Least-Squares Regression Models - Parameter Estimates and Boundaries Rules

Parameter Estimates and Boundaries Rules

Rules for using parameter estimates and boundaries in Indirect Response, Linear, Michaelis-Menten, PD, PK, PK/PD, and ASCII models include the following:

Using boundaries, either user- or WinNonlin-generated, is recommended.
Phoenix uses curve stripping only for single-dose data. If multiple dose data is fit to a PK model and Phoenix generates initial parameter estimates, a grid search is performed to obtain initial estimates. In this case, boundaries are required.
For linear regressions, it is recommended that users keep the Do Not Use Bounds option checked in the Parameter Options tab and select minimization method 3 (Gauss-Newton (Hartley)) in the Engine Settings tab.
The grid used in grid searching uses three values for each parameter. If four parameters are estimated, the grid contains 3⁴=81 possible points. The point on the grid associated with the smallest sum of squares is used as an initial estimate for the estimation algorithm. This option is included for convenience, but it might greatly slow the parameter estimation process if the model is defined in terms of differential equations.
If the data are fitted to a micro-constant model, Phoenix performs curve stripping for the corresponding macro-constant model then uses the macro-constants to compute the micro-constants. It is recommended that users specify the Lower and Upper boundaries, even when Phoenix computes the initial estimates.
If the Gauss-Newton methods (minimization methods 2 and 3) are used to estimate the parameters, then the boundaries are determined by applying a normit transform to the parameter space.
If the Nelder-Mead method (minimization method 1) is selected, the residual sum of squares is set to a large number if any of the parameter estimates go outside the specified constraints at any iteration, which forces the parameters back within the specified bounds.
Unlike the linearization methods, the use of bounds does not affect numerical stability when using the Nelder-Mead method. Using bounds with the Nelder-Mead method will keep the parameters within the bounds. The use of bounds, either user-supplied or Phoenix-supplied, is always recommended with the Gauss-Newton methods.

References

Beck and Arnold (1977). Parameter Estimation in Engineering and Science. John Wiley & Sons, New York.

Bard (1974). Nonlinear Parameter Estimation. Academic Press, New York.

Bates and Watts (1988). Nonlinear Regression Analyses and Its Applications. John Wiley & Sons, New York.

Davies and Whitting (1972). A modified form of Levenberg's correction. Chapter 12 in Numerical Methods for Non-linear Optimization. Academic Press, New York.

Dayneka, Garg and Jusko (1993). Comparison of four basic models of indirect pharmacodynamic responses. J Pharmacokinet Biopharm 21:457.

Draper and Smith (1981). Applied Regression Analysis, 2nd ed. John Wiley & Sons, NY.

Endrenyi, ed. (1981). Kinetic Data Analysis: Design and Analysis of Enzyme and Pharmacokinetic Experiments. Plenum Press, New York.

Gabrielsson and Weiner (2016). Pharmacokinetic and Pharmacodynamic Data Analysis: Concepts and Applications, 5^th ed. Apotekarsocieteten, Stockholm.

Gill, Murray and Wright (1981). Practical Optimization. Academic Press.

Hartley (1961). The modified Gauss-Newton method for the fitting of nonlinear regression functions by least squares. Technometrics 3:269–80.

Jennrich and Moore (1975). Maximum likelihood estimation by means of nonlinear least squares. Amer Stat Assoc Proceedings Statistical Computing Section 57–65.

Jusko (1990). Corticosteroid pharmacodynamics: models for a broad array of receptor-mediated pharmacologic effects. J Clin Pharmacol 30:303.

Kennedy and Gentle (1980). Statistical Computing. Marcel Dekker, New York.

Metzler and Tong (1981). Computational problems of compartmental models with Michaelis-Menten-type elimination. J Pharmaceutical Sciences 70:733–7.

Nelder and Mead (1965). A simplex method for function minimization. Computing Journal 7:308–13.

Ratkowsky (1983). Nonlinear Regression Modeling. Marcel Dekker, New York.

Shampine, Watts and Davenport (1976). Solving nonstiff ordinary differential equations - the state of the art. SIAM Review 18:376–411.

Tong and Metzler (1980). Mathematical properties of compartment models with Michaelis-Menten-type elimination. Mathematical Biosciences 48:293–306.