Anal. Chem. 1980, 52, 1187-1191
Potential Error in Single-Point-Ratio Calculations Based on Linear Calibration Curves with a Significant Intercept Mario J. Cardone” and Philip J. Palermo Pharmaceutical Research Division, Norwich-Eaton Pharmaceuticals, Box 19 1, Norwich, New York 138 15
Larry B. Sybrandt
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Stuart Pharmaceuticals, Division of ICI Americas Inc., Route 202 and New Murphy Roads, Wilmington, Delaware 19897
Calculations based on a single-point standard linear calibration curve which has a finite intercept are a potential source of bias error. The error has been shown to be a function of magnitude of the slope and intercept, and the magnitude of the difference in the standard and sample responses. For maximum accuracy, the ratio of the difference in responses to the standard response should be the lowest possible. For any response curve, a reliable calculation of the percent error resulting from making a single-point-ratio calculation (SPRC) where a finite intercept exists is shown so that a decision as to its proper use or non-use may be made. I n most cases, even when an analytically significant intercept is found in a linear calibration curve, the SPRC technique may stili be employed by calculating the sample signal to standard signal, S,/S, ratio range for an acceptable % error bias range. This calculation of the S,/S, ratio constitutes a trade-off between the use of a single standard and a shortened linear dynamic range with its possibility of more reanalyses against the use of the entire linear dynamic range with its mandatory requirement of more standards but the lesser possibility of reanalyses.
Calibration-curve-based methodology ( 1 ) is one of the most common quantitation techniques, and one which is increasing rapidly with the growing usage of instrumental methods. It is our concern that analytical chemists often fail to give due consideration to the potential error which may arise from a nonzero intercept in a linear calibration (response) curve. Other sources of error in the calibration process are not the subject of this discussion. For an excellent coverage of standardization procedures, complete with statistical evaluation techniques, the collection of papers in the National Bureau of Standards Publication 300, Vol. I should be consulted ( 2 ) . All analytical chemists should know that if a response curve has an intercept significantly different from zero, proper provision in the calculation must be made. The conventional, and always correct techniques, whether the function is linear or nonlinear, are graphical interpolation from the plot (3-5) or the use of an algebraic equation of the function (6,7). These techniques can be quite sophisticated as illustrated in the work of Govindaraju, Mevelle, and Chouard (8) who use computer-stored calibration curves as second-degree polynominals with three regression coefficients for each curve, and in the work of Mitchell et al. ( I ) who employ a multiple-curve procedure which uses a series of least-squares regression equations with confidence bands computed from the calibration data. These techniques do not require takiig any more experimental data than as normally practiced. For the special case where a response curve consistently is found to have a zero intercept, a calibration can be made by 0003-2700/80/0352-1187$01 .OO/O
measuring a single reference point standard and determining the sample content by use of a single-point-ratio calculation (SPRC). Although this is a common, well-understood fact, it has been difficult to find specific statements to this effect in the literature. Broughton (9) points out that “single-point calibration assumes a linear relationship passing through the origin and the calibration point. . .”. Some authors (10, 11) have alluded to this indirectly by stating that “the use of a calibration standard at a single concentration assumes not only that the detector response is linear, but also that the Calibration line goes through zero”. Wittmer and Haney (12) and Won et al. (13) state correctly after having shown that the intercept value was insignificantly low, that these data “indicate that the liquid chromatographic method can be used with a single-point standard”. Authors such as Sondack (14) who states that “the extrapolation results passed through the origin” and King et al. (5) who conclude “a standard graph rather than a one-point standard is required for maximum accuracy in the procedure” are examples of those who clearly understand the requirement. Unfortunately, too often authors simply state that the response curve was linear but fail to report whether a significant intercept was found (15-18). In one case where the response curve itself was reported, there is no indication of how the calculation was made (19). The decision as to whether or not to use the SPRC technique is an important one not only since its usage offers considerable time savings in simplicity but because of the limitation of many computer-based chromatographic data processing systems which do not allow for the possibility of a nonzero intercept in their calculation. However, usage of the SPRC must follow after full consideration of the potential error as demonstrated by Corder et al. (20). As will be shown, the intercept value must be reported, preferably as part of the regression analysis data especially since it is now the editorial policy of many analytical journals to omit straight-line calibration curves (21). In cases where the SPRC technique is used, and an author either makes only a simple linearity statement, or discloses by a response curve or by regression analysis data that an intercept value was observed, the reader cannot make any conclusion as to whether or not this potential error source was simply ignored as negligible or was unrecognized (15, 17,18, 22-24). It is only fair to point out, however, as this paper will show, that in either case, under certain fortuitous circumstances involving the sample and standard responses, no computational error in the result may, in fact, exist. Intercepts in response curves are often encountered with positive or negative values almost always of the algebraic sign as predictable from the type of measured signal: positivefluorometry (3,ZO), spectrophotometry ( 2 2 , 2 3 ) ,spectrofluorodensitometry or spectrophotodensitometry (3, 7 , 25); negative-polarography ( 4 ) , GLC (19),HPLC (6, 24). It is the purpose of this paper to evaluate both qualitatively and 0 1980 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 52, NO. 8, JULY 1980
Quantitative. Using Figure 1 to derive the error function E associated with assuming a zero intercept for a linear response function using the SPRC technique, we have:
E=C-T
(1)
where E is the absolute error in the same units or scalar value as the abscissa of the response curve; C is the calculated mass, concentration, or scalar value on the abscissa of the response curve as calculated from the response curve with assumed zero intercept; and T i s the true mass, concentration, or scalar value of the analyte species as plotted on the abscissa of the true response curve. Assuming a response curve with zero intercept, line b, the result for C is then:
c=~
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Figure 1. Response curve with significant intercept. (a)True response curve. (b) Assumed response curve
quantitatively the bias error caused by an SPRC calculation from an assumed linear response curve through zero for the case where the true response curve is a straight line with a significant intercept.
THEORETICAL Qualitative. Broughton (9) has presented a graphical treatment of single-point calibrations for both linear and nonlinear response curves, the latter which did or did not pass through the origin. We are interested here only in linear response curves with a significant intercept. The intercept may be real, that is the regression line includes the zero concentration response for the contiguous linear dynamic range of interest. This was the case for certain references previously cited ( 3 , 4 , 7, 22). Particularly for curvilinear response curves, the intercept may be only a mathematical parameter of the regression expression for the extrapolated linear dynamic range of interest (Figure 1in Ref. 25). In such a case, the zero concentration response value is analytically immaterial. T h e SPRC error associated with a significant intercept is shown in Figure 1. Line a is the response curve as established by measuring the responses, S,’s obtained from two or more different masses or concentrations taken of a pure reference standard. The term response as used here refers to that parameter by which an unknown will be determined, e.g., the relative fluorescence, absorbance, peak height, peak area, or sample to standard response ratio when using the internal standard technique. Point I is the intercept on the ordinate. Point G represents the response obtained for a given weight of reference standard. In a single-point-ratio calculation, SPRC, the assumption is automatically made that the line passes through point G and the origin as described by the dashed line b. When the sample and standard have the same response, S , or S,, the resulting point G for both falls on the line b, the assumed function, and thus no error results. For the case where the amount of the analyte species in the sample is greater than that for the standard, as represented by point D in Figure 1,then the falsely high result of D’ on line b will occur. The error is presented by D’ - D. Similarly, for the case where the amount of the analyte species in the sample is less than that for the standard, as represented by point F in Figure 1, then the falsely low result of F’ on line b will occur. The error is represented by F’ - F. The reversal in sign of the error for values above and below the coincident value was also noted in Broughton’s presentation (9),as was the observation that the absolute error increases as the difference in the sample and standard responses increases.
~ , / ~ , ~ (2) ~ ~
where S , is the response from the analyte species in the sample, S , is the response of analyte species in the standard and W , is the true mass or concentration of the standard analyte species. However, the value for T i s obtained from the expression of the true response curve, line a:
S , = mT
+I
(3)
where m is the slope and I is the intercept. Rearranging Equation 3,
T=(S,-I)/m
(4)
Substituting Equations 2 and 4 into 1 gives:
E = (Sx/S,)(W,)- ( S , - I ) / m
(5)
Now, it is also true that the reference standard point also falls on the true response curve, line a, so that
S , = mW, + I
(6)
W , = (SI- O / m
(7)
or, and substituting Equation 7 into 5 gives:
E = (S,/S,)[(S,- n / m l - ( S , - O / m
(8)
which reduces to,
DISCUSSION OF ERROR FUNCTION Equation 9 shows that when the intercept value is zero, the error is zero and that when the standard and sample responses are equal, the error is zero. Also that the error is directly proportional to the difference in the responses, the magnitude of the intercept value, and inversely proportional to the slope. All these points were only intuitively obvious in the graphical presentation. In addition, from Equation 9 it can be seen that the error is directly proportional to the term AS/S,, which is smallest for small values of A S and high values of S , which means that for optimum accuracy, the responses should be closely matched a t the high end of the linear dynamic range. Rewriting Equation 9 into the form,
E = (1 - S x / S r ) ( I / ~ )
(10)
readily permits prediction of the sign change in the error as mentioned in the graphical presentation when response values are mismatched, above or below the coincident value, that is as the ratio S,/S, varies from 1,thus: For positive
ANALYTICAL CHEMISTRY, VOL. 52, NO. 8, JULY 1980 ________.
Table I. Linear Regression Response Curve Statistics std error slope rn = 1.020 intercept I = -0.0368 std-err-est. = 0.00793
0.0130 0.0137
= 0.977
%variation ((std-err-est/u)X 1 0 0 ) = 0.81 relative intercept = -3.77% linear dynamic range = 2-12 pg injected nominal sample = 7 p g injected correlation coefficient R = 0.9997 R 2 = 0.9994 n= 5
values of I , E is negative if S,> SI, positive if S , < SI. For negative values of I , E is positive if S , > SI,negative if S,
