Weighted linear regression with errors

CA Weighted Linear Regression with Errors in Both x andy... 

FIGURE 13.29 Correlation between scratch counts and LPC determined for particles with diameter >0.469 pm (SO3 + SO5) with mixtures of slurry A (0 wt% of slurry B) and slurry B. Weight percentage of slurry B for the mixtures is labeled at the corresponding data point. The weighted linear regression fit to the data set and 95% confidence limits are represented by the solid line and dashed lines, respectively. Error bars correspond to 1 standard deviation (from Ref. 40). 

Fig. 1 shows such a plot and illustrates that the error in the value of log increases with time, i.e. as the concentration approaches its limiting value at / = oo, and is asymmetrically distributed. As the values of log 4 at long times are less reliable than those determined at short times, it is not formally correct to use a simple linear regression analysis to fit such a logarithmic plot and to determine k bs-Rather a non-linear regression with the data weighted to account for the increase in error with time and its asymmetric distributions should be employed. Nevertheless, it is often adequate to use a simple linear regression provided that only data collected over, say, the first 90% of the reaction (i.e. 3 half-lives) are used (see Fig. 

We wish to apply weighted linear least-squares regression to Eq. (6-2), the linearized form of the Arrhenius equation. Let us suppose that our kinetic studies have provided us with data consisting of Tj, and for at least three temperatures, where o, is the experimental standard deviation of fc,. We will assume that the error in T is negligible relative to that in k. For convenience we write Eq. (6-2) as... 

FIG. 3 Total (a) Cl, (b) N, and (c) weight uptakes as a funetion of the number of reaction cycles for ( ) gas and (+) liquid synthesis routes, with corresponding linear regressions. A deflection point is introduced in the regression of the (+) curve of (a). The experimental errors for the first three reaction cycles are smaller than the symbols used. 

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