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.
•Symmetrical confidence intervals for that parameter cannot be interpreted at face value. If the parameter is very asymmetrical, then that symmetrical confidence interval does not give a accurate picture of the uncertainty. Note that Prism (starting with version 7) can compute asymmetrical profile likelihood confidence intervals, and these work fine with asymmetrical parameters.
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 interpretations apply to the absolute value of the Hougaard's measure.
Hougaard's skewness with unequal weighting
While Prism 6 and 7 calculated Hougaard's skewness correctly for unweighted fits, they computed it incorrectly if you chose unequal weighting. This is fixed in Prism 8.
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 a robust fit, because the method is not defined for this situation.
•The values depend on the equation, the number of data points, the spacing of the X values, and the Y values.
•Hougaard's measure of skewness has no units.
•A positive value means that the asymmetry is to the right, with a longer confidence interval above the best-fit value than below it. A negative value means that the asymmetry is to the left.
•The SAS documentation does a great job of explaining Hougaard's measure (3).
•Prism can compute asymmetrical profile likelihood confidence intervals. These show the asymmetry directly, so reduce the need to ask Prism to compute Hougaard's skewness.
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
