The NLMIXED Procedure

Modeling Assumptions and Notation

PROC NLMIXED operates under the following general framework for nonlinear mixed models. Assume that you have an observed data vector $\mb {y}_ i$ for each of i subjects, $i=1,\ldots ,s$. The $\mb {y}_ i$ are assumed to be independent across i, but within-subject covariance is likely to exist because each of the elements of $\mb {y}_ i$ is measured on the same subject. As a statistical mechanism for modeling this within-subject covariance, assume that there exist latent random-effect vectors $\mb {u}_ i$ of small dimension (typically one or two) that are also independent across i. Assume also that an appropriate model linking $\mb {y}_ i$ and $\mb {u}_ i$ exists, leading to the joint probability density function

\[  p(\mb {y}_ i | \mb {X}_ i, \bphi , \mb {u}_ i) q(\mb {u}_ i | \bxi )  \]

where $\mb {X}_ i$ is a matrix of observed explanatory variables and $\bphi $ and $\bxi $ are vectors of unknown parameters.

Let $\btheta = [\bphi , \bxi ]$ and assume that it is of dimension n. Then inferences about $\btheta $ are based on the marginal likelihood function

\[  m(\btheta ) = \prod _{i=1}^ s \int p(\mb {y}_ i | \mb {X}_ i, \bphi , \mb {u}_ i) q(\mb {u}_ i | \bxi ) d \mb {u}_ i  \]

In particular, the function

\[  f(\btheta ) = - \log m(\btheta )  \]

is minimized over $\btheta $ numerically in order to estimate $\btheta $, and the inverse Hessian (second derivative) matrix at the estimates provides an approximate variance-covariance matrix for the estimate of $\btheta $. The function $f(\btheta )$ 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 $p(\mb {y}_ i | \mb {X}_ i, \bphi , u_ i)$ is normal with mean

\[  \frac{b_1 + u_{i1}}{1 + \exp [-(d_{ij} - b_2)/ b_3]}  \]

and variance $\sigma ^2_ e$; thus $\bphi = [b_1,b_2,b_3,\sigma ^2_ e]$. Also, $u_ i$ is a scalar and $q(u_ i | \xi )$ is normal with mean 0 and variance $\sigma ^2_ u$; thus $\xi = \sigma ^2_ u$.

The following additional notation is also found in this chapter. The quantity $\btheta ^{(k)}$ refers to the parameter vector at the kth iteration, the vector $\mb {g}(\btheta )$ refers to the gradient vector $\nabla f(\btheta )$, and the matrix $\mb {H}(\btheta )$ refers to the Hessian $\nabla ^2 f(\btheta )$. Other symbols are used to denote various constants or option values.