Extended Least Squares (ELS)
Another approach to choosing the most appropriate weighting scheme is to fit parameters of the variance model during the optimization process. The form of the variance versus data equation needs to be determined. Once the form and parameters involved are determined the optimization process can determine the variance and model parameters given good initial estimates and enough good data. There are some differences. Since there are more parameters more data are necessary. Different algorithms are necessary for the optimization. A different objective function, the function minimized during the optimization, is needed. Different optimization algorithms are necessary. The programs ADAPT and NONMEM include these algorithms.
Equation 13.7.1 Objective Function for Extended Least Squares Optimization
Note the inclusion of the lnV term in Equation 13.7.1. This will prevent the optimization algorithm from driving the value of V high and thus the weight low to minimize the objective function (WSS) without regard for the model parameter values.
The equation for V, the variance of each data point, may take a number of forms. Some of these are shown in Equation 13.7.2.
Equation 13.7.2 Example Variance Equations
- 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
- Peck, C.C., Beal, S.L., Sheiner, L.B. and Nichols, A.I. 1984 "Extended Least Squares Nonlinear Regression: A Possible Solution to the 'Choice of Weights' Problem in Analysis of Individual Pharmacokinetic Data", J. Pharmcokin. Biopharm., 12, pp 545-558
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