Non-linear regression programs

Data given in Tables 1-6 clearly show a significant dependence of P2 and p4 on amine concentration, that is, at least one of the apparent rate constants kj contains a concentration factor. Thus, according to the mathematical considerations outlined in the Analysis of Data Paragraph, both p2, P4 exponents and the derived variables -(P2 + p)4> P2 P4 ind Z (see Eqns. 8-12) are the combinations of the apparent rate constants (kj). To characterize these dependences, derived variables -(p2+p)4, P2 P4 and Z (Eqns. 8,11 and 12) were correlated with the amine concentration using a non-linear regression program to find the best fit. Computation resulted in a linear dependence for -(p2 + p)4 and Z, that is... 

The lipase activity unit (LU) is described in publication AF 95.1/2GB obtainable from Novo Nordisk A/S. The experimental data was fit to Eq. 6.1 with the non-linear regression program GraphPad Prism. 

The data can be evaluated using any commonly available non-linear regression program or with a linear regression, in which k,a is the slope from the plot of the natural log of the concentration difference versus time. Linearity of the logarithmic values over one decade is required for the validity of the measurement. Of course the assumptions inherent in the model must apply to the experimental system, especially in respect to completely mixed gas as well as liquid phases and reactions are negligible. Two common problems are discussed below. Other common pitfalls and problems are summarized in Table 3-3. 

For the determination of the initial rate constants and the kinetic parameters, the experimental data can be fitted using non-linear regression programs such as Origin 7.5SR6 (OriginLab Corporation, Northampton, Massachusetts). 

The minimization of the WSS value is achieved by systematic modification of the values of the adjustable parameters of the model chosen. Changing the parameters moves the calculated line and thus the WSS. Achieving the minimum WSS most efficiently is the objective of the minimization algorithms incorporated into the various non-linear regression programs. Fig. 2 illustrates how the WSS value changes with different parameter values. 

A typical pharmacokinetic analysis of data from a single individual results in estimates for the values and variance (or SD) of each parameter. The calculated variance gives an estimate of the closeness of fit, the adequacy of the number of data points, and the error in the data values. If this analysis is repeated in a number of individuals, it is also possible to obtain some information about the average population value and intersubject variance. With sufficient data available from each subject, this two-step approach can be efficient using any number of non-linear regression programs. 

Once that is done plug into POLYMATH S non-linear regression program with the following table. 

The non-linear regression program supplied with the Hewlett-Packard 9845 minicomputer was used to solve the kinetic uptake phase equation for a one-compartment open model operating under first order kinetics ... 

The other possibility to contend with this problem is to fit the value of cL to the data. This can be done per hand, or may be possible with the program used for non-linear regression. 

The Biodegradability Probability Program has been developed over a period of years and is still being refined. As indicated in Table 12.6, the original model was developed with 35 structural fragments whose coefficients were developed by linear and non-linear regression to an experimental "weight-of- evidence" evaluation for 264 chemicals from the BIO-DEG database (Howard et al., 1987). Chemicals were used only if two or more... 

For both the subdistribution and the GEX fit methods a Marquardt algorithm for constrained non-linear regression was used to minimize the sum of squares error (.10). The FORTRAN program CONTIN was used for the constrained regularization method. All computations were performed on a Harris H-800 super mini computer. 

There are many published computer programs and commercial software packages that perform non-linear regression analysis, but you can obtain the same results very easily by using the Solver. When applied to the same data set, the Solver gives the same results as commercial software packages. 

Non-linear regression analyses involve relatively complex calculations and thus are well suited to computer assistance. However, the program must have a well developed sequence of steps or algorithm to follow. Some methods are better than others. The program is asked to find the minimum point on a weighted sum of squares (objective or minimized function) surface. For two parameters, this can be represented as a three-dimensional surface (Fig. 4). 

NONMEM is provided as FORTRAN source code for UNIX, IBM, and other computers. The program performs non-linear regression of individual or population data. 

SAAMII is provided as a compiled program for Windows or Macintosh systems and performs non-linear regression analysis. It has a graphical user interface for model specification. 

The parameters of the above model were estimated using a non-linear regression computer program. The parameters obtained are ... 

---