Anal. Chem. 1988, 58, 433-438
transformed back to integer format for display by the frame buffer. Without calibration, the image directly represents the integrated relative intensity differences on the fluorescent screen from pixel to pixel. CONCLUSION The data presented here establish that video recording of ion images is a feasible and highly versatile method of data acquisition in ion microscopy. In our experience, the VTR system has proven invaluable to a wide array of experiments and has been a convenient tool even for experiments without strict requirements. Quantitative analysis is easily accomplished by recording images of known ion intensity under conditions identical to those used for experimental specimens. The current system, with its capability to precisely digitize single frames, is appropriate for use with any type of experiment where data are in the form of, or can be converted to, a video signal. LITERATURE CITED (1) Morrison, G. H.; Moran, MG. Proceedings of the 41st Annual Meeting of the Electron Microscopy Society of America, 1983;pp 14-17. (2) Morrison, G.H.; Moran, M. G. I n “Secondary Ion Mass Spectrometry
SIMS-IV”; Benninghoven, A., et al., Eds.; Springer Series in Chemical Physics; Springer: New York, 1983;Vol. 36,pp 178-182. Green, W. B. “Digital Image Processing: A Systems Approach”; Van (3) Nostrand Reinhoid: New York, 1983. (4) Rudenauer, F. G.; Steiger, W. Mikrochim. Acta Suppl. I I 1981,
375-389.
433
(5) Rudenauer, F. G.;S I A , Surf. Interface And. 1984, 6 , 132-139. (6) Lepareur, M. Revue Technique Thompson-CSF 1980, 12 (l), 225-265. (7) Furman, B. K.; Morrison, G. H. Anal. Chem. 1980, 52, 2305-2310. (8) Scilla, G. J.; Morrison, G. H. Anal. Chem. 1977, 49, 1529-1536. (9) Drummer, D. M.; Fassett, J. D.; Morrison, G. H. Anal. Chim. Acta 1978, 100, 15-22. (10)Patkin, A. J.; Morrison, 0. H. Anal. Chem. 1982, 54. 2-5. (11) Chu, P. K.; Harris, W. C., Jr.; Morrison, G. H. Anal. Chem. 1982, 5 4 , 2208-2210. (12) Harris, W. C.; Chandra, S.;Morrison, G. tl. Anal. Chem. 1983, 55, 1959-1963. (13) Patkin, A. J.; Chandra, S.; Morrison, G. H. Anal. Chem. 1983, 55, 2507-2510. (14) Fassett, J. D.; Roth, J. R.; Morrison, G. H. Anal. Chem. 1977, 49, 2322-2329. (15) Drummer, D. M.; Morrison, G. H. Anal. Chem. 1980, 52.2147-2152. (18) Fassett, J. D.; Drummer, D. M.; Morrison, G. H. Anal. Chim. Acta 1979, 172, 165-173. (17) Fassett, J. D.;Morrison, G. H. Anal. Chem. 1978, 50, 1881-1866. (18)Patkin, A. J. Ph.D. Thesis. Corneii University, Ithaca, NY. 1982. (19) Shuetzie, D.;Riley, T. L.; deVries, J. E.; Prater, T. J. Mass Spectrom. Rev. 1984, 3, 527-585. (20) Brenna, J. T.; Morrison, G. H. Anal. Chem. 1984, 58, 2791-2797. (21) Snedecor, G. W.; Cochran, W. G. “Statistical Methods”, 7th ed.; Iowa State University: Ames. Iowa.
(22) Bernius, M. T.; Ling, Y. C.; Morrison, G. H. Anal. Chem. 1988, 58, 94-10 1. (23) Brenna, J. T.; Morrison, G. H. in preperation.
RECEIVED for review May 29, 1985. Accepted September 16, 1985. This work was supported by The National Science Foundation, The Office of Naval Research, and The National Institutes of Health.
New Technique in Chemical Assay Calculations. 1. A Survey of Calculational Practices on a Model Problem Mario J. Cardone
Norwich Eaton Pharmaceuticals, Inc.,’ P.O.Box 191, Norwich, New York 13815
Assay data from a model containing hidden constant and proportional systematlc errors were recelved from more than 50 experienced analytical chemists engaged in the industrial and research analysis of sample matrices covering pharmaceutlcals, envlronmental samples, synthesis compounds, and manufactured and natural products. Data for two simple generic model problems are given as presented to the survey respondents. The crlterla by whlch the results were evaluated are presented and the patterns that emerged are discussed. The diverslty of the results shows that confusion and lack of consensus about what standard calculatlonal practice Is, or should be, exlsts. Thls underlying lack of agreement stems principally from confusion as to the nature of the true blank in an analysis and how It Is correctly and accurately determlned.
Calculation of quantitative analytical results is indeed a simple matter involving formulas of now untraceable origin, apparently based only on simple logic and arithmetic. However, the common notion that no problems exist in this area is wrong. Some real, albeit simple questions can be asked that elict no unambiguous, unified responses but rather a ‘ A Procter and Gamble Company. 0003-2700/86/0358-0433$01.50/0
variety of conflicting opinions. What is the true blank in an analysis? How is it correctly and accurately determined? Why do assay results too often show a trend, sometimes up, sometimes down, as a function of sample size? Is this trend symptomatic of a constant error or of a proportional error? What is the significance of the difference between the slope of the standard response curve and the slope of the method of standard additions (MOSA) curve? And, most intriguing of all, why do analysis results obtained from a standard response curve too often not agree with the results from the MOSA technique, even when the procedural steps are the same? Any measurement system, procedure or method should be free of any known or detected systematic error (1, 2) but despite the best experimental efforts, hidden systematic error may still be present. What is the quantitative effect of such undetected error on the results in a real sample analysis? Definitive answers t o all these questions are required if a unified practice for general use is to be developed. The purpose of these studies is to show that the development of such a practice is possible. SURVEY To demonstrate the effect of hidden systematic errors on the accuracy of calculated analytical results and to draw attention to this relatively unstudied phenomenon, two simple, practical analytical problems were designed. All of the data 0 1986 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 58, NO. 2, FEBRUARY 1986
Table I. Data for Problem 1 MOSA
standard curve WI SI 1.6667 5.0000 8.3333 9.5507 11.6667 18.1600 19.9333 "MB = 0.100.
0.2500 0.5000 0.7500 0.8413 1.0000 1.4870 1.6200
sample
set l A b
set lBC
WZ
s,D
WI
s,'
62.4746 82.7915 103.1085
0.8000 1.0000 1.2000
0 2.540 8.724 10.413
0.8000 1.0000 1.4870 1.6200
wr
s,'
3.1429 5.7143 7.8730
1.0000 1.2475 1.4500 1.6200
0
Wz,u= 62.4746. Wz,u= 82.7915.
in these problems are random-error free with an accuracy of better than 1ppt, so that calculated analytical results have a significance of at least 1 ppt. Method descriptions were generic, purposely simple procedures that would apply to all disciplines, (chromatographic,spectrophotometric,titrimetric, etc.). For the sake of simplicity, the dilution factor was unity, and no special procedural aspects in the methodology were intended. This was simply an exercise to determine what calculational practices are in use. None of the special mathematical problems discussed by Shukla and Rusling (3) and Meites (4)were intended. All the data that would be required for calculational techniques such as calibrationcurve-based (5-8), single-point-ratio (SPRC, i.e., a single standard) (9-11), and method of standard additions (MOSA), were given in the problems (12-14). These model problems were sent to about 200 experienced, practicing analytical chemists engaged in the industrial and research analysis of a gamut of sample matrices covering pharmaceuticals, environmental samples, synthesis compounds, and manufactured and natural products. The recipients of the respective problem were clearly told that the object of the survey was not to arrive at the one correct answer for the analyte concentration, W,, in the sample. Rather, the object was to sample and evaluate the diversity of calculational options possible from current analytical practices, almost all of which are acceptable and scientifically defensible based on the information in the literature. As anticipated, however, a number of respondents did try to obtain the correct answer. PROBLEMS: PROCEDURES A N D DATA Standard Curve Method. Standard. Weigh the pure standard, dissolve in a suitable solvent, dilute to specified volume, and subject an aliquot to the measurement procedure. Record the response as S, (peak area, peak height, absorbance, milliliters of titrant, etc.). Calculate effective standard mass or concentration taken for the measurement. This is W,. Sample. Weigh the sample, dissolve in the same solvent, dilute to specified volume, and subject an aliquot to the measurement procedure. Record the response as S,. Calculate effective sample mass or concentration taken for the measurement. This is W,. Method Blank. The method blank (MB) defined as a nominal-size analyte-free sample (e.g., a placebo) processed as above, was found to give an S, value of 0.100 (problem 1) and 0.095 (problem 2). See Tables I and 11. MOSA Method. Dissolve an exactly weighed sample in the same solvent as in the previous procedure and dilute to a specified volume. Subject an aliquot to the measurement procedure. This is the unspiked effective sample W,,", and the response is S,'. Also dissolve three samples of identical weight in the same solvent. Spike each with an exact, increasing volume of spike standard solution in the same solvent, and dilute each to the same specified volume as the unspiked sample. Then subject aliquots of the same specified volume of each spiked sample solution to the measurement procedure.
Table 11. Data for Problem 2 standard curve 3.3333 6.6667 10.0000 13.3333 16.6667 20.0000 26.6667
samole
0.250 0.500 0.750 1.0000 1.250 1.500 2.000
"MB = 0.095.
51.6491
MOSAb
0.400
147.0877
1.250
0
2.1053 10.5263 16.1404
0.800 0.950 1.550 1.950
* Wz,u= 96.5616. PROBLEM 1
Standard
,
MOSA
,
SET1
,
SET 1A
I
SETlB
Sr
ms 0075
i
mM 007875
-_--
-
t Wr
Wr (Added)
Wr (Added)
PROBLEM 2 Standard
MOSA
SET 2
SET 2
L -
Wr (Added)
WI
Flgure 1. and 11.
Standard and
MOSA
curves for problem data in Tables I
These are the effective spike standards W,; the respective responses are the S,' values. The data for the two problems are given in Tables I and 11, and the respective standard curve and MOSA plots are shown in Figure 1. (At this point, the reader is invited to try his own solution of either one or both of the problems.) SURVEY RESULTS A N D EVALUATION CRITERIA Both model problems contained hidden constant and proportional errors, each type of error at a level so that the effects of its bias upon calculated results would be significant and readily discernible under the random-error-free conditions. From a study of other models such as these and from the survey replies, the criteria for evaluation of the calculational approaches were drawn up and are shown in Table 111. Except for a few special calculations, almost all of the approaches reported are modifications of the conventional expressions in everyday use. Each approach was encoded to
ANALYTICAL CHEMISTRY, VOL. 58, NO. 2, FEBRUARY 1986
435
Table 111. Evaluation Criteria for Survey Data and Distribution of Respondents' Replies distribution*
criterion#
A. Constant Error/Blank 1.10 ignored experimental method blank, MB = S, = 0.100 (0.095), entirely 1.2 used the MB inconsistently in the various calculations as a response correction 2. calculated a sample blank from the sample data for use in a correction calculation 3. standard curve calculation 3.1 ignored standard curve calculation entirely 3.2 applied conventional standard curve calculation, no correction to S, values 3.2.1 used complete standard curve expression 3.2.2 used only the standard curve slope [set 1 only] 3.3 use& experimental MB, S, = 0.100 (0.095), to correct S, values 3.3.1 for use in standard curve expression 3.3.2 for use with a corrected Standard Curve 3.4 used experimental MB, S, = 0.100 (0.095), in place of the SB in the standard curve expression 3.5 extrapolated calculated assay values to the infinite sample size 3.6 averaged results from the sample values 4. single-point-ratio calculation (SPRC) 4.1 ignored SPRC entirely 4.2 applied conventional SPRC, no correction to S, (or S,) 4.3 used the experimental MB, S , = 0.100 (0.095), as the S, correction 4.4 used the SB as the S, correction 4.5 used a calculated MB (code A2) as the S, correction 4.6 corrected both the S, and S, responses with the SB and MB 4.6.1 in either combination of the MB and SB 4.6.2 either the SB or the MB used to correct both the S, and S, responses 4.6.3 used both the SB and MB to correct the S, and only the MB to correct the S, 4.7 ignored the difference in the SPRC result(s) from the standard curve result(s) 4.8 averaged results from the sample values 4.9.1 used the two equal sample and standard responses in the calculation 4.9.2 used only one of the standard responses with the sample responses 4.9.3 attempted to match the standard and sample responses as closely as possible 4.9.4 used arbitrarily selected standard responses for the respective sample responses 5. MOSA calculation 5.1 Ignored MOSA calculation entirely 5.2 applied conventional MOSA calculation, no correction to S,' (unspiked sample) values 5.3 used the experimental MB value, S, = 0.100 (0.095) as the S', correction 5.3.1 with the MOSA slope, m M ,in the calculation 5.3.2 with the standard slope, m,,in the calculation 5.4 used the SB as the correction to the S,' values 5.5 used a calculated method blank (code A2) as the S', correction 5.6 used both the MB and SB as corrections to the S,' values 5.7 used one or both of the equal S,' response pairs from the two sets for the calculation [set 1 only] 5.8 ignored the difference in the MOSA result(s) from the standard curve and/or SPRC result(s) 5.9 ignored the difference in the two MOSA values obtained [set 1 only] 5.10 averaged results from the two sample values [set 1 only]
18 17 7 5 17 1
6 6 12 1 21 26 8 9 2
5 1 15 13 4 15 4 1 5 21
14 1 4 3 2
2 31 28 13
B. Proportional Error/Slopes 1.1 attached no significance to difference between standard curve and MOSA slopes 1.2 assumed that the MOSA value(s) was (were) best because of the matrix effect compensation 2.1 used the ratio of the sample response slope (Youden regression) curve, my,to that of the standard response curve, m, 2.2 used the ratio of the sample response slope (Youden regression) curve, my,to that of the MOSA slope curve, m M 2.3 used the ratio of the slope of the MOSA curve, m M ,to that of the slope of the standard curve, m,,as the proportional error factor 3 accepted the constancy of the MOSA slopes in the MOSA sets [set 1 only]
42 13 3 2 1 31
C. Final Calculated Result(s) 1 concludes that no result is correct (includes those not making any selection) 2.1 selected best answer-incorrect 2.2 selected best answer-correct but for wrong reasons 3 calculated correct answer by one or more approaches
21 23 4 3
"Encoded formulas, see Figure 2. bTotal number of replies: problem 1, 36: problem 2, 15. I
correspond to the respective criteria listed in Table 111; these expressions are shown in Figure 2. Constant Error/Blank. As is true in real sample analysis, the first question the respondent had to answer was which blank to use. In the survey problems, an experimentally determined MB was provided, defined as the response from a nominal-size analyte-free sample (e.g., a placebo) processed through the procedure. An unexpectedly large percentage of the respondents, 35% (18/51), ignored the MB entirely, while 52% (17/33) of those
who applied the MB correction did so inconsistently; that is, the MB was applied to one or two of the three conventional calculations but not to all of the approaches attempted. One respondent applied this inconsistency to advantage. He used the MB for the MOSA calculation, the system blank (SB)in the normal standard curve calculation, and the uncorrected ratio of the W,/ W , a t the two equal sample and standard responses for the SPRC. Upon averaging the standard curve and MOSA results (Table IV, problem l),he obtained almost exactly the same numerical answers as obtained by the SPRC.
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ANALYTICAL CHEMISTRY, VOL. 58, NO. 2, FEBRUARY 1986
STANDARD CURVE A3.2.1
A3.2.2
A3.3.1
Table IV. Some Assay Answers by Formulas in Figure 2 A3.4
Problem 1 Standard Curve A3.2.1*"
A3.2.2
A3.3.1
A3.4**u
S, = 1.2
0.1707 0.1441 0 . 1 4 0 9 ~ ~ 0.1610 0.1390 0.1552
0.1227 0.1248 0.1261
0.1494 0.1449 0.1422
I
0.1413
0.1245
0.1455
S, = 0.8 S, = 1.0
wx/wz =
Sx - SB (or. MB)' Sr - M B (or. S B )
3 Wz
Sx - SB (or, M B ) ' Sr - SB (or. MB)
0.1623
SPRCd
3
A4.6.2
W,
A4.2
A4.6.3
A4.3
SB*"
A4.4
MB
A4.6.3
S, = 0.8 0.1494 0.1307 0.1261 0.1441 0.1452 0.1193 S, = 1.0 0 . 1 4 0 9 ~ ~0.1268 0.1233 0 . 1 4 0 9 ~ ~0 . 1 4 0 9 ~ 0.1213 ~ S, = 1.2 0.1358 0.1245 0.1216 0.1390 0.1383 0.1226 0.1420
I
0.1273 0.1237 0.1413
0.1415
0.1211
MOSA
Figure 2. Calculation formulas encoded according to criteria in Table 111. Note: conventional calculation expressions are in the dashed
A5.2
A5.3.1
A5.3.2**a
A5.4
A5.6
S,' = 0.8 S,' = 1.0
0.1626 0.1534
0.1423 0.1380
0.1494 0.1449
0.1372 0.1342
0.1169 0.1189
I
0.1580
0.1401
0.1472
0.1357
0.1179
boxes.
Identical functions. Standard curve value equals SPRC value at S , = 1.0. ' S , = SI;or, S," = S," where Sc is the corrected response. dStandard pair: S, = 1.0: W. = 11.6667 (code A4.9.2).
The respondent felt that this fact, namely the same result by the three techniques, was necessary for the answer to be correct. (This criterion will be demonstrated for the correct answer in part 2.) Sixty percent of the SPRC users (15/25) ignored this consideration, as did 67% (31/46) of the MOSA users. Failure of standard curve and MOSA results to agree is one of the most important design considerations that the model demonstrates and is one of the fundamental issues of this study. Several respondents stated that they had not applied the MB correction to the standard curve calculation because they felt that the intercept of the standard curve was the correct blank. This is reflected in the 37% rate (17/46), by far the largest of the standard curve calculations. On the other hand, they felt that a blank correction of some kind was needed for the SPRC and the MOSA calculation. Yet, curiously, no one used the SB as the S , correction, and only 9% (4/46) used the SB as the S,' correction. This can be interpreted as meaning that the SB is not favored as the blank correction for the SPRC and the MOSA calculation. Many respondents who ignored the SPRC entirely (51%, 26/51) stated that they did so because of the high S B intercept (10). Even so, the SPRC is apparently unfamiliar to many analysts, since only 49% (25/51) supplied an SPRC result as against the 88% (45/51) who supplied either a MOSA result or a standard curve result. This finding was not surprising, but that relatively few applied the SB as the correction in the MOSA calculation (13%, 6/46) is surprising, because the literature is quite positive on this point (13, 15, 16). The respondents who provided a standard curve calculation (88%, 45/51) were nearly split between usage of the standard curve equation (37%, 17/46) and the MB as the correction (39%, 18/46). Only 17% (8/46) wery so dissatisfied with either option that they attempted to calculate a blank that they felt would better represent the constant error in the sample data. These special calculations will be discussed in a later section. Those respondents who applied the MB correction for use with the standard curve in problem 1 obtained an average
value which is the correct value after rounding (22%)6/27). This situation arose because the value of the MB is such that the constant error exactly nullifies the proportional error. For the 48% who used the MB consistently for the various calculational approaches, several other options had to be faced. Thus, in the standard curve calculation, there was the choice of code A3.3.1 mentioned above or codes A3.4 and A3.3.2. For the SPRC, the corresponding choices were code A4.3 or one of the three variations of code A4.6. Similarly, for the MOSA, there were codes A5.3.1, A5.3.2, and A5.6. Each of these options was in fact exercised by the repondents, with the distribution shown in Table 111. Only one respondent used the MB correction with the slope of the standard curve, m,, in the MOSA calculation. Only a few criteria could not be directly reduced to a corresponding code. Six respondents (13%,6/46) who used the MB in the standard curve calculated a corrected standard curve expression, S, = m,W, (SB - MB), or, for problem 1, S,' = 0.075W1 + 0.025. Some did this by subtracting the MB from each S, response and then regressing the corrected SIvalues with the W, values. The resulting expression is more readily obtained by simply subtracting the MB from the SB. The assay calculation is then completed as usual except that the S , values for use in the corrected standard curve are themselves corrected by subtracting the MB value. This entire process reduces to [(S, - MB)/(S, - MB)] X WI/ W,, which is equivalent to the SPRC code A4.6.2. In this same group there was also one respondent who used a response factor defined as the corrected S, (Le., S, - MB) divided by the corresponding W,, with the pair of values selected at the average expected sample response on the standard curve, a process that also reduces to the same code. A choice faced by those who elected to perform the SPRC was the selection of which standard data pair, WJS, to use. Four variations were used by the respondents (Table 111). Table I11 shows that the most popular option uses the same standard pair for the respective sample pairs. Most of the respondents who received problem 2 commented that since
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ANALYTICAL CHEMISTRY, VOL. 58, NO. 2, FEBRUARY 1986
the SB was zero, the conventional standard curve calculation and the SPRC were the same. In both problems, the sample and standard data each included a pair in which S , = s,. Only four respondents used these pairs in spite of the published fact that for equal standard and sample responses, the bias error due to a significant standard curve intercept is zero (10). Proportional Error/Slopes. A surprisingly high percentage of the respondents (82%, 42/51) attached no significance to the difference between the standard curve and MOSA slopes. To be sure, the literature contains only a few references to this (17, 18) but the layout of both problems attempted to set these slopes into relief. However, 25% (13/51) of the respondents either assumed or flatly stated that the calculated MOSA results were best because the MOSA technique provides for an in situ normalization of proportional error bias. Since this percentage is larger than the 18% who did attach a qualitative significance to the difference in the slope, one may infer that many of its users realize the usefulness of the MOSA technique, because of its normalization of proportional error, but that they have not yet discerned the full quantitative significance of the MOSA slope. All of the respondents (31/31) who supplied a MOSA calculation for problem 1 accepted without comment the constancy of the two MOSA slopes (Figure 1,sets 1A and 1B). If we subtract from these 31 the 3 who made the special calculation using a calculated MB, we have 28 respondents who made the MOSA calculation with no correction, or with either the SB or the experimental MB as the correction. In every case, each respondent obtained two different MOSA results, which they all also accepted without comment. Special Techniques and Final Calculated Results. Two distinct types of special techniques were utilized by the respondents. In the first, as mentioned earlier, the respondents felt the sample data indicated that neither the SB nor the experimental MB was satisfactory, and so they attempted to calculate a true sample blank from the sample data. In the second type, the respondents used unique approaches specifically to solve an algebraic problem without regard to any analytical practicality or generality. The number of both types totals 12, contributed by six respondents. Of these six, three arrived at the exact, correct answer; one respondent calculated the proportional error factor, P; another used three different calculational techniques. One other respondent also did the code A5.7 calculation correctly but did not select it as the correct answer, choosing an incorrect ressult instead. Five respondents made use of the two ratios of two response curve slopes, but one of these (code B2.1) yields a result corrected only for the constant error bias. Four respondents got a correct but not exact result using code A3.3.1, as explained earlier, but only three of these selected the result as their best answer. One respondent did an incorrect algebraic derivation, which gave the correct result on one sample data pair only, but he named it as the correct result. This respondent also tried problem 2, but his algorithm failed on both sample data pairs. The majority of the respondents, (86%, 44/51), failed to arrive at the correct solution to their respective problems. It is worth mentioning that code A3.3.1, when applied to any other problem with different values for SB and MB, would almost always fail except by chance, so the percentage of incorrect solutions is really 94% (48/51). (An example of this is in problem 2, in which code A 3.3.1 reduces to code A3.4, which yields an incorrect model result.)
DISCUSSION What do the results of the survey show as a whole? Certainly there is an ambiguous and widely conflicting array of opinions about the approach to use in analytical calculations as shown by the 33 different types of calculational approaches
437
submitted by the 51 respondents. The number of numerical answers possible by utilizing all of these options is several hundred (as shown by the examples in Table IV for those that could be reduced to simple formulas). The models designed for this survey were based on procedures of the simplest type commonly referred to as “shake-out” or “dilute-and-shoot”methods in which a sample is dissolved, clarified, and measured directly in exactly the same manner as for simple standard solutions. The problems contain hidden bias error; the degree of error is dependent upon the extent of hidden bias error in the sample data. The main point of this exercise was to demonstrate that the formulas in the boxes of Figure 2 are correct only for bias-free data. The respondents’ attempts to correct the basic formulas, as shown by the other formulas in Figure 2, were not successful. Without question, the degree of bias error designed into the survey problems was purposely high to make their effects easily visible. The constant error biases were 18.5% and -7.35% for problems 1and 2, respectively, and 5% and -5% for the proportional error biases, respectively. Clearly, these levels of bias would not be acceptable under normal circumstances. But how does one know when a method does not have a 2-5% bias error? Most applications of methodology do not warrant an exhaustive “first principles” error-source search techniques unlike those applications for which ultimate accuracy is absolutely essential, as exemplified by Standard Reference Materials (SRMs) assay certification (19). It is therefore of basic importance that analysts know how to detect, characterizq, and measure bias errors and to make the corrigible corrections. Many of the respondents recognized the presence of such errors from their results, e.g., as shown in Table IV, code A3.2.1, in which the individual sample values decreased with increasing sample size. Although one might attribute this, for example, to extraction efficiency on real samples, it is not the case here. But would one know that on a set of real sample data? In Table IV, note that the data assay for code A4.6.3 increases with sample size even though these are the same data and only the calculation is different. Many respondents handled this problem by averaging the sample values, calculating the relative standard deviation, and accepting the mean value because the relative standard deviation was l % or less. However, in th8 problems these are random error-free data, therefore, the trend is real, indicating bias error, not random error (Table IV, codes A3.6 A4.8 + A5.10). The six respondents who tried to calculate the true sample blank recognized the bias problem, but as previously mentioned, only three of them succeeded in obtaining the correct answer. For these six respondents, another indicator was the varying S,/ W,ratio for the various sample sizes. By algebraic calculations or computer algorithms,the true sample blank for the sample curve was calculated. In part 2, a simple general calculation will be shown. Two respondents rejected such “lousy” data, confirmingthe premise that no method with such a high level of bias would normally be accepted. In the short term, this may be good, but in the long term, these analysts would fail to detect corrigible bias error and would too often be burdened with unnecessary development effort. Attempts to solve the model problems by the various calculational formulas shown in Figure 2 and the results in Table IV involve the correction of the sample response, S, or S,’ (for the MOSA). These attempts demonstrate the recognized need for a true sample blank. In part 2, a definition of the true sample blank and its accurate measure will be presented. Part 2 will also demonstrate the criterion that the same answer for the sample data must be obtained by each of the calculational techniques for the correct solution of the model problems. Further, part 2 will discuss the concepts that led to the design of the model problems, and the relationship of the model to
+
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Anal. Chem. 1986, 58,438-445
some real sample applications will be indicated. A general practice that satisfies all the pertinent criteria will be recommended.
LITERATURE CITED (1) Eckslager, K. ”Errors, Measurement and Results in Chemical Analysls”; Chaimers, R. A., English Translatlon, 1972; Van Nostrand Reinhold: London, 1969, p 20. (2) Currie, L. N.; DeVoe, J. R. In “Validation of the Measurement Process”; American Chemical Society: Washington, DC, 1977; ACS Syrnp. Ser. No. 63, Chapter 3, p 117. (3) Shukla, S. S.; Rusling, J. F. Anal. Chem. 1984, 56, 1347A-1368A. (4) Meltes, L. CRC Cflt. Rev. Anal. Chem. 1979, 8, 1-53. (5) Mltchell, D. G.; Mills, W. N.; Garden J. S.; Zdeb, M. Anal. Chem. 1977, 49, 1655-1660. (6) Kaiser, H. Pure Appl. Chern. 1973, 34, 35-61. (7) Kalser, H. Spectrochlm. Acta, Part 8 1978, 336,551-576. (8) Wlnefordner, J. D. “Trace Anaiysls: Spectroscopic Methods for Elements”, Wlley-Interscience: New York, 1976; pp 38-43. (9) Broughton, P. M. 0.; Gowenlock, A. H.; McCormack, J. J.; Nelll, D. W. Ann. Clin. Blochem. 1974, 1 1 , 207-218.
(IO) Cardone, M. J.; Palermo, P. J.; Sybrandt, L. B. Anal. Chern. 1980, 52,
1187-1 191. (11) Blshara, R. H.; Rutherford, B. S.; Dlnner, A. J . Pherrn. Scl. 1975, 64, 12 10-121 2. (12) Foster, J. S.; Langstroth, G. 0.; McRae, D. R. R o c . R . Soc. London, A 1935, 153, 141-152. (13) Larsen, I . L.; Hartmann, N. A.; Wagner, J. J. Anal. Chern. 1973, 45, 1511-1513. (14) Klein, R., Jr.; Hach, C. Am. Lab. (FalrfieM, Conn.) 1977, 0(7), 21-27. (15) Dvorak, J.; Rubeska I.; Rezac, 2. “Flame Photometry: Laboratory Practice”; Butterworth & Co.: London, England, 1971; p 154. (16) Ahrens, L. H.; Taylor, S. R. “Spectrochemical Analysis”, 2nd ed.; Addison-Wesley: London, England, 1961; p 159. (17) Henning, S.; Jackson, T. L. At. Absofpt. News/. 1973, 12, July-Aug. (18) Massart. D. L.; Dijkstra, A.; Kaufman, L. “Evaluation and Optimizatlon of Laboratory Methods and Procedures”; Elsevier: New York, 1978; pp 55-57. (19) Kelly, W. R.; Fassett, J. D. Anal. Chem. 1983, 55, 1040-1044.
RECEIVED for review July 17, 1985. Accepted September 20, 1985.
New Technique in Chemical Assay Calculations. 2. Correct Solution of the Model Problem and Related Concepts Mario J. Cardone Norwich Eaton Pharmaceuticals, Inc.,’P.O. Box 191, Norwich, New York 13815
The analytlcai calculations made by analysts have been shown to be correct only for data that Is free of bias errors. Wlth simple arlthmetlc corrections for the hidden constant and proportional systematlc errors, the formulas normally used produce the correct assay result wlth any of the model sample data palrs by any calculatlonal technlque. The correct calculations are based on the determlnatlon of the true Sample blank by the technlque suggested by Youden In 1947. Both the constant and proportional corriglble error$ are detected and corrected for on the sample under analysis. This overall technique has been named the corrigible error correction (CEC) procedure. The method of standard additions (MOSA) is reexamined and Its true functionality Is dlsciosed. Certain special techniques that also yield correct results are descrlbed. The CEC procedure has been shown to be based entirely on theoretical principles and the correct caicuiatlon equations are derivable solely from the standard and sample response curve functions.
In part 1,it was demonstrated by the conflicting and varied results obtained from over 50 experienced analytical chemists representing most aspects of analytical practice on either of two simple, generic model problems, that confusion and lack of consensus as to what correct calculational practice is, exists. The multiplicity of calculational approaches and the extremely large number of answers obtained in the survey stem principally from an underlying lack of agreement as to the nature of the true blank in an analysis and how it is correctly determined and used. It was also shown in Part 1that the solutions presented, except for a few special cases, could all be reduced to simple formulas that are only slight modifications of the basic forA Procter a n d Gamble Company.
mulas used in common practice (1). Three of these formulas, namely, the standard curve, the single-point ratio (SPRC, or single standard), and the method of standard additions (MOSA) were used in the survey (and are shown in part 1, Figure 2 in the dashed boxes). It now remains to be shown that the one correct answer (for either problem) can be obtained with these conventional formulas, using any calculational technique, simply by incorporating the proper correction constants as shown in Figure 1. The explanation of these constants-their definition, measure, and use-will follow. The special approaches to the model problems applied by the respondents along with others will also be discussed, since these add valuable insight into the nature of the systematic errors. The concepts that were used in designing the model will also be discussed. Through these discussions, all of the questions raised in part 1 will be answered.
CONSTANT ERROR CORRECTION Blank. There is no disagreement in the literature that a constant error is independent of the size of the sample taken for analysis (3). Nor is there any disagreement that a “blank” should be used to correct for this constant error ( 4 ) . Also, there is no confusion concerning the use of blanks that are an integral part of the measurement technique, e.g., the drop error in titrimetry (5))the optical reference in spectrophotometry, or the “noise” in a response tracing. Such blank responses are automatically either physically cancelled or algebraically subtracted from all sample and standard responses. The blank, variously called a reagent blank ( 2 , 4 ) , chemical blank (6),or analytical blank (7), is the response from a solution containing all constituents of the sample except the analyte processed through a procedural step of the method under study or through the entire method. There is general agreement that the chemical blank is valuable as a trouble-shooting diagnostic device for tracking signals from all sources external to the sample, such as from reagents, apparatus, and the environment. A thorough study and
0003-2700/86/0358-0438$01.50/00 1986 American Chemical Society
