Zero-intercept model

If linear, is the intercept (b ) significantly different from zero, l.e., is a zero-intercept model suitable ... 

Linear Model With Intercept. There are two distinct linear first-order regression models that are generally encountered in analytical calibration. The non-zero intercept model is the most familiar, and it is given by Equation 1. 

The estimates of Intercept (b ) and slope (b ) are calculated so as to minimize the sum of squares (SS) of the deviations of the observed signals (y ) from the predicted value (y) at any concentration (x) without constraints. For some determinations, however, theory predicts that the response of the instrument should be linear with concentration and should also be zero when there is no analyte present. Thus, if the Instrument has been calibrated correctly, the calculated line should pass through the origin by definition. The proper regression model would then be the zero Intercept model shown as Equation 2. 

Before the equation of the line calculated using the zero intercept model is employed to evaluate unknowns, it must be tested to determine if the model is adequate to describe the experimental data. Regression analysis tables are constructed prior to testing the statistical validity of the assumption that the intercept of the line is zero. The format for calculation of the regression analysis tables is shown in Table Ill-a and the analyses of the Table I data are shown in Table Ill-b. 

Inspection of these tables shows that the LOF test results are very similar to those for the models with intercepts. Comparison of Tables Il-b and Ill-b reveals that the SS residuals are somewhat larger for the zero intercept models than for the models with an intercept. This difference can be used to test the hypothesis that the intercept is zero. First, it must be demonstrated that the LOF is not significant since it would not make good sense to test the zero intercept hypothesis for linear models shown not to fit the data. Furthermore, the SS error and SS(LOF) should not be combined as SS residuals when LOF is significant. These requirements are met by the Case I results. To test the hypothesis that the intercept does not differ significantly from zero, calculate ... 

Using The Calibration Data in Table I and a Zero Intercept Model... 

The widths of confidence intervals around calibration curves depend not only on the variability in the data, but also on the regression model chosen QiL). For the non- ero intercept model, the best absolute precision occurs at x, y which is the centroid of the regression ine (Figure la). This circumstance pertains largely because x, y is the axis of rotation for the uncertainty in the fitted slope of the calibration curve. In contrast, the best absolute precision for the zero Intercept model occurs when the concentration (x) is zero (Figure 1b). 

In our experience, the unweighted zero-intercept model is often justified. As shown in Figure 1b, CL s for this model reflect our intuitive expectations and the DL estimates are usually consistent with our experimental evidence. Thus, we recommend careful study of the suitability of this model as described earlier. 

This suggests that a plot of r0 vs pA (Fig. 4) or r0/pA vs pA should be linear. It is very difficult to reject this model on the basis of data curvature, even though it is evident that some curvature could exist in Fig. 4. However, Eq. (16) demands that Fig. 4 also exhibit a zero intercept. In fact, the 99.99% confidence interval on the intercept of a least-square line through the data does not contain zero. Hence the model could be rejected with 99.99% certainty. 

In addition, rearranging Eq. (21) and combining with Eq. (19) illustrates that i = k" y. Therefore, a two parameter (one term) model would require that a plot of the enthalpies of adduct formation of one acid versus the enthalpies of adduct formation of another acid for the same series of bases be linear with a zero intercept. The enthalpies of adduct formation for 12 and phenol with a wide series of bases does not give rise to such a plot as can be seen in Fig. 5. These acids have very different C/E ratios and their enthalpies of adduct formation cannot be correlated by a one term model. Furthermore, a one term model could riot incorporate systems in which reversals in donor-acceptor strength are observed 32). However, it is possible to correlate enthalpies of adduct formation for acids with very similar C/E ratios such as hydrogenbonding acids using a one term equation. Correlations restricted to one particular type of acid are, of-course, only a subset of the overall E and C correlation. 

Concentration Residual vs. Predicted Concentration Plot (Model and Sample Diagnostic) Examination of the concentration, residuals versus predicted concentration plot provides a different perspective of the prediction performance, as shown in Figure 5-95. Ideal residuals fall randomly about the line with zero slope and zero intercept. See also Section 5.2.1.1 for more discussion of this diagnostic, hi Figure 5.95. sample number 3 is dearly po dh-u d. more poorly than the other samples. Howci er. there is not enough information to determine the root cause of the problem. Furthermore, it is not advisable to assign root causes when the validity of the model is in question. 

One of the imderlying assumptions of the least squares method is that there is no error in the measurement of the independent variable (the y values). This assumption is often not valid, and one of the most obvious cases of this is found in method-comparison analysis. A typical example of method-comparison analysis involves the comparison of two different instruments, a current production instrument and an improved model. Measurements made on a series of samples with the two instruments, and plotted by using current instrument readings as the x values and new instrument readings as the y values, should ideally result in a straight line of unit slope and zero intercept. The actual slope and intercept of the line can provide estimates of the proportional and constant error between the two methods. 

Prerequisites for the Calibration Types. It depends on the design of the analytical procedure as to which regression parameters are meaningful and which results are acceptable. In other words, the model to be used for quantitation must be justified. For a singlepoint calibration (external standardization), a linear function, zero intercept, and the homogeneity of variances are required. The prerequisites for a linear multiple-point calibration are a linear function and in case of an unweighted calibration also the homogeneity of variances. A non-linear calibration requires only a continuous function. With respect to the 100%... 

The zero intercept of this equation indicates that there is no coupling when the s character is zero, in agreement with the Fermi contact model. 

Therefore, it is well established that topological entanglements dominate and control the modulus of polymer networks with long network strands. The Edwards tube model explains the non-zero intercept in plots of network modulus against number density of strands (see Figs 7.11 and 7.12). The modulus of networks with very long strands between crosslinks approaches the plateau modulus of the linear polymer melt. The modulus of the entangled polymer network can be approximated as a simple sum. 

Figure 1. Confidence Limits For (a) Unweighted Non-Zero Intecept, (b) Unweighted Zero Intercept, and (c) Weighted Non-Zero Intercept Regression Models.

The molecular structure of PFOA is indicated in fig. (1) and polymer chain dimensions were derived using both Zimm plots [fig. (la)] and also by fitting the data to a Debye random coil model [9], with good agreement between the two approaches. As the pressure is reduced, the solubility decreases and the PFOA falls out of solution at the critical, "neutron cloud point" (T = 65 300 bar), as indicated [fig. (lb)] by a zero intercept... 

The classical lattice statistical model only considers the mixtures of two incompressible fluids. Flory recognized that the line of the actual mixing interaction parameter versus the reciprocal temperature does not have zero intercept, i.e. 

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