Modeling and Multivariate Methods - SAS

Chapter 9 Performing Nonlinear Regression 269
Additional Examples
The loss function is used in this example, so the second derivative as well as the first one is calculated for the
optimization. With least squares, the second derivatives are multiplied by residuals, which are usually near
zero. For custom loss functions, second derivatives can play a stronger role. Follow these steps to fit the
model:
1. Select Analyze > Modeling > Nonlinear.
2. Assign Model2 Y to the X, Predictor Formula role.
3. Assign Loss2 to the Loss role.
4. Check the Second Derivatives option.
5. Click OK.
6. Type 1000 for the Iteration Stop Limit.
7. Click Go.
The Solution report shows the same Loss and parameter estimates as before.
Probit Model with Binomial Errors: Numerical Derivatives
The Ingots2.jmp file in the Sample Data folder records the numbers of ingots tested for readiness after
different treatments of heating and soaking times. The response variable, NReady, is binomial, depending
on the number of ingots tested (Ntotal) and the heating and soaking times. Maximum likelihood estimates
for parameters from a probit model with binomial errors are obtained using
• numerical derivatives
• the negative log-likelihood as a loss function
• the Newton-Raphson method.
The average number of ingots ready is the product of the number tested and the probability that an ingot is
ready for use given the amount of time it was heated and soaked. Using a probit model, the P column
contains the model formula:
Normal Distribution[b0+b1*Heat+b2*Soak]
The argument to the Normal Distribution function is a linear model of the treatments.
To specify binomial errors, the loss function, Loss, has the formula
-[Nready*Log[p] + [Ntotal - Nready]*Log[1 - p]]
Follow these steps to fit the model:
1. Select Analyze > Modeling > Nonlinear.
2. Assign P to the X, Predictor Formula role,
3. Assign Loss to the Loss role.
4. Select the Numeric Derivatives Only option.
5. Click OK.
6. Click Go.