Correlation coefficients, additive equations

Lines in Figure 30.12 were drawn with parameters obtained when fitting data with Equation 30.3. It is fairly obvious that, outside the experimental window, data would not necessarily conform to such a simple model, which in addition cannot meet the inflection at 100% strain. All results were nevertheless fitted with the model essentially because correlation coefficient were excellent, thus meaning that the essential features of G versus strain dependence are conveniently captured through fit parameters. Furthermore any data can be recalculated with confidence within the experimental strain range with an implicit correction for experimental scatter. Results are given in Table 30.1 note that 1/A values are given instead of A. 

There are two statistical assumptions made regarding the valid application of mathematical models used to describe data. The first assumption is that row and column effects are additive. The first assumption is met by the nature of the smdy design, since the regression is a series of X, Y pairs distributed through time. The second assumption is that residuals are independent, random variables, and that they are normally distributed about the mean. Based on the literature, the second assumption is typically ignored when researchers apply equations to describe data. Rather, the correlation coefficient (r) is typically used to determine goodness of fit. However, this approach is not valid for determining whether the function or model properly described the data. 

In a similar approach to Hamaker, Timme et al. proposed six functions that are also empirically based. However, they took the additional step of suggesting that the choice of the equation should be based on the regression correlation coefficient (r). 

As more wavelengths are added, the standard error diminishes and approaches some minimum value, and the correlation coefficient increases and approaches unity. The equation becomes more complicated with the addition of more coefficients so that it may better define the analyte s concentration. 

Equation (15) has a standard deviation of 0.0871 and a correlation coefficient of 0.979. It is based on a data set containing 61 polymers chosen in such a way as to avoid incorporation of the effects of additives and fillers. Further analysis of the data set gave the following results ... 

Regression equations do not indicate the accuracy and spread of the data. Consequently, they are normally accompanied by additional data, which as a minimum requirement should include the number of observations used (n), the standard deviation of the observations (s) and the correlation coefficient (r). 

The thermal fading rates were measured in toluene at 10°C for a series of 6-nitroBIPS having additional substituents (nitro, methoxy, chloro, bromo, carbo-methoxy, and carboethoxy) in the 5 - and 8-positions. The rate constants were fitted to an equation of the form log (k/k0) =piOi + p2a2 with a correlation coefficient / =().85. This relatively low correlation coefficient, which is not improved by attempts to include the specific steric effects of the 8 substituent or by including the cross-term p12ai02, implies the complexity of the substituent effects on the fading rate.113... 

Many studies indicated that in the presence of DOM, the metal sorption capacity decreased markedly for most soils, and the effect on the calcareous soil was greater than on the acidic sandy loam. Figure 10.4 shows the metal sorption equilibrium isotherms onto soils with or without the addition of 400 mg C/l of DOM. The equilibrium isotherms could be better depicted according to the linear Freundlich equation with the high value for the correlation coefficient of determination (r2) ... 

Table III. Excerpt from RSQUARE Output Showing Correlation Coefficients for Additive Equations Containing Selected Independent Variables...

Additional AES foaming experiments were performed in the presence of a Gulf Coast stock tank oil (Table 5). The correlation coefficient of equation 2 was 0.995 and 0.983 at pH 7 and pH 2.5 respectively. When the foaming data at both pH values were combined and the data analyzed using equation 3, the correlation coefficient was 0.990. 

Values of measured for reversible reactions under conditions permitting absorption equilibration between gas and solid products are often comparable with the reaction enthalpy [1,47]. Under such conditions the identification of the rate expression, g(nr) = kt, on the single criterion that this equation gives the most acceptable correlation coefficient is not a sufficient foimdation to characterize a geometric reaction model. The use of additional information, for example microscopy, can provide confirmatory evidence concerning interface development. Similarly, the value of studies which conclude that kinetic results are satisfactorily described by equations based on diffusion models is increased considerably if the identity of the migrating species is established [53]. 

According to the rules of statistical analysis, the fraction of the variation of a property accounted for by a correlation is equal to the square of the correlation coefficient. The square of the correlation coefficient is hence sometimes referred to as the "coefficient of determination". A correlation coefficient of 0.9957 indicates that Equation 3.9 accounts for approximately 99.15% of the variation of the Vw values. The addition of extra terms proportional to °%v and... 

The use of equations 3.13 and 3.14 results in a standard deviation of only 3.2 cc/mole relative to, and a very large correlation coefficient of 0.9989 with, the experimental values of V. This correlation therefore accounts for 99.78% of the variation of the 152 experimental V values in the dataset. The standard deviation is only 2.2% of the average V of 145.4 cc/mole in this dataset. It is quite possible that most or all of the remaining 0.22% of the variation is caused by the experimental error margins. If there is any remaining systematic portion in the deviation which should be accounted for in the correlation, this systematic portion can only be uncovered after a considerable amount of additional experimental data become available. 

In the equations, n represents the number of datum points used to derive the equation, r is the multiple correlation coefficient, s is the standard deviation from regression, F is the F statistic for variance of each additional variable, the values in parentheses after the equation coefficients are for construction of confidence intervals and the Roman numbers in subscript refer to the substituent position.)... 

The ion-exchange data reported by Haddad and Cowie for a series of anions with phthalate/biphthalate additive at pH 5.3 (63) comprise an opportunity for testing equation 39 (as well as equations 41, 42a, and 42c see below). Accordingly, we provide in Table VI log hias a function of log for the systems they considered, as well as the mope m, the y-intercept b, and the absolute value of the linear least-squares correlation coefficient r, the data being derived from Figure 1 of their work. 

The addition of log k values at pH 7.0 in ion-exchange liquid chromatography using a triethylphenylammonium phase (IXi) to log lvalues of the above results improved the correlation coefficient to 0.959 ( = 7). The relationship is given by the following equation ... 

---