Straight-line calibration

C.I Linear Regression of Straight-Line Calibration Curves... 

Boque, R. Rius, F. X. Massart, D. L. Straight Line Calibration Something More Than Slopes, intercepts, and Correlation Coefficients, /. Chem. Educ. 1993, 70, 230-232. 

For the determination of DNA size suitable markers are included in one of the lanes. A straight line calibration graph can be obtained by plotting distance migrated against log10 base pairs of the markers. 

Decision and Detection — Linear Calibration Curves. Before examining the actual Fenvalerate GC data, let us consider the basic linear calibration relations. (What follows was inspired in part by Hubaux and Vos (14), to which the reader might refer for supplemental detail.) If we represent a straight-line calibration as... 

In this experiment one highly conducting ion is replaced by a less conducting ion. Since the initial and final conductivities are known, this enables a straight line calibration to be prepared. 

As the motor-driven syringe slowly injected 1 yL of pure vinyl acetate into the closed recirculating loop at 3.89 nL/s, the change in absorbance was recorded. This procedure was repeated five times and the results were averaged. The calculated concentration plotted against the absorbance gave a straight line calibration curve. 

A straight-line calibration curve over many orders of magnitude can often be found in the case... 

The worksheet functions SLOPE known ys, known xs) and INTERCEPT(/f/ioiv/i ys, known xs) return the slope, m, and intercept, b, respectively of the least-squares straight line through a set of data points. For example. Figure 11-1 illustrates some spectrophotometric calibration curve data (concentration of potassium permanganate standards in column B, absorbance of the standards in column C). The formula =SLOPE(C4 C8,B4 B8) in cell C10 was used to obtain the slope of the straight-line calibration curve. The SLOPE and INTERCEPT functions should be used with some caution, since they do not provide a measure of how well the data conforms to a straight line relationship. 

Very commonly, as in the calibration curve example, we use the measured y-value to estimate an x-value. Once the slope and intercept of a straight-line calibration curve have been established, it is easy to calculate an x-value (e.g., a concentration) from a measured y-value. The estimation limits of the estimated x-value are given by the equation... 

Two x-ray fluorescence spectrometers were used for the analyses a General Electric XRD-6 for iron, copper, tin, and antimony, and a General Electric XRD-5 for nickel, silver, and lead (the latter machine has updated electronics and gave superior results for these three elements). Four certified standards from the National Bureau of Standards were used for each element to obtain a straight line calibration curve using linear regression (10). The experimental conditions used for the determination of each element were given by Carter et al. (10). 

Fig. 15. Straight line calibration with confidence interval of calibration defined in y direction a and confidence intervals of the analytical result defined in x-direction b applying variing amounts of repeated measurements in the analytical procedure S8>...

A least-squares method gave a straight-line calibration graph with the relationship... 

In atomic absorption spectrometry, the use of inverse sensitivity is commonly defined by the mass or concentration per 1% absorption (0.00436 absorbance units). The sensitivity can be given as a constant value for the whole concentration range only in the case of a straight-line calibration curve. In the case of curved calibration graphs, the sensitivity varies with concentration. 

Figure 4.2 Sensitivity for a straight-line calibration as functional dependence of signal on concentration.

To introduce univariate regression analysis for straight-line calibration and empirical model building of analytical relationships... 

In subsequent sections we shall assume that straight-line calibration graphs take the algebraic form ... 

Figure 6.4 Straight-line calibration graph calculated by Theil s method (-------], and by the...

In all cases, standards are trapped and chromatographed in exactly the same manner as reaction products. The best straight-line calibration curves are determined by a least-square regression curve fitting computer program. 

However, when an analytical chemist says a linear calibration it usually should be understood as a straight line calibration function of the form ... 

Finally, we must recall that, in mathematical jargon, linearity refers to the coefficients of the mathematical function, not to the straight line appearance of the plot of the experimental points. Hence, a quadratic model (a parabola) may be (mathematically speaking) a linear calibration, but would not describe a straight line calibration. 

Once the requirements of a straight-line calibration by the ordinary least squares criterion (LS) have been met, we proceed with the method itself. LS leads to the estimates bq and bj of the population parameters (the intercept) and Pi (the slope) in eqn (2.1), respectively i.e. the straight line that, hopefully, describes best the experimental data. ... 

Application of the method with a curved calibration, although possible in principle, should be avoided as extrapolation from a curve is prone to unknowable inaccuracy. The analyst should ensure in advance that a straight-line calibration is appropriate. The almost invariable approach is to make up the final volume of the solutions in steps 1 and 2 equal, although some other possibilities exist. ... 

While a deviation from a straight line calibration is often predictable in principle from physical theory, a quantitative account is usually lacking there is no known reason why a true calibration graph should be a quadratic function or higher order polynomial. Indeed they are often of a somewhat different shape. This leads to a degree of lack of fit between the true function and the fitted function. Figure 2.23 shows an example, where a quadratic function has been fitted to closely spaced data of the slightly different shape typical of inductively coupled plasma atomic emission spectroscopy (ICP-AES) calibrations. 

In summary, polynomials of order 2 (quadratic) provide the analyst with a readily available means of matching points from a curved calibration to a function suitable for interpolation. Tests for lack of fit based on residuals are available in much the same way as for straight-line calibration. Extra caution is necessary to avoid lack of fit, and even small extrapolations beyond the calibrated range are unwise. It is unlikely that polynomials of order greater than quadratic would be appropriate for analytical calibration. 

J. O. De Beer, T. R. De Beer and L. Goeyens, Assessment of quality performance parameters for straight line calibration curves related to the spread of the abscissa value around their mean. Anal. Chim. Acta, 2007, 584, 57-65. 

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