# Statistical Results

Nonlinear regression programs provide a variety of statistical information that can be used to evaluate the fit to the data.
## Weighted Sum of Squared Residual (WSS)

WSS =
**Equation 9.4.1 Weighted sum of squared residuals**

With the same weighting scheme WSS can be used to compare different models and data sets. The lower the value the better. For a series of subjects the WSS values should be similar especially if the assay errors are similar between data sets. Generally the WSS should get smaller (or at least not get larger) with a bigger model. The question becomes; is the improvement in WSS a significant improvement?

## Coefficient of Determination (R^{2}) and Correlation Coefficient (R)

The coefficient of determination (R^{2}) and the correlation coefficient also indicate information about the goodness of the fit. Values close to one are best. These criteria don't seem to be very sensitive as it can be easy to get good values.
## Akaike Information Criterion (AIC) and Schwarz Criterion (SC)

The AIC (1, 2) and BIC (3) values were designed to help in the selection of the better of two alternate models. The lower value is better. It is important that these comparisons are made using the same weighting scheme.

AIC = N • ln(WSS) + 2 • M
**Equation 9.4.2 Akaike Information Criterion (AIC)**

SC = N • ln(WSS) + M • ln(N)

**Equation 9.4.3 Schwarz Criterion (SC)**

where N is the number if data points, M is the number of adjustable parameters, and WSS is the weighted sum of squared residuals.

References

- Akaike, H. 1974 "A New Look at the Statistical Model Identification",
*IEEE Trans. Automat. Control*, **19**, p 716-723
- Yamaoka, K., Nakagawa, T., and Uno, T. 1978 "Application of Akaike's Information Criterion (AIC) in the Evaluation of Linear Pharmacokinetic Equations",
*J. Pharmacokin. Biopharm.*, **6**, p 165-175
- Schwarz, G. 1978 "Estimating the Dimension of a Model",
*Ann. Stat.*, **6**, p 461-464

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