Regression curvilinear

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The determination of errors in the slope a and the intercept b of the regression line together with multiple and curvilinear regression is beyond the scope of this book but references may be found in the Bibliography, page 156. 

There are a number of ways to model calibration data by regression. Host researchers have attempted to describe data with a linear function. Others ( 4,5 ) have chosen a higher order or a polynomial method. One report ( 6 ) compared the error in the interpolation using linear segments over a curved region verses using a curvilinear regression. Still others ( 7,8 ) chose empirical or spline functions. Mixed model descriptions have also been used ( 4,7 ). 

In the case of polynomial or curvilinear regression, as given by the model ... 

In general, the functional range of a curvilinear regression model is narrower than that of a linear model the assay range is extended by the dilution of samples with concentrations higher than the ULOQ. Dilution linearity also demonstrates the lack of a prozone or hook effect of the assay. A hook (prozone) effect is the phenomenon where the responses of higher concentrations of analyte are lower than expected. 

Curvilinear regression should not be confused with the nonlinear regression methods used to estimate model parameters expressed in a nonlinear form. For example, the model parameters a and b in y = axb cannot be estimated by a linear least-squares algorithm. Information in Chapter 7 describes nonlinear approaches to use in this case. Alternatively, a transformation to a linear model can sometimes be used. Implementing a logarithmic transformation on yt = ax/ produces the model log y = log a + blog x which can now be utilized with a linear least-squares algorithm. The literature [4, 5] should be consulted for additional information on linear transformations. 

The residual plot in Figure 6.2a represents a linear relationship between the response and the independent variable. The residuals are randomly scattered around the centerline. The U-shaped residual plot of Figure 6.2b indicates that a curvilinear regression model should be fitted through the data points. Any departure from the symmetric bar shape of Figure 6.2a may indicate that the chosen regression model is inappropriate. The residual plots are also used to detect violations of other basic assumptions. This is discussed further in the text. 

A second-order regression curve (V, = ao + ci Xi + a2Xf) is in most cases sufficient to deal with curvilinear regression curves. The concentration of the analyte in the sample can be back-calculated from the response. Nowadays, most instrument software packages include the use of different types of regression models and the concentration of the samples is automatically back-calculated. 

The deposition process of the major elements was strongly affected by the SST. Fig. 5.30 illustrates that the deposition processes of 11 major elements except K and Ba decreased exponentially with an increase in SST, while the deposition processes of K and Ba were SST free. The correlation between SST and the deposition fluxes was obtained based on the curvilinear regression as follows ... 

Quadracity (0 to inf) If two variables show a strong linear relationship, they also produce a small error for curvilinear regression because the linear... 

Figure 5.14 Curvilinear regression identification of the linear range. The data in Example 5.13.1 are used the unweighted linear regression lines through all the points (—), through the first five points only (-------), and through the first four points only (.) are shown.

Here M = N = 6, and the number of runs is three. Table A. 10 shows that, at the T = 0.05 level, the number of runs must be <4 if the null hypothesis is to be rejected. So in this instance we can reject the null hypothesis, and conclude that the sequence of + and - signs is not a random one. The attempt to fit a straight line to the experimental points is therefore unsatisfactory, and a curvilinear regression plot is indicated instead. 

To monitor a patient s body temperature, thermocouples and thermistors are typically used. Thermocouples are based on Seebeck s electromotive force (emf) across a junction of two dissimilar metals. Typical empirical calibration data can be fit by a curvilinear regression equation E = a T + + where E is Seebeck emf in... 

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