Model and Estimation

Let $f$ be a function of multivariate variables $t_1, \cdots, t_d$. Let $\mbox{\boldmath$t$}=(t_1,\cdots,t_d)$ and assume that $\mbox{\boldmath$t$}\in {\cal T}={\cal T}_1\otimes \cdots \otimes {\cal T}_d$. Suppose that we model $f$ using an SS ANOVA decomposition. Specifically, we assume $f\in{\cal H}$ with ${\cal H}$ given in ().

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

$$\mbox{\boldmath$y$}=\mbox{\boldmath$f$}+ X\mbox{\boldmath$\beta$}+ Z\mbox{\boldmath$b$}+\mbox{\boldmath$\epsilon$},$$ (27)

where $\mbox{\boldmath$y$}=(y_1,\cdots,y_n)^T$, $\mbox{\boldmath$f$}=(f(\mbox{\boldmath$t$}_1), \cdots, f(\mbox{\boldmath$t$}_n))^T$, $\mbox{\boldmath$t$}_1,\cdots,\mbox{\boldmath$t$}_n$ are design points, $X$ is the design matrix for some fixed effects with parameters $\mbox{\boldmath$\beta$}$, $Z$ is the design matrix for the random effects $\mbox{\boldmath$b$}$, $\mbox{\boldmath$b$}\sim N(0, \sigma^2 D)$, $\mbox{\boldmath$\epsilon$}$ are random errors which are independent of $\mbox{\boldmath$b$}$ and $\mbox{\boldmath$\epsilon$}\sim N(0, \sigma^2\Lambda)$. The covariance matrices $D$ and $\Lambda$ are assumed to depend on a parsimonious set of covariance parameters $\mbox{\boldmath$\tau$}$. Regarding the fixed effects part $\mbox{\boldmath$f$}+X\mbox{\boldmath$\beta$}$ 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 $\mbox{\boldmath$y$}$ is $\mbox{\boldmath$y$}\sim N(\mbox{\boldmath$f$}+X\mbox{\boldmath$\beta$}, \sigma^2 W^{-1})$ where $W^{-1}=Z D Z^T+\Lambda$. Given $\mbox{\boldmath$\tau$}$ and $\lambda/\theta_1,\cdots,\lambda/\theta_p$, we estimate fixed parameters $f$ and $\mbox{\boldmath$\beta$}$ as the minimizers of the following penalized weighted least squares

$$(\mbox{\boldmath$y$}-\mbox{\boldmath$f$}-X\mbox{\boldmath$\beta$}... ...math$\beta$})+n\lambda\sum_{k=1}^p \theta_k^{-1} \vert\vert P_{k}f\vert\vert^2.$$ (28)

Denote the estimates as $\hat{f}$ and $\hat{\mbox{\boldmath$\beta$}}$. We estimate $Z\mbox{\boldmath$b$}$ as the posterior mean $ZDZ^TW(\mbox{\boldmath$y$}-\hat{\mbox{\boldmath$f$}}-X\hat{\mbox{\boldmath$\beta$}})$, where $\hat{\mbox{\boldmath$f$}}=(\hat{f}(\mbox{\boldmath$t$}_1),\cdots,\hat{f}(\mbox{\boldmath$t$}_n))^T$.

Again, the solution of $f$ has the form (). Similar to Section 2.4, we use connections between a SLM and a LMM to estimate $\mbox{\boldmath$\tau$}$ and $\lambda/\theta_1,\cdots,\lambda/\theta_p$. Consider the following LMM

$$\mbox{\boldmath$y$}=T\mbox{\boldmath$d$}+ \sum_{k=1}^p Z_k \mbox{... ...$b$}_p \\ \mbox{\boldmath$b$} \end{array}\right) + \mbox{\boldmath$\epsilon$},$$ (29)

where $Z_k$'s and $\mbox{\boldmath$b$}_k$'s are defined in (), $\mbox{\boldmath$\epsilon$}\sim N(0, \sigma^2\Lambda)$, and $\mbox{\boldmath$b$}_1,\cdots,\mbox{\boldmath$b$}_p$, $\mbox{\boldmath$b$}$ and $\mbox{\boldmath$\epsilon$}$ are mutually independent. Then the GML estimates of $\mbox{\boldmath$\tau$}$ and $\lambda/\theta_1,\cdots,\lambda/\theta_p$ 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 $\mbox{\boldmath$\tau$}$ and $\lambda/\theta_1,\cdots,\lambda/\theta_p$. Then we transform the data and call dsidr.r to calculate $\mbox{\boldmath$c$}$, $\mbox{\boldmath$d$}$, and $\mbox{\boldmath$\beta$}$ 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.