Consequences of an asymmetrical parameter
Even though nonlinear regression, as its name implies, is designed to fit nonlinear models, some of the inferences actually assume that some aspects of the model are close to linear, so that the uncertainty about each parameter's value is symmetrical. This means that if you analyzed many data sets sampled from the same system, the distribution of the best-fit values of the parameter would be symmetrical and Gaussian.
If the distribution of a parameter is highly skewed, there are two consequences:
•The SE of that parameter will not be a very useful measure of uncertainty. The SE is interpreted as a plus-minus assessment of how sure you are of the parameter value. But if the parameter is very asymmetrical, then a single SE cannot really describe the uncertainty.
•The confidence interval for that parameter cannot be interpreted at face value. Prism always computes the CI from the SE of that parameter (and the number of degrees of freedom), and the CI is always symmetrical around the best-fit value. If the parameter is very asymmetrical, then that symmetrical confidence interval does not give a accurate picture of the uncertainty.
Hougaard's measure of skewness
Hougaard (1) developed a way to assess the skewness of a parameter used in nonlinear regression without doing any simulations. Prism will compute this value for each parameter when you check the option the Diagnostics tab of nonlinear regression in the section labeled "Are the parameters intertwined, redundant or skewed?". The results are tabulated along with the other results of nonlinear regression.
Ratkowsky has proposed the following interpretation:
Absolute value |
Interpretation |
<0.10 |
Ideal. Almost linear. Confidence intervals can be interpreted at face value |
0.10 - 0.25 |
Adequate |
0.25 - 1.00 |
Noticeable skewness. Consider alternative parameterizations of the equation |
> 1.00 |
Considerable skewness. Strongly consider alternatives |
Note that these values are for the absolute value of the Hougaard's measure.
For the simulated data set for the example above, Hougaard's skewness is 0.09 for Khalf and 1.83 for Kprime. This one value (no simulations needed) tells you to choose the form of the model that fits Khalf rather than the form that fits Kprime.
Notes
•Hougaard's measure of skewness is measured for each parameter in the equation (omitting parameters fixed to constant values).
•Prism does not compute Hougaard's skewness if you chose unequal weighting, or a robust fit, because the method is not defined for these situations.
•The values depend on the equation, the number of data points, the spacing of the X values, and the values of the parameters.
•Hougaard's measure of skewness has no units.
•The SAS documentation does a great job of explaining Hougaard's measure (3).
References
1. P. Hougaard. The appropriateness of the asymptotic distribution in a nonlinear regression model in relation to curvature. Journal of the Royal Statistical Society. Series B (Methodological) (1985) pp. 103-114
2. David A. Ratkowsky, Nonlinear Regression Modeling: A Unified Practical Approach (Statistics: a Series of Textbooks and Monogrphs). IBSN:0824719077
