B5 Weighting

As in linear regression, the relation between random errors in experimental values of dependent and independent variables dictates the choice of weighting factors Wj in the error sum in eq. (2.9). For a model with the general form yj = F(xj,bu...,bk) (2.28)

standard errors in both the dependent variable y and the independent variable x may be significant. However, a useful simplifying assumption that we have used thus far is that standard error in x is negligible. That is, the Xj are known with absolute precision. This commonly used assumption is reasonably accurate in many cases, such as when y is a slowly varying instrumental response measured vs. time as x. If random errors in x can be neglected, only the random errors in y need be considered in the error sum and the weighting factor.

In the example of Section B.4, the error in y was independent of the size of y and the unweighted error sum was minimized. In such cases, we say that there is absolute error in y. Suppose we measure the optical absorbance of a reacting species vs. time with a conventional spectrophotometer. As long as the sensitivity scale is the same for all measurements, the standard error in absorbance (y) is likely to be the same whether the absorbance is small or large. Therefore, we have absolute error in y. This can be checked experimentally by measuring standard deviations of the absorbance. If the errors are absolute, the standard deviations should be approximately the same at different absorbance values in the range of the experimental data.

In other cases, standard errors in y may be proportional to y, and the sum of squares of the relative errors should be minimized. Standard errors in y may be neither absolute nor strictly relative, and alternative weighting factors are required. This occurs for data obtained by counting detectors, for which the random error in y(meas) is Poisson distributed [10]; that is, proportional to [_y(meas)]1/2. Appropriate weighting factors for these common cases are listed in Table 2.6.

Table 2.6 Common Weighting Factors for Errors in Assuming Error-Free

Error distribution

Absolute, independent of y, Relative, proportional to yt Proportional to y)n

SD"

" n = number of data; p = number of parameters

The use of weighting factors in the Mathcad nonlinear regression program simply requires its inclusion in the expression in S as SSE^, b2), using the form in eq. (2.9). In general, for a Poisson error distribution in y,

For weighted regression however, the SD has a different relation to the root mean square error. These are also listed in Table 2.6.

Finally, eq. (2.29) expressed the sum of squares of the differences in calculated and experimental values of y. If random errors in x are also significant, an appropriately weighted function reflecting the distribution of errors in both x and y should be minimized [6].

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