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## Model and Estimation

Let be a function of multivariate variables . Let and assume that . Suppose that we model using an SS ANOVA decomposition. Specifically, we assume with given in ().

A semi-parametric linear mixed-effects (SLM) model assumes that

 (27)

where , , are design points, is the design matrix for some fixed effects with parameters , is the design matrix for the random effects , , are random errors which are independent of and . The covariance matrices and are assumed to depend on a parsimonious set of covariance parameters . Regarding the fixed effects part as a partial spline, model () is essentially the same as the non-parametric mixed-effects model introduced in Wang (1998a).

For model (), the marginal distribution of is where . Given and , we estimate fixed parameters and as the minimizers of the following penalized weighted least squares

 (28)

Denote the estimates as and . We estimate as the posterior mean , where .

Again, the solution of has the form (). Similar to Section 2.4, we use connections between a SLM and a LMM to estimate and . Consider the following LMM

 (29)

where 's and 's are defined in (), , and , and are mutually independent. Then the GML estimates of and in () are the REML estimates of the variance components in () (Opsomer et al., 2001; Wang, 1998a). As in Section 2.4, we use lme to calculate the GML (REML) estimates of and . Then we transform the data and call dsidr.r to calculate , , and with smoothing parameters fixed at the GML estimates. Formulae for calculating posterior variances were provided in Wang (1998a). Thus Bayesian confidence intervals can be constructed.

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