Hougaard's Measure of Skewness

The NLIN Procedure

Hougaard's Measure of Skewness

A "close-to-linear" nonlinear regression model, first described by Ratkowsky (1990), is a model that produces parameters having properties similar to those produced by a linear regression model. That is, the least squares estimates of the parameters are close to being unbiased, normally distributed, minimum variance estimators.

A nonlinear regression model sometimes fails to be close to linear due to the properties of a single parameter. When this occurs, bias in the parameters can render inferences using the reported standard errors and confidence limits invalid. You can often fix the problem with reparameterization, replacing the offending parameter by one with better estimation properties.

You can use Hougaard's measure of skewness, g_1i, to assess whether a parameter is close to linear or whether it contains considerable nonlinearity. Specify the HOUGAARD option in the PROC NLIN statement to compute Hougaard's measure of skewness.

According to Ratkowsky (1990), if |g_1i| < 0.1, the estimator $\hat{\theta}_i$ of parameter $\theta_i$ is very close-to-linear in behavior and, if 0.1 < |g_1i| < .25, the estimator is reasonably close-to-linear. If |g_1i| > .25, the skewness is very apparent. For |g_1i| > 1, the nonlinear behavior is considerable.

Hougaard's measure is computed as follows

$E[ \hat{\theta}_i - E(\hat{\theta}_i)]^3 = -(mse)^2 \sum_{jkl}^{np} L^{ij}L^{ik}L^{il}(W_{jkl}+W_{kjl}+W_{lkj})$

where the sum is a triple sum over the number of parameters and

L = X'X^-1

$W_{jkl} = \sum_{m=1}^n J_{m}^j H_m^{kl}$

In the preceding equation, J_m is the Jacobian vector and H_m is the Hessian matrix evaluated at observation m. This third moment is normalized using the standard error as

$g_{1i} = E[ \hat{\theta}_i - E(\hat{\theta}_i)]^3 /( mse * L^{ii} )^{3/2}$

Chapter Contents
Previous
Next
Top