%\chapter{Appendix}

%% --------- DEFINITIONS ----------------------
%\def\bB{\mathbf{B}}
%\def\bb{\boldsymbol{\beta}}
%\def\bPsi{\boldsymbol{\Psi}}
%\def\invPsi{\boldsymbol{\Psi}^{-1}}
%\def\brho{\boldsymbol{\rho}}
%\def\bxi{\boldsymbol{\xi}}
%\def\bphi{\boldsymbol{\phi}}
%\def\b0{\mathbf{0}}
%\def\bI{\mathbf{I}}
%\def\bJ{\mathbf{J}}
%\def\bX{\mathbf{X}}
%\def\br{\mathbf{r}}
%\def\Wg{\mathbf{W}_g}
%\def\Wgs{\mathbf{W}_{gs}}
%\def\bM{\mathbf{M}}
%\def\Mgs{\mathbf{M}_{gs}}
%\def\tr{\text{tr}}
%\def\tV{\text{V}}
%\def\V{\text{Var}}

%% --------- Greek Letters ----------------------
%\def\al{\alpha}
%\def\De{\Delta}
%\def\de{\delta}
%\def\ga{\gamma}
%\def\Ga{\Gamma}
%\def\ka{\kappa}
%\def\la{\lambda}
%\def\La{\Lambda}
%\def\om{\omega}
%\def\Om{\Omega}
%\def\Th{\Theta}
%\def\th{\theta}
%\def\si{\sigma}
%\def\Si{\Sigma}
%\def\vp{\varphi}
%\def\be{\beta}
%\def\e{\epsilon}
%\def\ve{\varepsilon}
%
%\def\p{\partial}
%\def\ra{\rightarrow}
%\def\Ra{\Rightarrow}

\section{Chapter 2 Appendices}
\subsection{Appendix A: Spatio-temporal Model in Matrix Form}
\label{ss:STM_matrix_form}

\noindent
Our stage-two spatio-temporal model \eqref{2nd_stage} can be written in matrix form as follows:
\begin{equation*}
\bB_{igs} = \bb_{gs} +\rho_{gs} \mathbf{W}^* (\bB_{igs} -\bb_{gs}) + \xi_{gs} \mathbf{M}^*
(\bB_{igs} -\bb_{gs}) +\boldsymbol{e}_{igs}\, ,
\end{equation*}
where $\boldsymbol{e}_{igs} \sim \text{MVN} (\mathbf{0},\phi_{gs}^2 \mathbf{I})$ and
where $\mathbf{W}^*=\bI_q \otimes \mathbf{W}_g$, $\mathbf{M}^*=1/(q-1) (\bJ_q \otimes \bI_{V_g} - \bI_q \otimes \bI_{V_g})$
and $\mathbf{W}_g=1/(V_g-1) (\bJ_{V_g}-\bI_{V_g})$.
We can rewrite the above equation as
\begin{equation}
\bB_{igs} = \bb_{gs} +(\mathbf{I}-\rho_{gs} \mathbf{W}^* - \xi_{gs} \mathbf{M}^*)^{-1}\boldsymbol{e}_{igs}=
\bb_{gs} +\bPsi_{gs}\boldsymbol{e}_{igs}
\end{equation}
Therefore, $\bB_{igs} \sim \text{MVN}(\bb_{gs},\phi_{gs}^2 \boldsymbol{\Psi}_{gs} \boldsymbol{\Psi}_{gs})$.

\subsection{Appendix B: Exploratory Data Analysis}
\label{ss:exploratory_analysis}
\noindent
%----------------------------------------------------------------------------------------------------
B.1. {\it Crude estimates of the spatial and temporal correlations.}
%\subsection{Crude estimates of the spatial and temporal correlations.}
%----------------------------------------------------------------------------------------------------

\noindent
To motivate our approach, we first performed an exploratory data analysis.
We obtained crude empirical estimates of the spatial as well as temporal
correlations in our data. To estimate the spatial correlations, we randomly selected a
small percentage ($< 1$\%) of voxel-pairs from each anatomically defined brain region and calculated
Pearson correlation coefficients between the response variable values for these voxel pairs,
across the subjects. Then we plotted the histogram of medians of these estimated correlations of each region.

To estimate the temporal correlations, we randomly selected a number of voxels from each region, calculated
correlations between pre- and post treatment measurements, across the subjects, and plotted the medians of
the region estimates.
The estimates, shown in Figure~\ref{f:emp_ests}, suggest that the voxels within the same regions
have correlated brain activity and that there is correlated brain activity %in the same voxels
between pre- and post-treatment sessions.

\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.5]{Figures/Spat_Emp_Ests_Pats.eps}
\includegraphics[scale=0.5]{Figures/Temp_Emp_Ests_Pats.eps}
%\includegraphics[scale=0.5]{Figures/CrudeEstsSpatialCorr_pats.png} % new: 10-26-08
%\includegraphics[scale=0.5]{Figures/CrudeEstsTemporal_30perc.png} % new: 8-15-08
\caption[Empirical estimates of the spatial and temporal correlations in the Inhibitory Control in Cocaine Addicts Data.]
{Empirical estimates of spatial correlations between randomly selected intra-regional
voxel pairs for patients for the 6 mm FWHM smoothed data (left);
empirical estimates of temporal (repeated measurement: pre- and post-treatment)
correlations between 30\% randomly selected voxels from each cluster (right).}
\label{f:emp_ests}
\end{center}
\end{figure}

\noindent
B.2. {\it Exploring spatial correlations within anatomically defined regions.}

%\subsection{Exploring spatial correlations within anatomically defined regions.}

\noindent
{\it  Descriptive covariance plots.} To investigate the spatial correlations for our
 experimental data, we calculated the empirical correlation matrix separately for each
of the regions in a Brodmann area-based parcellation of the brain.
For every voxel in the region, the mean and the standard error
of the correlations with each of the other voxels in that region were calculated (across the subjects) and plotted.
The mean values were smoothed using the `loess' curve to get a better idea of the overall
spatial correlation for that region (see Figure \ref{f:emp_ests_Brodmann}). We performed this step for both patients and controls.
We found that the means of the correlations did not fluctuate much within a region.
Figure \ref{f:emp_ests_Brodmann} shows such plots of several regions from the anatomical parcellation.
%
%In our model we assume the compound symmetry as a covariance structures between voxels in the same
%region. Compound symmetry assumes equal variances % for all repeated measures
%and equal covariances between all pairs of voxels within the region.
% scans and results from fitting a single random effect (intercept) for each individual.
%
Figure \ref{f:emp_ests_Brodmann} provides support for the compound symmetric spatial covariance structure that we assume in our model.
%Analyzing the plots of all 76 regions, we found that, for most regions, mean correlations
%across the voxels within a region are higher in patients than in controls.

\begin{figure}
\begin{center}
\includegraphics[scale=0.5]{Figures/Spatial_corr_est_BA3.eps}
\includegraphics[scale=0.5]{Figures/Spatial_corr_est_BA3_hist.eps}
\includegraphics[scale=0.5]{Figures/Spatial_corr_est_BA18.eps}
\includegraphics[scale=0.5]{Figures/Spatial_corr_est_BA18_hist.eps}
\includegraphics[scale=0.5]{Figures/Spatial_corr_est_BA48.eps}
\includegraphics[scale=0.5]{Figures/Spatial_corr_est_BA48_hist.eps}
\caption[Empirical estimates of spatial correlations for 3 Brodmann regions: BA 3(L), BA 18 (L), and BA 48(L).]
{Empirical estimates of spatial correlations for 3 Brodmann regions (BA 3 (L) (part of the primary somatosensory cotrex),
BA 18 (L) (V2-visual association cortex), and BA 48 (L) (retrosubicular area) with 334, 1017 and 2914 voxels respectivelly.}
\end{center}
\label{f:emp_ests_Brodmann}
\end{figure}

%===========================

\newpage
\subsection{Appendix C. Computing Variance-Covariance Matrix: Coefficients of $\bPsi_g$.}
\label{ss:coefficients_of_psi}
\vspace{2ex}
\noindent

%A.1. {\it Model formulation in matrix form.}
%
%We arrive at the basic mean, covariance, and distributional assumptions of our
%stage-two model (2) (section 3.1) by first considering its matrix form:
%\begin{equation*}
%\bB_{igs} = \bb_{gs} +\rho_{gs} \mathbf{W}^* (\bB_{igs} -\bb_{gs}) + \xi_{gs} \mathbf{M}^*
%(\bB_{igs} -\bb_{gs}) +\boldsymbol{e}_{igs}\, ,
%\end{equation*}
%where $\boldsymbol{e}_{igs} \sim \text{MVN} (\mathbf{0},\phi_{gs}^2 \mathbf{I})$ and
%where $\mathbf{W}^*=\bI_q \otimes \mathbf{W}_g$, $\mathbf{M}^*=1/(q-1) (\bI_q \otimes \bJ_{V_g} - \bI_q \otimes \bI_{V_g})$
%and $\mathbf{W}_g=1/(V_g-1) (\bJ_{V_g}-\bI_{V_g})$.
%We can rewrite the above equation as
%\begin{equation}
%\bB_{igs} = \bb_{gs} +(\mathbf{I}-\rho_{gs} \mathbf{W}^* - \xi_{gs} \mathbf{M}^*)^{-1}\boldsymbol{e}_{igs}=
%\bb_{gs} +\bPsi_{gs}\boldsymbol{e}_{igs}
%\end{equation}
%
%Therefore, $\bB_{igs} \sim \text{MVN}(\bb_{gs},\phi_{gs}^2 \boldsymbol{\Psi}_{gs} \boldsymbol{\Psi}_{gs})$.
%%($\otimes$ denotes a Kronecker product).
%
%
%\vspace{2ex}
%\noindent

%A.2. {\it Coefficients of $\bPsi_g$.}

%======================== Coefficients of Psi ========================
Let $a=1+\frac{\rho_g}{V_g-1}$, $ b=-\frac{\rho_g}{V_g-1}$ and $k=-\frac{\xi_g}{q-1}$. (For simplicity we drop the group
index $s$  here.) $\bPsi_g$ (see \eqref{f:Psi} in section \ref{ss:covar_model}) is a partitioned matrix with the {\it diagonal blocks}
determined by
{\small
\begin{equation*}
\begin{split}
d & = \frac{a+(q-2)k}{a^2+(q-2)ak-(q-1)k^2} \\
f & = \frac{-b\{a^2+[1+(q-1)(q-2)]k^2+V_g [ab+(q-2)bk]+2(q-2)ak\} }
{[a^2+(q-2)ak-(q-1)k^2] \{ a^2+(q-2)ak-(q-1)k^2+V_g b[2a+(q-2)k+V_g b\} }
\end{split}
\end{equation*}
}

\vspace{2ex}
\noindent
The {\it off-diagonal blocks} are determined by
{\small
\begin{equation*}
\begin{split}
u &=\frac{-k}{a^2+(q-2)ak-(q-1)k^2} \\
z &=\frac{bk[2a+(q-2)k+V_g b]}
{[a^2+(q-2)ak-(q-1)k^2] \{ a^2+(q-2)ak-(q-1)k^2+V_g b[2a+(q-2)k+V_g b\} }
\end{split}
\end{equation*}
}

\vspace{2ex}
This result was derived recursively on $q$, the number of blocks (sessions represented) in $\bPsi_g^{-1}$.


\subsection{Appendix D. Score Functions.}
\label{ss:score_fcts}
%======================== SCORE functions ========================

\noindent

We present the score functions, which facilitate implementation of our model. For simplicity, we drop the group notation $s$.
Let $\br_{ig} = \bB_{ig}-\bX_i \bb_g$ and we denote $\phi_g^2$ by $w_g$.
Also, from now on $\bI$ will denote an identity matrix of size $V_g$ and $\bJ$ will denote a $V_g \times V_g$ matrix of ones. In addition,
$d=d(\rho_g,\xi_g,V_g,q)$, $f=(\rho_g,\xi_g,V_g,q)$, $u=(\rho_g,\xi_g,V_g,q)$, and $z=z(\rho_g,\xi_g,V_g,q)$ are the entries in matrix $\bPsi_g$, defined in section 2.3.2. $A=A(d,f,u,z,V_g,q)$, $B=B(d,f,u,z,V_g,q)$, $C=C(d,f,u,z,V_g,q)$, and $D=D(d,f,u,z,V_g,q)$ are
the entries in $\bPsi_g \bPsi_g$, also defined in section \ref{ss:covar_model}.

The {\it score functions} are as follows:

%\begin{itemize}
%---------------------------- S_beta ------------------------
% \item[(1)]
\noindent (1)
\begin{equation*}
S_{\bb_g}  = \frac{\p \log L}{\p \bb_g} =
  \frac 1{w_g} \sum_{i=1}^K \bigl\{\bB_{ig}'\bPsi_g^{-1}\bPsi_g^{-1}\bX_i - \bX_i' \bPsi_g^{-1}\bPsi_g^{-1}
\bX_i \bb_g \bigr\}
\end{equation*}
From this we get (since in our case $\bX_i=\bI_{qV_g})$):
$$
\hat{\bb}_{g_{\text{OLS}}} = \frac 1K \sum_{i=1}^K \bB_{ig} \quad g=1,\ldots,G.
$$
%---------------------------- S_rho ------------------------
%\item[(2)]
\noindent (2)
\begin{equation*}
\begin{split}
S_{\rho_g} &= \frac{\p \log L}{\p \rho_g} =\sum_{i=1}^K \tr \biggl\{ \frac{\p \bPsi_g^{-1}}{\p \rho_g}\cdot \bPsi_g \biggr\}
-\frac{1}{2 w_g} \sum_{i=1}^K \br_{ig}' \frac{\p}{\p \rho_g} \bigl(\bPsi_g^{-1} \bPsi_g^{-1}\bigr)
\br_{ig} =\\
& =-K \cdot q \cdot f \cdot V_g -\frac{1}{w_g}\sum_{i=1}^K \tfrac1{V_g-1} \br_{ig}'
\begin{bmatrix}
k_1 \bI + k_2 \bJ & k_3\bI -k_3\bJ & \cdots & k_3\bI -k_3\bJ \\
k_3\bI -k_3\bJ & k_1 \bI + k_2 \bJ & \cdots & k_3\bI -k_3\bJ \\
\vdots & \vdots & \ddots & \vdots \\
k_3\bI -k_3\bJ & k_3\bI -k_3\bJ & \cdots & k_1 \bI + k_2 \bJ\\
\end{bmatrix}\br_{ig}
\end{split}
\end{equation*}
where
$$
k_1 = 1+\frac{\rho_g}{V_g-1} , \quad
k_2 = \rho_g-1 -\frac{\rho_g}{V_g-1} , \quad \text{and } \;
k_3  = -\frac{\xi_g}{q-1}.
$$

%---------------------------- S_xi ------------------------
% \item[(3)]
\noindent (3)
\begin{equation*}
\begin{split}
S_{\xi_g} &= \frac{\p \log L}{\p \xi_g} =\sum_{i=1}^K  \tr \biggl\{ \frac{\p \bPsi_g^{-1}}{\p \xi_g}\cdot \bPsi_g \biggr\}
- \frac 1{2w_g}\sum_{i=1}^K \br_{ig}' \frac{\p}{\p \xi_g} \bigl(\bPsi_g^{-1}
\bPsi_g^{-1}\bigr)\br_{ig} =\\
& = - K \cdot q \cdot (u+z) V_g - \frac 1{w_g}\sum_{i=1}^K \br_{ig}'\! \begin{bmatrix}
                                                                                    m_1 \bI & m_2 \bI +m_3\bJ & \cdots & m_2 \bI +m_3\bJ\\
                                                                                    m_2 \bI +m_3\bJ & m_1 \bI & \cdots & m_2 \bI +m_3\bJ \\
                                                                                    \vdots & \vdots & \ddots & \vdots \\
                                                                                    m_2 \bI +m_3\bJ & m_2 \bI +m_3\bJ & \cdots & m_1\bI \\
                                                                                  \end{bmatrix}
\! \br_{ig}
\end{split}
\end{equation*}


where
$$
m_1 =\frac{\xi_g}{q-1}, \;
m_2 = \frac{(q-2)\xi_g -(q-1)}{(q-1)^2} -\frac{\rho_g}{(q-1)(V_g-1)}, \; \text{and } \;
m_3 = \frac{\rho_g}{(q-1)(V_g-1)}\,.
$$
%---------------------------- S_w ------------------------
%\item[(4)]
\noindent (4)
\begin{equation*}
\begin{split}
S_{w_g} & = \frac{\p \log L}{\p w_g} =\sum_{i=1}^K \biggl\{ -\frac{q V_g}{2w_g} +\frac{1}{2w_g^2}  \br_{ig}' \bPsi_g^{-1} \bPsi_g^{-1} \br_{ig} \biggr\} =\\
&=\! -\frac{K q V_g}{2w_g}+\frac{1}{2w_g^2} \sum_{i=1}^K \br_{ig}' \bPsi_g^{-1} \bPsi_g^{-1} \br_{ig} =\\
&=\! -\! \frac{K q V_g}{2w_g}+\frac{1}{2w_g^2} \sum_{i=1}^K \br_{ig}'
\begin{bmatrix}
u_1\bI + u_2 \bJ  & u_3\bI +u_4 \bJ & \cdots & u_3\bI +u_4 \bJ \\
u_3\bI +u_4 \bJ & u_1\bI + u_2 \bJ  & \cdots & u_3\bI +u_4 \bJ \\
\vdots & \vdots & \ddots& \vdots \\
u_3\bI +u_4 \bJ & u_3\bI +u_4 \bJ & \cdots & u_1\bI + u_2 \bJ  \\
\end{bmatrix}
 \br_{ig}
\end{split}
\end{equation*}

where
\begin{align*}
u_1 & =\Bigl(1+\frac{\rho_g}{V_g-1}\Bigr)^2 +\frac{\xi^2_g}{q-1} & u_2 &= \frac{\rho_g}{V_g-1} \bigl( \rho_g-2-\frac{\rho_g}{V_g-1}\bigr) \\
 u_3 & = -\frac{\xi_g}{q-1}\Bigl(2+\frac{2\rho_g}{V_g-1} -\frac{(q-2)\xi_g}{q-1} \Bigr) &
u_4 &= \frac{2\xi_g \rho_g}{(q-1)(V_g-1)} \; .
\end{align*}

%\end{itemize}

\vspace{4ex}
\noindent
In a similar fashion, one can derive second derivatives and their
expected values.

\vspace{4ex}
\noindent


%----------------------------------------------------------------------------------
%{\bf Appendix D. Simulation Details and Results.}
%----------------------------------------------------------------------------------

%\noindent
%{\it Simulation 1.}
%The goal of the first simulation study was to evaluate the accuracy of our estimation procedure.
%% {\it Description of the simulated data.}
%We selected regions (Brodmann areas) with voxel sizes ranging from 100 to 2000.
%To generate data for the simulations, we used our experimental data from the inhibitory control
%fMRI study to specify {\it true} values for $\bb_{gs}=(\bb_{gs}^{(1)}, \bb_{gs}^{(2)})$ (where the superscripts
%denote sessions 1 and 2), $\rho_{gs}$, $\xi_{gs}$ and
%$\phi_{gs}$. We drew 200 samples from the multivariate normal probability distribution  ${\rm MVN}(
%\bb_{gs}, \phi_{gs}^2 \mathbf{\Psi}_{gs} \mathbf{\Psi}_{gs})$, %with $\bPsi_{gs} =(\bI -\rho_{gs} \Wgs^* - \xi_{gs} \Mgs^*)^{-1}$,
%%as defined previously,
%and applied our spatio-temporal model to estimate the true parameters $\bb_{gs}$, $\rho_{gs}$, $\xi_{gs}$, and $\phi_{gs}$.
%
%{\it Simulation 2.}
%In the second simulation study, we evaluated the relative efficiency of our spatio-temporal model compared to the GLM,
%both for voxel-level and region level estimators of secondary parameters of interest
%($\boldsymbol{\th}_g^*$ and $\th_g$).
%We selected two regions, region A with 100 randomly selected voxels
%from BA 24, and region B with 1,000 randomly selected voxels from BA 19. We again set the true parameters
%for $\bb_{gs}$, $\rho_{gs}$, $\xi_{gs}$, $\phi_{gs}$ from our experimental data and drew 200 samples from
%${\rm MVN}(\bb_{gs}, \phi_{gs}^2 \mathbf{\Psi}_{gs} \mathbf{\Psi}_{gs})$.
%We estimated the secondary parameters of interest
%$\boldsymbol{\theta}_g^*$ (voxel-level) and $\theta_g$ (region-level), defined to address the stated objective of our study
%(see Section 2 and Section 3.4 in the main paper).
%We calculated the estimated variances for $\hat{\boldsymbol{\theta}}_g^*$ and $\hat{\theta}_g$,
%both applying the GLM and our spatio-temporal model, and obtained the voxel-level and region-level relative efficiencies
%(spatio-temporal model vs. GLM) (averaging over 200 samples in each case.)
%
%{\it Simulation Results.}
%The first simulation study revealed extremely small biases in the estimator of the mean model parameters
%$\bb_{gs}$ (not shown). Table 3 shows the simulation results for  the covariance parameters for the
%generated data. The following regions were selected: BA 24 (L), BA 32 (R), BA 19 (L) and BA 48 (R).
%The differences between the median values of the estimated parameters and the true values, from 200 generated samples
%are shown for each of the regions (1)-(4). we found that estimates for both $\rho_{gs}$ and $\xi_{gs}$ (spatial and temporal
%parameters) are quite accurate, while the between subject variability $\phi^2_{gs}$
%was slightly underestimated (roughly 6\%), which likely stems from the well-known
%downward bias of ML estimates of variance components. The increased accuracy for the controls group (in particular for the between
%subject variance parameter) reflects the larger sample size (15 control subjects, 12 addicts).
%
%
%\begin{table*}
% \centering
% % \def\~{\hphantom{0}}
% \caption{Estimation accuracy of the spatio-temporal model obtained from 200 simulated data sets for the cocaine addicts
%and the healthy control subjects, for four selected regions. The difference between the median of the estimated values from the generated samples and the true value is shown.}
%
%\vspace{2ex}
%% Starting value for $\phi^2$ was estimated from the simulated data.}
%\label{t:table2}
%{\small
%\begin{tabular*}{\textwidth}{@{}l@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c@{\extracolsep{\fill}}}
%% {|c|c|c|c|c|c|c|c|c|}
%\hline \hline
%\multirow{2}{*}{Group} & Est.  & True  & \multicolumn{4}{c}{Region$^*$ (no. voxels)} \\
%          \cline{4-7} \\ [-24pt]
%& param. & value & 1 (100) & 2 (500) & 3 (1000) & 4 (2000)  \\
%\hline
%\multirow{3}{*}{\underline{Addicts}} & & & & & & \\ [-20pt]
%& $\rho$   & \bf{0.5} & -0.0097 (0.0621) & -0.0001 (0.0646) & -0.0015 (0.0581) & -0.0097 (0.0650)\\
%& $ \xi$   & \bf{0.2} &  0.0001 (0.0146) &  0.0001 (0.0063) & -0.0001 (0.0044) &  0.0001 (0.0032)\\
%& $\phi^2$ & \bf{1.8} & -0.1592 (0.0490) & -0.1537 (0.0213) & -0.1483 (0.0148) & -0.1501 (0.0120)\\
%\hline
%\multirow{3}{*}{\underline{Controls}} & & & & & & \\ [-20pt]
%& $\rho$   & \bf{0.5} & -0.0075 (0.0531) & -0.0011 (0.0547) & -0.0034 (0.0574) &  0.0009 (0.0517)\\
%& $ \xi$   & \bf{0.2} &  0.0012	(0.0111) &  0.0009 (0.0056) & -0.0005 (0.0039) &  0.0000 (0.0028)\\
%& $\phi^2$ & \bf{1.8} & -0.1137	(0.0461) & -0.1209 (0.0191) & -0.1196 (0.0140) & -0.1191 (0.0107)\\
%\hline \hline
%\end{tabular*}
%}
%
%%\vspace*{-2pt}
%{\small $^*$ 1=BA 24(L), 2=BA 32(R), 3=BA 19(L), 4=BA 48(R). Corresponding region sizes are 100, 500, 1000, 2000 voxels, respectively.
%Starting value for $\rho$: 0.7, starting value for $\xi$: 0.1.}
%\end{table*}

%The second simulation study revealed similar results for regions A and B, and we only present the region B results for brevity.
%To summarize the voxel-level relative efficiency results, we plot the distribution of relative efficiencies across 1,000
%voxels from region B (Figure \ref{f:voxel_lev_rel_eff}). We see that the GLM-based voxel-level variance estimates are on
%average higher than the corresponding estimates obtained from our spatio-temporal model.
%The simulation-based estimate of the region-level relative efficiency is 1.72 for region B.
%This demonstrates that, on the region level, our model also leads to more precise estimates compared to GLM, and therefore
%may yield increased statistical power.
%
%\begin{figure}[!h]
%\begin{center}
%\includegraphics[scale=0.7]{Figures/Voxel_Rel_Eff_regB_new.eps}
%\caption{The distribution of relative efficiency values from 1,000 voxels
%randomly selected from left BA 19. Each voxel-level quantity is calculated
%as the mean relative efficiency across 200 simulated data sets. The relative efficiency
%was calculated based on the estimates from our model, relative to GLM.}
%\label{f:voxel_lev_rel_eff}
%\end{center}
%\end{figure}

\section{Chapter 4 Appendices}
\subsection{Appendix E: Bayesian Spatial Hierarchical Model - region level}
\label{ss:BSPM_region_level}
Let  $\bY_{ig}=\bigl(\bY_{ig}^T(v_1), \dots, \bY_{ig}^T(v_{V_g}) \bigr)^T_{(2V_g \times 1)}$,
and $v_1$, \ldots, $v_{V_G}$ denote the voxels in region $g$.
The Kronecker product $(\bI_{V_g} \otimes \bX_{ig})\cdot(\textbf{1}_{V_g}\otimes\bga_{g})$, i.e.

$\begin{bmatrix}
         \bX_{ig} &  &   &   \\
           & \bX_{ig} &   &    \\
            &   & \ddots &   \\
           &   &   & \bX_{ig} \\
       \end{bmatrix}
       \begin{bmatrix}
         \bga_{g} \\
         \bga_{g}\\
         \vdots \\
         \bga_{g} \\
       \end{bmatrix}=(\bI_{V_g} \textbf{1}_{V_g}) \otimes(\bX_{ig} \bga_{g})
     =\textbf{1}_{V_g} \otimes (\bX_{ig} \bga_{g})$.

Let also $\bb_g=\bigl(\bb_g^T(v_1),\ldots,\bb_g^T(v_{V_g})\bigr)^T$.
Since $\bb_{gj}|\la_{gj}^2 \sim \text{N}(\bb_{0gj},\la_{gj}^2\bI)$
are assumed to be independent, then $\bb_g \sim \text{N}(\bb_{0g},\bLa_g)$, where  \\
$\bb_{0g}=\begin{bmatrix}
            \bb_{0g}^{(1)}(v_1) \\
            \bb_{0g}^{(2)}(v_1) \\
            \vdots \\
            \bb_{0g}^{(1)}(v_{V_g}) \\
            \bb_{0g}^{(2)}(v_{V_g})
          \end{bmatrix}$,
          $\bLa_g=\begin{bmatrix}
                  \la_{g1}^2 &  & & & \\
                  & \la_{g2}^2 &  & & \\
                  &   & \ddots & &  \\
                  &  &   & \la_{g1}^2  &  \\
                  &  &   &   & \la_{g2}^2 \\
                  \end{bmatrix}=\bI_{V_g} \otimes \begin{bmatrix}
                                                    \la_{g1}^2  & 0 \\
                                                    0 & \la_{g2}^2  \\
                                                  \end{bmatrix}=\bI_{V_g} \otimes \bLa_v$.

 Finally, let $\bV_g=\bI_{V_g} \otimes \bZ_g$, where $\bZ_g=\begin{bmatrix}
                                                              Z_g^{11} & Z_g^{12} \\
                                                              Z_g^{21} & Z_g^{22} \\
                                                            \end{bmatrix}$ (same for all $v \in g$),
                                                            and let
 $\bphi_g=[\bphi_g^T(v_1) \cdots \bphi_g^T(v_{V_g})]^T$.
%
Then model \eqref{model_voxel} can be re-expressed as

\vspace{2ex}
\noindent {\bf Model (region level):}
\begin{equation}
\begin{split}\label{model_region}
\bY_{ig} | \bb_g,\bphi_g,\ba_{ig},\bga_{gv},\bV_g & \sim \text{N} \bigl(\bb_g+\bphi_g+\textbf{1}_{V_g} \otimes \ba_{ij}+\textbf{1}_{V_g} \otimes (\bX_{ig} \bga_{g}),\bV_g\bigr) \\
\bb_{gj} | \la_{gj}^2 & \sim \text{N}(\textbf{1}_{V_g} \bb_{0gj},\la_{gj}^2\bI_{V_g}) \\
\bphi_g | \bSi_{\phi_g^{-1}} &\sim \text{N}(\mathbf{0},\bSi_{\phi_g^{-1}} ), \;\; \bSi_{\phi_g^{-1}}=(D_{W_g} -\rho W_g)^{-1} \otimes \Sigma \\
%\text{ (we will update voxel by voxel)}\\
\ba_{ij}|\bGa_j &\sim \text{N}(\mathbf{0}, \bGa_j), \;\text{ where } \ba_i=[\ba_{i1}^T,\ldots,\ba_{iG}^T]^T,\; j=1,2 \\
\bga_{gj}| \boldsymbol{\tau}^2_{gj} & \sim \text{N}(\mathbf{0},\text{diag}(\tau_{gj1}^2,\ldots,\tau_{gjQ}^2) ) \\
(\bGa_j)^{-1} & \sim \text{Wishart}\{(h_j H_j)^{-1},h_j \}, \;\; j=1,2\\
\la_{gj}^{-2} &\sim \text{Gamma}(a_j,b_j) \\
\tau_{gjq}^{-2} & \sim \text{Gamma}(e_{0q},f_{0q}) \\
\mathbf{Z}_g^{-1} & \sim \text{Wishart}\bigl( (c_1 \boldsymbol{\Omega}_1)^{-1}, c_1 \bigr) \\
\Sigma^{-1} & \sim \text{Wishart}\bigl( (c_2 \boldsymbol{\Omega}_2)^{-1}, c_2 \bigr) \\
%\rho & \sim \text{Uniform} (\{0.05, 0.1, \ldots, 0.8, 0.82, \ldots, 0.9, 0.91, \ldots, 0.99 \})
\rho & \sim \text{Uniform} (\{0, 0.05, 0.1, \ldots, 0.8, 0.81, \ldots, \\
     & \quad  0.9, 0.91, \ldots, 0.99 \})
\end{split}
\end{equation}

\subsection{Appendix F: Full Conditional Distributions for the \\ Bayesian Spatial Hierarchical Model}
\label{ss:full_conditionals}
Let for each $g=1,\ldots,G$
$$
\bY_{ig}=\begin{bmatrix}
           \bY_{ig}(v_1) \\
           \vdots \\
           \bY_{ig}(v_{V_g}) \\
         \end{bmatrix}_{(2V_g \times 1)},
         %\quad
\bphi_g=\begin{bmatrix}
           \bphi_g(v_1) \\
           \vdots \\
           \bphi_g(v_{V_g}) \\
         \end{bmatrix}_{(2V_g \times 1)},
         \quad
\ba_{ig}=\begin{bmatrix}
           \ba_{ig}(v_1) \\
           \vdots \\
           \ba_{ig}(v_{V_g}) \\
         \end{bmatrix}_{(2V_g \times 1)}\!\!\!\!\!\!\!\!\!\!\!\! = \mathbf{1}_{V_g}\otimes \ba_{ig}
$$
Then we can write the model \eqref{model_region} as
\begin{equation}
\bY_{ig} =\bb_g+ \bphi_g + \b1_{V_g} \otimes \ba_{ig} + \b1_{V_g} \otimes \bX_{igv}\bga_{gv}+\bve_{ig}
\end{equation}
where $\bb_g=(\bb_g^T(v_1),\ldots,\be_g^T(v_{V_g}))$.

Let $\bb_{0g}=(\bb_{0g}^T(v_1),\ldots,\bb_{0g}^T(v_{V_g}))$.

Prior for $\bb$ (combined pre and post):
\begin{equation*}
\begin{split}
\bb_g^{(1)} &\sim \text{N}(\bb_{0g1}^{(1)},\la_{g1}^2\bI_{V_g}) \\
\bb_g^{(2)} &\sim \text{N}(\bb_{0g2}^{(2)},\la_{g2}^2\bI_{V_g})
\end{split}
\end{equation*}
Under the independence assumption
$$
\bb_g \sim \text{N}\bigl(\bb_{0g},\bLa_g \bigr)
$$
where
$$
\bLa_g=\begin{bmatrix}
       \la_{g1}^2 & 0 &   &   &   &   \\
       0 & \la_{g2}^2 &   &   &   &   \\
         &   & \ddots &   &   &   \\
         &   &  & \ddots &   &   \\
         &   &  &   & \la_{g1}^2 & 0 \\
         &   &   &   & 0 & \la_{g2}^2 \\
     \end{bmatrix}=\bI_{V_g} \otimes \begin{bmatrix}
                                       \la_{g1}^2 & 0 \\
                                       0 & \la_{g2}^2  \\
                                     \end{bmatrix}
$$
%Let $\bV_g=\bI_{V_g} \otimes \bZ_g$, a $(2V_g \times 2V_g)$ matrix where $\bZ_g=\begin{bmatrix}
%                                                                        Z_{g,11} & Z_{g,12} \\
%                                                                        Z_{g,21} & Z_{g,22} \\
%                                                                      \end{bmatrix}$.
%$$
%\Ra \bY_{ig} \sim \text{N}(\bX_{ig} \bb_g +\bphi_g+\ba_{ig}, \bV_g)
%$$

\vskip 2ex
\noindent
{\bf Deriving Full Conditional Distributions.}

Denote by $\bLa(\ba)$ the set of all parameters, excluding parameter $\ba$. The full conditional distributions for
each of the parameters in the model are as follows.

%\begin{enumerate}
%-- beta_g ----
%\item
\noindent (1)
\begin{eqnarray*}
[\bb_g | \bLa(\bb_g), \bY] & \propto & [\bb_g | \bLa_g] \times \prod_{i=1}^n [Y_{ig}|\bb_g,\bphi_g,\ba_{ig},\bga_g,\bV_g]\\
& \propto & \exp\bigl\{-\tfrac 12 (\bb_g-\bb_{0g})^T \bLa_g^{-1}(\bb_g-\bb_{0g})\bigr\} \times \\
& & |\bV_g|^{-\frac{n}{2}V_g} \exp \Bigl\{ \! -\tfrac 12 \sum_{i=1}^n [\bY_{ig}-\bb_g -\bphi_g -\b1\otimes \ba_{ig} -\b1 \otimes \bX_{ig}\bga_g]^T
\bV_g^{-1}\\
& & [\bY_{ig}-\bb_g -\bphi_g -\b1\otimes \ba_{ig} -\b1 \otimes \bX_{ig}\bga_g] \Bigr\} \\
& \propto & \exp\bigl\{ -\tfrac 12 [\bb_g^T\bLa_g^{-1}\bb_g - 2\bb_g^T\bLa_g^{-1}\bb_{0g}]\bigr\} \times \\
& & \exp \Bigl\{\! -\tfrac 12 \sum_{i=1}^n [\bb_g^T\bV_g^{-1}\bb_g - 2 \bb_g^T \bV_g^{-1} \!
\underbrace{(\bY_{ig} \bphi_g -\b1\otimes \ba_{ig} -\b1 \otimes \bX_{ig}\bga_g}_{:=\textbf{r}_{ig}})]\!\Bigr \} \\
& \propto & \exp\bigl\{ -\tfrac 12 [\bb_g^T[\bLa_g^{-1}+n\bV_g^{-1}]\bb_g - 2\bb_g^T[\bLa_g^{-1}\bb_{0g}+\bV_g^{-1}\sum_{i=1}^n\textbf{r}_{ig}]\bigr\} \\
& \propto & \text{N}\bigl( \bOm_g \mathbf{T}_g,\bOm_g \bigr)
\end{eqnarray*}
where $\bOm_g$, $\bT_g$, $\bLa_g$ are as defined in Section \ref{full_conditionals}

%--- phi_g ------
%\item
\noindent (2) The full conditional distribution for $\bphi_v$ is a likelihood adjusted version of the conditional distribution
for MCAR$(\rho,\Si)$:

\begin{eqnarray*}
%\begin{split}
\lefteqn{[\bphi_v | \bLa(\bphi_v),\bY] \propto } \\
& \propto &  \text{N} \bigl(\rho \sum_{v' \ne v} \frac{w_{vv'}}{w_{v+}} \bphi_{v'}, w_{v+}^{-1} \Si \bigr)
\cdot \prod_{i=1}^n \text{N}(\bmu_{igv},\bZ_g) \;\; (\text{denote } b_{v'}=\tfrac{w_{vv'}}{w_{v+}} )\\
& \propto &    \exp \Bigl\{ -\frac 12 \bigl( \bphi_v -\rho \sum_{v'} b_{v'} \bphi_{v'}\bigr)^T w_{v+} \Si^{-1} \bigl(\bphi_v -\rho \sum_{v'} b_{v'} \bphi_{v'}\bigr) \Bigr\} \cdot \\
& &   \exp \Bigl\{ -\frac 12 \sum_{i=1}^n [\bY_{ig}(v) -(\underbrace{\bb_g(v) + \bphi_G(v) +\ba_{ig}(v)+\bX_{ig}\bga_g}_{\textbf{r}})]^T \bZ_g^{-1} [\bY_{ig}- \textbf{r}]  \Bigr\} \\
& \propto & \exp \Bigl\{-\frac 12 w_{v+} [ \bphi_v^T\Si^{-1}\bphi_v - 2 \bphi_v^T\Si^{-1} \rho \sum_{v'} b_{v'} \bphi_{v'}]  \Bigr\} \cdot \\
& & \exp \Bigl\{ -\frac 12 \sum_{i=1}^n [\bphi_v-(\underbrace{\bY_{ig}(v)-\bb_g(v) - \bphi_G(v) -\ba_{ig}-\bX_{ig}\bga_g)}_{\textbf{s}_{ig}}]^T [\bphi_v -\textbf{s}_{ig}] \Bigr\} \\
& \propto &  \exp \Bigl\{ \text{ same as above }  \Bigr\} \cdot \\
& & \exp \Bigl\{ -\frac 12 \Bigl[ \sum_{i=1}^n \bphi_v^T \bZ_g^{-1} \bphi_v - 2 \bphi_v^T \bZ_g^{-1} \sum_{i=1}^n \textbf{s}_{ig} \Bigr] \Bigr\} \\
& \propto & \exp \Bigl\{ \text{ same as above }  \Bigr\} \cdot \\
& &  \exp \Bigl\{ n \bphi_v^T \bZ_g^{-1} \bphi_v - 2\bphi_v^T\bZ_g^{-1} n \cdot \bigl( \bar{\bY}_g(v) -\bb_g(v) -\bar{\ba}_g - \bar{\bX}_g \bga_g \bigr)\Bigr\}
%\end{split}
\end{eqnarray*}
where
\begin{align*}
\bar{\bY}_g(v) &=\frac 1n \sum_{i=1}^n \bY_{ig}(v) \\
\bar{\ba}_g    &=\frac 1n \sum_{i=1}^n \ba_{ig} \\
\bar{\bX}_g    &=\frac 1n \sum_{i=1}^n \bX_{ig}
\end{align*}
Then
\begin{eqnarray*}
\lefteqn{ [\bphi_v | \bLa(\bphi_v),\bY] \propto
\exp \Bigl\{- \frac 12 \Bigl[ \bphi_v^T (\underbrace{w_{v+} \Si^{-1} + n\bZ_g^{-1}}_{\textbf{H}_v^{-1}}) \bphi_v \Bigr] \Bigr\} } \\
& & - 2\bphi_v^T \bigl(
\underbrace{w_{v+}\rho\Si^{-1} \sum_{v' \ne v}\frac{w_{v v'}}{w_{v+}}\bphi_{v'} + n\bZ_g^{-1} ( \bar{\bY}_g(v) -\bb_g(v) -\bar{\ba}_g - \bar{\bX}_g \bga_g) \bigr)}_{\textbf{P}_v} \Bigr] \Bigr\}\\
& \propto & \text{N}(\textbf{H}_v \textbf{P}_v, \textbf{H}_v).
\end{eqnarray*}
%
%--- alpha_ig ------
%\item
\noindent (3)
\begin{eqnarray*}
\lefteqn{
[\ba_i | \bLa(\ba_i),\bY) \propto \text{N}(\textbf{0},\bGa) \cdot \prod_{g=1}^G \rN( \bb_g+\bphi_g+\b1_{V_g}\otimes\ba_{ig}
   +\b1_{V_g}\otimes \bX_{ig}\bga_g, \bV_g) }\\
& \propto & \exp \bigl\{ -\tfrac 12 \ba_i^T \bGa^{-1} \ba_i \bigr\} \cdot \\
& &  \exp \bigl\{ -\tfrac 12 \sum_{g=1}^G [\bY_{ig} -\bb_g -\bphi_g -\b1_{V_g}\otimes\ba_{ig}
   -\b1_{V_g}\otimes \bX_{ig}\bga_g]^T \bV_g^{-1} \cdot \\
& & \qquad [\bY_{ig} -\bb_g -\bphi_g -\b1_{V_g}\otimes\ba_{ig} -\b1_{V_g}\otimes \bX_{ig}\bga_g] \bigr\} \\
& \propto & \exp \bigl\{ -\tfrac 12 \ba_i^T \bGa^{-1} \ba_i \bigr\} \cdot \\
& &  \exp \bigl\{ -\tfrac 12 \sum_{g=1}^G [\b1_{V_g}\otimes\ba_{ig} \!\! - \!
(\underbrace{\bY_{ig} -\bb_g -\bphi_g -\b1_{V_g}\otimes \bX_{ig}\bga_g)}_{=:\textbf{t}_{ig}}]^T \bV_g^{-1} [\b1_{V_g}\otimes\ba_{ig}-\textbf{t}_{ig}] \bigr\} \\
& \propto & \exp \bigl\{ -\tfrac 12 \ba_i^T \bGa^{-1} \ba_i \bigr\} \cdot \\
& & \exp \Bigl\{ -\tfrac 12 \Bigl[ \underbrace{\sum_{g=1}^G (\b1_{V_g}\otimes\ba_{ig})^T \bV_g^{-1}(\b1_{V_g}\otimes\ba_{ig})}_{\textbf{I}}
- 2\underbrace{\sum_{g=1}^G (\b1_{V_g}\otimes\ba_{ig})^T \bV_g^{-1} \textbf{t}_{ig} }_{\textbf{II}} \Bigr] \Bigr\}\\
\end{eqnarray*}
%
\begin{equation*}
\begin{split}
\textbf{I} &=\sum_{g=1}^G (\b1_{V_g}\otimes\ba_{ig})^T \bV_g^{-1}(\b1_{V_g}\otimes\ba_{ig})
=[\underbrace{\ba_{i1}^T,\ldots,\ba_{i1}^T}_{V_1 \text{ times}}]\begin{bmatrix}
                                 \bZ_1^{-1} &  &  \\
                                  & \ddots &  \\
                                  &  & \bZ_1^{-1} \\
                               \end{bmatrix}
                               \begin{bmatrix}
                                 \ba_{i1} \\
                                 \vdots \\
                                 \ba_{i1} \\
                               \end{bmatrix} + \cdots + \\
& \qquad  [\underbrace{\ba_{iG}^T,\ldots,\ba_{iG}^T}_{V_G\text{ times}}]\begin{bmatrix}
                                 \bZ_G^{-1} &  &  \\
                                   & \ddots &   \\
                                  & & \bZ_G^{-1} \\
                               \end{bmatrix}
                               \begin{bmatrix}
                                 \ba_{iG} \\
                                 \vdots \\
                                 \ba_{iG} \\
                               \end{bmatrix} =\\
& = V_1\ba_{i1}^T\bZ_1^{-1}\ba_{i1}+\ldots+ V_G\ba_{iG}^T\bZ_G^{-1}\ba_{iG} \quad (V_g \text{ is the number of voxels in region } g) \\
&= \underbrace{[\ba_{i1}^T,\ldots,\ba_{iG}^T]}_{\ba_i^T}
                                \underbrace{\begin{bmatrix}
                                 V_1\bZ_1^{-1} &  &  \\
                                   & \ddots &  \\
                                  &  & V_G\bZ_G^{-1} \\
                               \end{bmatrix}}_{\bOm_\al^{-1}}
                               \begin{bmatrix}
                                 \ba_{i1} \\
                                 \vdots \\
                                 \ba_{iG} \\
                               \end{bmatrix}
\end{split}
\end{equation*}
%
\begin{equation*}
\begin{split}
\textbf{II} &=\sum_{g=1}^G (\b1_{V_g}\otimes\ba_{ig})^T \bV_g^{-1} \textbf{t}_{ig}
=[\ba_{i1}^T,\ldots,\ba_{i1}^T]\begin{bmatrix}
                                 \bZ_1^{-1} &  &  \\
                                   & \ddots &  \\
                                  &  & \bZ_1^{-1} \\
                               \end{bmatrix}
                               \textbf{t}_{i1}+ \cdots + \makebox[0.5in]{} \\
& \qquad [\ba_{iG}^T,\ldots,\ba_{iG}^T]\begin{bmatrix}
                                 \bZ_G^{-1} & &  \\
                                   & \ddots &   \\
                                  &  & \bZ_G^{-1}
                               \end{bmatrix}
                               \textbf{t}_{iG}=\\
&=\underbrace{[\ba_{i1}^T,\ldots,\ba_{iG}^T]}_{\ba_i^T}
                                \begin{bmatrix}
                                 \underbrace{\bZ_1^{-1}\ldots \bZ_1^{-1}}_{V_1 \text{ times}} & &  \\
                                  & \ddots & \\
                                   &   & \underbrace{\bZ_G^{-1}\ldots \bZ_G^{-1}}_{V_G \text{ times}}
                               \end{bmatrix}
                               \begin{bmatrix}
                                 \textbf{t}_{i1} \\
                                 \vdots \\
                                 \textbf{t}_{iG} \\
                               \end{bmatrix} \\
& =\underbrace{\ba_i^T}_{(2\times 2G)}\underbrace{\begin{bmatrix}
                                       \b1_{V_1}\otimes \bZ_1^{-1} &   &  \\
                                        & \ddots &   \\
                                         &   & \b1_{V_G}\otimes \bZ_G^{-1}  \\
                                     \end{bmatrix}
                               \begin{bmatrix}
                                 \textbf{t}_{i1} \\
                                 \vdots \\
                                 \textbf{t}_{iG} \\
                               \end{bmatrix}}_{\textbf{P}}.\\
\end{split}
\end{equation*}

Hence, it follows that
\begin{equation*}
\begin{split}
[\ba_i | \bLa(\ba_i),\bY) &\propto \exp\bigl\{ -\tfrac 12 \ba_i^T \bGa^{-1}\ba_i\bigr\} \times
\exp \bigl\{-\tfrac 12 \ba_i^T \bOm_\al^{-1} \ba_i - 2\ba_i^T \textbf{P}\bigr\} \\
& \propto \exp \bigl\{ -\tfrac 12 \ba_i^T\underbrace{(\bGa^{-1}+\bOm_\al^{-1})}_{\bPsi^{-1}}\ba_i -2\ba_i^T \textbf{P}\bigr\} \\
& \propto \text{N}(\bPsi \textbf{P}, \bPsi),
\end{split}
\end{equation*}
where \\
$\bPsi=\bigl(\bGa^{-1}+\bOm_\al^{-1}\bigr)^{-1}$, \\
$\textbf{P}=\begin{bmatrix}
              1_{V_1}\otimes \bZ_1^{-1} & \ldots & \ldots \\
              \vdots & \ddots & \vdots \\
              \ldots & \ldots & 1_{V_G}\otimes \bZ_G^{-1} \\
            \end{bmatrix}
            \cdot
            \begin{bmatrix}
              \bY_{i1}-\bb_1-\bphi_1-1_{V_1}\otimes \bX_{i1}\bga_1 \\
              \vdots \\
              \bY_{iG}-\bb_G-\bphi_G-1_{V_G}\otimes \bX_{iG}\bga_G \\
            \end{bmatrix}$,
            and \\
$\bGa^{-1}=\bGa_1^{-1} \otimes \begin{bmatrix}
                       1 & 0 \\
                       0 & 0 \\
                     \end{bmatrix}+ \bGa_2^{-1} \otimes
                     \begin{bmatrix}
                       0 & 0 \\
                       0 & 1 \\
                     \end{bmatrix}$,
                     while
$\bOm_{\al}^{-1}=\begin{bmatrix}
                   V_1 \bZ_1^{-1} & \ldots & \ldots \\
                   \vdots & \ddots & \vdots \\
                   \ldots & \ldots & V_G \bZ_G^{-1} \\
                 \end{bmatrix}$.

%----- Gamma_inv ---
% \item
\noindent (4)
\begin{eqnarray*}
%\begin{split}
[\bGa_j^{-1} | \bLa(\bGa_j^{-1}),\bY] &\propto & [\bGa_j^{-1}] \cdot \prod_{i=1}^n [\ba_i^{(j)} | \bGa_j^{-1}]\\
& \propto & \text{Wishart}\bigl( (h_j H_j)^{-1}, h_j\bigr) \cdot \prod_{i=1}^n \text{N} (\textbf{0},\bGa_j) \\
& \propto & |\bGa_j|^{(h_j-G-1)/2} \exp \bigl\{ -\frac{h_j}{2} \tr (H_j\bGa_j^{-1})\bigr\} \cdot \\
& & |\bGa_j|^{n/2} \exp \bigl\{-\frac 12 \sum_{i=1}^n \ba_{ij}^T \bGa_j^{-1} \ba_{ij}\bigr\} \\
& \propto & |\bGa_j|^{(h_j+n-G-1)/2} \exp \bigl\{ -\frac 12 \tr [(h_j H_j +\sum_{i=1}^n \ba_{ij}\ba_{ij}^T) \bGa_j^{-1}] \bigr\} \\
& \propto & \text{Wishart}\bigl( (h_j H_j +\sum_{i=1}^n \ba_{ij}\ba_{ij}^T)^{-1}, h_j+n \bigr)
% \end{split}
\end{eqnarray*}

%------ Zg_inv ----
%\item
\noindent (5)
\begin{eqnarray*}
\lefteqn{[\bZ_g^{-1} | \bLa(\bZ_g^{-1}),\bY] \propto [\bZ_g^{-1}] \cdot \prod_{i=1}^n \prod_{v \in g} \text{N}(\bY_{ig}(v)|\bb_g(v),\bphi_g(v),\ba_{ig},\bX_{ig}\bga_g,\bZ_g)}\\
& & (\text{ Let } \textbf{u}_{igv}= \bY_{ig}(v)-\bb_g(v)-\bphi_g(v)-\ba_{ig}-\bX_{ig}\bga_g ) \\
& \propto & |\bZ_g^{-1}|^{(c_1-2-1)/2} \exp \bigl\{ -\tfrac 12 \tr (c_1\Om_1 \bZ_g^{-1}) \bigr\} \cdot\\
& & |\bZ_g^{-1}|^{nV_g/2} \exp \bigl\{-\tfrac 12 \sum_{i=1}^n \sum_{v\in g} \textbf{u}_{igv}^T \bZ_g^{-1} \textbf{u}_{igv}\bigr\} \\
& \propto & |\bZ_g^{-1}|^{(c_1+nV_g-2-1)/2} \exp \bigl\{ -\tfrac 12 \tr(c_1\Om_1\bZ_g^{-1}) -
  \tfrac 12 \underbrace{\sum_{i=1}^n \tr\bigl( (\sum_{v\in g} \textbf{u}_{igv} \textbf{u}_{igv}^T)\bZ_g^{-1}\bigr)}_
  {=\text{tr}(\sum_{i=1}^n (\sum_{v\in g} u_{igv} u_{igv}^T )\bZ_g^{-1})} \bigr\} \\
& \propto & |\bZ_g^{-1}|^{(c_1+nV_g-2-1)/2} \exp \bigl\{ -\tfrac 12 \tr[(c_1\Om_1+\sum_{i=1}^n \sum_{v\in g} \textbf{u}_{igv} \textbf{u}_{igv}^T) \bZ_g^{-1}] \bigr\} \\
& \propto & \text{Wishart}\bigl( (c_1\Om_1+\sum_{i=1}^n \sum_{v\in g} \textbf{u}_{igv} \textbf{u}_{igv}^T)^{-1}, c_1+nV_g \bigr)
\end{eqnarray*}

%--- Sigma_inv ----
%\item
\noindent (6)
\begin{eqnarray*}
\lefteqn{[\bSi^{-1} | \bLa(\bSi^{-1}),\bY]
\propto [\bSi^{-1}] \cdot \prod_{v=1}^V \text{N} (\rho\sum_{v' \ne v} \frac{w_{v v'}}{w_{v+}} \bphi_{v'}, \frac 1{w_{v+}} \bSi)
\;\; (\text{ Denote } b_{v'}:=\frac{w_{v v'}}{w_{v+}}) } \\
& \propto & |\bSi^{-1}|^{(c_2-2-1)/2} \exp \bigl\{ -\tfrac 12 \tr (c_2 \Om_2 \bSi^{-1}) \bigr\} \cdot \\
& & \prod_{v=1}^V w_v |\bSi^{-1}|^{V/2} \exp \Bigl\{ \tfrac 12 \sum_{v=1}^V [ \bphi_v -\rho \sum_{v'\ne v} b_{v'}\bphi_{v'}]^T w_{v+}\bSi^{-1}
[ \bphi_v -\rho \sum_{v'\ne v} b_{v'}\bphi_{v'}]\bigr\} \\
& \propto & |\bSi^{-1}|^{(c_2+V-2-1)/2} \exp \bigl\{ -\tfrac 12 \tr \bigl[c_2 \Om_2 + \\
& & \sum_{v=1^V} [w_{v+}^{1/2}(\bphi_v-\rho\sum_{v'\ne v} b_{v'}\bphi_{v'})]
[w_{v+}^{1/2}(\bphi_v-\rho\sum_{v'\ne v} b_{v'}\bphi_{v'})]^T \bigr] \bSi^{-1}) \bigr\}\\
& \propto & \text{Wishart}\bigl( (c_2\Om_2 +\sum_{v=1}^V w_{v+}[\bphi_v-\rho\sum_{v'\ne v} b_{v'}\bphi_{v'}][\bphi_v-\rho\sum_{v'\ne v} b_{v'}\bphi_{v'}]^T),c_2+V \bigr),
\end{eqnarray*}
where $V$ is the total number of voxels included in the analysis (which have at least one within-region neighbor).

%---- lambda_g_inv ----
%\item
% \newpage
\noindent (7) Recall the notation: $\bb_{gj}=\bb_g^{(j)}$, where $j=1,2$ denote the baseline and follow-up sessions. The independence assumption implies

$\bb_{gj} \sim \text{MVN}(\bb_{0gj},\la_{gj}^2)$, $\bb_g \sim \text{MVN}\biggl(\bb_{0g}, \underbrace{\bI_{V_g}\otimes \begin{bmatrix}
                                                                                                           \la_{g1}^2 & 0 \\
                                                                                                           0 & \la_{g2}^2 \\
                                                                                                         \end{bmatrix}}_{\bLa_g} \biggr)$.

\noindent Then
\begin{eqnarray*}
\lefteqn{[\la_{gj}^{-2} | \bLa(\la_{gj}^{-2}),\bY] \propto [\la_{gj}^{-2} ] \cdot [\bb_{gj} | \la_{gj}^{-2}]}\\
& \propto & \text{Gamma}(a_j,b_j) \cdot \text{Normal}(1_{V_g} \bb_{ogj},\la_{gj}^2\bI_{V_g}) \\
& \propto & (\la_{gj}^{-2})^{a_j-1} \exp \Bigl( \frac{\la_{gj}^{-2}}{b_j} \Bigr) (\la_{gj}^2)^{-V_g/2}
\exp \Bigl\{-\frac{1}{2\la_{gj}^2}(\bb_{gj}-\textbf{1}_{V_g} \bb_{0gj})^T (\bb_{gj}-\textbf{1}_{V_g} \bb_{0gj})\Bigr\} \\
& \propto & (\la_{gj}^{-2})^{a_j+V_g-1} \exp \Bigl\{-\la_{gj}^2\Bigl[ \frac 1{b_j}+\frac{(\bb_{gj}-\textbf{1}_{V_g} \bb_{0gj})^T (\bb_{gj}-\textbf{1}_{V_g} \bb_{0gj})}{2}\Bigr]^{-1} \Bigr\} \\
& \propto & \text{Gamma}\Bigl( a_j+V_g/4,\Bigl[ \frac 1{b_j}+\frac{(\bb_{gj}-\textbf{1}_{V_g} \bb_{0gj})^T (\bb_{gj}-\textbf{1}_{V_g} \bb_{0gj})}{2}\Bigr]^{-1}\Bigr)
\end{eqnarray*}

%---- rho ----
%\item
\noindent (8)
%\begin{equation*}
%\begin{split}
%[\rho|\bLa(\rho),\bY] & \propto [\rho] [\bphi_v|\rho] \\
%& \propto \text{Uniform}(\{ 0.05,0.1,\ldots,0.99\}) \cdot \underbrace{\prod_{v=1}^V \text{N} (\rho\sum_{v'\ne v} b_{v'} \bphi_{v'},w_{v+}^{-1}\bSi)}_{\textbf{I}} \\
%\end{split}
%\end{equation*}
%The second term
%\begin{equation*}
%\begin{split}
%\textbf{I} & \propto \exp \Bigl \{-\tfrac 12  \sum_{v=1}^V[\bphi_v - \rho \sum_{v'\ne v} \frac{w_{vv'}}{w_{v+}} \bphi_{v'}]^T w_{v+}\bSi^{-1}[\bphi_v - \rho \sum_{v'\ne v} \frac{w_{vv'}}{w_{v+}} \bphi_{v'}]\Bigl\}  \\
%& \propto \exp \Bigl\{-\tfrac 12 \sum_{v=1}^V [\rho\underbrace{\sum_{v'\ne v} \frac{w_{vv'}}{w_{v+}}\bphi_{v'}}_{A}-\bphi_v]^T w_{v+} \bSi^{-1}[\rho\sum_{v'\ne v} \frac{w_{vv'}}{w_{v+}}\bphi_{v'}-\bphi_v] \Bigr\} \\
%&\propto \exp \Bigl\{-\tfrac 12 \sum_{v=1}^V(\rho A^T-\bphi_v^T)w_{v+}\bSi^{-1}(\rho A-\bphi_v) \Bigr\} \\
%& \propto \exp \Bigl\{ -\tfrac 12 \sum_{v=1}^V (\rho A^T w_{v+}\bSi^{-1}\rho A - \bphi_v^T w_{v+} \bSi_{-1}\rho A -\rho A^T w_{v+} \bSi^{-1} \bphi_v\Bigr\}\\
%& \propto \exp \Bigl\{ -\tfrac 12 \sum_{v=1}^V \rho^2 A^T w_{v+}\bSi^{-1} A -2\rho\sum_{v=1}^V \bphi_v^T w_{v+} \bSi^{-1} A \Bigr\}\\
%& \propto \exp \Bigl\{ -\tfrac 12 \rho^2 \Bigl(\underbrace{\sum_{v=1}^V A^T w_{v+}\bSi^{-1} A}_{\eta^{-1}}\Bigr) -2 \rho
%\Bigl(\underbrace{\sum_{v=1}^V\bphi_v^Tw_{v+} \bSi^{-1}A }_{\xi}\Bigr)\Bigr\}\\
%&\propto \text{N}(\eta\xi,\eta)
%\end{split}
%\end{equation*}
%where
%\begin{equation*}
%\begin{split}
%\eta&=\Bigl(\sum_{v=1}^V \bigl( \sum_{v'\ne v}\frac{w_{vv'}}{w_{v+}}\bphi_{v'}^T \cdot w_{v+} \bSi^{-1} \cdot \sum_{v'\ne v} \frac{w_{vv'}}{w_{v+}}\bphi_{v'}\Bigr)^{-1} \\
%& = \Bigl(\sum_{v=1}^V \frac{1}{w_{v+}} M^T\bSi^{-1} M\Bigr)^{-1},\;\; M=\sum_{v'\ne v}w_{vv'} \bphi_{v'}\\
%\xi&=\sum_{v=1}^V\bphi_v^T \bSi^{-1} M.
%\end{split}
%\end{equation*}

Let $\{\rho_1,\ldots, \rho_n\}$ be an array of values on which we specify a discrete uniform prior for $\rho$. ($n=36$ in our analysis.)
Let $L(\rho)=\prod_{v=1}^V \text{N} (\rho\sum_{v'\ne v} b_{v'} \bphi_{v'},w_{v+}^{-1}\bSi)$. Then

\begin{equation*}
\begin{split}
L(\rho) & \propto \exp \Bigl \{-\tfrac 12  \sum_{v=1}^V[\bphi_v - \rho \sum_{v'\ne v} \frac{w_{vv'}}{w_{v+}} \bphi_{v'}]^T w_{v+}\bSi^{-1}[\bphi_v - \rho \sum_{v'\ne v} \frac{w_{vv'}}{w_{v+}} \bphi_{v'}]\Bigl\}  \\
& \propto \exp \Bigl\{-\tfrac 12 \sum_{v=1}^V [\rho\underbrace{\sum_{v'\ne v} \frac{w_{vv'}}{w_{v+}}\bphi_{v'}}_{A}-\bphi_v]^T w_{v+} \bSi^{-1}[\rho\sum_{v'\ne v} \frac{w_{vv'}}{w_{v+}}\bphi_{v'}-\bphi_v] \Bigr\} \\
&\propto \exp \Bigl\{-\tfrac 12 \sum_{v=1}^V(\rho A^T-\bphi_v^T)w_{v+}\bSi^{-1}(\rho A-\bphi_v) \Bigr\} \\
& \propto \exp \Bigl\{ -\tfrac 12 \sum_{v=1}^V (\rho A^T w_{v+}\bSi^{-1}\rho A - \bphi_v^T w_{v+} \bSi_{-1}\rho A -\rho A^T w_{v+} \bSi^{-1} \bphi_v\Bigr\}\\
& \propto \exp \Bigl\{ -\tfrac 12 \sum_{v=1}^V \rho^2 A^T w_{v+}\bSi^{-1} A -2\rho\sum_{v=1}^V \bphi_v^T w_{v+} \bSi^{-1} A \Bigr\}\\
& \propto \exp \Bigl\{ -\tfrac 12 \rho^2 \Bigl(\underbrace{\sum_{v=1}^V A^T w_{v+}\bSi^{-1} A}_{\eta^{-1}}\Bigr) -2 \rho
\Bigl(\underbrace{\sum_{v=1}^V\bphi_v^Tw_{v+} \bSi^{-1}A }_{\xi}\Bigr)\Bigr\}
\end{split}
\end{equation*}
where
\begin{equation*}
\begin{split}
\eta&=\Bigl(\sum_{v=1}^V \bigl( \sum_{v'\ne v}\frac{w_{vv'}}{w_{v+}}\bphi_{v'}^T \cdot w_{v+} \bSi^{-1} \cdot \sum_{v'\ne v} \frac{w_{vv'}}{w_{v+}}\bphi_{v'}\Bigr)^{-1} \\
& = \Bigl(\sum_{v=1}^V \frac{1}{w_{v+}} M^T\bSi^{-1} M\Bigr)^{-1},\;\; M=\sum_{v'\ne v}w_{vv'} \bphi_{v'}\\
\xi&=\sum_{v=1}^V\bphi_v^T \bSi^{-1} M.
\end{split}
\end{equation*}

The corresponding set of normalized likelihood weights is then given by $\{p1,\ldots,p_n\}$, where
$$
p_k=\frac{L(\rho_k)}{L(\rho_1)+\ldots+ L(\rho_n)}=\frac{1}{\frac{L(\rho_1)}{L(\rho_k)}+\ldots+\frac{L(\rho_n)}{L(\rho_k)}}, \; k=1,\ldots,n.
$$
For example, for $n=1$
\begin{equation*}
\begin{split}
\frac{L(\rho_1)}{L(\rho_k)} &=\frac{\exp\{ -\tfrac 12 (\rho_1^2\eta^{-1} - 2 \rho_1\xi)\} }{\exp\{-\tfrac 12 (\rho_k^2\eta^{-1}-2\rho_k\xi)  \}}\\
& = \exp\bigl\{ -\tfrac 12 (\rho_1^2\eta^{-1}-2\rho_1\xi-\rho_k^2\eta^{-1}+2\rho_k\xi)\bigr\} \\
&= \exp \bigl\{ -\tfrac 12 [(\rho_1^2 - \rho_k^2)\eta^{-1}-2\xi(\rho_1-\rho_k)]\bigr\} \\
&= \exp \bigl\{ -\tfrac 12 [(\rho_1-\rho_k)(\rho_1+\rho_k)\eta^{-1} - 2 \xi (\rho_1 -\rho_k)]\bigr\} \\
&= \exp\bigl\{ -\tfrac 12 \bigl( (\rho_1-\rho_k) [(\rho_1+\rho_k)\eta^{-1} -2 \xi]\bigr) \bigr\}.
\end{split}
\end{equation*}
%---- gamma_g ---
% \item
\noindent (9) $[\bga_{g} | \bLa(\bga_g),\bY]$ =?
Recall the notation:
$\bga_g=[\ga_{g1}^{(1)}, \ga_{g1}^{(2)},\ldots,\ga_{gQ}^{(1)},\ga_{gQ}^{(2)}]^T$, where
$Q$ denotes the number of covariates in the model and the superscripts $(1)$ and $(2)$
denote the baseline and follow-up sessions. We assume independence between the covariates, i.e.,
$$
\bga_g|\Lambda_{\ga_g} \sim \text{MVN} \bigl(0,\text{diag}(\tau_{g11}^2,\tau_{g21}^2,\ldots,\tau_{g1Q}^2,\tau_{g2Q}^2)\bigr)
$$
\begin{eqnarray*}
\lefteqn{[\bga_{g} | \bLa(\ga_g),\bY] \propto [\bga_g | \bLa(\bGa_g)]\cdot \prod_{i=1}^n \bY_{ig}| \bb_g,\bphi_g,\ba_{ig},\bga_g,\bV_g } \\
& \propto & \exp\bigl\{ -\tfrac 12 \bga_g^T \bLa_{\ga_g}^{-1}\bga_g\bigr\} \cdot\\
& & \exp\bigl\{ -\tfrac 12 \sum_{i=1}^n(\bY_{ig}-\bb_g-\bphi_g-\b1_{V_g}\otimes \ba_{ig} -\b1_{V_g} \otimes \bX_{ig}\bga_g)^T\bV_g^{-1} \cdot \\
& & (\bY_{ig}-\bb_g-\bphi_g-\b1_{V_g}\otimes \ba_{ig} -\b1_{V_g} \otimes \bX_{ig}\bga_g)\bigr\}.
\end{eqnarray*}
The sum in the second exponential term can be written as
\begin{eqnarray*}
\lefteqn{\sum_{i=1}^n [\b1_{V_g} \otimes \bX_{ig}\bga_g-(\underbrace{\bY_{ig}-\bb_g-\bphi_g-\b1_{V_g}\otimes \ba_{ig})}_{:=\textbf{u}_{ig}})]^T \bV_g^{-1} [\b1_{V_g} \otimes \bX_{ig}\bga_g - \textbf{u}_{ig}] } \\
& = & \sum_{i=1}^n \bigl[ (\b1_{V_g} \otimes \bX_{ig}\bga_g)^T-\textbf{u}_{ig}^T\bigr]\bV_g^{-1}\bigl[ (\b1_{V_g} \otimes \bX_{ig}\bga_g)-\textbf{u}_{ig}\bigr]= \\
& & \biggl[(\b1_{V_g} \otimes \bX_{ig}\bga_g)^T=\b1_{V_g}^T \otimes \bga_g^T\bX_{ig}^T=[\bga_g^T \bX_{ig}^T \ldots \bga_g^T \bX_{ig}^T ] =
\bga_g^T[\bX_{ig}^t \ldots \bX_{ig}^T]= \\
& & \quad \bga_g^T(\b1_{V_g}^T \otimes \bX_{ig}^T)\biggr] \\
&= & \sum_{i=1}^n \bigl[ \bga_g^T (\b1_{V_g}^T\otimes \bX_{ig}^T) -\textbf{u}_{ig}\bigr] \bV_g^{-1} \bigl[\b1_{V_g}\otimes \bX_{ig}\bga_g-\textbf{u}_{ig} \bigr] \\
&=& \sum_{i=1}^n \bigl[ \bga_g^T (\b1_{V_g}^T \otimes \bX_{ig}) \bV_g^{-1} (\b1_{V_g}\otimes \bX_{ig}) \bga_g -2\bga_g^T (\b1_{V_g}^T \otimes \bX_{ig}) \bV_g^{-1} \textbf{u}_{ig}\bigr] \\
&= & \bga_g^T \Bigl( \sum_{i=1}^n (\b1_{V_g}^T \otimes \bX_{ig}) \bV_g^{-1} (\b1_{V_g}\otimes \bX_{ig}) \Bigr)\bga_g
-2 \bga_g^T\sum_{i=1}^n (\b1_{V_g}^T \otimes \bX_{ig}) \bV_g^{-1} \textbf{u}_{ig} \\
\end{eqnarray*}
Hence,
\begin{equation*}
\begin{split}
[\bga_{g} | \bLa(\ga_g),\bY] & \propto \exp \Bigl\{ -\tfrac 12 \bga_g^T
\bigl[\underbrace{ \bLa_{\ga_g}^{-1}+\sum_{i=1}^n (\b1_{V_g}^T\otimes \bX_{ig}^T) \bV_g^{-1} (\b1_{V_g} \otimes \bX_{ig}) }_{\bOm_{\ga_g}^{-1}}\bigr] \bga_g \\
& -2\bga_g^T\underbrace{\sum_{i=1}^n (\b1_{V_g}^T \otimes \bX_{ig}^T) \bV_g^{-1} \textbf{u}_{ig}}_{\textbf{T}_{\ga_g}} \Bigr\} \\
& \propto \text{N} (\bOm_{\ga_g} \bT_{\ga_g},\bOm_{\ga_g})
\end{split}
\end{equation*}
where $\bOm_{\ga_g}^{-1}=\bLa_{\ga_g}^{-1}+\sum_{i=1}^n (\textbf{1}_{V_g}^T\otimes \bX_{ig}^T) \bV_g^{-1}(\textbf{1}_{V_g}\otimes \bX_{ig})$,
$\bT_{\ga_g}=\sum_{i=1}^n (\textbf{1}_{V_g}^T\otimes \bX_{ig}^T)\bV_g^{-1} \textbf{u}_{ig}$ and
$\textbf{u}_{ig}= \bY_{ig}-\bb_g-\bphi_g-\textbf{1}_{V_g}\otimes \ba_{ig}$.

{\it Computational Details}:
$\bOm_{\ga_g}$ and $\textbf{T}_{\ga_g}$ can further be simplified for computational reasons.
\begin{equation*}
\begin{split}
\bOm_{\ga_g}^{-1} &= \bLa_{\ga_g}^{-1} +\sum_{i=1}^n (\b1_{V_g}^T \otimes \bX_{ig}^T) (\bI_{V_g} \otimes \bZ_g^{-1}) (\b1_{V_g} \otimes \bX_{ig}) =\\
&= \bLa_{\ga_g}^{-1}+V_g \sum_{i=1}^n \bX_{ig}^T \bZ_g^{-1}\bX_{ig}
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
\textbf{T}_{\ga_g} &= \sum_{i=1}^n [\bX_{ig} \cdots \bX_{ig}]\begin{bmatrix}
                                                               \bZ_g^{-1} &   &  \\
                                                                & \ddots &   \\
                                                                 &   & \bZ_g^{-1} \\
                                                             \end{bmatrix} \textbf{u}_{ig} =\\
&= \sum_{i=1}^n (\b1_{V_g} \otimes \bX_{ig}^T \bZ_g^{-1}) \textbf{u}_{ig}
\end{split}
\end{equation*}

%---- tau_gjq^2 ----
%\item
\noindent (10)
\begin{equation*}
\begin{split}
[\tau_{gjq}^2|\bLa(\tau_{gjq}^2),\bY] & \propto [\tau_{gjq}^2][\bga_{gjq}|\tau_{gjq}^2] \\
& \propto \text{Gamma}(e_0,f_0) \cdot\text{N}(0,\tau_{gjq}^2)\\
& \propto (\tau_{gjq}^{-2})^{e_0-1} \exp\Bigl( -\frac{\tau_{gjq}^{-2}}{f_0}\Bigr) (\tau_{gjq}^2)^{-1/2} \exp \Bigl\{ -\tfrac{1}{2\tau_{gjq}^2}\bga_{gjq}^2\bigr\} \\
& \propto (\tau_{gjq}^{-2})^{e_0+1/2-1} \exp \Bigl\{-\tau_{gjq}^{-2} \bigl[\tfrac{1}{f_0}+\tfrac 12 \bga_{gjq}^2 \bigr] \Bigr\}\\
&\propto \text{Gamma}\Bigl( e_0+\tfrac 12,\bigl[\tfrac{1}{f_0}+\tfrac 12 \bga_{gjq}^2 \bigr]^{-1} \Bigr)
\end{split}
\end{equation*}

%\end{enumerate}


\vspace{2ex}
\noindent
\subsection{Appendix G: Additional results from the analysis of ADNI data}
\label{ss:additional_results}

\noindent
{\bf Regions included in analysis of ADNI data}
%\begin{center}
%\begin{tabular}{|c|c|c|c|c|c|c|c|}
%\hline
%1	&	L Cau	    &	71	&	877	&	24	&	1760	&	R Insula 	&	30	\\
%2	&	R Cau	    &	72	&	950	&	25	&	281	&	L Pal 	&	75	\\
%3	&	L A Cing	&	31	&	1352	&	26	&	280	&	R Pal 	&	76	\\
%4	&	R A Cing	&	32	&	1352	&	27	&	982	&	R PHG 	&	39	\\
%5	&	L Mid Cing	&	33	&	1845	&	28	&	3726	&	L Po 	&	57	\\
%6	&	R Mid Cing	&	34	&	2098	&	29	&	3698	&	R Po 	&	58	\\
%7	&	L Inf Op	&	11	&	970	&	30	&	3331	&	L Pr 	&	1	\\
%8	&	R Inf Op	&	12	&	1321	&	31	&	3206	&	R Pr 	&	2	\\
%9	&	L Inf OF	&	15	&	1537	&	32	&	976	&	L Put 	&	73	\\
%10	&	R Inf OF	&	16	&	1580	&	33	&	1058	&	R Put 	&	74	\\
%11	&	L Inf Tr	&	13	&	2376	&	34	&	963	&	L Rol Op	&	17	\\
%12	&	R Inf Tr	&	14	&	1891	&	35	&	1278	&	R Rol Op	&	18	\\
%13	&	L Mid Fr	&	7	&	4618	&	36	&	1769	&	L SMA 	&	19	\\
%14	&	R Mid Fr	&	8	&	4896	&	37	&	2193	&	R SMA 	&	20	\\
%15	&	L Sup Fr	&	3	&	3208	&	38	&	1225	&	R SMG 	&	63	\\
%16	&	R Sup Fr	&	4	&	3864	&	39	&	4655	&	L Mid Temp	&	85	\\
%17	&	L Sup Med	&	23	&	2763	&	40	&	4337	&	R Mid Temp	&	86	\\
%18	&	R Sup Med	&	24	&	2059	&	41	&	1189	&	L Sup TP	&	83	\\
%19	&	L He	    &	79	&	234	&	42	&	1171	&	R Sup TP	&	84	\\
%20	&	R He	    &	80	&	252	&	43	&	2278	&	L Sup Temp	&	81	\\
%21	&	L Hippo	    &	37	&	929	&	44	&	3112	&	R Sup Temp	&	82	\\
%22	&	R Hippo	    &	38	&	941	&	45	&	1076	&	L Thal 	&	77	\\
%23	&	L Insula	&	29	&	1898	&	46	&	1055	&	R Thal 	&	78	\\
%\hline
%\end{tabular}
%\end{center}

We include 39 regions from the AAL map \citep{tzourio:02} in our analysis of the PET data from the study Alzheimer's disease study (ADNI).
The regions are listed in Table \ref{ADNI_regions}.

\begin{table}\label{ADNI_regions}
\centering
\begin{tabular}{|c|l|r|c|l|r|}
\hline \hline
No.	& 	Region Name	& 	$V_g$	& 	No.	& 	Region Name	& 	$V_g$\\
\hline
1	&	Caudate L	&	877	&	21	&	Insula L	&	1898	 \\
2	&	Caudate R	&	950	&	22	&	Insula R	&	1760	 \\
3	&	Cingulum Ant L	&	1352	&	23	&	ParaHippocampal R	&	982 \\
4	&	Cingulum Ant R	&	1352	&	24	&	Putamen L	&	976	 \\
5	&	Cingulum Mid L	&	1845	&	25	&	Putamen R	&	1058	 \\
6	&	Cingulum Mid R	&	2098	&	26	&	Rolandic Oper L	&	963	 \\
7	&	Frontal Inf Oper L	&	970	&	27	&	Rolandic Oper R	&	1278	 \\
8	&	Frontal Inf Oper R	&	1321	&	28	&	Temporal Mid L	&	4655	 \\
9	&	Frontal Inf Orb L	&	1537	&	29	&	Temporal Mid R	&	4337	 \\
10	&	Frontal Inf Orb R	&	1580	&	30	&	Temporal Pole Sup L	&	1189	 \\
11	&	Frontal Inf Tri L	&	2376	&	31	&	Temporal Pole Sup R	&	1171	 \\
12	&	Frontal Inf Tri R	&	1891	&	32	&	Temporal Sup L	&	2278	 \\
13	&	Frontal Sup L	&	3208	&	33	&	Temporal Sup R	&	3112	 \\
14	&	Frontal Sup R	&	3864	&	34	&	Thalamus L	&	1076	 \\
15	&	Frontal Mid L	&	4618	&	35	&	Thalamus R	&	1055	 \\
16	&	Frontal Mid R	&	4896	&	36	&	Cingulum Post L	&	466	 \\
17	&	Frontal Sup Medial L	& 2763  &	37	&	Cingulum Post R	& 281 \\
18	&	Frontal Sup Medial R	&	2059	&	38	&	Precuneus L	&	3475 \\
19	&	Hippocampus L	        &	929	&	39	&	Precuneus R	&	2832	 \\
20	&	Hippocampus R	        &	941	&		&		&		\\
\hline
\end{tabular}
\caption[Selected 39 AAL regions of interest considered in the Alzheimer's disease study. ]
{Selected 39 AAL regions of interest considered in the Alzheimer's disease study. $V_g$ denotes the regions size, in voxels.
Notation is as follows: R=right, L=left, Inf=inferior, Med=medial, Mid=middle, Orb=orbital, Oper=operculum, Post=posterior, Sup=superior, Temp=temporal, Tri=triangularis.}
\end{table}



Figures \ref{trace_plot1} and \ref{hist_plot_1} give the trace plot and the corresponding histogram plot of
the posterior distribution for parameter $\bb_g$. The figures show the trace plot and histogram for one selected voxel in region $g$.

\begin{figure}[!h]
\centering
%\includegraphics[width=1\textwidth]{Figures/trace_beta_g2_v600.eps}
\includegraphics[width=1\textwidth]{Figures/appendix/trace_beta_g2_v200.eps}
\includegraphics[width=1\textwidth]{Figures/appendix/trace_phi_g16_v1400.eps}
\caption{Trace plot for $\bb_g$ parameter for a randomly selected voxel in region 2 (top) and
for $\bphi_g$ parameter for a randomly selected voxel in region 16 (bottom).}
\label{trace_plot1}
\end{figure}

\begin{figure}[!h]
\centering
\includegraphics[scale=0.5]{Figures/appendix/hist_beta_g2_v200.eps}
\includegraphics[scale=0.5]{Figures/appendix/hist_phi_g16_v1400.eps}
\caption{Histogram plot for $\bb_g$, for a randomly selected voxel in region 2 (left) and
for $\bphi_g$, for a randomly selected voxel in region 16 (right).}
\label{hist_plot_1}
\end{figure}


\vspace{2ex}
\noindent
{\bf Individualized prediction maps for subjects from ADNI data set.}
Figure \ref{pred-BSPM-AD-appendix} shows the individualized prediction maps for four additional AD patients from the test data set.
Figure \ref{pred-BSPM-ctrl-appendix} shows the individualized prediction maps for two additional HC subjects from the test data set.

\begin{figure}\label{pred-BSPM-AD-appendix}
\centering
{\small (a) predicted month 6 maps} \\
\includegraphics[scale=0.34]{Figures/appendix/AD_s4_pred.eps}
\hspace{-2em} \includegraphics[scale=0.34]{Figures/appendix/AD_s10_pred.eps}
\hspace{-2em} \includegraphics[scale=0.34]{Figures/appendix/AD_s25_pred.eps}
\hspace{-2em} \includegraphics[scale=0.34]{Figures/appendix/AD_s32_pred.eps}

{\small (b) observed month 6 maps}\\
\includegraphics[scale=0.34]{Figures/appendix/AD_s4_obs.eps}
\hspace{-2em} \includegraphics[scale=0.34]{Figures/appendix/AD_s10_obs.eps}
\hspace{-2em} \includegraphics[scale=0.34]{Figures/appendix/AD_s25_obs.eps}
\hspace{-2em} \includegraphics[scale=0.34]{Figures/appendix/AD_s32_obs.eps}
\caption[Individualized predicted and observed 6 month follow-up glucose uptake measurements for 4 AD patients from the test data set.]
{Individualized predicted and observed 6 month follow-up glucose uptake measurements for 4 AD patients from the test data set.
Axial slice 40 is shown in radiological view.}
\end{figure}


\begin{figure}[!h]
\label{pred-BSPM-ctrl-appendix}
\centering
{\small (a) predicted month 6 maps} \\
\includegraphics[scale=0.33]{Figures/ch4_final/ctrl_s1_pred.eps}
\hspace{-2em} \includegraphics[scale=0.33]{Figures/ch4_final/ctrl_s13_pred.eps}
\hspace{-2em} \includegraphics[scale=0.33]{Figures/ch4_final/ctrl_s17_pred.eps}
\hspace{-2em} \includegraphics[scale=0.33]{Figures/ch4_final/ctrl_s21_pred.eps}

{\small (b) observed month 6 maps}\\
\includegraphics[scale=0.33]{Figures/ch4_final/ctrl_s1_obs.eps}
\hspace{-2em} \includegraphics[scale=0.33]{Figures/ch4_final/ctrl_s13_obs.eps}
\hspace{-2em} \includegraphics[scale=0.33]{Figures/ch4_final/ctrl_s17_obs.eps}
\hspace{-2em} \includegraphics[scale=0.33]{Figures/ch4_final/ctrl_s21_obs.eps}
\caption[Individualized predicted and observed 6 month follow-up glucose uptake measurements for 4 HC subjects from the test data set.]
{Individualized predicted and observed 6 month follow-up glucose uptake measurements for 4 HC subjects from the test data set.
Axial slice 40 is shown in radiological view.}
\end{figure}

\vspace{2ex}
\noindent
\subsection{Appendix H: Simulation Results.}
\label{ss:simulation_results}
Tables \ref{t:sim_results_beta1} through \ref{t:sim_results_alpha} give results from the simulation study described in Section
\ref{s:ch4_sim_study}. Tables \ref{t:sim_results_beta1} and \ref{t:sim_results_beta2} list the estimated and the true values, as well as the bias, for parameter $\bb$ from the mean of the top level of our model (BSPM). The results are shown for all 5 regions included in the simulation study and for 25 randomly selected voxels from each region. Table \ref{t:sim_results_beta1} gives the simulation results for the baseline session, and Table \ref{t:sim_results_beta2} for the follow-up session.
%Table \ref{t:sim_results_phi} gives results for the spatial parameter $\bphi$, for selected 20 voxels in region 1; results are given for both baseline and follow-up session.
Table \ref{t:sim_results_alpha} gives the simulation results for the random effect parameters $\ba_i$, for baseline session (results for the follow-up session are similar).

\begin{sidewaystable}
\centering
\begin{tabular}{c | r r r| r r r| r r r | r r r | r r r |}
\hline \hline
Voxel & \multicolumn{15}{c}{\bf Region }  \\
& \multicolumn{3}{c}{{\bf 1}} & \multicolumn{3}{c}{{\bf 2}} & \multicolumn{3}{c}{{\bf 3}} & \multicolumn{3}{c}{{\bf 4}} & \multicolumn{3}{c}{{\bf 5}} \\
 & {Est.} & {True} & Bias & {Est.} & {True} & Bias & {Est.} & {True} & Bias & {Est.} & {True} & Bias & {Est.} & {True} & Bias \\
1	&	14.42	&	12.95	&	1.47	&	20.50	&	20.93	&	-0.43	&	37.26	&	34.18	&	3.08	&	62.55	&	60.16	&	2.39	&	 63.848	&	63.02	&	0.83	\\
2	&	11.53	&	10.35	&	1.18	&	15.77	&	15.32	&	0.45	&	31.06	&	28.66	&	2.40	&	63.44	&	61.75	&	1.69	&	 62.2453	&	61.95	&	0.29	\\
3	&	14.07	&	12.20	&	1.87	&	15.63	&	17.00	&	-1.37	&	22.97	&	22.11	&	0.86	&	63.86	&	62.62	&	1.24	&	 60.8592	&	59.62	&	1.24	\\
4	&	14.00	&	13.61	&	0.39	&	22.71	&	22.48	&	0.23	&	24.48	&	22.93	&	1.55	&	64.32	&	62.90	&	1.42	&	 56.8206	&	55.89	&	0.93	\\
5	&	16.20	&	14.53	&	1.67	&	26.29	&	27.40	&	-1.11	&	31.97	&	29.74	&	2.23	&	65.45	&	62.90	&	2.55	&	 51.2923	&	50.55	&	0.75	\\
6	&	16.53	&	14.95	&	1.57	&	30.89	&	31.70	&	-0.81	&	37.20	&	35.21	&	1.99	&	64.03	&	62.63	&	1.40	&	 43.5581	&	43.90	&	-0.34	\\
7	&	16.36	&	14.89	&	1.47	&	32.75	&	35.23	&	-2.48	&	41.85	&	39.72	&	2.13	&	63.24	&	61.78	&	1.46	&	 38.7694	&	36.57	&	2.20	\\
8	&	13.88	&	12.01	&	1.87	&	36.43	&	37.82	&	-1.38	&	45.41	&	43.36	&	2.05	&	61.25	&	59.88	&	1.37	&	 28.7784	&	29.39	&	-0.61	\\
9	&	15.14	&	13.85	&	1.29	&	37.20	&	39.40	&	-2.20	&	47.60	&	46.22	&	1.38	&	57.60	&	56.55	&	1.05	&	 23.6408	&	22.79	&	0.85	\\
10	&	16.76	&	15.28	&	1.48	&	39.33	&	40.21	&	-0.88	&	49.55	&	48.27	&	1.28	&	53.41	&	51.47	&	1.95	&	 24.191	&	23.43	&	0.76	\\
11	&	17.50	&	16.21	&	1.29	&	40.13	&	40.49	&	-0.35	&	51.18	&	49.35	&	1.83	&	45.85	&	44.67	&	1.17	&	 31.0402	&	30.27	&	0.77	\\
12	&	18.51	&	16.61	&	1.90	&	38.96	&	40.43	&	-1.48	&	52.09	&	49.29	&	2.80	&	38.29	&	36.73	&	1.55	&	 37.2314	&	37.14	&	0.10	\\
13	&	17.16	&	16.52	&	0.65	&	38.40	&	39.99	&	-1.59	&	49.76	&	48.15	&	1.62	&	29.10	&	27.93	&	1.17	&	 43.9114	&	43.91	&	0.00	\\
14	&	16.33	&	16.00	&	0.33	&	38.52	&	39.09	&	-0.57	&	47.91	&	46.23	&	1.68	&	18.60	&	17.88	&	0.71	&	 49.7671	&	49.87	&	-0.10	\\
15	&	13.03	&	11.05	&	1.98	&	35.81	&	37.73	&	-1.93	&	46.02	&	43.64	&	2.38	&	15.45	&	15.00	&	0.46	&	 54.8177	&	54.54	&	0.28	\\
16	&	14.54	&	13.37	&	1.17	&	35.99	&	36.42	&	-0.43	&	42.07	&	40.23	&	1.84	&	26.76	&	24.97	&	1.78	&	 57.9509	&	57.83	&	0.13	\\
17	&	17.48	&	15.28	&	2.21	&	33.89	&	35.43	&	-1.54	&	38.37	&	36.00	&	2.37	&	35.16	&	34.31	&	0.85	&	 59.7718	&	59.77	&	0.00	\\
18	&	18.33	&	16.75	&	1.58	&	33.51	&	34.70	&	-1.19	&	33.39	&	31.07	&	2.32	&	44.32	&	43.18	&	1.14	&	 60.6768	&	60.53	&	0.15	\\
19	&	19.36	&	17.63	&	1.73	&	28.43	&	28.92	&	-0.49	&	43.93	&	41.93	&	2.00	&	51.92	&	51.13	&	0.79	&	 61.4144	&	60.45	&	0.96	\\
20	&	20.34	&	17.95	&	2.38	&	28.12	&	28.75	&	-0.63	&	48.47	&	45.31	&	3.16	&	58.40	&	57.33	&	1.07	&	 58.9549	&	59.75	&	-0.79	\\
21	&	18.98	&	17.82	&	1.15	&	30.45	&	31.15	&	-0.70	&	48.67	&	47.71	&	0.96	&	62.89	&	61.56	&	1.33	&	 59.6355	&	58.62	&	1.01	\\
22	&	18.32	&	17.39	&	0.93	&	33.73	&	33.19	&	0.54	&	50.91	&	49.18	&	1.73	&	65.36	&	64.22	&	1.14	&	 58.7166	&	57.38	&	1.33	\\
23	&	17.97	&	16.74	&	1.22	&	33.61	&	34.58	&	-0.98	&	51.78	&	49.88	&	1.90	&	67.10	&	65.87	&	1.23	&	 58.1904	&	56.33	&	1.86	\\
24	&	12.08	&	9.82	&	2.26	&	32.62	&	35.45	&	-2.83	&	52.50	&	50.08	&	2.42	&	68.38	&	66.77	&	1.61	&	 57.539	&	55.69	&	1.84	\\
25	&	11.00	&	8.82	&	2.17	&	34.55	&	36.15	&	-1.60	&	52.01	&	49.65	&	2.36	&	69.19	&	67.04	&	2.15	&	 56.4307	&	55.49	&	0.94	\\
\hline
\end{tabular}
\caption{Estimated and true values of $\bb$ for selected 25 voxels from each region in the Simulation study, baseline session.}
\label{t:sim_results_beta1}
\end{sidewaystable}


\begin{sidewaystable}
\centering
\begin{tabular}{c | r r r| r r r| r r r | r r r | r r r |}
\hline \hline
Voxel & \multicolumn{15}{c}{\bf Region }  \\
& \multicolumn{3}{c}{{\bf 1}} & \multicolumn{3}{c}{{\bf 2}} & \multicolumn{3}{c}{{\bf 3}} & \multicolumn{3}{c}{{\bf 4}} & \multicolumn{3}{c}{{\bf 5}} \\
	&	est.	&	true	&	bias	&	est.	&	true	&	bias	&	est.	&	true	&	bias	&	est.	&	true	&	bias	&	 est.	&	true	&	bias	\\
1	&	27.03	&	27.95	&	-0.92	&	25.75	&	27.44	&	-1.69	&	28.45	&	28.89	&	-0.44	&	53.78	&	52.62	&	1.16	&	 64.42	&	62.24	&	2.18	\\
2	&	24.21	&	24.78	&	-0.57	&	19.41	&	20.68	&	-1.27	&	25.25	&	25.52	&	-0.26	&	56.05	&	54.74	&	1.31	&	 62.84	&	60.55	&	2.29	\\
3	&	26.88	&	27.54	&	-0.66	&	19.00	&	20.78	&	-1.78	&	21.33	&	21.58	&	-0.25	&	56.73	&	56.09	&	0.63	&	 60.00	&	57.42	&	2.58	\\
4	&	27.70	&	29.37	&	-1.67	&	25.02	&	26.09	&	-1.07	&	19.45	&	20.48	&	-1.03	&	58.06	&	56.92	&	1.14	&	 55.33	&	52.88	&	2.45	\\
5	&	28.82	&	30.51	&	-1.69	&	29.41	&	31.40	&	-1.99	&	25.04	&	25.45	&	-0.41	&	59.35	&	57.22	&	2.13	&	 49.50	&	47.13	&	2.38	\\
6	&	31.20	&	30.97	&	0.23	&	34.53	&	36.33	&	-1.80	&	29.70	&	29.76	&	-0.05	&	57.62	&	56.93	&	0.69	&	 42.53	&	40.36	&	2.17	\\
7	&	30.84	&	31.01	&	-0.17	&	39.39	&	40.68	&	-1.28	&	32.90	&	33.76	&	-0.86	&	57.02	&	56.00	&	1.02	&	 35.73	&	33.08	&	2.64	\\
8	&	26.63	&	26.94	&	-0.31	&	43.23	&	44.08	&	-0.85	&	36.46	&	37.22	&	-0.75	&	55.11	&	54.23	&	0.88	&	 27.71	&	26.12	&	1.59	\\
9	&	28.94	&	30.01	&	-1.07	&	44.24	&	46.48	&	-2.23	&	38.98	&	39.91	&	-0.93	&	52.32	&	51.35	&	0.97	&	 22.87	&	20.04	&	2.83	\\
10	&	31.13	&	31.91	&	-0.79	&	45.54	&	47.89	&	-2.35	&	40.45	&	41.49	&	-1.04	&	48.55	&	47.10	&	1.46	&	 24.68	&	22.83	&	1.85	\\
11	&	32.47	&	33.12	&	-0.66	&	46.52	&	48.36	&	-1.84	&	41.00	&	42.09	&	-1.09	&	40.99	&	41.35	&	-0.36	&	 30.90	&	28.54	&	2.36	\\
12	&	33.36	&	33.84	&	-0.48	&	46.82	&	48.13	&	-1.31	&	41.22	&	41.70	&	-0.48	&	35.52	&	34.33	&	1.19	&	 37.87	&	35.14	&	2.72	\\
13	&	33.04	&	34.01	&	-0.97	&	46.10	&	47.56	&	-1.46	&	39.51	&	40.36	&	-0.85	&	27.85	&	26.79	&	1.05	&	 44.23	&	41.94	&	2.29	\\
14	&	32.24	&	33.64	&	-1.40	&	45.65	&	46.94	&	-1.30	&	37.69	&	38.13	&	-0.45	&	20.79	&	19.45	&	1.34	&	 50.70	&	48.00	&	2.70	\\
15	&	23.91	&	24.66	&	-0.75	&	43.79	&	46.22	&	-2.43	&	34.78	&	35.12	&	-0.34	&	16.91	&	15.42	&	1.48	&	 54.86	&	52.81	&	2.04	\\
16	&	27.62	&	28.20	&	-0.58	&	44.54	&	45.51	&	-0.96	&	31.34	&	31.54	&	-0.20	&	24.42	&	23.04	&	1.38	&	 58.25	&	56.41	&	1.84	\\
17	&	31.38	&	31.56	&	-0.18	&	42.73	&	45.14	&	-2.41	&	27.01	&	27.45	&	-0.44	&	31.57	&	30.88	&	0.69	&	 60.90	&	58.96	&	1.93	\\
18	&	33.26	&	34.03	&	-0.76	&	43.25	&	45.18	&	-1.93	&	22.57	&	23.41	&	-0.84	&	40.01	&	38.65	&	1.36	&	 62.62	&	60.68	&	1.94	\\
19	&	34.78	&	35.66	&	-0.88	&	40.01	&	42.14	&	-2.13	&	32.96	&	33.31	&	-0.34	&	47.47	&	45.91	&	1.56	&	 64.05	&	61.67	&	2.38	\\
20	&	35.72	&	36.59	&	-0.87	&	39.34	&	40.67	&	-1.33	&	36.64	&	36.62	&	0.01	&	52.68	&	51.95	&	0.73	&	 64.14	&	62.06	&	2.08	\\
21	&	35.51	&	36.88	&	-1.37	&	41.05	&	42.81	&	-1.75	&	38.69	&	39.48	&	-0.80	&	57.65	&	56.27	&	1.38	&	 64.32	&	61.84	&	2.49	\\
22	&	35.06	&	36.65	&	-1.59	&	43.62	&	44.40	&	-0.79	&	40.24	&	41.56	&	-1.31	&	60.34	&	59.15	&	1.19	&	 63.64	&	61.29	&	2.34	\\
23	&	34.44	&	35.82	&	-1.38	&	43.70	&	45.22	&	-1.52	&	42.07	&	42.68	&	-0.61	&	62.10	&	60.80	&	1.30	&	 63.28	&	60.74	&	2.54	\\
24	&	33.23	&	34.33	&	-1.09	&	43.23	&	45.72	&	-2.49	&	42.22	&	42.76	&	-0.53	&	62.76	&	61.66	&	1.10	&	 61.83	&	60.39	&	1.45	\\
25	&	29.40	&	30.23	&	-0.83	&	44.92	&	46.33	&	-1.41	&	41.82	&	41.88	&	-0.07	&	63.18	&	62.09	&	1.09	&	 62.93	&	60.26	&	2.67	\\
\end{tabular}
\caption{Estimated and true values of $\bb$ for selected 25 voxels from each region in the Simulation study, follow-up session.}
\label{t:sim_results_beta2}
\end{sidewaystable}

% don't show
%\begin{table}
%\centering
%\begin{tabular}{c | r r r|r r r| }
%\hline \hline
%Voxel & \multicolumn{3}{c}{\bf Baseline} & \multicolumn{3}{c}{\bf Follow-up}  \\
%	&	est.	&	true	&	bias	&	est.	&	true	&	bias	\\
%1	&	-0.04	&	-0.43	&	0.38	&	-0.02	&	-0.45	&	0.43	\\
%2	&	-0.05	&	-0.46	&	0.41	&	-0.04	&	0.01	&	-0.05	\\
%3	&	-0.04	&	-1.34	&	1.29	&	-0.02	&	-0.01	&	-0.01	\\
%4	&	-0.06	&	-0.02	&	-0.04	&	-0.03	&	0.33	&	-0.36	\\
%5	&	-0.04	&	0.20	&	-0.24	&	-0.03	&	-0.78	&	0.75	\\
%6	&	-0.05	&	-0.39	&	0.34	&	-0.02	&	-0.52	&	0.49	\\
%7	&	-0.04	&	-2.00	&	1.95	&	-0.02	&	-0.54	&	0.52	\\
%8	&	-0.03	&	-0.81	&	0.78	&	-0.03	&	-0.66	&	0.62	\\
%9	&	-0.05	&	-0.01	&	-0.03	&	-0.02	&	0.01	&	-0.03	\\
%10	&	-0.04	&	-0.62	&	0.58	&	-0.04	&	-0.12	&	0.09	\\
%11	&	-0.03	&	-0.15	&	0.11	&	-0.03	&	-0.60	&	0.57	\\
%12	&	-0.03	&	0.71	&	-0.74	&	-0.03	&	0.24	&	-0.27	\\
%13	&	-0.02	&	-0.27	&	0.25	&	-0.02	&	0.21	&	-0.23	\\
%14	&	-0.03	&	0.51	&	-0.54	&	-0.03	&	0.49	&	-0.52	\\
%15	&	-0.02	&	-0.37	&	0.35	&	-0.02	&	0.02	&	-0.04	\\
%16	&	-0.03	&	0.19	&	-0.22	&	-0.02	&	-0.08	&	0.07	\\
%17	&	-0.01	&	-0.22	&	0.21	&	-0.03	&	0.08	&	-0.10	\\
%18	&	-0.02	&	-0.21	&	0.19	&	-0.02	&	0.44	&	-0.45	\\
%19	&	-0.02	&	-0.08	&	0.06	&	-0.02	&	0.88	&	-0.90	\\
%20	&	-0.02	&	-0.23	&	0.20	&	-0.02	&	0.20	&	-0.22	\\
%\hline
%\end{tabular}
%\caption{Estimated and true values of $\phi$ for randomly selected 20 voxels in one of the regions from the Simulation study, baseline and
%follow-up sessions.}
%\label{t:sim_results_phi}
%\end{table}


\begin{sidewaystable}
\centering
\begin{tabular}{c | r r r|r r r|r r r|r r r|r r r|}
\hline \hline
Voxel & \multicolumn{15}{c}{\bf Region }  \\
& \multicolumn{3}{c}{{\bf 1}} & \multicolumn{3}{c}{{\bf 2}} & \multicolumn{3}{c}{{\bf 3}} & \multicolumn{3}{c}{{\bf 4}} & \multicolumn{3}{c}{{\bf 5}} \\
	&	est.	&	true	&	bias	&	est.	&	true	&	bias	&	est.	&	true	&	bias	&	est.	&	true	&	bias	&	 est.	&	true	&	bias	\\
1	&	5.63	&	6.65	&	-1.02	&	9.94	&	8.46	&	1.48	&	-1.18	&	1.30	&	-2.48	&	-1.68	&	0.00	&	-1.68	&	 -1.9278	&	-1.20	&	-0.72	\\
2	&	-21.20	&	-23.68	&	2.48	&	1.39	&	0.14	&	1.25	&	5.91	&	-4.67	&	10.58	&	-0.47	&	1.75	&	-2.22	&	 -2.7562	&	-2.38	&	-0.37	\\
3	&	-29.16	&	-19.74	&	-9.43	&	-8.01	&	-10.56	&	2.55	&	9.95	&	-7.40	&	17.34	&	2.54	&	3.21	&	-0.67	&	 -0.4261	&	1.47	&	-1.90	\\
4	&	14.32	&	23.95	&	-9.63	&	5.23	&	2.75	&	2.48	&	-8.01	&	10.48	&	-18.49	&	-2.46	&	-2.00	&	-0.46	&	 1.332	&	3.17	&	-1.84	\\
5	&	-0.57	&	-19.71	&	19.14	&	-5.89	&	-4.52	&	-1.37	&	-0.28	&	-1.09	&	0.81	&	-1.07	&	2.22	&	-3.29	&	 -1.3073	&	-3.92	&	2.61	\\
6	&	-8.26	&	-8.63	&	0.38	&	10.00	&	8.93	&	1.08	&	2.95	&	-1.88	&	4.83	&	1.52	&	2.96	&	-1.44	&	 0.2527	&	0.57	&	-0.32	\\
7	&	19.37	&	29.75	&	-10.38	&	4.19	&	1.57	&	2.62	&	-1.42	&	4.04	&	-5.46	&	-1.54	&	-0.94	&	-0.60	&	 -2.9279	&	-0.97	&	-1.95	\\
8	&	1.20	&	-5.15	&	6.34	&	4.25	&	4.72	&	-0.47	&	1.52	&	-1.98	&	3.50	&	0.59	&	1.61	&	-1.02	&	 0.6604	&	-0.60	&	1.26	\\
9	&	2.38	&	3.00	&	-0.63	&	3.98	&	3.47	&	0.51	&	-1.96	&	2.47	&	-4.44	&	-0.74	&	-0.47	&	-0.26	&	 -0.9299	&	-1.14	&	0.21	\\
10	&	-17.23	&	-16.06	&	-1.16	&	-16.26	&	-18.02	&	1.76	&	-1.32	&	3.08	&	-4.40	&	0.81	&	2.55	&	-1.74	&	 1.7195	&	2.54	&	-0.82	\\
11	&	14.58	&	17.68	&	-3.09	&	-4.06	&	-5.02	&	0.96	&	-0.35	&	1.31	&	-1.66	&	0.99	&	1.29	&	-0.30	&	 0.8902	&	1.05	&	-0.16	\\
12	&	5.86	&	0.07	&	5.79	&	2.11	&	2.16	&	-0.05	&	-1.14	&	1.09	&	-2.22	&	1.15	&	2.67	&	-1.52	&	 2.4952	&	1.66	&	0.84	\\
13	&	17.38	&	19.58	&	-2.19	&	-4.90	&	-6.78	&	1.88	&	-7.64	&	9.52	&	-17.16	&	-0.75	&	1.10	&	-1.85	&	 3.0931	&	4.14	&	-1.05	\\
14	&	-17.23	&	-7.53	&	-9.70	&	-8.57	&	-10.88	&	2.31	&	3.06	&	-0.75	&	3.80	&	0.28	&	0.35	&	-0.07	&	 -0.2636	&	1.35	&	-1.62	\\
15	&	12.86	&	17.16	&	-4.29	&	6.13	&	3.49	&	2.63	&	0.44	&	2.20	&	-1.76	&	1.32	&	3.83	&	-2.51	&	 0.974	&	2.85	&	-1.88	\\
\hline
\end{tabular}
\caption{Estimated and true values of $\alpha_i$ (random effects), baseline session.}
\label{t:sim_results_alpha}
\end{sidewaystable}

