438
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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. Gowenlock, A. H.; McCormack, J. J.; Nelll, D. W. (9) Broughton, P. M. 0.; 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
ANALYTICAL CHEMISTRY, VOL. 58, NO. 2, FEBRUARY 1986
A
STANDARD
439
B
II.
I.
= 0.1250
= 0.1313 (problem 1) =0.1188 (problem 2)
p4= 1 0.009844
0.008906
MB = 0.100 wz
SPRC
MOSA IV.
111.
0.1313 (problem 1)
= 0.1250
= 0.1188 (problem 2)
Figure 1. General recommended practice formulas: TYB, the total Youden blank (see text); P , the proportional error factor (see text); other symbols, as in texts, parts 1 and 2.
discussion of all these contributing sources have been detailed by Murphy (7). A good example of some procedural chemical blank measurements in the various steps of a complex method is found in the work of Kelly and Fassett on a standard reference material (SRM) certification analysis (8). . However, what a sample blank is and how it should be measured is quite another matter. A good listing of several kinds of sample blanks and their use in specialized sample matching or compensation is given by Winefordner (2). As defined by Winefordner ( Z ) , “the blank is a solution that contains everything in the sample solution except the analyte” (in pharmaceutical analytical parlance, a “placebo”). When processed through the entire method, the response is the method blank (MB) (6). How good the method blank is depends on how well the sample can be simulated, if it can be simulated at all. Almost everyone believes that the actual sample cannot be used since it normally contains analyte (except for special cases such as biological fluids from unmedicated, healthy patients). The sample blank, in which the sample is present but the analytical reagent is omitted, and the internal blank, in which the sample is present but the fluorophore, for example, has been destroyed, are special cases in which an approximated actual sample is present in the blank solution (2). How good these matrix approximations are is debatable since the actual sample and blank sample solutions are not exactly the same, but they have unquestionably been serviceable. True Blank. The use of approximated sample matrices, however, is not the pathway to answering the question, “what is the true blank in an analysis?” In the standard response curve, no matrix is present, so the system blank, SB (the intercept of the standard response curve), cannot be the true blank except for a special case to be explained below. In both chemical and method blanks (the latter only if and when a suitable “placebo” is available), analyte is not present. The true blank must be determined on the actual sample, so that both matrix and analyte will be present. But how is this possible, since the response from an actual sample is comprised of signals from three possible sources-the analyte, the sample blank, and, if the measurement system has a deficient specificity (selectivity), also a contributory signal from direct interferents (9)? For the purposes of this paper, this direct interferents factor shall be assumed to be absent, since whatever specificity a method possesses is locked in at the usage stage and cannot be changed. Proper provision for it
Flgure 2. Youden sample plots for survey model data: (A) problem 1, part 1, Table I; (B)problem 2, part 1, Table 11.
is a requirement of prior method development; whenever it is detected, it must be eliminated. In order to separate the first two of the three signals mentioned above, at least two sample responses are necessary. From these responses, the true sample blank may be determined by a suitable algebraic calculation. Measurement of the True Sample Blank. In the survey, part 1,six respondents recognized from the sample data that a bias was present, and so they tried to calculate a blank that would be the same for both the sample and the standard-curve data. Analysts logically expect to obtain the same result, within the variance of the method, from different sampling sizes. Since there was no random error in the model problems, exactly the saine result must be obtained. One respondent did get the same result by averaging and rounding and by using the experimental MB arbitrarily across the different calculational techniques. However, one other respondent achieved the same result correctly by using the high and low sample data in problem 1and performing the required algebraic calculation mentioned in the section above. This calculation was done in the same manner that was first published by Kimball and Tufts (10) in 1947 and it has been described in detail (6). It is now mostly of historical interest because in that same year, Youden (11-13) first published his general, simple solution to this dilemma. Algebraic solutions such as the one by Kimball and Tufts (10) are only special cases of Youdenh general solution. Youden’s general approach answers the question in part 1,“How is the true blank correctly determined?”, by applying the linear regression analysis technique to the sample data (6). The Youden sample plot is defined as the sample response curve; the sample responses, S , (ordinate), are plotted as a function of the respective sample masses or concentrations, W, (abscissa), as shown in Figure 2. By linear regression of the sample data, the intercept and slope statistics are obtained wherein the intercept, termed the total Youden blank (TYB) represents the constant error of the method extrapolated to the zero sample level. This is the true sample blank. This Youden sample plot with Youden slope, my, is the simpler technique mentioned in the Discussion, part 1 (6, 14-16). (With real sample data, remote extrapolation to the ordinate as shown in Figure 2A would be extremely inaccurate (17,18) and data across the entire linear dynamic range are required. For these problems, with their random error free data, the extrapolation technique is both illustrative and accurate.) Relationship of the True Sample Blank to the Method Blank. It must be remembered that the intercepts of both the standard response curve, SB, and the Youden sample curve, TYB, are mathematical parameters of the functions over the linear dynamic range of analytical interest. They may well not represent real physical entities. The measured responses will represent real values only if the functions are shown by actual experimental data to project linearly to the respective zero concentration value. That standard or sample
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responses at low concentrations often do not extrapolate linearly to the ordinate is well-known. This very property of the standard and sample response curves in the two survey problems is depicted in Figure 1of part 1and in Figure 2, in which the MB values as given are the “real” physical values as measured on a simulated sample. It should now be clear why experimentally measured MB values cannot be used as the true sample blank except in the special instance when the linear function includes the MB value at the ordinate. For the survey problems, and generally when real samples are analyzed, the mathematical intercept parameter of the function over the linear dynamic range covering the sample responses must be used. That is the functional requirement of the true sample blank and that is why the TYB value was designed into the model. The TYB constant error bias is a relative error whose characteristics are well-known (4). At large sample sizes, the relative error becomes insignificant; at small sample sizes, it can be very significant. This is shown in Figure 3, ref 6, for the problem 1model full range sample data and is generally seen to be a Y = k / X function (14). This, then, is one of the reasons why some data sets show a trend, as in the respondents’ solutions to the model problems when the wrong blank was used. The trend is due to residual relative error, and the trend can be upward, or downward, depending upon whether the incorrect blank over- or undercompensated for the correction. Using the true sample blank yields exactly the same result independent of sample size within the random error of the data. True Sample Blank (TYB) in Real Sample Data. In model data, the intercept and slope of a response curve behave as though the intercept and the slope do not interact (see Figure 1,ref 6). If the intercept is increased or decreased, the entire line moves up or down with no change in the slope. Analogously, changes in the slope do not affect the intercept. Real data, however, do not behave in this simplistic fashion. Mandel and Linnig (19) showed that errors of slope and intercept are always negatively correlated, regardless of the precision of the data. Thus, a positive error in slope will result in a negative error in the intercept and vice versa. This property is o f considerable importance when comparing the standard curve and Youden sample curve intercepts obtained from experimental data and when assessing their relative significance since a negative correlation effect on a regression line can be mistakenly attributed to the effect of concomitant constant and proportional errors. Youden Blank. The TYB response is comprised of the signal due to the nonanalytes in the sample and the solvent, the pseudo signal from the negative correlation bias error, and, most importantly, the signal from any analyte/matrix interaction bias. The SB, on the other hand, can only contain the signals from the nonanalytes in the solvents and the negative correlation bias. Both the SB and TYB also contain any measurement-system constant-error bias that is indigenous to the technique and presumably not removable. A well-understood and familiar example would be the effect of the signal scatter in thin-layer-chromatographic densitometry that results in a significant positive intercept (SB) in the standard response curve. There is quite often a significant difference between the SB and TYB values that can be sample-matrix related, particularly if the measurement system biases, the negativecorrelation effect, and the indigenous system bias error are the same in both the standard curve and the Youden sample curve measurements. This difference between the TYB and SB is termed the Youden blank (YB) from which it follows that SB + YB = TYB (6,15,16). In the special case referred to above for the SB, when the YB is zero, then the SB is also the true sample blank, TYB.
The YB is a valuable diagnostic parameter because it is a significant indicator of sample-related biases. Examining standard curve repeatability is common practice and repeatability may be found to be quite reproducible, as in spectrophotometry, or quite variable, either from day to day or from data set to data set because of system instability, as is common in gas chromatography. It may even vary within a system component as in the plate-to-plate variability in TLC densitometry. These problems are acceptable and are controlled by maintaining all standard and sample responses within the data set limitations that are required by the system instability or component variability. Both SB and TYB confidence intervals at the selected confidence level may be examined to determine whether the expected zero value is included in the interval. Also, it is helpful to interpret the significance of these blanks when they are expressed as the relative SB or TYB values, SB/P or TYB/Y (20). The YB is, in actuality, the true blank for the S , sample responses to be used in the standard curve calculation. However, this correction results in the term, W, = ((S,- YB) - SB)/rn,, which reduces to WJW, = (S, - TYB)/(rn,)(W,). This two-step standard curve calculation was proposed in an earlier paper (6) and was referred to again as such later (14). Thus, it is an unnecessary arithmetical step and may be truncated as in formula 1, Figure 1 in final calculations.
PROPORTIONAL ERROR CORRECTION Method of Standard Additions. Another significant bias error that contributes to the multiplicity of calculational approaches in the survey is the proportional error hidden in problems 1 and 2 at the +5% and -5% relative levels, respectively. Less than 18% of the respondents recognized the principal diagnostic for the detection of proportional error, namely, the ratio of the slope of the MOSA to that of the slope of the standard curve. Only one respondent utilized this P value to correct the proportional error. The use of the proportional error factor, P, has already been proposed in an earlier paper (6). Although the P factor has been referred to in the literature (21,22),its explicit use has not been recognized. One reason for this has been the confusion in the literature concerning the MOSA technique. In a recent example, Delaney (23) observed that the MOSA technique is the worst of eight possible techniques (15) for “detecting interferences caused by unresolved sample components.” This is because an interferent and analyte simply contribute additively to the apparent (i.e., total) signal and cannot be differentiated. The hidden bias in assay results from any technique including the MOSA due to a direct interferent may be positive or negative depending upon the type of signal interference. For example, the bias is positive for an interferent that contributes an additive absorbance and negative for an interferent that behaves as a fluorescence quencher. However, Delaney has correctly pinpointed a limitation of the MOSA technique; it cannot correct for a failure of the required degree of specificity (selectivity) in the method. Delaney states, “The best time to address the possibility of interference is when the method is originally developed,”using another approach (15). Delaney also recognized the advantages of the MOSA technique for “demonstrating the linearity of the quantitation”, which is to say, the linearity of the sample system either at the method development or the validation stage. “Spiked placebos” is also a MOSA technique that is Delaney’s first choice when a placebo is available with satisfactory sample simulation. I agree. The power of the MOSA when it is applicable lies in its ability to be used as a freestanding technique, independent of a standard response curve (part 1, Figure 1 and Figure 1, formula IV) because of its
ANALYTICAL CHEMISTRY, VOL. 58, NO. 2, FEBRUARY 1986 Y
A
1 YoudenPlotSample y 1 I
I
441
B
MOSA Plot
.+/
s,
''
----
Analyie
------
Uncorrected Sample
TYB WZ
Corrected Sample Concentration
Wr ( A d d )
Figure 4. Relationship of MOSA to the Youden sample plot with correction for the true sample blank, TYB: a = A - TYB. W, (added)
W, (added)
Figure 3. Conventional concepts of MOSA plot.
inherent, in situ normalization of the proportional error in the method from the point in the procedure at which the spike addition is made (6,15). Further, in the case of spiked placebo analysis, direct recovery measurements can be made (6). Functionality of the MOSA. Another important reason for the confusion in the literature is the misunderstanding of the functionality of the MOSA technique. Although it can be and is used as a freestanding technique, its functionality is not freestanding. The straight-line equation of the typical MOSA plot shown in Figure 3A, Y = mX + A, is not its true functionality despite its long-standing, deep-rooted status (1, 22, 24). In part 1, all 31 respondents who made the MOSA calculation did so by solving the Y = mX + A straight line function, S,' = (W,)(m) A for the condition that S,' = 0 when Y = 0 at the intersection of the line at the abscissa, so that (Wr)(m) A = 0 yielding -Wr = A/m, precisely the procedure described in the literature (1,22,24). No one expressed concern over the negative and physically meaningless result. However, if one simply subtracts the value of the intercept A (the unspiked sample response) from all values, the Y = mx function line in Figure 3A is generated. This line is in fact the analyte/matrix response curve and the W , value is simply the Y = A value divided by the slope as reported by Chow and Thompson (25,26)in 1955, indicated by Barnett and Youden (27) in 1970 and Massart et al. (22) in 1978, and detailed in 1983 (6, 15, 16). This confusion of a simple calculation is a trivial point when contrasted with the graver misconceptions prompted by the extrapolation calculation technique above. Regarding the question in part 1about the frequent nonagreement between MOSA and standard curve reeults, the concept shown in Figure 3B raises a problem. If Q is the correct result, then different sample sizes 1,2, and 3 in Figure 3B for spiking must result in different MOSA slopes as shown. Direct experimental data for different sample sizes yield different intercept values, in agreement with expectation, A, A', and A". Yet experimentation on this point has proven that the MOSA slope is constant, regardless of sample size (6). There was no disagreement on this point from the survey respondents, who all accepted without comment the constancy of the MOSA slopes (part 1, problem 1)even when the two MOSA results they obtained were significantly different. Clearly, the functionality depicted in Figure 3B is wrong. If the experimentally observed proportional changes in the MOSA intercepts A' and A" are obtained for the respective different sample sizes while maintaining a constant MOSA slope as shown in Figure 3C, then the results, Q and Q , will be biased accordingly. Larsen (28) has presented a model (Figure 3C) for two constant error components of the total
+
+
sample system to show the bias displacement on the abscissa caused by the additivity of the ordinate responses. It is a correct model illustrating the MOSA slope constancy for components of constant error but the bias displacement as shown for the sample-size effect is clearly wrong. True Functionality of the MOSA. For the development of the correct model, in addition to the constancy of the MOSA slope with the different sample sizes taken for spiking, provision for the true sample blank correction must also be made (as shown in Figure 4B). Analogously to the discussion on constant error, the MOSA curve responses also include the analyte signal and the true sample blank, TYB, that in turn also contains any pseudo negative correlation signal effect and any nonanalyte contributions. The unspiked sample, W,,+, is a sample datum point from the Youden sample response line usually from 40% to 70% of the normal sample response range. Thus, the representation of the true sample blank correlation, TYB, can be depicted as in Figure 4, parts A and B, which lead to formula IV, Figure 1, Wx/W, = (A TYB)/(~M)(W,,~). Conceptualization of the true functionality of the MOSA stems from the spiking process itself; that is, the addition of known increments of standard to a known weight of a sample. The amount of the analyte (standard) in the sample is unknown but depends upon the sample's weight and analyte concentration. Different weights of a sample may be taken for spiking, and each spike series in turn generates a dynamic range for the analyte. The result of this rationale is shown in Figure 5B, a merger of the Youden sample plot and the MOSA plot previously shown in Figure 4. The sample chosen for spiking is the highest sample point in the Figure 5B Youden sample plot, with analyte spike additions merely extending the analyte (standard) curve range. The size of the sample taken is shown by the selected sample size slider C. Each level of sample size chosen clearly declares the magnitude of the intercept along the line at the W,, point, and the normal appearance and structure of the MOSA is evident. Why the slope of the MOSA is independent of sample size is also now evident. The abscissa is simply an overall standard scale, and the overall response curve is simply a standard response curve in the matrix (6). The slope of the overall response curve is the slope of the MOSA since this portion of the overall curve is directly in the standard units. Since both the Youden sample plot and the MOSA are functions in the same matrix, the slope of both is the same. The magnitudes are different because the slope of the Youden sample plot contains the analyte concentration term that is, in fact, the ratio of the two slopes, mY/mM. The true sample blank, TYB, a mathematical parameter of the overall functional calibration line S, = (W,)(mM) + TYB is also evident. Finally, the correct MOSA calculation using the MOSA plot is seen to be Sxr = (W,)(mM) + TYB that leads to the formula W,/ W, = (A - TYB)/
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ANALYTICAL CHEMISTRY, VOL. 58, NO. 2, FEBRUARY 1986
Standard Plot
Y
Youdensample Plot
Y
y’
MC:.
each respective sample data pair, the result, corrected for both proportional and constant error, is 0.1250. Formula I1 may, of course, be applied directly. Formula I is the special case of formula I1 where P = 1.00, i.e., there is no proportional error. SPRC. The proper correction constants to be applied to the basic single-point-ratio-calculationformula (part 1, Figure 2) is given in Figure 1, formula 111, wherein the sample response, is corrected by using the true sample blank (the total Youden blank) TYB, and the standard response, S,, is corrected by using the system blank, SB. With this formula, any sample data pair and any standard data pair (part 1, Tables I and 11)will yield the one value of 0.1313 for problem 1 and 0.1188 for problem 2. Again, only correction for the constant error is made since the SPRC (as in the case of the standard curve calculation) uses a calibration curve without a matrix. The proportional error is removed as before by dividing the SPRC result by the P factor. Again, the result for every case is 0.1250. In part 1,most of the respondents who received problem 2 commented that since the SB was zero, the conventional standard-curve calculation and the SPRC were the same. This is true and is due to the fact that when SB = 0, the codes in part 1, Figure 2, codes A 3.2.1 and A4.2, are mathematical identities. (The slope, m,, in code 3.2.1 is equal to the SI/W, term in code A4.2 from the standard curve expression, S, = (mJ(W,) + 0.) However, these identities both yield the same incorrect result. Only when the YB or the TYB is zero will a result corrected for constant error be produced. This can happen in two ways: one, when the SB is equal to the TYB, and two, when the TYB is zero, as may be seen for both cases from the expression,SB + YB = TYB. When the TYB is zero, there is no net constant error in the sample system, but this is the special case in which the YB has been nullified by the SB bias. The usage for the basic uncorrected SPRC, W,/ W, = S,/S, X W,/ W, offers considerable practical advantages in applied analysis, e.g., the requirement of a single standard datum point rather than a full calibration curve. This is especially valuable when running a series of samples in which an intermittent standard serves as a satisfactory running monitor. In many cases in which the methodology is simple, a direct ratio of the sample response to the standard response provides the analytical concentration. However, this approach is possible only because the bias error of the SPRC technique is either ignored or considered acceptable (20, 30). If the constant error in a single standard calibration is totally removed by using SPRC formula I11 (Figure 1))then all the advantages mentioned above disappear, because the correction constants, TYB and SB, must be appropriately determined from their respective ‘responsecurves. Note that formula I11 and formula I are mathematical identities, since the term in formula 111, (S, - SB)/ W,) comes directly from the standard response curve, S, = (m,)(W,) + SB. This simply means that the SPRC technique should neuer be used when calculating results on research data, unknown samples, or unknown systems in general since the degree of acceptable error is most likely not known. This mistake is, unfortunately prevalent in the use of data from instrumental methodology. MOSA Calculation. The MOSA technique provides an in situ normalization of the proportional error. This normalization occurs because the same procedural operations are performed on both the unspiked and the spiked samples, so that a constant (i.e., reproducible) proportional bias is introduced onto the samples across the dynamic range. Correction for the constant error is provided by the total Youden blank, TYB, as shown in formula IV (Figure l),and Figures 4 and 5. The result for all the sample data is 0.1250 for both problems.
s,,
I
wz, u
X
Wr
-X
I
1
W.
-
Wr (added
w, (blall
Figure 5. Response curves: S , and S, (Sx’) are the anaiyte responses from the standard and sample, respectively: W , and W , are the standard and sample concentrations, respectively; W , , is the
unspiked sample concentration.
( m ~ W,,,J ) ( (formula IV, Figure 1). The picture of functionality can be completed with an example from what is one of the most easily simulated sample matrices, urine. If a urine sample containing no analyte is spiked, this MOSA technique will generate a real sample matrix with a response curve such as the overall line shown in Figure 5B,with the functionality S, = ( Wr)(mM)+ TYB. The same result can be conceived by pushing the slider C all the way to the ordinate (since the analyte in the sample is zero). In work with trace analytes, the slider C is a little higher; for samples such as pharmaceutical dosage forms, it is higher still. In every case, the total “after” spiking analyte concentration must be kept below the demonstrated linear dynamic range of the three curves-the standard response curve (Figure 5A), the Youden sample plot (Figure 4A), and the MOSA plot (Figure 4B). Within this constraint, calibrations with an external analyte standard are MOSA techniques simply dependent upon the matrix. If the standard addition is made to either a simulated sample matrix, whether the analyte content before spiking is zero, as in a placebo, very low, as in trace analysis, or a t nominal concentrations, as in environmental analyses or in pharmaceuticals dosage forms, then the familiar MOSA techniques and calculation become evident. If the three functional curves are all linear over the dynamic range of interest, and the MOSA plot does not exhibit any stoichiometriclimitations such as those discussed by Klein and Hach (29),then the MOSA technique, hence correction for the proportional error, is applicable. The overall technique for the correction of the constant and proportional error has been termed the corrigible error correction (CEC) technique (6,14-16).
CORRECT SOLUTION OF THE SURVEY MODEL PROBLEMS Corrigible Error Correction Technique. The first step is to determine the slope, my,and intercept, TYB, of the Youden sample response curve, performed most easily by simple linear regression (Figure 2). (The tests for linearity of the response curves (Le., goodness of fit) and other statistical considerations are not discussed in this paper.) Standard Curve Calculation. Using formula I, Figure 1, and substituting the appropriate values of the standard curve slope, m,, and TYB for the respective problems leads to for problem 1, W,/ W, = (S, - 0.185)/(0.075)(W2)and for problem 2, W,/ W, = (S, + 0.060)/(0.075)(Wz). Substitution of the respective sample data (part 1,Tables I and 11),W, and S,,leads in each case for problem 1to a W,/ W, result of 0.1313 and for problem 2,0.1188. These resulta have been corrected for the constant error. Proportional error is corrected for by dividing by the P factor as in formula 11, Figure 1, where P = mM/ms. For problems 1 and 2, respectively, P = 0.07875/0.075, or 1.05, and P = 0.07125/0.075, or 0.950. For
ANALYTICAL CHEMISTRY, VOL. 58, NO. 2, FEBRUARY 1986
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Table I. Data for Reciprocal Weight Technique Calculations Problem 1 regression
62.4746 82.7915 103.1085
0.8 1.0 1.2
0.01601 0.01208 0.009699
0.13720 0.13421 0.13239
0.14406 0.14092 0.13901
1.0000 0.1250 0.762d
1.0000 0.1313
r intercept slope
0.8OOc
Problem 2 regression
sx
wz
0.400 1.250
51.6491 147.0877
1/w, (X) 0.01936 0.006799
W,/W,'
(Y)
0.10326 0.11331
x,y
x,Yr
-1.0000 0.1188 -0.8008
-1.0000 0.1250 -0.842h
WJWJ ( 9 0.10870 0.11927
r intercept slope I
W,/ W, = (S, - 0.125)/(0.075) (WJ. W,/W, = (S, - 0.125)/(0.07875)(W2). CSlope(theory) = YB/m,or slope = 0.060/0.075 = 0.800. dSlope (theory) = YB/mM or slope = 0.060/0.07875 = 0.762. e WJW, = S,/(0.075)(Wz). f W,/W, = S,/(0.07125)(Wz). 8Slope (theory) = YB/m, or slope = -0.060/0.075 = -0.800. hSlope (theory) = YB/mM or slope = -0.060/0.07125 = -0.842.
Again, formula IV and formula I1 (Figure 1) are mathematical identities as seen when the value of P = mM/mais substituted for P in formula 11, which then reduces to formula IV. In formula IV, the sample for spiking is W,,,; in formula 11, it is simply W, but this is a simple matter of the designated sample size on the overall functional response curve in Figure 5B. Special Calculations, All three calculational techniques produce the correct result of 0.1250 using the CEC technique. This criterion of obtaining one result was established in part 1as a necessity for an answer to be considered correct. In addition, some special techniques will also produce the correct answer. Extrapolation to Infinite Sample Size. In part 1,it was reported that one respondent calculated the correct answer by extrapolating calculated assay results to the asymptotic value at the infinite sample size (Figure 3, ref 6). This was done by determining the sample response curve expression by linear regression of the sample data and obtaining S, = 0.009844 W, 0.185 (Figure 2A). From this expression,a series of increasing S, response values was generated for selected W , values that were then used in the expression W, = (S, MB) /mM to generate the respective analyte concentrations in the sample (which are corrected for the proportional error since the MOSA slope was used). The increasing values of W,/ W, thus obtained clearly approached the value of 0.1250 asymptotically; a value of 0.1254 resulted from a W, value of 3000, which represents almost a 30-fold increase in sample size from the highest sample data provided (part 1,Table I, problem 1). This is a correct solution, but it has no analytical practicality or generality. The technique is tedious, and the final asymptote value is somewhat subjective. However, it is resourceful, and, most important, it utilizes the concepts developed here. For example, the TYB value was determined and a correct usage of the MOSA slope value was recognized, so the correct recommended solution was only a hairbreadth away. Thus, instead of reverting to the experimental MB for correction of the S, use of the correctly determined TYB value would have provided the answer on any of the sample data pairs in the expression = (s,- 0.185)Im~.For example, W, = (1.000 - 0.185)/0.07875 = 10.349 and W,/W, = 10.349182.7915 = 0.1250. Reciprocal Weight. Extrapolation to an infinite sample size can be done in a very practical fashion that does not even require the determination of a true sample blank. None of the respondents used this technique, but it has been reported (14). In this simple procedure, the conventional standard-
+
w,
0.10
01 0
1
I
I
0.005
0.010
0.015
I
0.020
IIW,
Flgure 6. Reciprocal weights plots: problem 1, (0)X, Y; problem 2, (0)X, Y; (0)X, Y'. Data are from Table I.
(m) X, Y';
curve W,/ W, assay values are calculated from the sample data (which should be spread across the linear dynamic range). The reciprocals of the sample weights, 1/W,, are also calculated. These values are regressed, 1/ W,= X and W,/ W , = Y, and the intercept of this function is the corrected assay value. The data for this calculation are shown in Table I and plotted in Figure 6. If the slope of the standard response curve, m,, is used to calculate the W,/ W ,values, the intercept value is only corrected for the constant error. If the slope of the MOSA curve is used, however, the result is corrected for both the proportional and constant error. This technique relies on the true functional relationship in analysis, that is, that the analyte concentration is a function of the reciprocal sample weight, not the sample weight. This function is fundamental and has been derived theoretically (14). The slope of the function is not an arbitrary value but can be calculated from the response-curve constants. Thus, for data calculated using the ma, the slope is equal to the YB divided by the ma; for data Calculated using the mM, the slope is equal to the YB divided by the mM. See Table I. This technique is especially practical whenever an accurate assay value for a sample is the predominant issue, as in, for example, a troublesome out-of-spec production lot in QA or a standard reference material (SRM) in which an absolute, accurate certificationvalue is the objective (8). The technique does not provide any diagnostic indication of proportional error, but if the functions (see Figure 6), for either me- or mM-calculated W,/ W , values, do not have a zero slope, a source of constant error is indicated. [The slope can be subjected to a test for statistical significance by comparing the confidence interval of the slope (at the selected confidence level), as easily obtained from the linear regression statistics, against the zero value that the interval is expected to include. If the interval does not include zero, then the slope is sta-
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ANALYTICAL CHEMISTRY, VOL. 58, NO. 2, FEBRUARY 1986
tistically significantly different from zero at that confidence level.] Equal Analyte, Equal Signals. Two respondents noted that in the two MOSA data sets in problem 1, two equal signal values were listed in both sets (part 1, code A5.7). Thus, in sets 1A and 1B (part 1, Table I) there are two values of S, = 1.000 and also two values of S, = 1.6200. Both respondents correctly assumed that equal signal values mean that for the two respective samples, there were equal analyte concentrations. If the unknown analyte concentration in the unspiked sample is x , then for S, = 1.000,2.540 + 62.4746~= 82.7915~. For S, = 1.6200, 10.413 + 62.4746~= 7.873 + 82.7915~.In both expressions, x = 0.1250. The correct answer is obtained because the true sample response curve is employed in the interpolation of the sample data. In the model design, the placement of the S, = 1.6200 pair was deliberate; the S, = LOO0 pair was an oversight. This option demonstrated the resourcefulness of the two respondents, but it is not a practical technique and has no possible general applicability. Slope-RatioAssay. Five respondents utilized one of the two slope ratio techniques (part 1,Figure 2, codes B2.1 and B2.2), either the ratio of the Youden sample curve to that of the standard response curve, my/m,, or the ratio of the Youden sample curve to that of the MOSA curve, mY/mM. These responses were quite unexpected because there is apparently only one reference to this technique in the literature (6) and none of the respondents otherwise indicated that they were aware of that paper. The ratio my/m, corrects only for the constant error, yielding the value of 0.1313 for problem 1 (0.009844/0.075) and 0.1188 for problem 2 (0.008906/0.075); hence, its applicability is somewhat restricted. The other ratio, my/mM corrects for both constant and proportional error and yields the correct answer, 0.1250, for both problems (0.009844/0.07875 and 0.008906/0.07125). The potential practicality and applicability of this ratio are important. One characteristic that should make this technique more interesting is that the respective slopes need be over only a restricted range of the sample curve and not over the entire linear dynamic range. An accurate determination of the intercept parameter is not necessary. In this respect, then, the technique enjoys the same advantages as the bracketing technique (1) as discussed below, increased precision and compensation for any nonlinearity in the response curves. Its advantage over the conventional bracketing technique is that correction for both constant and proportional error is achieved. Bracketing. Although none of the respondents utilized this technique, it is important to place it into perspective with the concepts developed here. This technique has been discussed by Winefordner (1) so only its limitations concerning corrigible bias errors need be elucidated here. When executed as described (briefly, a sample datum point between two adjacent standard points), the technique we9 a relatively small portion of the standard-response-curve range. As such, there is no correction for either constant or proportional error (unless the standard response cume happens to be the same functional line as the sample response curve), although it does still have the advantages of increased precision and compensation for nonlinearity as Winefordner describes. Correction of the bracketing technique for this deficiency by corrigible error correction simply leads one to the sloperatio assay technique discussed above. However, if a standard reference material (SRM) or a standard reference method is available so that the two adjacent reference points would be on the sample response curve, then the constant and proportional corrigible errors would be corrected while retaining the unique advantages of the technique. In summation, the bracketing and the slope ratio assay techniques are quite
comparable if an SRM is used with the former. Dichotomous Search. Two respondents used the dichotomous search technique (31): one, by decreasing the difference in the SI/W, ratio of the low and high sample values to zero with successive estimates of an error correction to the S, responses; the other, by decreasing the difference in the calculated W,/ W, assay results of the low and high sample values with successive estimates of an error correction to the S, - SB responses. The former determined the true sample blank to be the TYB and the final calculation was made using formula 1, Figure 1. The latter determined the true sample blank to be the YB; the final calculation was made as a correction to code A3.2.1, Figure 2, part 1;i.e., W,/ W, = [(S,SB) - YB]/(m,)(W,), which reduces to formula 1,Figure 1. For both, the assay value of 0.1313 was obtained for all sample data pairs. This result is correct but is uncorrected for the proportional error. This resourceful use of this technique is tedious, but it does produce the correct result, though in a different mathematical manner than the more efficient Youden sample curve regression. (Note: the powerful dichotomous search technique (31) has many other uses for which it is uniquely suited.) Kimble and Tufts Two Sample Data Pairs. Earlier, in True Blank section, it was observed that a minimum of two sample responses was necessary for the true sample blank to be suitably calculated algebraically. An example of this was published by Kimble and Tufts (8). One respondent made the Kimble and Tufts calculation in a more elegant manner from the low and high sample data pairs in problem 1 as follows: (0.8 - x)/62.4746 = (1.2 - x)/103.1085, where x = 0.185, Le., the value of true sample blank, TYB, the same value as obtained by linear regression of all the sample data. For the random-error-free model data, the exact TYB value is obtained; however, with real data, the regression technique is recommended because all of the data points are used; hence it is the more accurate procedure. Although, in principle, they are equivalent procedures for calculating the TYB value, in today’s practice, the linear regression technique is the simpler concept with universality since the regression technique lends itself to functional linearity testing whereas a two point function does not.
CONCLUSION The solutions to the model problems provided by the survey respondents (part 1)are clearly inadequate because the role of systematic constant and proportional errors in sample systems has not been properly elucidated in the literature. Therefore, each analyst relates to his sample data in a different manner. The conventional calculations (1)have been shown to be correct only for bias-error-free systems and do not provide either for diagnostic detection of these errors or for their correction, if they are present. This paper has described the role of corrigible constant and proportional errors and techniques for their detection, measure, and use. Formulas incorporating the proper constant and proportional error constants for calculations on the sample under analysis have been provided for any calculational technique that utilizes an external standard. For clarity and simplicity, the concepts in this paper have been discussed largely in the frame of a random error free model. In part 3, application of the model calculations and the related conuepts to real samples will be shown together with a recommended general practice that detects bias error and determines its nature. With this information, a better independent search for the root source of bias error in a method from “first principles” can be made for its possible elimination. The recommended practice provides for calculations with in situ corrections on the sample under analysis. Lastly, the general recommended formulas will be shown by
Anal. Chem. 1906, 58,445-448
theoretical derivations to be based solely on the fundamental standard and samde remonse curve relationshim. * GLOSSARY A intercept of MOSA curve a intercept of MOSA curve corrected for constant error (=A - TYB) b intercept term in generic slope-intercept form of straight line equation, Y = mX b C analyte concentration in sample (=mY/mM) C’ constant error-corrected analyte concentration in sample (=CP) CEC corrigible error correction m slope term in generic slope-intercept form of straight line equation, Y = mX + b slope of standard curve m, slope of MOSA curve (=m,P) mM slope of Youden sample curve (=m,PC) my MB method blank, experimentally measured on an analyte-free simulated sample or placebo MOSA method of standard additions P proportional error factor ( = m M / m s ) standard (signal) response sr sample (signal) response SX MOSA (signal) response sx’ SB system blank, intercept of standard curve (system constant error) SPRC single-point-ratio calculation (single reference standard) TYB total Youden blank, intercept of Youden sample curve (sample and system constant error) mass or concentration of standard wr sample analyte concentration relative to sample W X response, S, or Sx‘ mass or concentration of sample wz mass or concentration of unspiked MOSA sample ?,u independent variable in generic slope-intercept form of straight line equation, Y = mX b Y dependent variable in generic slope-intercept form of straight line equation, Y = mX b YB Youden blank, analyte-matrix interaction constant error (=TYB - SB)
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LITERATURE CITED Wlnefordner, J. D. “Trace Analysis: Spectroscopic Methods for Elements”; Wlley Interscience: New York, 1976; pp 38-42. Wlnefordner, J. D. ”Trace Analysis: Spectroscopic Methods for Elements”; Wlley Interscience: New York, 1976; p 33. Skoog D. A.; West, D. M. “Fundamentals of Analytical Chemistry”, 3rd ed.; Holt, Rinehart and Winston: New York, 1976; p 48. Skoog D. A,; West, D. M. “Fundamentals of Analytical Chemistry”, 3rd ed.; Holt, Rinehart and Winston: New York, 1976; p 51, Figure 3.1. Kolthoff, I.M.; Stenger, V. A. “Volumetric Analysis”, 2nd Revised ed., Interscience: New York, 1942; Vol. I, p 143. Cardone, M. J. J. Assoc. Off. Anal. Chem. 1983, 6 6 , 1283-1294. Murphy, T. J. NBS Spec. Publ. (U.S.) 1976, No. 422, 509-539. Kelly W. R.; Fassett, J. D. Anal. Chem. 1983, 5 5 , 1040-1044. Wilson, A. L. Talanta 1974, 2 1 , 1109-1121. Klmball R. H.; Tufts, L. E. Anal. Chem. 1947, 19, 150-153. Youden, W. J. Anal. Chem. 1947, 19, 946-950. Youden, W. J. Biometrics 1947, 3, 61. Youden, W. J. Mater. Res. Stand. 1961, 1, 268-271. Cardone M. J.; Lehman, J. G. J. Assoc. Off. Anal. Chem. 1985, 6 8 , 199-202. Cardone, M. J. J. Assoc. Off. Anal. Chem. 1983, 6 6 , 1257-1282. Cardone, M. J. J. Assoc. Off. Anal. Chem. 1984, 6 7 , 12A. Daniel C.; Heerema, N. J. Am. Stat. Assoc. 1950, 45, 546-556. Neter J.; Wasserman, W. “Applied Linear Statistical Models”; Richard D. Irwin, Inc.: Homewood, IL, 1974; p 61. Mandel, J.; Llnnlg, F. J. Anal. Chem. 1957, 2 9 , 743-749. Cardone, M. J.; Palermo, P.J.; Sybrandt, L. B. Anal. Chem. 1980, 52, 1187-1 191. Henning, S.; Jackson, T. L. At. Absorpt. Newsl. 1973, 12, July-Aug. Massart, D. L.; Dljkstra, A.; Kaufman, L. “Evaluation and Optimization of Laboratory Methods and Procedures”; Elsevier: New York, 1978; pp 55-57. Delaney, M. F. Llq. Chromatogr. 1984, 3 , 85-86. Kolthoff, I. M.; Sandell, E. B.; Meehan E. J.; Bruckensteln, S. Qualltatlve Chemical Analysis”, 4th ed.; The Macmlllan Co.: Toronto, Ontario, Canada, 1969; p 419. Chow, T. J.; Thompson, T. G. Anal. Chem. 1955, 2 7 , 18-21. Chow, T. J.; Thompson, T. G. Anal. Chem. 1955, 2 7 , 910-913. Barnett R. N.; Youden, W. J. Am. J. Clln. Pathol. 1970, 5 4 , 454-456. Larsen, 1. L.; Hartmann, N. A.; Wagner, J. J. Anal. Chem. 1973, 45, 1511-1513. Klein R.; Hach, C. Am. Lab. (Falrfldd, Conn.) 1977, 9 , 21-27. Delaney, M. F. Llq. Chromatogr. 1985, 3 , 264-268. Wllde, D. J. “Optimum Seeking Methods”; Prentlce-Hall: Englewood Cliffs, NJ, 1964; pp 23-24.
RECEIVED for review July 17, 1985. Accepted September 20, 1985.
Modification and Evaluation of a Thermally Desorbable Passive Sampler for Volatile Organic Compounds in Air Robert W. Coutant*
Battelle Columbus Division, Columbus, Ohio 43201 Robert G. Lewis and James D. Mulik
Environmental Monitoring Systems Laboratory, U.S. Environmental Protection Agency, Research Triangle Park, North Carolina 27711
The effect of fluctuating concentration on the sampling rate of a thermally reversible passlve sampler is discussed, and guidelines are presented for mlnlmlzlng the error associated with this phenomenon. A modified passlve sampler developed using these guidelines Is described, and results of laboratory tests of this device are given.
In two previous papers (1,2),we described the development and evaluation of a thermally desorbable passive sampling 0003-2700/86/0358-0445$01.50/0
device, PSD, and we presented both a generalized performance model for such devices and a simplified model that is applicable to thin PSD’s. In brief, the simplified model describes the time dependence of the time-weighted average sampling rate, R, as a function of the initial sampling rate, Ro,the weight of sorbent, W, and the retention volume for the sorbate/ sorbent pair, V b , viz. R / R o = (1- e-kt)/kt (1) where
k = Ro/ WV, 0 1986 American Chemical Society
(2)
