Modeling Assumptions and Notation
PROC NLMIXED operates under the following general framework for
nonlinear mixed models. Assume that you have an observed data vector
yi for each of i subjects, i = 1, ... ,s. The yi are
assumed to be independent across i, but within-subject covariance
is likely to exist because each of the elements of yi are
measured on the same subject. As a statistical mechanism for
modeling this within-subject covariance, assume that there exist latent
random-effect vectors ui of small dimension (typically one or
two) that are also independent across i. Assume also that an
appropriate model linking yi and ui exists, leading to the
joint probability density function
where Xi is a matrix of observed explanatory variables and and are vectors of unknown parameters.
Let and assume that it is of dimension n.
Then inferences about are based on the marginal likelihood
In particular, the function
is minimized over numerically in order to estimate
, and the inverse Hessian (second derivative) matrix at the
estimates provides an approximate variance-covariance matrix for the
estimate of . The function is referred to both
as the negative log likelihood function and as the objective function
As an example of the preceding general framework, consider the
nonlinear growth curve example in the "Getting Started"
section. Here, the conditional distribution is normal with mean
and variance ; thus .Also, ui is a scalar and is normal with mean 0 and
variance ; thus .
[(b1 + ui1)/(1 + exp[-(dij - b2)/ b3])]
The following additional notation is also found in this chapter.
The quantity refers to the parameter vector at the
kth iteration, the function refers to the gradient
vector , and the matrix refers to the
Hessian . Other symbols are used to denote
various constants or option values.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.