Straight-line calibration curves, linear

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

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). 

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... 

A calibration curve is a model used to predict the value of an independent variable, the analyte concentration, when only the dependent variable, the analytical response, is known. The normal procedure used to establish a calibration curve is based on a linear least-squares fit of the best straight line for a linear regression, as indicated in... 

A calibration curve shows us the relationship between the measured signal and the analyte s concentration in a series of standards. The most useful calibration curve is a straight line since the method s sensitivity is the same for all concentrations of analyte. The equation for a linear calibration curve is... 

According to Beer s law, a calibration curve of absorbance versus the concentration of analyte in a series of standard solutions should be a straight line with an intercept of 0 and a slope of ab or eb. In many cases, however, calibration curves are found to be nonlinear (Figure 10.22). Deviations from linearity are divided into three categories fundamental, chemical, and instrumental. 

It must be noted, however, that a value of r close to either + 1 or — 1 does not necessarily confirm that there is a linear relationship between the variables. It is sound practice first to plot the calibration curve on graph paper and ascertain by visual inspection if the data points could be described by a straight line or whether they may fit a smooth curve. 

In case of a linear calibration function (see Chap. 6), the sensitivity becomes SAA = AyA/AxA and corresponds to the slope b of the calibration straight line see Fig. 7.4a. If the calibration function is a curved line, then the sensitivity will vary according to the analyte amount or concentration as Fig. 7.4b shows. 

The limit of determination is commonly estimated by finding the intercept of extrapolated linear parts of the calibration curve (see point L.D. in fig. 5.1). However, it is often difficult to construct a straight line through the experimental potentials at low concentrations and, moreover, the precision of the potential measurement cannot be taken into consideration. Therefore, it has been recommended that, by analogy with other analytical methods, the determination limit be found statistically, as the value differing with a certain probability from the background [94]. 

In the calibration process proper the straight-line curve representing the numerically correct, linear correlation between the gas flow per unit of time and the leak rate is defined by two points the zero point (no display where no emissions are detected) and the value shown with the test leak (correct display for a known leak). 

Linear equations of the type v = ct — C, where c and C are constants, relate kinematic viscosity to efflux time over limited time ranges. This is based on the fact that, for many viscometers, portions of the viscosity—time curves can be taken as straight lines over moderate time ranges. Linear equations, which are simpler to use in determining and applying correction factors after calibration, must be applied carefully as they do not represent the true viscosity—time relation. Linear equation constants have been given (158) and are used in ASTM D4212. 

The linear response range of the glucose sensors can be estimated from a Michaelis-Menten analysis of the glucose calibration curves. The apparent Michaelis-Menten constant KMapp can be determined from the electrochemical Eadie-Hofstee form of the Michaelis-Menten equation, i = i - KMapp(i/C), where i is the steady-state current, i is the maximum current, and C is the glucose concentration. A plot of i versus i/C (an electrochemical Eadie-Hofstee plot) produces a straight line, and provides both KMapp (-slope) and i (y-intercept). The apparent Michaelis-Menten constant characterizes the enzyme electrode, not the enzyme itself. It provides a measure of the substrate concentration range over which the electrode response is approximately linear. A summary of the KMapp values obtained from this analysis is shown in Table I. 

Because pressure transducers from different manufacturers can vary significantly, it is important to understand their performances such as accuracy. An ideal device would have a direct linear relationship between pressure and output voltage. In reality, there will always be some deviations this is referred to as nonlinearly. The best straight line is fitted to the nonlinear curve. The deviation is quoted in their specifications and expressed as a percent of full scale. The nonlinear calibration curve is determined in ascending direction from zero to full rating. This pressure will be slightly different from the pressure measured in descending mode. This difference is termed hysteresis it can be reduced via electrical circuits. 

In order to minimize the possibility of errors as a result of variable injection volumes, the internal standard (IS) method should be used. It involves the addition of a compound, the IS, which is not already present in the sample. This is normally a substance that elutes at a position near the sample component of interest and should be well resolved and readily detected under the given chromatographic conditions. The IS is added in constant amount to solutions that contain varying amounts of the analytical standard. Calibration curves are then constructed by plotting the ratio of either the areas or peak heights of the two peaks (analyte/IS) versus amount or concentration of analyte. The amount or concentration of each chromatographed sample can then be obtained by interpolation of the calibration curve. The calibration plot should be a straight line with an intercept of zero. However, non-linear standard curves may result when... 

When an assay presents a nonlinear calibration curve (Fig. 16.4), the data can be linearized using standard functions.4 The log-logit function transforms a sigmoid curve with a single point of inflection into a straight line, and is used extensively with data from competitive immunoassays. 

Although the linear model is the model most commonly encountered in analytical science, not all relationships between a pair of variables can be adequately described by linear regression. A calibration curve does not have to approximate a straight line to be of practical value. The use of higher-order equations to model the association between dependent and independent variables may be more appropriate. The most popular function to model non-linear data and include curvature in the graph is to fit a power-series polynomial of the form... 

Not all calibration curves generate a perfect straight line due to indeterminate or random errors. Most scattered points can be corrected using linear regression analysis to... 

When the log(Mn) determined from the H NMR spectrum in each file is plotted against the elution time, the plots for all three PMMA samples fall on a single straight line. The linear relation of the log(Mn) versus elution time obtained by the on-line GPC/NMR experiment should be the most accurate calibration curve for highly isotactic PMMA. The Mw/Mn values calculated from the GPC/NMR method agreed well with those obtained from the normal GPC/RI method using standard polystyrenes. 

Figure 13.4 Examples of calibration graphs in AAS. Left, a straight calibration line at sub-ppb concentrations obtained with an instrument equipped with a Zeeman effect device (see section 13.7) for the quantification of sodium. Right, a quadratic cirrve for the measurement for zinc at concentrations in the ppm range with a burner type instnrment. This second graph reveals that when concentrations increase, the absorbance is no longer linear. The quantitative analysis software for AAS provides several types of calibration curves.

The relationship between concentration and secondary measurement is not always linear. Whether it is a straight line or a curved line that describes the calibration, an equation is needed to be able to predict concentrations for future samples. A quantitative calibration curve can also be used to calculate a number of important analytical properties (sensitivity, linearity, offset or baseline, detection limit [Currie 1995]). The calibration line would be... 

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