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Unpaired Group Difference

We assume that the individual subjects are grouped in two groups and that within each group the first-level parameters are normally distributed around a group-specific mean. That is

$\displaystyle \beta_k\sim{\cal N}(\mu_1,\sigma^2_{s_1}), k\in G_1$    and $\displaystyle \quad\beta_k\sim{\cal N}(\mu_2,\sigma^2_{s_2}), k\in G_2.
$

In order to simplify further notation and without loss of generality we assume that the subjects $ 1,\dots,r$ belong to the first group and subjects $ r+1,\dots,N$ belong to the second group. We do not make any assumption about the first-level covariance structure and simply set

\begin{displaymath}
V=\left[
\begin{array}{ccc}
V_1& & 0 \\
& \ddots & \\
0...
...}=(2\,\,\,\,0)^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}.
\end{displaymath}

Then $ V_{\mbox{\tiny\textit{\sffamily {$\!$G2}}}}^{\mbox{}}$ is block diagonal with elements $ u_k = \left(X_k^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}V_k^{\mbox{\scri...
... {-1}}}}X_k\right)^{\mbox{\scriptsize\textit{\sffamily {-1}}}}+ \sigma^2_{s_k}.$ If we define $ s_1=\sum_{{_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}}_1}\!\!u_k^{\mbox{\scriptsize\textit{\sffamily {-1}}}}$ and $ s_2=\sum_{{_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}}_2}\!\!u_k^{\mbox{\scriptsize\textit{\sffamily {-1}}}}$, the group parameter estimate writes as

$\displaystyle \widehat{b_{_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}}}$ $\displaystyle =$ \begin{displaymath}\!\!\!C_{_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}}^{\mbox{\s...
...}}}}^{\mbox{\scriptsize\textit{\sffamily {-1}}}}\widehat{\beta}\end{displaymath}  
  $\displaystyle =$ \begin{displaymath}\!\!\!\frac{C_{_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}}^{\m...
...}}}}}_2}\!\!\frac{\widehat{\beta_k}}{u_k}\\
\end{array}\right]\end{displaymath}  
  $\displaystyle =$ $\displaystyle \left(\sum_{{_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}}_1}\!\!\fr...
...ny\textit{\sffamily {$\!$G}}}}}_2}\!\!\frac{\widehat{\beta_k}}{s_2u_k} \right),$  

where the variance, as usual, is calculated from the first term as

$\displaystyle \textrm{Var}(\widehat{b_{_{\mbox{\tiny\textit{\sffamily {$\!$G}}}...
...}}C_{_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}} = \frac{1}{s_1}+\frac{1}{s_2}.
$

Under the same assumptions as before, of equal covariance at the first level and normalised designs (i.e. homoscedastic model), these equation simplify to

$\displaystyle u_k=\sigma^2_w+\sigma^2_{s_k},\quad s_1=\frac{r}{\sigma^2_w+\sigma^2_{s_1}}, \quad s_2=\frac{N-r}{\sigma^2_w+\sigma^2_{s_2}},$   and thus$\displaystyle \quad
\widehat{b_{_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}}}=\fr...
...c{1}{N-r}\sum_{{_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}}_2}\widehat{\beta_k}
$

with

$\displaystyle \textrm{Var}(\widehat{b_{_{\mbox{\tiny\textit{\sffamily {$\!$G}}}...
...frac{\sigma^2_{s_1}}{r}+\frac{\sigma^2_{s_2}}{N-r}+\frac{N\sigma^2_w}{r(N-r)}.
$

Note that the second level contrast includes an appropriate scaling constant. This factor becomes irrelevant once the group parameter of interest is combined with its variance to form a test statistic.



Subsections
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Next: Numerical simulation Up: Examples Previous: Numerical simulation
Christian Beckmann 2003-07-16