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## Model and Estimation

Semi-parametric nonlinear mixed-effects (SNM) models extend current statistical nonlinear models for grouped data in two directions: adding flexibility to a nonlinear mixed-effects model by allowing the mean function to depend on some non-parametric functions, and providing ways to model covariance structure and covariates effects in an SNR model. An SNM model assumes that

 (42) (43)

where the first-stage model () is the same as a SNR model (), and the second-stage model is the same as one for a nonlinear mixed-effect model. Specifically, is the response of subject at design point , are independent variables in a general domain , is a known function of which depends on a vector of parameter and a vector of unknown non-parametric function ; random errors ; is a -vector of fixed effects, is -vector of random effects associated with subject ; and are design matrices of sizes and for the fixed and random effects respectively. It is assumed that the random effects and random errors are mutually independent. Each function is modeled using an SS ANOVA model ().

Let , , , , , and . The SNM model () and () can then be written in a matrix form

 (44)

where , and . In the following we assume that and depend on an unknown parameter vector .

For fixed and , we estimate , , , as the minimizers of the following double penalized log-likelihood

 (45)

Denote , and as solutions to (). Let , and . We estimate and as minimizers of the approximate profile log-likelihood
 (46)

Since may interact with and in a complicated way, we have to use iterative procedures to solve () and (). We proposed two procedures in Ke and Wang (2001) for the case when is linear in . It is not difficult to extend these procedures to the general case. In the following we describe the extension of Procedure 1 in Ke and Wang (2001).

Procedure 1: estimate , , , and iteratively using the following three steps:

(a) given the current estimates of , and , update by solving ();

(b) given the current estimates of and , update and by solving ();

(c) given the current estimates of , and , update and by solving ().

Note that step (b) corresponds to the pseudo-data step and step (c) corresponds to part of the LME step in Lindstrom and Bates (1990). Thus the nlme can be used to accomplish (b) and (c). In step (a) () is reduced to () after certain transformations. Then depending on if is linear in , the ssr or nnr function can be used to update . We choose smoothing parameters using a data-adaptive criterion such as GCV, GML or UBR at each iteration.

To minimize () we need to alternate between steps (a) and (b) until convergence. Our simulations indicate that one iteration is usually enough. Figure shows the flow chart of Procedure 1 if we alternate (a) and (b) only once. Step (a) can be solved by ssr or nnr. It is easy to see that steps (b) and (c) are equivalent to fitting a NLMM with fixed at the current estimate using the same methods proposed in Lindstrom and Bates (1990). Therefore these two steps can be combined and solved by S program nlme (Pinheiro and Bates, 2000). Figure suggests an obvious iterative algorithm by calling nnr and nlme alternately. It is not difficult to use other options in our implementation. For example, we may alternate steps (a) and (b) several times before proceeding to step (c). In our studies these approaches usually gave the same results. For details about the estimation methods and procedures, see Ke and Wang (2001).

Approximate Bayesian confidence intervals can be constructed for (Ke and Wang, 2001).

Next: The snm Function Up: Semi-parametric Nonlinear Mixed-Effects Models Previous: Semi-parametric Nonlinear Mixed-Effects Models
Yuedong Wang 2004-05-19