Linear regression defined

Data Analysis The results were plotted at first glance a linear regression of absorbance versus concentration appeared appropriate. The two dilution series individually yielded the figures of merit given in Table 4.17, bottom. The two regression lines are indistinguishable, have tightly defined slopes. 

The r coordinates of the variable point which traces out the trajectory of the yth column-item in the r-dimensional biplot are compiled in the r-vector Sj. The elements of the latter can be estimated by means of linear regression of the nxr factor scores S upon an n-vector which is defined as ... 

Note that the lipophilicity parameter log P is defined as a decimal logarithm. The parabolic equation is only non-linear in the variable log P, but is linear in the coefficients. Hence, it can be solved by multiple linear regression (see Section 10.8). The bilinear equation, however, is non-linear in both the variable P and the coefficients, and can only be solved by means of non-linear regression techniques (see Chapter 11). It is approximately linear with a positive slope (/ ,) for small values of log P, while it is also approximately linear with a negative slope b + b for large values of log P. The term bilinear is used in this context to indicate that the QSAR model can be resolved into two linear relations for small and for large values of P, respectively. This definition differs from the one which has been introduced in the context of principal components analysis in Chapter 17. 

It should be noted that the above definition of Xj is different from the one often found in linear regression books. There X is defined for the simple or multiple linear regression model and it contains all the measurements. In our case, index i explicitly denotes the i"1 measurement and we do not group our measurements. Matrix X, represents the values of the independent variables from the i,h experiment. 

The linearity of a method is defined as its ability to provide measurement results that are directly proportional to the concentration of the analyte, or are directly proportional after some type of mathematical transformation. Linearity is usually documented as the ordinary least squares (OLS) curve, or simply as the linear regression curve, of the measured instrumental responses (either peak area or height) as a function of increasing analyte concentration [22, 23], The use of peak areas is preferred as compared to the use of peak heights for making the calibration curve [24],... 

Fig. 14 Experimental (a) and calculated (b) conductance values of Au-n-alkanedithiol-Au junctions vs number n of methylene units in a semilogarithmic representation. The three sets of conductance values - high (H), medium (M), and low (L) - are shown as squares, circles, and triangles. The straight lines were obtained from a linear regression analysis with decay constants (3n defined per methylene (CH2) unit. The conductances of many different, nonequivalent gauche isomers cover the window below the medium values in (b) [64]...

Once suitable parameters are available the values of g can be correlated with them by means of either simple linear regression analysis if the model requires only a single variable, or multiple linear regression analysis if it requires two or more variables. Such a correlation results in a SPQR. In this work we consider only those parameters that are defined directly or indirectly from suitable reference sets or, in the case of steric parameters, calculated from molecular geometries. 

This problem may be solved by linear regression using equations 3.4-11 (n = 1) and 3.4-9 (with n = 2), which correspond to the relationships developed for first-order and second-order kinetics, respectively. However, here we illustrate the use of nonlinear regression applied directly to the differential equation 3.4-8 so as to avoid use of particular linearized integrated forms. The method employs user-defined functions within the E-Z Solve software. The rate constants estimated for the first-order and second-order cases are 0.0441 and 0.0504 (in appropriate units), respectively (file ex3-8.msp shows how this is done in E-Z Solve). As indicated in Figure 3.9, there is little difference between the experimental data and the predictions from either the first- or second-order rate expression. This lack of sensitivity to reaction order is common when fA < 0.5 (here, /A = 0.28). 

The slope of the linear array defined by the olivine-rich lavas (olivine control line) with more than 100 ppm Ni is found by linear regression to be... 

Note also that we can use the correlation test statistic (described in the correlation coefficient section) to determine if the regression is significant (and, therefore, valid at a defined level of certainty. A more specific test for significance would be the linear regression analysis of variance (Pollard, 1977). To so we start by developing the appropriate ANOVA table. 

It is important to stress that for this to work, the independently known matrix A of absorptivity coefficients needs to be square, i.e. it has previously been determined at as many wavelengths as there are chemical species. Often complete spectra are available with information at many more wavelengths. It would, of course, not be reasonable to simply ignore this additional information. However, if the number of wavelengths exceeds the number of chemical species, the corresponding system of equations will be over determined, i.e. there are more equations than unknowns. Consequently, A will no longer be a square matrix and equation (2.22) does not apply since the inverse is only defined for square matrices. In Chapter 4.2, we introduce a technique called linear regression that copes exactly with these cases in order to find the best possible solution. 

We repeat, the task of linear regression is to determine those values of the vector a for which the product vector yCaic=Fa is as close as possible to the actual measurements y. Closeness of course is defined by the sum of the squared differences between y and yCaic. 

One way to compare data to predicted fractionation laws is to plot the data on the three isotope plot in which 5 "Mg is the ordinate and 5 Mg is the abscissa, and examine how closely the data fall to the different curves defined by the exponent p. However, the differences between the different P values are often evident only with careful attention to the statistics of the data. Ideally, the values of P should be obtained by a best fit to the data. This is most easily accomplished if the problem can be rewritten so that P is the slope in a linear regression. 

Further analysis of linearity data typically involves inspection of residuals for fit in the linear regression form and to verify that the distribution of data points around the line is random. Random distribution of residuals is ideal however, non-random patterns may exist. Depending on the distribution of the pattern seen in a plot of residuals, the results may uncover non-ideal conditions within the separation that may then help define the range of the method or indicate areas in which further development is required. An example of residual plot is shown in Figure 36. There was no apparent trend across injection linearity range. 

There are several figures of merit that can be used to describe the quality of a linear regression model. One very common figure of merit is the correlation coefficient, r, which is defined as ... 

Another figure of merit for the fit of a linear regression model is the root mean square error of estimate (RMSEE), defined as ... 

---