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Final Parameter Values

Final Values provided by the Program

During the fitting process there may be intermediate result printed out. Review of this information can indicate problems in the fitting process.

Intermediate results from Boomer (DGN Method)

Figure 9.2.1 Intermediate results from Boomer (DGN Method)

The parameter to watch is the objective function (the number to be minimized) which in this case is the weighted sum of the squared residuals (WSS). It should continue to fall in value as a successful fit continues.

Intermediate result from Boomer (Simplex method)

Figure 9.2.2 Intermediate results from Boomer (Simplex Method)

Once the nonlinear regression program has converged to the best fit solution (maybe) the final parameter values with their uncertainty will be presented.

Final parameter values from Boomer

Figure 9.2.3 Final parameter values from Boomer

Are the final results reasonable?

Parameter Uncertainty

Standard Deviation, Coefficient of Variation (CV), Confidence Interval

Figure 9.2.3 provides information about the uncertainty in the final values as well the values themselves. The parameter standard deviation and coefficient of variability give good information about how well the data have been fit to the model. If these numbers are too large (CV > 20% lower is even better) there may be a problem with the model, too much error in the data or not enough data. Problems with the model would suggest trying a different model and refitting the data. However, problems with the error in the data or the number of data points would require redoing the experiment! Some programs provide an estimate of the parameter (95%) confidence intervals. It is best if these don't cross zero.

Figure with good CV values

Figure 9.2.4 Linear plot of Cp versus time - Good CV values

Figure 9.2.4 illustrates a good fit to the provided data. Note the values for the parameter CV are all below 20%.

Plot of Cp versus time - bad CVs

Figure 9.2.5 Linear plot of Cp versus time - High CV values

Figure 9.2.5 looks the same as 9.2.4, why the higher CV values? Have a closer look at the early time points!

Close-up of early data point

Figure 9.2.6 Enlarged view of the early time points

Notice the second and third points are nearly superimposed. Maybe there is data entry error and even an assay error.

Another example. Again the fit looks good but the CV values for the distribution rate constants, k12 and k21, are quite large. Why?

Figure with too few data points

Figure 9.2.7 Linear plot of Cp versus time - fewer data

In Figure 9.2.7 the CV are large because there aren't enough data points (at the right times) to clearly define the chosen model. More, better times data points should help. Or, maybe a smaller model.

It is important to note that the standard deviation and CV values provide by nonlinear regression programs when fitting a single subject data set have no correlation with the population variability (standard deviation) in the parameter values.

Parameter Limit Error Messages

When preforming nonlinear regression analysis I like to specify limit on all the adjustable parameters. For example, a lower limit of zero on rate constants to stop them from going to zero. With poor initial estimates and/or complex models it is possible that the nonlinear regression program may get 'lost'. An error message relating to parameter limits may be the result.

SAAM II error message - hit limit

Figure 9.2.8 Error message from SAAM II - Hit limit on two parameters

This may result in a less than ideal convergence.

Plot of Cp versus time - fit with hit limit

Figure 9.2.9 Linear plot of Cp versus time with fit constrained by parameter limit hit

Notice the poor fit. It is time to consider the parameter limits originally specified. It may be possible to simply expand the limits and refit. It might be an error in the way the model is specified however. This should be fixed. There may be a data entry error. It could be a scaling error. A dose in milligram may be confused with concentration measured in ng/L. In the present case refitting with a slighter wider limit gives a more satisfactory result.

Plot of Cp versus time with better limits

Figure 9.2.10 Linear plot of Cp versus time with better parameter limits

Correlation Matrix

Some nonlinear regression programs provide the additional output of a correlation matrix between the adjustable parameters.

Correlation matrix from SAAM II

Figure 9.2.11 Correlation matrix from SAAM II

A high correlation between two parameters (absolute value close to 1) might suggest that a smaller model would work. That is, it may not be necessary to include both parameters in the model.

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