Modeling and Multivariate Methods - SAS

Chapter 9 Performing Nonlinear Regression 271
Statistical Details
Figure 9.20 Solution Table for the Poisson Loss Example
Statistical Details
This section provides statistical details and other notes concerning the Nonlinear platform.
Profile Likelihood Confidence Limits
The upper and lower confidence limits for the parameters are based on a search for the value of each
parameter after minimizing with respect to the other parameters. The search looks for values that produce
an SSE greater by a certain amount than the solution’s minimum SSE. The goal of this difference is based
on the F-distribution. The intervals are sometimes called likelihood confidence intervals or profile likelihood
confidence intervals (Bates and Watts 1988; Ratkowsky 1990).
Profile confidence limits all start with a goal SSE. This is a sum of squared errors (or sum of loss function)
that an F test considers significantly different from the solution SSE at the given alpha level. If the loss
function is specified to be a negative log-likelihood, then a Chi-square quantile is used instead of an F
quantile. For each parameter’s upper confidence limit, the parameter value is increased until the SSE reaches
the goal SSE. As the parameter value is moved up, all the other parameters are adjusted to be least squares
estimates subject to the change in the profiled parameter. Conceptually, this is a compounded set of nested
iterations. Internally there is a way to do this with one set of iterations developed by Johnston and DeLong.
See SAS/STAT 9.1 vol. 3 pp. 1666-1667.
Figure 9.21 shows the contour of the goal SSE or negative likelihood, with the least squares (or least loss)
solution inside the shaded region:
• The asymptotic standard errors produce confidence intervals that approximate the region with an
ellipsoid and take the parameter values at the extremes (at the horizontal and vertical tangents).
• Profile confidence limits find the parameter values at the extremes of the true region, rather than the
approximating ellipsoid.