Straight weighted regression

It is often assumed in regression calculations that the experimental error only affects the y value and is independent for the concentration, which is typically placed on the x axis. Should this not be the case, the data points used to estimate the best parameters for a straight line do not have the same quality. In such cases, a coefficient Wj is applied to each data point and a weighted regression is used. A variety of formulae have been proposed for this method. 

The normalized patterns for another seven samples (B.5, B.IO, B.ll, B.18, B.19, B.25 and B.26) can best be fitted by single straight lines. The slopes of these lines have been calculated by the weighted regression analysis described previously. Because the pattern of B.2 could be fitted by a straight line too, this sample was included with this set of seven samples. The results of the regression analysis are summarized in Table V. Of these eight samples only samples B.ll, B.19 and B.25 do not differ... 

The intercept guarantees that the straight line need not pass the origin of the coordinate system, but passes the centroids of the variables, that is, x and y. For estimation of the parameters for a weighted regression, first the weighted centroids are calculated as follows ... 

The standard errors for estimation of the two parameters of the straight-line model were obviously decreased by applying weighted regression. 

The lead-alpha dates of zircon in the Thiel Mountains Porphyry appeared to be confirmed by Eastin (1970) who obtained a whole-rock Rb-Sr date of 632 102 Ma and an initial Sr/ Sr ratio of 0.7086 0.0059 for specimens of the Porphyry. This date and the initial Sr/ Sr was derived by an error-weighted regression procedure applied to the analytical data in Appendix 8.5.3. The same data were used by Faure et al. (1977) to calculate a date of 660 79 Ma and an initial ratio of 0.7069. The large uncertainty ( 102 x 10 year) of the date calculated by Eastin (1970) arose because the data points do not constrain the slope and intercept of a straight line well enough to calculate a geologically useful crystallization date of the Thiel Mountains Porphyry. 

The simplest procedure is merely to assume reasonable values for A and to make plots according to Eq. (2-52). That value of A yielding the best straight line is taken as the correct value. (Notice how essential it is that the reaction be accurately first-order for this method to be reliable.) Williams and Taylor have shown that the standard deviation about the line shows a sharp minimum at the correct A . Holt and Norris describe an efficient search strategy in this procedure, using as their criterion minimization of the weighted sum of squares of residuals. (Least-squares regression is treated later in this section.)... 

Fig. 6.1 Plot of logio/li + 4Z) vs. 7 for Eq. (6.13). The straight hne shows the result of the weighted linear regression, and the area between the dashed lines represents the uncertainty range of logioi i and As.

If unit weights are employed, aU Wj are deleted and S Wj is replaced by m.) These equations may be useful when a simple straight-line fit ( linear regression ) is being done with a spreadsheet program or a pocket calculator. Many calculators accumulate most or aU of the sums required in Eqs. (20) to (22) some complete the calculation and offer both the refined parameters and their estimated standard deviations. 

For all mathematical models that are not naturally straight lines, non-linear regression analysis is often the best approach. The observed data and the corresponding dependent variable can be analyzed without transformation. Thus, the data and the error or variance are not distorted during the analysis. If necessary, clearly defined weighting schemes can be applied. Furthermore, multiple observation sets can be readily accommodated. 

Figure B-1 Plot of log,o + AD versus / , for Reaction (B.12), at 25°C and 1 bar. The straight line shows the result of the weighted linear regression, and the dotted lines represent the uncertainty range obtained by propagating the resulting uncertainties at / = 0 back to / = 4 m.

In classic calculations the experimental error is considered to affect the y value exclusively and not the concentration recorded in x. If this is not the case, the data points will not have the same quality for the regression line hence comes the idea of according less value to data more distant from the line. Through iterative calculations the equation of a straight line is reached which takes account of the weighting of each point. 

Fig. 3. Calculated relative intramembrane diffusion coefficients across the Nitella cell membrane as a function of molecular weight of the permeant. Ordinate logarithm of the ratio of the permeability coefficient (in 10 cm/s) to the olive oil/water partition coefficient for the permeants of Fig. 1. Abscissa logarithm of their molecular weights. The solid straight line is the Unear regression of log(P/K) on log M with slope of — 1.22. The dashed lines are at a distance of one standard deviation away from the regression line.

The complete (shown in the following inclusive of weighting) equations can be found in Ref. [8] or papers cited therein. Here, the two square roots should be replaced by a factor of 1.2, which leads (without weighting) to the approximation of two constant confidence lines parallel to the straight line. Visualized, these confidence lines would be elastically fixed at the ends and slightly pressed in the middle. This is exactly the effect of the real square-root term, for which using ordinary linear regression (iv ) must be set to 1 and Ziv = n. 

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