Linear least squares calibration

Calibration Ojrve Linear least squares calibration Cross Reactivity Dscrimination... 

Linear least squares calibration is a calibration obtained using the method of minimizing the least squares. Sometimes calibration curves of higher order are used. 

If the linear least squares calibration does not yield results that meet the 5 % specification for the reference material above, then try forcing the calibration through the origin, that is, = 0, using the GC/MS quantitation software and recalculate the results for the reference material. For components present in high concentrations, such as toluene, try using a quadratic fit as described in 8.3.6. If the results for the reference material are still in error, verify the calibration and instrument set-up. 

The amounts of EMA and HEMA (derivatized as MEMA) are determined based upon external standard calibration. A nonweighted linear least-squares estimate of... 

The amount of NIPA is determined based upon external standard calibration. A non-weighted linear least-squares estimate of the calibration curve is used to calculate the amount of NIPA in the unknowns. The response of any given sample must not exceed the response of the most concentrated standard. If this occurs, dilution of the sample will be necessary. 

Figure 3 Least squares calibration line for photometric detector. (From Dorschel, C. A., Ekmanis, J. L., Oberholtzer, J. E., Warren, Jr., F. V., and Bidlingmeyer, B. A., LC detectors evaluations and practical implications of linearity, Anal. Chem., 61, 951 A, 1989. Copyright American Chemical Society Publishers. With permission.)...

Non-linear curves may be treated using Equation 9 directly, using the techniques of non-linear least squares, when appropriate. (Note that a non-linear calibration curve does not necessarily imply non-linear least squares. The latter is necessary only if the problem is non-linear in the estimated parameters (16). For example, y = a+bx+cx and y = a+bx are both non-linear functions, but only the latter is non-linear in the parameters.)... 

This is perhaps the "best solution for the given data set, and it is certainly the most interesting. It is not offered as a rigorous solution, however, for the lack of fit (x /df -[9.64]2) implies additional sources of error, which may be due to additional scatter about the calibration curve (oy -"between" component), residual error in the analytic model for the calibration function, or errors in the "standard" x-values. (We believe the last source of error to be the most likely for this data set.) For these reasons, and because we wish to avoid complications introduced by non-linear least squares fitting, we take the model y=B+Axl 12 and the relation Oy = 0.028 + 0.49x to be exact and then apply linear WLS for the estimation of B and A and their standard errors. 

Figure 1. Calibration curve for a series of narrow MWD PS standards. Linear least squares fit for log (MW) versus elution volume.

Precision can be increased if several injections of the sample and the reference solutions are made, always using equal volumes. In a multilevel calibration several different amounts of the standard are prepared and analyzed. A regression method is used (e.g. linear least-square, or quadratic least square) and this leads to a more precise value for Cunk- This quantitative method is the only one adapted to gaseous samples. 

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 stock solution of NH4C1 dissolved in deionized-distilled water was prepared for calibration. Aliquots of this solution were further diluted to obtain the solutions for the calibration curve. Each solution was then subjected to the treatment just described. The calibration curve (Figure 2) was fit by the linear least-squares regression equation... 

The procedure described for calibration of K and a is laborious because of the required fractionation process. The two constants are derived as described from the intercept and slope of a linear least squares fit to values for a series... 

Then, the parameters of the calibration curve are computed by mean of a linear least-squares regression ... 

In anal3rtical chemistry, developii a calibration curve or modelling a phenomenon often requires the use of a mathematical fitting procedure. Probably the most familiar of these procedures is linear least-squares fitting [1]. Criteria other than least-squares for defining the best fit have been developed for linear parameters when the data possibly contain outliers [2,3]. Sometimes, the model equation to be fit is nonlinear in the parameters. This requires appeal to other fitting methods [4]. 

The RATIO method table (Table I) includes provision for specifying upper and lower limits of integration for both primary and reference bands with the peak area evaluation procedure. The practical limits of the integration can be determined empirically by evaluating a set of spectra stored on microfloppy disks with varying limits set in the appropriate locations in the method table. Optimum limits can be determined from the calibration plots and related error parameters. The calibration plots shown in Figures 4 and 5 indicate that both evaluation procedures, peak height and peak area provide essentially the same level of precision for the linear least squares fit of the data. The error index and correlation coefficients listed on each table are both indicators of the relative scatter in the data from the least squares fit line. The correlation coefficient is calculated as traditionally defined in statistics. 

Using a plotting program, plot absorbance vs [Co(II)]. This is your Beer s law calibration plot. Obtain a linear least-squares fit to the data points. 

S Spreadsheet Summary In the final exercise of Chapter 13 of Appli-"II cations of Microsoft Excel in Analytical Chemistry, the initial-rate method is explored for determining the concentration of an analyte. Initial rates are determined from a linear least-squares analysis and are used to establish a calibration curve and equation. An unknown concentration is determined. 

Figure 19 Standard addition calibration curves. Equal volumes of solvent containing varying amounts of standard are added (spiked) into the sample. The samples are analyzed and the analyzer response (e.g., area under the TIC or selected ion chromatogram) is plotted against the amount of standard added. The analyte concentration is estimated by extrapolating a linear least-squares regression toy = 0 (a). An alternative approach is to plot the difference between the spiked samples and the unspiked sample. The same calibration curve now passes through the origin and the sample analyte concentration can now be determined by interpolation with improved confidence limits164 (b).

The calibration curve for the oxygen sensor operated at —0.55 V fits a linear least-squares fit well (R = 0.993), but exhibits a slight sublinearity. Interestingly, the calibration curve obtained at a potential of —0.65 V does not show this sublinearity. This evidence points towards the rate of the electrochemical reaction itself being the rate limiting step in the gas sensing process. 

A calibration curve for the colorimetric determination of phosphorous in urine is prepared by reacting standard solutions of phosphate with molybdenum(VI) and reducing the phosphomolybdic acid complex to produce the characteristic blue color. The measured absorbance A is plotted against the concentration of phosphorous. From the following data, determine the linear least-squares line and calculate the phosphorous concentration in the urine sample ... 

The linear least-squares line gives a slope of 0.861 and an intercept of —0.002 (using Options under Chart, Add Trendline, when highlighting the chart or line). Hence, the concentration of the unknown is equal to (0.463 — 0.002)70.861, as given by the formula in the spreadsheet (below). The sample concentration is 0.540 ppm. We will now perform the same calculation without charting the calibration curve, and including the standard deviation of the sample concentration. 

You are applying linear least-squares analysis to a set of 20 data pairs. The model is a linear first-order polynomial with slope b (i.e., the first-order coefficient) and intercept c (i.e., the zeroth-order coefficient). It is necessary to force the intercept c to be zero, analogous to some of your laboratory calibration curves. Hence, c = 0. 

---