Intercept term

The apparent viscosity, defined as du/dj) drops with increased rate of strain. Dilatant fluids foUow a constitutive relation similar to that for pseudoplastics except that the viscosities increase with increased rate of strain, ie, n > 1 in equation 22. Dilatancy is observed in highly concentrated suspensions of very small particles such as titanium oxide in a sucrose solution. Bingham fluids display a linear stress—strain curve similar to Newtonian fluids, but have a nonzero intercept termed the yield stress (eq. 23) ... 

The symbol k or K is the rate or equilibrium constant, respectively, for a side-chain reaction of a meta- or para-substituted benzene derivative, and k° or K° denotes the statistical quantity (intercept term) approximating to k or K for the parent or unsubstituted compound. The substituent constant a measures the polar (electronic) effect of replacing H by a given substituent (in the meta- or para-position) and is, in principle, independent of the nature of the reaction. The reaction constant p depends on the nature of... 

Oj and aR are respectively the inductive and resonance constants of Taft s analysis of ordinary composite Hammett constants (values obtained by Charton were used) and v is the steric substituent constant developed by Charton161,162. The intercept term, h, notionally corresponds to log k for H as an orfho-substituent, but is not found in practice always to agree closely with the observed value of log k for the parent system. 

As an intercept, 1 t-f = 0, so the only sensible temperature to include as T in the intercept term would be T = oc, which means that AS = —oo. Again, this is not realistic. 

Multiple regression on j and og-type parameters employs the dual substituent-parameter equation, which may be written as in equation 891. (The combining of the k and k° terms implies that there is no intercept term allowed, and k° is now the actual value for the parent system, cf below.) For any given reaction series the equation is applied to meta- and para-substituents separately, and so values of pi and pr characteristic both of reaction and of substituent position are obtained. The various op-type scales are linearly related to each other only approximately. In any given application the scale which gives the best correlation must be found92. 

A slightly different procedure for carrying out multiple regression on cr/ and og-type parameters employs the extended Hammett equation of Charton95, which may be written as in equation 9. For the substituent X, Q is the absolute value of the property to be correlated (logfc or logX in the case of reactivity), i.e. not expressed relative to X = H, h is introduced as the appropriate intercept term, and the regression coefficients are a and fi. (Charton has used various symbols at various times.)... 

The symbols were defined in Section IV.B.) A full discussion of the ortfio-effect as revealed in this work would be inappropriate here. We must restrict ourselves to the more limited task of indicating the role of o-N02. We discuss first the work involving alcohols as solvents. To apply the extended Hammett equation, i.e. to determine the regression coefficients a, fi and

conformational effects, and internal hydrogen-bonding may occur... 

Taft and Topsom s article151 and also Topsom s171 should be consulted for details of the setting up of the scales of substituent parameters. The equation has been applied to a wide range of gas-phase reactivities. (In the multiple regressions an intercept term is often permitted, but usually this turns out to be indistinguishable from zero, as it should be if equation 20 is valid.) For aliphatic and alicyclic saturated systems the resonance term is duly negligible. The roles of field, resonance and polarizability effects are discussed and the interpretat of the various p values is attempted. 

X has been mean-centered b includes no intercept term... 

The difference between interval and ratio scales can be important for including or not including an intercept term in mathematical models for the correct calculation of the correlation coefficient for deciding to mean center or not in principal component analysis and for a host of other decisions in data treatment and modeling. 

Hawkins, D.M. (1980), A Note on Fitting a Regression Without an Intercept Term, American Statistician, 34, 233. 

The intercept term C Ar/ Ar)o, which accounts for igneous, metamorphic, or atmospheric sources, is regarded as the excess contribution present at time = 0, whereas the second term is the radiogenic component accumulating in the various minerals of the isochron by decay of If all the minerals used to construct the isochron underwent the same geologic history and the same sort of contamination by excess " Ar, the slope of equation 11.100 would have a precise chronological... 

Symbol for the dissociation constant of an inhibitor with respect to a particular form of the enzyme. This dissociation constant is associated with the intercept term in the double-reciprocal form of the initial-rate equation. For example, consider an inhibitor that can bind to either the free enzyme, E, or the binary central complex, EX, of a Uni Uni mechanism. Ka would be the dissociation constant for the EX -t 1 EXl step and is equal to [EX][1]/[EX1]. The binding of 1 to the free enzyme (i.e., E -t 1 El) is governed by Kis (equal to [E][1]/[E1]). 

Data sets aic often mean centered to account for an intercept term in regression modefe. To decide if mean centering is ncccssar>, compare the results when the analysis is performed using centered and noncentered data. 

The next step is to construct the regression model from the calibration data using one ariable plus an intercept. Tlie results of this (now univariate) model (see Table 5.10) indicate that the intercept is not a significant term in the regression. This is expected because the data are known to obey Beer s Law. Traditionally, in statistics the intercept term is left in the model even when found to be nonsignificant. 

Note that the coefficients are different from those of Section 2.1.2. One reason is that there are still a number of interferents, from the other PAHs, in the spectrum at 336 nm, and these are modelled partly by the intercept term. The models of Sections 2.1.1 and 2.1.2 force the best fit straight line to pass through the... 

It is easy to show algebraically that the value of b is identical with bi obtained for the uncentred data ( = 4.391 in this example), but includes the intercept, whereas the old value of b0 is given by (c — b X), so the two methods are related. It is common to centre both sets of variables for this reason, the calculations being mathematically simpler than including an intercept term. Note that the concentrations must be centred at the same time as the response, and the predictions are of the concentrations minus their mean. 

Near-infrared (NIR) spectra of water-methanol mixtures are examined to demonstrate the fundamental aspects of calibration. These spectra are used because they present unique challenges to calibration. Another reference NIR data set is also briefly evaluated. The reader should remember that the information presented is generic and applies to all calibration situations, not just spectroscopic data. Additionally, for discussion purposes, the quantitative information for the target analyte will be concentration. However, other chemical or physical properties can also be modeled. Throughout this chapter, unless noted otherwise as in Sections 5.3 and 5.4, it will be assumed that the described models have had the intercept term eliminated. The easiest way to accomplish this is to mean-center the data. 

FIGURE 5.2 Diagram of three different types of linear models with n standards. Left the simplest model has a slope and no intercept. The center model adds a nonzero intercept. The right model is typically noted in the literature as the multiple linear regression (MLR) model because it uses more than one response variable, and n>(m+ 1) with an intercept term and n> m without an intercept term. This model is shown with a nonzero intercept. 

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