SNR Models for Grouped Data

Next: Estimation Up: Semi-parametric Nonlinear Regression Models Previous: SNR Models for Non-grouped

SNR Models for Grouped Data

Grouped data include repeated measures data, longitudinal data, functional data and multilevel data as special cases. For such data, we define SNR models as

$\displaystyle y_{ij}= \eta(\mbox{\boldmath$\phi$}_i, \mbox{\boldmath$f$}; \mbox{\boldmath$t$}_{ij}) + \epsilon_{ij}, ~~~~i=1, \cdots, m, ~~j=1,\cdots, n_i,$

(38)

where $y_{ij}$ is the response of subject

at design point $\mbox{\boldmath$t$}_{ij}$ , $\mbox{\boldmath$t$}_{ij}=(t_{1ij},\cdots,t_{dij})$ is a covariate in a general domain ${\cal T}$ , $\eta$ is a known function of $\mbox{\boldmath$t$}_{ij}$ which depends on a vector of parameter $\mbox{\boldmath$\phi$}_i=(\phi_{i1},\cdots,\phi_{ir})^T$ and a vector of unknown non-parametric function $\mbox{\boldmath$f$}=(f_1,\cdots,f_q)^T$ ; and $\mbox{\boldmath$\epsilon$}=(\epsilon_{11}, \cdots, \epsilon_{1n_1}, \cdots,\epsilon_{m1}, \cdots\epsilon_{mn_m}) \newline \sim \mbox{N} (0, \sigma^2 W^{-1})$ . For grouped data, usually observations are correlated within a subject but are independent between subjects. In this case W is in the form of block diagonal. Again, each function

is modeled using an SS ANOVA model (

). Again, assume that

depends on a parsimonious set of parameters $\mbox{\boldmath$\tau$}$ .

For model (), we denote $n=\sum_{i=1}^m n_i$ , $\mbox{\boldmath$y$}_{i}=(y_{i1},\cdots,y_{in_{i}})^{T}$ , $\mbox{\boldmath$y$}=(\mbox{\boldmath$y$}_{1}^{T}, \cdots,\mbox{\boldmath$y$}_{m}^{T})^{T}$ , $\mbox{\boldmath$\eta$}_{i}(\mbox{\boldmath$\phi$}_{i}, \mbox{\boldmath$f$})=(\e... ...x{\boldmath$\phi$}_{i}, \mbox{\boldmath$f$}; \mbox{\boldmath$t$}_{in_{i}}))^{T}$ , $\mbox{\boldmath$\phi$}=(\mbox{\boldmath$\phi$}_{1}^{T}, \cdots, \mbox{\boldmath$\phi$}_{m}^{T})^{T}$ , and $\mbox{\boldmath$\eta$}(\mbox{\boldmath$\phi$}, \mbox{\boldmath$f$})=(\mbox{\bol... ...ox{\boldmath$\eta$}_{m}(\mbox{\boldmath$\phi$}_m, \mbox{\boldmath$f$})^{T})^{T}$ . Then model () can be written in the same vector form as ().

Yuedong Wang 2004-05-19