Chapter 29 Weighting Data

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Iteratively Reweighted Least Squares (IRWLS)

On the previous page all of the weighting schemes involved variance as a function of the value of the observed data point, except for the equal weight scheme. This is computationally efficient since each weight only needs to be calculated once at the beginning of the optimization step. It also ties the weight to the variance of the observed data points. However, when a weighting scheme is applied to a series of data points, data points with very low values may be given more emphasis than is appropriate. An alternative weighting has been used to try to overcome this disadvantage. This method is the iteratively reweighted least squares method. Effectively, this is identical to the methods on the previous pages except that observed data is replaced with calculated data. Thus, the weight is recalculated during each phase of the optimization process. Thus very low observed data points would not have the emphasis on the overall analysis. However, it is possible that the optimization may drive calculated values low, giving these points more emphasis and potentially distorting the final analysis.

Weighting Schemes Based on IRWLS

Equation 29.6.1 Variance Proportional to Calculated Data

Equation 29.6.2 Variance Proportional to the Square of Calculated Value

Equation 29.6.3 Variance as a Function of Calculated Value

Equation 29.6.4 Variance as a Function of Calculated Value with an Assay Sensitivity Term

Equation 29.6.5 Variance as a Function of Calculated Value with Time Information

The above equations are just a collection of possible forms for the variance and thus weight equation that might be useful. Various software packages will have similar and other options for weighting the data.

References

Peck, C.C., Sheiner, L.B. and Nichols, A.I. 1984 "The Problem of Choosing Weights in Nonlinear Regression Analysis of Pharmacokinetic Data", Drug Metabolism Reviews, 15, pp 133-148

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