WNL - Core output file

Core Output File

The Core Output is an ASCII text file that contains a complete summary of the model commands, options, parameters, and values for a model, as well as any errors that occurred during modeling. This file is generated as output for Indirect Response, Linear, Michaelis-Menten, PD, PK, PKPD, and ASCII Model objects.

Sections of the Core Output file include:

List of input commands
Minimization process
Final parameters
Variance-covariance matrix, correlation matrix, and eigenvalues
Residuals
Secondary parameters

List of input commands

The first section lists the input commands used to run the model. See “Commands, Arrays, and Functions” for an alphabetic listing of commands.

Minimization process

This section varies depending on the selected minimization method. If a Gauss-Newton method is used (methods 2 or 3), this page shows the parameter values and the computed weighted sum of squared residuals at each iteration. In addition, the following two values are listed for each iteration:

RANK: The rank of the matrix of partial derivatives of the model parameters. If the matrix is of full rank, then the rank equals the number of parameters. If the rank is less than the number of parameters, then the problem is ill-conditioned. That means there is not enough information contained in the data to precisely estimate all of the parameters in the model.

CONDITION NO.: Condition number of the matrix of partial derivatives. The condition number is the square root of the ratio of the largest to the smallest eigenvalue of the matrix of partial derivatives. If the condition number gets to be very large, for example greater than 10E+06, then the estimation problem is very ill-conditioned. If minimization methods 2 or 3 are used, then using lower and upper parameter boundaries can help reduce the condition number.

If the Nelder-Mead method is used, this section only shows the parameter values and the weighted sum of squares for the initial point, final point (the best or smallest sum of squares), and the next-to-best point.

Final parameters

This section lists the parameter estimates, the asymptotic estimated standard error of each estimate, and two intervals based on this estimated standard error. The intervals labeled UNIVAR_CI_LOW and UNIVAR_CI_UPP are the parameter estimates plus and minus the product of the estimated standard error and the appropriate value of the t-statistic. Univariate intervals should be applied individually to the relevant single parameters, and have the interpretation that there is a 95% probability that the interval contains the true value of the individual parameter.

The intervals labeled PLANAR_CI_LOW and PLANAR_CI_UPP are obtained from the tangent planes to the joint 95% ellipsoid of all the parameter estimates. The intervals are defined by the parameter estimates plus and minus the product of the standard error and the appropriate value of an F-statistic. The PLANAR intervals in general are larger than the corresponding UNIVARIATE intervals and jointly define a region in the shape of a rectangular box that contains the joint ellipsoid. This PLANAR region has the interpretation there is at least a 95% probability that this box contains the true parameter vector formed from the individual parameter values.

For an introductory discussion of the issues involved in UNIVARIATE and PLANAR intervals see page 95 (Draper and Smith (1981)). Details of the appropriate F-distribution statistic used in the PLANAR ellipsoidal and box region computations can be found in most advanced statistical texts.

The estimated standard errors and intervals are only approximate, because they are based on a linearization of a nonlinear model. The closer the model is to being linear, the better the approximation. See Chapter 10 of Draper and Smith (1981) for a complete discussion of the true intervals for parameter estimates in nonlinear models.

The estimated standard errors are valuable for indicating how much information about the parameters is contained in the data. Many times a model provides a good fit to the data in the sense that all of the deviations between observed and predicted values are small, but one or more parameter estimates have standard errors that are large relative to the estimate.

Variance-covariance matrix, correlation matrix, and eigenvalues

The next three sections of the Core Output show the correlation matrix of the estimates and the eigenvalues of the linearized form of the model. High correlations among one or more pairs of the estimates indicate that the UNIVAR limits are underestimates of the uncertainty in the parameter estimates and, although the data may be well fit by the model, the estimates may not be very reliable. Also, datasets that result in highly correlated estimates are often difficult to fit, in the sense that the Gauss-Newton algorithm will have trouble finding the minimum residual sum of squares.

The eigenvalues are another indication of how well the data define the parameters. If the parameter estimates are completely uncorrelated, then the eigenvalues are equal. A large ratio between the largest and smallest eigenvalue may indicate that there are too many parameters in the model. However, it is usually not possible to remove one parameter from a nonlinear model. For discussion of the use of eigenvalues in modeling see (Belsley, Kuh and Welsch (1980)).

Residuals

This section of the output lists the observed data, calculated predicted values of the model function, residuals, weights, the standard deviations (S) of the calculated function values and standardized residuals. Runs of positive or negative deviations indicate non-random deviations from the model and are indicators of an incorrect model and/or choice of weights.

Also in this section are the sum of squared residuals, the sum of weighted squared residuals, an estimate of the error standard deviation, and the correlation between observed and calculated function values. In nonlinear models the correlation is not a particularly good measure of fit. In most problems it is greater than 0.9, and anything less than 0.8 probably indicates serious problems with the data and/or model. Two statistical criterion for model selection and comparison, the AIC (Akaike (1978)) and SBC (Schwarz (1978)), are also listed in this section.

If the data is fit to a model for extravascular or constant infusion input, area under the curve (AUC) is listed at the end of this section of the Core Output. The AUC is computed by the linear trapezoidal rule. If the first time value is not zero, then WinNonlin generates a data point with time and concentration values of zero, to compute AUC from zero to the last time. Note that this AUC in the Core Output differs from the AUC in the Secondary Parameters results worksheet. The secondary parameter AUC is calculated using the model parameters.

Note:AUC values are always associated with the first dose and might not be meaningful if the data were obtained after multiple dosing or IV dosing, due to time zero extrapolation problems.

Secondary parameters

Any secondary parameters and their estimated asymptotic standard errors are listed in the last section. The secondary parameters are functions of the primary parameters listed in the Core Output. In the case of multiple-dose data, secondary parameters that depend on dose use the first dose in their computation. The standard errors of the secondary parameters are obtained by computing the linear term of a Taylor series expansion of the secondary parameters. Secondary parameters are the third level of approximation and their accuracy should be regarded with caution.

References

Akaike (1978). Posterior probabilities for choosing a regression model. Annals of the Institute of Mathematical Statistics 30:A9 –14.

Belsley, Kuh and Welsch (1980). Regression Diagnostics. John Wiley & Sons, New York.

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

Schwarz (1978).Estimating the dimension of a model. Annals of Statistics 6:461–4.