Straight Line Regression

Consider straight line regression where theory dictates that the data should lie on a straight line (or in the absence of a theoretical justification, a plot of the data suggests that a straight line should suffice). The fitting function is then 

If the initial guess is taken to be zero (this is always suggested for linear problems), then 

The following Excel spreadsheet shows the calculation of all quantities required in these equations  

FIGURE 7.3 Scatter plot of evaporation coefficient data. 

The cs are identical to those found by Excel when a trendline was added to the plot of the data. The cs were determined using the inverse of the matrix C. The C matrix and its inverse have other powerful uses as well, as will be covered in later discussions. 

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Schwartz, L. M., Rejection of a Deviemt Point from a Straight-Line Regression, Analytica Chimica Acta 178, 1985, 355-359. 

The model of eq. (36.3) has the considerable advantage that X, the quantity of interest, now is treated as depending on Y. Given the model, it can be estimated directly from Y, which is precisely what is required in future application. For this reason one has also employed model (36.3) to the controlled calibration situation. This case of inverse calibration via Inverse Least Squares (ILS) estimation will be treated in Section 36.2.3 and has been treated in Section 8.2.6 for the case of simple straight line regression. 

Forsythe, A. B. (1972). Robust estimation of straight line regression coefficients by minimizing p-th power deviations. Technometrics 14, 159-166. 

The angular coefficient of straight-line regressions in figure 9.13 thus depends on the stoichiometry of the silicate melt component (moles of oxygen per mole of anhydrous melt) and on the value of constant K72. 

A straight-line model is the most used, but also the most misused, model in analytical chemistry. The analytical chemist should check five basic assumptions during method validation before deciding whether to use a straight-line regression model for calibration purposes. These five assumptions are described in detail by MacTaggart and Farwell [6] and basically are linearity, error-free independent variable, random and homogeneous error, uncorrelated errors, and normal distribution of the error. The evaluation of these assumptions and the remedial actions are discussed hereafter. 

In Table 5.2 and Figure 5.5, data for a quantification comparison between the two techniques referred to some compounds important for the wine quality such as esters, alcohols and fatty acids relative to analysis of different wines, are reported. Figure 5.5 shows the straight line regressions between the concentrations measured after XAD-2 extraction and the ratios of GC-FID area of each compound to that of the internal standard applying the Kaltron method. 

The DeJongh model is similar to the Lachance-Trail model 16), but includes correction for self-absorption effects. The alpha coefficients shown in Table 5 were used as starting values for the straight line regression procedure executed by the PW-1400 software package. 

It is important for all kinds of modeling that the estimated parameters are tested for their statistical significance. To derive appropriate tests for the adequacy of a regression model, we need to generalize the straight-line regression. 

In the case of conventional regression, we obtain the following model, with the corresponding standard errors of the parameters, for the straight-line regression ... 

Extreme values make a great difference to correlation coefficients. The linear interrelationship between two variables can be described functionally by a straight-line regression curve. However, this type of description does not always lead to satisfactory results, because deviations from the... 

Schwartz, L. M. 1985. Rejection of a deviant point from a straight-line regression. Anal. Chim. Acta 178 355-359. 

Linear regression models a linear relationship between two variables or vectors, x and y Thus, in two dimensions this relationship can be described by a straight line given by tJic equation y = ax + b, where a is the slope of tJie line and b is the intercept of the line on the y-axis. 

How do we find the best estimate for the relationship between the measured signal and the concentration of analyte in a multiple-point standardization Figure 5.8 shows the data in Table 5.1 plotted as a normal calibration curve. Although the data appear to fall along a straight line, the actual calibration curve is not intuitively obvious. The process of mathematically determining the best equation for the calibration curve is called regression. 

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

The plot of AEi/2 as a function of the log of the ligand concentration is shown in Figure 11.43. Finear regression gives the equation for the straight line as... 

Earlier we noted that a response surface can be described mathematically by an equation relating the response to its factors. If a series of experiments is carried out in which we measure the response for several combinations of factor levels, then linear regression can be used to fit an equation describing the response surface to the data. The calculations for a linear regression when the system is first-order in one factor (a straight line) were described in Chapter 5. A complete mathematical treatment of linear regression for systems that are second-order or that contain more than one factor is beyond the scope of this text. Nevertheless, the computations for... 

The variables that are combined hnearly are In / 17T, and In C, Multilinear regression software can be used to find the constants, or only three sets of the data smtably spaced can be used and the constants found by simultaneous solution of three linear equations. For a linearized Eq. (7-26) the variables are logarithms of / C, and Ci,. The logarithmic form of Eq. (7-24) has only two constants, so the data can be plotted and the constants read off the slope and intercept of the best straight line. 

Figure 3-24 shows the relationship between 1/C as a function of time t. The graph is a straight line, therefore, the assumed order of the reaction is correct. The slope of the line from the regression analysis is the rate constant k. 

The simplest procedure is merely to assume reasonable values for A and to make plots according to Eq. (2-52). That value of A yielding the best straight line is taken as the correct value. (Notice how essential it is that the reaction be accurately first-order for this method to be reliable.) Williams and Taylor have shown that the standard deviation about the line shows a sharp minimum at the correct A . Holt and Norris describe an efficient search strategy in this procedure, using as their criterion minimization of the weighted sum of squares of residuals. (Least-squares regression is treated later in this section.)... 

The most widely used method for fitting a straight line to integrated rate equations is by linear least-squares regression. These equations have only two variables, namely, a concentration or concentration ratio and a time, but we will develop a more general relationship for use later in the book. 

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