Extended least-squares

The aim of parameter estimation is an adaptation of the model function to the observations made to gain model parameters which describe the observed data best. In NONMEM this is done by the minimization of the extended least square objective Oels function, which provides maximum likelihood estimates under Gaussian conditions [13]. The equation calculating the Oels function is given in the following ... 

Assuming that we have measured a series of concentrations over time/ we can define a model structure and obtain initial estimates of the model parameters. The objective is to determine an estimate of the parameters (CLe, Vd) such that the differences between the observed and predicted concentrations are comparatively small. Three of the most commonly used criteria for obtaining a best fit of the model to the data are ordinary least squares (OLS)/ weighted least squares (WLS)/ and extended least squares (ELS) ELS is a maximum likelihood procedure. These criteria are achieved by minimizing the following quantities/... 

With normal weighted non-linear regression, the objective is to minimize this objective function. As it is described later additional terms may be added to the objective function when performing extended least-squares Eq. (23) or Bayesian analyses Eq. (24). 

Another approach to weighting the data uses the data itself to develop the variance equation. This is the extended least squares (ELS) method. Parameters for the variance equations are included in the fitting... 

ADAPT II is supplied as FORTRAN code for VAX VMS, MS DOS, and SUN UNIX systems. It performs simulations, fitting, and optimal sampling and includes extended least-squares and Bayesian optimization. Models can be... 

MULTI programs by K. Yamaoka and col-leaguesf" are provided as BASIC source code within the manuscripts. Different versions include fitting to integrated or differential equations, Bayesian analysis, and extended least squares. 

Peck, C.C. Beal, S.L. Sheiner, L.B. Nichols, A.I. Extended least squares non-linear regression a possible solution to the choice of weights problem in analysis of individual pharmacokinetic data. J. Pharmacokinet. Biopharm. 1984, 12, 545-558. 

Sheiner, L.B. Beal, S.L. A note on confidence intervals with extended least squares parameter estimates. J. Pharmacokinet Biopharm 1987, 15, 93-98. 

L. B. Sheiner and S. L. Beal, Pharmacokinetic parameter estimates from several least squares procedures superiority of extended least squares. J Pharmacokinet Biopharm 13(2) 185-201 (1985). 

Equation (3.11) is sometimes called the objective function, although it should be stressed that there are many different types of objective functions, such as extended least squares. For purposes of this book, an objective function will be defined as any quadratic function that must be minimized to obtain a set of parameter estimates. For this chapter the focus will be on the residual sum of squares as the objective function, although in later chapters more complex objective functions may be considered. 

Another common fitting algorithm found in the pharmacokinetic literature is extended least-squares (ELS) wherein 0, the structural model parameters, and 4>, the residual variance model parameters, are estimated simultaneously (Sheiner and Beal, 1985). The objective function in ELS is the same as the objective function in PL... 

Extended least-squares with estimated < and weights equal to l/Y , denoted as ELS. 

---