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Modeling Assumptions and Notation
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PROC NLMIXED operates under the following general framework for nonlinear mixed models. Assume that you have an observed data vector
for each of
subjects,
. The
are assumed to be independent across
, but within-subject covariance is likely to exist because each of the elements of
is measured on the same subject. As a statistical mechanism for modeling this within-subject covariance, assume that there exist latent random-effect vectors
of small dimension (typically one or two) that are also independent across
. Assume also that an appropriate model linking
and
exists, leading to the joint probability density function
where
is a matrix of observed explanatory variables and
and
are vectors of unknown parameters.
Let
and assume that it is of dimension
. Then inferences about
are based on the marginal likelihood function
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 for optimization.
As an example of the preceding general framework, consider the nonlinear growth curve example in the section Getting Started: NLMIXED Procedure. Here, the conditional distribution
is normal with mean
and variance
; thus
. Also,
is a scalar and
is normal with mean 0 and variance
; thus
.
The following additional notation is also found in this chapter. The quantity
refers to the parameter vector at the
th iteration, the vector
refers to the gradient vector
, and the matrix
refers to the Hessian
. Other symbols are used to denote various constants or option values.
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