next up previous
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 $i$ 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 $f_j$ is modeled using an SS ANOVA model ([*]). Again, assume that $W$ 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