Modeling and Multivariate Methods - SAS

272 Performing Nonlinear Regression Chapter 9
Statistical Details
Figure 9.21 Diagram of Confidence Region
Confidence Limit for Parameters
95% confidence
interval for b1
approximate 95%
confidence ellipse
likelihood 95%
confidence region
95% confidence
interval for b0
Likelihood confidence intervals are more trustworthy than confidence intervals calculated from approximate
standard errors. If a particular limit cannot be found, computations begin for the next limit. When you have
difficulty obtaining convergence, try the following:
• use a larger alpha, resulting in a shorter interval, more likely to be better behaved
• use the option for second derivatives
• relax the confidence limit criteria.
How Custom Loss Functions Work
The nonlinear facility can minimize or maximize functions other than the default sum of squares residual.
This section shows the mathematics of how it is done.
Suppose that f(β) is the model. Then the Nonlinear platform attempts to minimize the sum of the loss
functions written as
L =
n
i = 1
ρ( f( β)
)
The loss function ρ (•)
for each row can be a function of other variables in the data table. It must have
nonzero first- and second-order derivatives. The default ρ (•)
function, squared-residuals, is
ρ( f( β)
) = ( y–
f( β)
) 2
To specify a model with a custom loss function, construct a variable in the data table and build the loss
function. After launching the Nonlinear platform, select the column containing the loss function as the loss
variable.