H. Food Research and Data Analysis; Appiled Science: New York, 1982; pp 5-38. Metzler, D.; Harris, C. M.; Reeves, R. L.; Lawton, W. H.; Maggio, M. S. Anal. Chem. 1977, 4 9 , 864A-874A. Cochran, R. N.; Horne, F. H. Anal. Chem. 1977, 49, 846-853. Kawata, S.; Komeda, H.; Sasaki, K.; Mlnami, S. Appi. Spectrosc. 1985. 39. 610-614. Naes; T.;'Martens, H. Commun. Statista .-Sirnula. Computa 1985, 14. 545-576. Naes, T.; Martens, H. Trends Anal. Chem. 1984, 3 , 266-271. Cowe, I.A.; McNicol, J. W. Appl. Spectrosc. 1985, 3 9 , 257-266. Cowe, I.A.; McNicoi, J. W.; Cuthbertson, D. C. Analyst (London) 1985. 110, 1227-1232. Cowe, I. A.; McNicoi, J. W.; Cuthbertson, D. C. Analyst (London) 1985, 110, 1233-1240. Massie, D. R.; Norris, K. H. Trans. Am. SOC.of Agric. Eng. 1968, 8 , 598. McClure, W. F. "Status of Near-Infrared Technology in the Tobacco Industry"; Proceedings of an International Symposium on Near-Infrared Reflectance Spectroscopy; heid at Old Melbourne Hotel, Melbourne, Victoria, Australia, Oct 15-16, 1984; Miskeliy, Diane, Law, Donald P., Ciucas, Tony, Eds.; Cereal Chemistry Division, Royal Australian Chemical Institute. Williams, P.; Norris, K. H. Cereal Chem. 1983, 6 0 , 202-207. Wold, S.; Martens, H.; Wold, H. Lecture Notes in Mathematics, Spring er-Verlag: Heidelberg, 1982; pp 286-293. Mark, H.; Workman, J. Spectroscopy 1986, 1(5), 39-46. Mark, H.; Workman, J. Anal. Chem. 1988, 58, 1454-1459. Hruschka, W.; Norris, K. H. Appl. Spectrosc. 1982, 3 6 , 261-265. Peng, K. C. The Design and Analysis of Scientific Experiments ; Addison-Wesley: Reading, MA, 1967; p 67. Draper, N.; Smith, H. Applied Regresslon Analysis, 2nd ed.;Wiley: New York, 1981. Honigs, D. E.: Hieftje, G. M.; Hirschfeid, T. Appl. Spechosc. 1984, 38, 317-322. Mandel, J. The Statistical Analysis of Experimental Data ; Interscience: New York, 1964; p 72. Mark, H.; Tunneii, D. Anal. Chem. 1985, 57, 1449-1456. Norris, K. H. I n Food Research and Data Analysis; Applied Science: New York, 1976; p 98. Martens, H.; Naes, T. Trends Anal. Chem. 1984, 3 , 204-210. Honigs, D. E.; Freelin, J. M.; Hieftje, G. M.; Hirschfeid, T. 6 . Appl. Spechosc;,1983, 3 7 , 491-497. Mark, H. Specifying Wavelength Searches for the Combinations Program"; Infranote 104; Technlcon Instrument Corp., Tarrytown, NY.
RECEIVED for review March 18, 1986. Resubmitted June 2, 1986. Accepted July 15, 1986.
