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\title{AFNI: Population-Level Modeling}
\author{Gang Chen}
\institute{SSCC/NIMH, National Institutes of Health, USA}
\date{\today}
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\begin{document}
\maketitle

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\begin{frame}{Three subsections}
%\transglitter<2-4>
\begin{itemize}
%\item Why population-level modeling?
%\begin{itemize}https://www.overleaf.com/project/5f2f024f304e5e00012b2958
%    \item Two-stage methodology
%    \item Generalizability
%\end{itemize}
%\pause
\item \textcolor{blue}{Part 1 - Overview, perspectives and concepts}
\bigskip
\item Part 2 - Basic modeling approaches
\bigskip
\item Part 3 - Advanced modeling approaches
\end{itemize}
\end{frame}


\begin{frame}{Themes}
%\transglitter<2-4>
\begin{itemize}
%\item Why population-level modeling?
%\begin{itemize}https://www.overleaf.com/project/5f2f024f304e5e00012b2958
%    \item Two-stage methodology
%    \item Generalizability
%\end{itemize}
%\pause
\item Modeling considerations
\begin{itemize}
    \item Spatial unit: voxel, surface node or ROI level
    \item Input data: effect estimates with/without uncertainty
    \item Data reduction: trial- vs condition-level effects
    \item BOLD response: presumed vs estimated HDR
    \item Handling quantitative variables: linear vs nonlinear
    \item Interaction: homogeneity vs heterogeneity
\end{itemize}
\medskip
\pause
\item Model types
\begin{itemize}
    \item Conventional: Student's \textit{t}, GLM, AN(C)OVA, LME
%    \item Complex: LME
    \item Adventurous: Bayes, multilevel smoothing splines
\end{itemize}
\medskip
\pause
\item Focus
\begin{itemize}
    \item \textcolor{cyan}{Model} vs effects of interest or no interest 
    \item \textcolor{cyan}{Estimation (full results)} vs inference (dichotomization)
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}{Program list}
\begin{table}
\begin{tabular}{|c||l|l|}
\hline
Spatial Unit & Program & Model \\\hline\hline
& \textsf{3dttest++} & \textit{t}-tests, GLM \\
\cline{2-3}
&\textsf{3dMEMA} & %similar to \textsf{3dttest++} with 
effect + \textit{t}-stat as input \\
\cline{2-3}
voxel, node, ROI&\textsf{3dMVM} & GLM, AN(C)OVA \\
\cline{2-3}
&\textsf{3dLME} & simple LME \\
\cline{2-3}
&\textsf{3dLMEr} & LME, test-rest reliability \\
\cline{2-3}
massively univariate&\textsf{3dMSS} & multilevel smoothing splines \\
\cline{2-3}
&\textsf{3dICC} & intra-class correlation \\
\cline{2-3}
&\textsf{3dISC} & inter-subject correlation \\
\cline{2-3}
\hline
ROI & \textsf{RBA} & region-based analysis \\
\cline{2-3}
Bayesian multilevel&\textsf{MBA} & matrix-based analysis \\
\hline
\end{tabular}
\end{table}
\end{frame}

\begin{frame}{Data hierarchy}
\begin{center}
\tikzset{edge from parent/.style=
{draw, edge from parent path={(\tikzparentnode.south)
-- +(0,-8pt)
-| (\tikzchildnode)}},
blank/.style={draw=none}}
\noindent\begin{tikzpicture}
\fontsize{12}{4}
\matrix (mat) [matrix of nodes,ampersand replacement=\&]
{
\node{\Tree%\.style={draw,rectangle,fill=white}
    [.\textbf{Population} \edge[blank];
    [.\textbf{Subject}  \edge[blank]; 
    [.\textbf{Condition} \edge[blank]; 
    [.\textit{Trial} ]]]]};
\&
\node{\Tree
[.Group
[.$S_1$
        [.$C_1$  {$T_1$}  $\cdots$ $T_m$ ]
        $\cdots$
        [.$C_k$  {$T_1$}  $\cdots$ $T_m$ ] 
] 
$\cdots$
[.$S_n$
        [.$C_1$  {$T_1$}  $\cdots$ $T_m$ ]
        $\cdots$
        [.$C_k$  {$T_1$}  $\cdots$ $T_m$ ] 
] ]
$\cdots$
};\\
}; 
\end{tikzpicture}
\end{center}
\end{frame}

\begin{frame}{Why population-level modeling?}
\begin{itemize}
\item Ideal but impractical: one model that incorporates  everything
\medskip\pause
\item Two-stage methodology
\begin{itemize}
    \item Splitting
    \begin{itemize}
        \item Subject level: time series regression with GLS
        \item Population level
    \end{itemize}
    %\item Alternative terms: 2-, 3- or 4-level analysis
    \item Good but challenging: subject-level effect estimates with reliability (e.g. std dev)
    \item Common: subject-level effect estimates only; ignoring reliability

\end{itemize}
\medskip
\pause
\item Generalizability: part of scientific endeavor
\begin{itemize}
    \item Prior assumption: cross-subject variability $\sim \mathcal{N}(0, \sigma^2$)
    \item Equally applicable to trials? Aggregation vs cross-trial variability%: complete pooling %$\sim \mathcal{N}(0, \theta^2$)
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}{Perspectives of population-level modeling}
\begin{itemize}
\item Data structure
\begin{itemize}
    \item Categorical variables: factors
    \item Quantitative variables
    \item Within- or between-subject? Crossed or nested?
\medskip    
\end{itemize}
\pause
\item Effects of interest vs no interest
\begin{itemize}
    \item Interest: contrasts (A vs B), simple effects (A, B)
    \item No interest
    \begin{itemize}
        \item No-love treatment (nuisance variables): "covariates"
        \item Additive effects w/o interactions
        \item No mention in publications
    \end{itemize} 
\end{itemize}
\medskip
\item Model structure
\begin{itemize}
    \item Student's $t$, GLM, AN(C)OVA, LME, MSS, BML
\end{itemize}
\medskip
\pause
\item Multiple testing adjustment
\begin{itemize}
    \item Voxel-wise vs ROI-based
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}<presentation:0>{Issues of population-level modeling}
\begin{itemize}
\item Simplification
\begin{itemize}
    \item One HDR curve for all scenarios: trials, conditions, regions, subjects
\medskip    
\end{itemize}
\pause
%\item Fragmentation
%\begin{itemize}
%    \item Not one, but two or more models across runs, sessions, subjects
%\medskip    
%\end{itemize}
\item Linearity
\begin{itemize}
    \item Covariate: quantitative predictors
\end{itemize}
\medskip
\pause
\item Summarization
\begin{itemize}
    \item Condition-level modeling: cross-trial variability largely ignored
\end{itemize}
\medskip
\item Uncertainty
\begin{itemize}
    \item Subject-level effect estimates: uncertainty ignored
\end{itemize}
\medskip
\pause
\item Overfitting: massively univariate
\begin{itemize}
    \item Unrelatedness: assumption of no commonality across the brain
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}{Some concepts}
\begin{itemize}
\item Factors: within- vs between-subjects
\begin{itemize}
    \item Between-subjects (patient vs control): independence
    \item Within-subject (positive vs negative): relatedness (e.g. variance-covariance)
\medskip    
\end{itemize}
\pause
\item Factors: fixed- vs random-effect
\begin{itemize}
    \item Fixed: constant; effects of interest (e.g., positive vs negative)
    \item Random: sample size; exchangeable (e.g., subjects, trials); \textcolor{red}{generalizeability}
    \item Clear dichotomy: conventional statistics 
\end{itemize}
\medskip\pause
\item Model structure
\begin{itemize}
    \item Student's $t$, GLM, AN(C)OVA, LME, MSS, BML
\end{itemize}
\medskip
\pause
\item Multiple testing adjustment
\begin{itemize}
    \item Overfitting: assuming no commonality
    \item Voxel-wise vs ROI-based
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}{Three subsections}
\begin{itemize}
\item Part 1 - Overview, perspectives and concepts
\bigskip
\item \textcolor{blue}{Part 2 - Basic modeling approaches}
\bigskip
\item Part 3 - Advanced modeling approaches
\end{itemize}
\end{frame}

\begin{frame}{Student's \textit{t}-test}
\begin{itemize}
\item One-sample: \textcolor{purple}{\textsf{3dttest++}}
\begin{itemize}
    \item Data at each spatial unit: $y_i, ~i=1,2,...,n$
    \item Special GLM with 2 parameters: $y_i \sim \mathcal{N}(m, \sigma^2)$
    \item \textcolor{ForestGreen}{Estimation}
    \begin{itemize}
        \item $\widehat{m}=\frac{1}{n}\sum_{i=1}^n y_i, ~\widehat{\sigma}=\frac{1}{n-1}\sum_{i=1}^n (y_i-\widehat{m})^2$
    \item Uncertainty interval: $(\widehat{m}- 2\widehat{\sigma}, ~\widehat{m} + 2\widehat{\sigma})$
    \end{itemize}
    \item \textcolor{Magenta}{Inference} - imprimatur: $t(n-1)$-statistic; $p$-value
    %\item program: \textcolor{purple}{\textsf{3dttest++}}
\end{itemize}
\medskip
\pause
\item Paired: \textcolor{purple}{\textsf{3dttest++ -paired}}
\begin{itemize}
    \item 1 group with 2 conditions - data at each spatial unit: $(y_{i1},y_{i2}), ~i=1,2,...,n$
    \item Reducing to one-sample: $y_{i1}-y_{i2} \sim \mathcal{N}(m, \sigma^2)$
    %\item program: \textcolor{purple}{\textsf{3dttest++ -paired}}
\end{itemize}
\medskip
\pause
\item Two-sample: extension of one-sample; special univariate GLM
\item Handling missing voxel values: \textcolor{purple}{-zskip}
\end{itemize}
\end{frame}

%\begin{frame}{Student's %\textit{t}-test}
%\begin{itemize}
%\item Two-sample
%\begin{itemize}
%    \item 2 groups with data at each spatial unit: $y_{i1}, ~i=1,2,...,n_1; ~y_{i2}, ~i=1,2,...,n_2$
%    \item Special GLM with 3 parameters: $y_{ij} \sim \mathcal{N}(a+bx_{ij}, \sigma^2)$
%    \item Imprimatur: $t(n_1+n_2-2)$-statistic; $p$-value
%    \item program: \textcolor{purple}{\textsf{3dttest++}}
%\end{itemize}
%\end{itemize}
%\end{frame}

\begin{frame}{Univariate GLM}
\begin{itemize}
    \item $\geq 1$ groups; $\geq 0$ quantitative variables
    \item \textcolor{red}{AN(C)OVA without within-subject variables}
    \item Data at each spatial unit: $(y_i, x_{i1}, ...), ~i=1,2,...,n$
    \item Formulation: $y_i \sim \mathcal{N}(a+b_1x_{i1}+..., \sigma^2)$
    \item Effects of interest: $a, b_1,...$
    \item When an explanatory variable $x$ is quantitative
    \begin{itemize}
        \item Centering: not needed for $x$ effect; crucial for some effects
        \item Linearity assumption: too strong?
    \end{itemize}
    \item Special GLMs
    \begin{itemize}
        \item Two-sample $t$-test
        \item AN(C)OVA w/o within-subject variables
    \end{itemize}
%    Two-sample $t$-test: a special GLM
%    \item AN(C)OVA w/o within-subject variables: 
    \item programs: \textcolor{purple}{\textsf{3dttest++}}, \textcolor{purple}{\textsf{3dMVM}}
\end{itemize}
\end{frame}

\begin{frame}{Multivariate GLM}
\begin{itemize}
    \item AN(C)OVA with \textcolor{red}{$\geq 1$ within-subject factors}
    \begin{itemize}
        \item Extension of paired $t$-test
        \item $\geq 1$ groups, $\geq 0$ quantitative variables (between-, not within-, subject)
    \end{itemize}
    \item No quantitative within-subject variables
    \begin{itemize}
        \item Yes, go with LME
    \end{itemize}
    \item Data at each spatial unit: $\boldsymbol{y}$
    \item Formulation: $\boldsymbol{Y} \sim \mathcal{N}(\boldsymbol{X}\boldsymbol{\beta}, \boldsymbol{\Sigma})$
    \item \textcolor{red}{Problematic approach via univariate GLM}: popular
    \item When an explanatory variable $x$ is quantitative
    \begin{itemize}
        \item Centering: not needed for $x$ effect; crucial for some effects
        \item Linearity assumption: too strong?
    \end{itemize}
\end{itemize}
\end{frame}

\begin{frame}{Multivariate GLM (cont.)}
\begin{itemize}
    \item Special cases: Student's $t$, univariate GLM
    \item Omnibus inferences through $F$-statistic
    \begin{itemize}
        \item Main effect - overall assessment about the differences among levels of a factor: emotion valences (positive, negative, neutral)
        \item Interaction - overall assessment about the relationship between $\geq$2 explanatory variables: group (patients, controls) and emotion (positive, negative, neutral)
    \end{itemize}
    \item Effect partitioning
    \begin{itemize}
        \item contrasts: positive vs negative
        \item simple effects: positive
    \end{itemize}
    \item Programs: \textcolor{purple}{\textsf{3dMVM}}, \textcolor{purple}{\textsf{3dLME}}, \textcolor{purple}{\textsf{3dLMEr}}
\end{itemize}
\end{frame}

\begin{frame}{LME}
\begin{itemize}
    \item \textcolor{red}{$\geq 1$ within-subject variables}; multivariate GLM: a special case
%    \begin{itemize}
%        \item Extension of paired $t$-test and within-subject ANOVA
%    \end{itemize}
    \item Differentiation of fixed and random effects
    \begin{itemize}
        \item Fixed: population effects (groups, tasks, slopes)
        \item Random: lower-level effects (cross-subject, cross-trial, cross-family)
    \end{itemize}
    \item Data at each spatial unit: $\boldsymbol{y}$
    \item Hierarchical or multilevel structure
        \item Complex random effects
    \begin{itemize}
        \item \textcolor{red}{$\geq 2$ levels: cross- and within-subject; cross- and within-family}
        \item Crossed random-effects structure: subject + trial
    \end{itemize}
    \item Formulation: $\boldsymbol{Y} \sim \mathcal{N}(\boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{Z}\boldsymbol{b}, \boldsymbol{\Sigma})$
    \begin{itemize}
        \item Fixed effects $\boldsymbol{\beta}$
        \item Random effects $\boldsymbol{b} \sim \mathcal{N}(\boldsymbol{0}, \boldsymbol{R})$: Varying intercept, varying slope
    \end{itemize}
%    \item \textcolor{red}{Problematic model as univariate GLM}: popular
\end{itemize}
\end{frame}

\begin{frame}{LME (cont.)}
\begin{itemize}
    \item When an explanatory variable $x$ is quantitative
    \begin{itemize}
        \item Centering: not needed for $x$ effect; crucial for some effects
        \item Linearity: too strong?
    \end{itemize}
    \item Specialities
    \begin{itemize}
        \item Relatedness among varying effects: within-subject quantitative variables
        \item Missing data: missing at random
        \item Complex random effects: crossed structure; ICC; ISC
    \end{itemize}
    \item When an explanatory variable $x$ is quantitative
    \begin{itemize}
        \item Centering: not needed for $x$ effect; crucial for some effects
        \item Linearity: too strong?
    \end{itemize}
    \item Special cases: paired $t$-test and within-subject AN(C)OVA
    \item programs: \textcolor{purple}{\textsf{3dLME}}, \textcolor{purple}{\textsf{3dLMEr}}
    \item Gaussianity, point estimate, measurement error and numerical issues.
\end{itemize}
\end{frame}

\begin{frame}{Accounting for effect uncertainty}
\begin{itemize}
    \item Uncertainty of subject-level effect estimates
    \begin{itemize}
        \item Largely ignored in the field
        \item Impact: mostly (not always) negligible
    \end{itemize}
    \medskip
    \item Incorporation of uncertainty in response variable
    \begin{itemize}
        \item Weighting: differentiation based on reliability
        \item Similar to meta analysis
        \item Program: \textcolor{purple}{\textsf{3dMEMA}}
        \item Input: effect estimate ($\beta$) and $t$-statistic from each subject
        \item Applicability: similar to \textcolor{purple}{\textsf{3dttest++}}
        \item Missing data at voxel level: \textcolor{purple}{-missing\_data 0}
    \end{itemize}
\end{itemize}
\end{frame}

\begin{frame}{Handling quantitative predictors}
\begin{itemize}
    \item Quantitative predictors
    \begin{itemize}
        \item Examples: age, RT, gray-matter volume, ...
        \item Types: between-subject, within-subject
        \item Longitudinal vs cross-sectional
    \end{itemize}
    \medskip
    \item Linearity
    \begin{itemize}
        \item Popular, easy implementation
        \item Between-subject predictor: \textcolor{purple}{\textsf{3dttest++}} \textcolor{purple}{\textsf{3dMEMA}}, \textcolor{purple}{\textsf{3dMVM}}, \textcolor{purple}{\textsf{3dLME}},
        \textcolor{purple}{\textsf{3dLMEr}}
        \item Within-subject predictor: \textcolor{purple}{\textsf{3dLME}},
        \textcolor{purple}{\textsf{3dLMEr}}
        \medskip
    \end{itemize}
    \item Nonlinearity
    \begin{itemize}
        \item Polynomials: difficulty with order selection and model validation
        \item Smoothing splines: adaptive and flexible 
        \item Program: \textcolor{purple}{\textsf{3dMSS}}
    \end{itemize}
\end{itemize}
\end{frame}

\begin{frame}{Estimating hemodynamic response}
\begin{itemize}
    \item Presumed HDR
    \begin{itemize}
        \item Convenient, popular
        \item Large variations across regions, tasks, subjects, groups        
        \item Inflexibility, lackluster fitting, compromised detection
    \end{itemize}
    \medskip
    \item Estimating HDR
    \begin{itemize}
        \item Subject level: tent, cubic splines
        \item Population level: smooth splines 
        \item Programs: \textcolor{purple}{\textsf{3dMVM}},  \textcolor{purple}{\textsf{3dMSS}}
        \medskip
    \end{itemize}
\end{itemize}
\end{frame}

\begin{frame}{Three subsections}
\begin{itemize}
\item Part 1 - Overview, perspectives and concepts
\bigskip
\item Part 2 - Basic modeling approaches
\bigskip
\item \textcolor{blue}{Part 3 - Advanced modeling approaches}
\end{itemize}
\end{frame}


\begin{frame}{Accounting for cross-trial variability}
\begin{itemize}
    \item Subjects: samples for population
    \begin{itemize}
        \item Representatives from a hypothetical pool
        \item Each subject's effects expressed in the model
        \item \textcolor{cyan}{Generalizability} - reason for various models: GLM, AN(C)OVA, LME 
    \end{itemize}
    \medskip
    \item How about trials?
    \begin{itemize}
        \item Representatives from a hypothetical pool of experimental \textcolor{cyan}{condition}
        \item Subject level: one regressor per condition
        \item Cross-trial variability: fully ignored!
        \item Consequences: loss of generalizability legitimacy; distortion of effect estimates and statistical evidence
        \medskip
    \end{itemize}
    \item Better approach: modeling trials %(Chen et al. (2020))
    \begin{itemize}
        \item Subject level: estimate trial effects
        \item Population level: accounting for cross-trial effects (e.g., \textcolor{purple}{\textsf{3dLMEr}})
    \end{itemize}
\end{itemize}
\end{frame}

\begin{frame}{Inter-subject correlation analysis}
\begin{columns}
\column{0.55\textwidth}
\begin{itemize}
    \item Naturalistic scanning
    \begin{itemize}
        \item Task-related FMRI: too far-fetched from real life experience
        \item Movie watching, speech/music listening
    \end{itemize}
    \medskip
    \item ISC analysis
%    \begin{columns}
%    \column{0.4\textwidth}
    \begin{itemize}
        \item Data structure complexity: $n$ subjects leads to $\frac{1}{2}n(n-1)$ ISC pairs
        \item How to disentangle the hierarchical structure? \textcolor{cyan}{LME}
        \item Program:  \textcolor{purple}{\textsf{3dISC}}
    \end{itemize}
\end{itemize}
\column{0.45\textwidth}
$n=5$ subjects: 10 ISC pairs\\
\medskip
    \includegraphics[width = 0.975 \linewidth]{matrix.png}
\end{columns}
\end{frame}

\begin{frame}{Test-retest reliability}
\begin{itemize}
    \item Intra-class correlation (ICC)
    \begin{itemize}
        \item Same conditions repeated with the same subjects
        \item How repeatable or consistent of subjects' BOLD response across repetitions?
        \item ICC computation: ANOVA, LME
        \item Program: \textcolor{purple}{3dICC}
        \item Poor ICC: strong effects (e.g., Stroop, Flanker) in behavior measure and FMRI
    \end{itemize}
    \medskip
    \item Modeling problem with classical ICC
    \begin{itemize}
        \item Reliability: subject-level metric
        \item Not suited for data with multiple trials
        \item Cross-trial variability not accounted for
    \end{itemize}
    \medskip
    \item New modeling framework
    \begin{itemize}
        \item Subject level: obtain trial-level effects
        \item Population level: disentangle trial-level effects
        \item LME approach: not ideal (\textcolor{purple}{\textsf{3dLMEr}})
        \item Bayesian multilevel (BML): \textcolor{purple}{TRR}
    \end{itemize}
\end{itemize}
\end{frame}

\begin{frame}{Handling multiplicity}
\begin{itemize}
    \item Massively univariate analysis
    \begin{itemize}
        \item Treat each spatial unit as an isolated entity: no commonality with peers
        \item As many models as spatial units
        \item Staple methodology over 30 years in neuroimaging
        \item Intuitive and straightforward
    \end{itemize}
    \medskip
    \item Multiple testing adjustment: two approaches
    \begin{itemize}
        \item Leverage among neighboring spatial units
        \item \textcolor{cyan}{Cluster-based adjustment}
        \begin{itemize}
            \item \textcolor{purple}{\textsf{3dttest++ -Clustsim}}
            \item Other programs (e.g., \textsf{3dMVM}, \textsf{3dLME}, \textsf{3dLMEr}): \textcolor{purple}{\textsf{3dClustSim}}
        \end{itemize} 
        \item \textcolor{cyan}{Permutation-based adjustment}: \textcolor{purple}{\textsf{3dttest++ -ETAC}}
    \end{itemize}
\end{itemize}
\end{frame}

\begin{frame}{Handling multiplicity (cont.)}
\begin{columns}
\column{0.5\textwidth}
\begin{itemize}
    \item Problems
    \begin{itemize}
        \item Overfitting
        \item Information waste
        \item Heavy penalty
        \item Dichotomization
        \item Discrimination against anatomically small regions
        \item Vulnerability to data manipulations
    \end{itemize}
\end{itemize}
\column{0.5\textwidth}
Ignorant information across brain
\includegraphics[width = 0.55 \linewidth]{uniform.png}\\
Really no prior knowledge?
\includegraphics[width = 0.55 \linewidth]{voxel.png}
\end{columns}
\end{frame}

\begin{frame}{Region-based analysis}
\begin{columns}
\column{0.5\textwidth}
\begin{itemize}
    \item One model integrating all ROIs: Bayesian multilevel model (BML)
    \begin{itemize}
        \item Multiplicity dissolved
        \item Likely high efficiency
        \item Transparency: full results
        \item Region specificity
        \item No dichotomization
        \item No discrimination against anatomically small regions
        \item Less vulnerability to data manipulations
    \end{itemize}
    \item Program: \textcolor{purple}{RBA}
\end{itemize}
\column{0.5\textwidth}
\includegraphics[width = 0.95 \linewidth]{effect.pdf}
\end{columns}
\end{frame}

\begin{frame}{Matrix-based analysis}
%\begin{columns}
%\column{0.5\textwidth}
\begin{itemize}
    \item Complexity of data structure
    \begin{itemize}
        \item Conventional: massively univarate analysis + multiplicity
        \item Hierarchical structure: BML
        \item Multiplicity dissolved
    \end{itemize}
    \item Program: \textcolor{purple}{MBA}
\end{itemize}
%\column{0.5\textwidth}

\includegraphics[width = 0.95 \linewidth]{matrix2.png}
%\end{columns}
\end{frame}

\begin{frame}{Matrix-based analysis (cont.)}
%\begin{columns}
%\column{0.5\textwidth}
\begin{itemize}
    \item BML applied to a matrix dataset
\end{itemize}
%\column{0.5\textwidth}

  \includegraphics[width=0.45\linewidth]{BML.pdf}
 \raisebox{9mm}{\includegraphics[width=4.5mm]{colorbar2.pdf}}
 \includegraphics[width=.45\linewidth]{ROI.pdf}
%\end{columns}
\end{frame}

\end{document}