2019_srp.tex

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\begin{document}

\begin{poster}
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%-------------------------------------------------------------------------------
% TITLE SECTION
%-------------------------------------------------------------------------------
%
{\includegraphics[height=6.5em]{logo_cal.jpg}} % university/lab logo on left
{\textbf{\LARGE Semiparametric Estimation with Robust Empirical Bayes Inference
  and Supervised Clustering in High-Dimensional Biological Exposure
  Studies\vspace{0.5em}}}
{\textsc{Nima S.~Hejazi, Mark J.~van der Laan, Martyn T.~Smith \& Alan
  E.~Hubbard} \hspace{12pt} \textit{}}
{\includegraphics[height=9em]{logo_sph.jpg}} % university/lab logo on right

%-------------------------------------------------------------------------------
% OVERVIEW
%-------------------------------------------------------------------------------

\headerbox{Overview \& Motivations}{name=overview,column=0,row=0}{

\begin{itemize}
  \itemsep0.5pt
  \item A general approach for deriving stable variance estimates for
    data-adaptive semiparametric estimators is introduced.
  \item Variance moderation uniformly improves variance estimates, with a
    negligible effect asymptotically.
    \begin{itemize}
      \itemsep0pt
      \item curbing the error rate of tests relative to classical approaches
      \item and facilitating \textit{supervised clustering} from derived
        association profiles.
    \end{itemize}
  \item Illustrate how the proposal applies for any asymptotically linear
    estimator through the lens of targeted maximum likelihood.
  \item The \underline{\texttt{biotmle} R package} \cite{hejazi2017biotmle}
    implements the variance moderation procedure, leveraging state-of-the-art
    machine learning.
\end{itemize}

}

%-------------------------------------------------------------------------------
% INTRODUCTION
%-------------------------------------------------------------------------------

\headerbox{Data: Benzene Biomarkers}
{name=introduction,column=1,row=0,bottomaligned=overview}{

\begin{itemize}
  \itemsep1pt
  \item There is a pressing need for model-free, technology-agnostic statistical
    methods for analyzing multiple kinds of exposome data.
  \item We consider data generated by a study of occupational exposure, using
   the \textit{Illumina Human Ref-8 BeadChips} platform.
  \item Data on baseline confounders and exposure collected for $n = 125$
   subjects/participants, with $22,000$+ gene expression measures.
  \item Covariates ($W$): age, sex, smoking status.
  \item Exposure ($A$): degree of Benzene exposure (none, $<1$ppm, $>5$ppm).
  \item Outcome ($Y = (Y_b: b = 1, \ldots, B)$): gene expression measures
    vector.
\end{itemize}

}

%-------------------------------------------------------------------------------
% METHODS 2: Supervised Clustering
%-------------------------------------------------------------------------------
\headerbox{Methodology II: Supervised Distance Matrices}{name=results,column=2,span=2,row=0}{

\begin{itemize}
  \itemsep0.75pt
  \item Let $\phi: (W, A, Y) \mapsto D_b(P_0)(O)$ be the EIF transformation,
    where $D_b(P_0)(O_i)$ is the contribution of subject $i$ to the estimate of
    the biomarker-specific target parameter $\Psi_{b,n}$.
  \item $Z = \phi(W, A, Y)$ is then a $B \times N$ matrix, where each entry
    $(b,i)$ may be interpreted as the degree to which subject $i$ deviates from
    the target parameter $\Psi_{b,n}$ \cite{hejazi2018+supervised}, and is thus
    an \textit{association profile}.
  \item A \textit{supervised distance matrix} \cite{pollard2008supervised}
    may be constructed by applying an appropriate distance metric of choice
    (e.g., Euclidean, correlation) to the transformed values $Z$.
  \item $\widetilde{T}(Z)$, the resultant $B \times B$ empirical distance
    matrix, encodes the dissimilarity between pairs of biomarker association
    profiles.
  %\item When $\widetilde{T}(b,b')$ is small, the biomarkers $b$ and $b'$ have
    %similar contriubtions to the target parameter $\Psi$, across the $n$
    %subjects.
  \item \textit{Supervised clustering} may be performed by applying standard
    unsupervised clustering algorithms to the matrix $\widetilde{T}$, thereby
    finding groups of biomarkers that share an association profile
    w.r.t.~$\Psi$.
  \item In the case of the average treatment effect, a supervised cluster in
    $\widetilde{T}$ of biomarkers is a group whose causal differential
    expression profiles varies similarly with the treatment $A \in \{0, 1\}$.
\end{itemize}
}

%-------------------------------------------------------------------------------
% REFERENCES
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\headerbox{\small References}{name=references,column=2,above=bottom}{
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  \bibliography{2019_srp}
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%-------------------------------------------------------------------------------
% Contact Information
%-------------------------------------------------------------------------------

\headerbox{\small Contact Information}{name=ack,column=3,aligned=references,above=bottom}{
% This block is as tall as the references block
\begin{itemize}
\itemsep0.25pt
  \item \textbf{N.S.~Hejazi}: \texttt{nhejazi@berkeley.edu};
      \textbf{M.J.~van der Laan}: \texttt{laan@berkeley.edu};
    \textbf{M.T.~Smith}: \texttt{martynts@berkeley.edu};
    \textbf{A.E.~Hubbard}: \texttt{hubbard@berkeley.edu}
  \item \url{https://bioconductor.org/packages/biotmle}
  \item \url{https://arxiv.org/abs/1710.05451}
\end{itemize}
}

%-------------------------------------------------------------------------------
% CONCLUSION
%-------------------------------------------------------------------------------

\headerbox{Numerical Study \& Results}
{name=conclusion,column=2,span=2,row=0,below=results,above=references}{

\vspace{0.25em}

\begin{multicols}{2}

\begin{center}
\vspace*{-0.5cm}
\includegraphics[scale=0.48,trim={1cm 1cm 1cm 3cm},clip]{supervised_heatmap}
\captionof{figure}{\textit{Supervised heatmap} of the top 25 biomarkers
  visualizes groups with a shared exposure response.}
\end{center}

\begin{center}
\vspace*{-0.55cm}  % CHANGE THIS TO ALIGN IMAGES
\includegraphics[scale=0.173]{biotmle_fdr_eg}
\captionof{figure}{Enhanced control of the False Discovery Rate (FDR) with
  variance-moderated efficient estimator.}
\end{center}

\end{multicols}
}


%-------------------------------------------------------------------------------
% METHODS
%-------------------------------------------------------------------------------

\headerbox{Methodology I: Semiparametric Variance Moderation}{name=method,column=0,span=2,below=overview,bottomaligned=references}{
% This block's bottom aligns with the bottom of the conclusion block

\begin{itemize}
  %\compresslist
  \itemsep0.50pt
  \item Let observed data $O = (W, A, Y) \sim P_0 \in \M$, where $W$ represents
      potential baseline confounders, $A$ the exposure of interest, and
      $Y = ({Y_b}, b = 1, \dots, B)$ a vector of potential biomarkers.
  \item We consider, as an example, the \textit{average treatment effect} (ATE),
      as the causal parameter of interest, which is identified by the observed
      data parameter:
      \begin{equation}\label{ate}
        \Psi_b(P_0) = \E_W[ Q_0^b(A = 1, W) - Q_0^b(A = 0, W)],
      \end{equation}
        where $Q_0^b(A, W) \equiv \E_{P_0}(Y_b \mid A, W)$ and may be estimated
        via \textit{ensemble machine learning}
        \cite{vdl2007super,breiman1996stacked,wolpert1992stacked}.
  %\item To estimate $\Psi$ using the construction from Eqn.~\ref{ate}, a plug-in
    %estimator may be constructed
    %\begin{equation}
      %\Psi_b(P_n) = \frac{1}{n} \sum_{i = 1}^n Q_n^b(A_i = 1, W_i) -
      %Q_n^b(A_i = 0, W_i),
    %\end{equation}
      %where ${Q_n}^b$ is an initial estimate of ${Q_0}^b$ constructed via
      %\textit{ensemble machine learning} \cite{vdl2007super,breiman1996stacked,
      %wolpert1992stacked}.
  \item Like the estimator $\hat{\beta}$ in a linear model $m(A,W \mid \beta)$,
      $\Psi_b(P_n)$ is \textit{asymptotically linear} (for $\Psi_b$)
      \cite{vdl2011targeted}:
      \begin{equation}\label{asymp_lin}
        \sqrt{n} (\Psi_b(P_n) - \Psi_b(P_0)) = \frac{1}{\sqrt{n}} \sum_{i=1}^n
        D_b(O_i) + o_p(1).
      \end{equation}
  \item $\Psi_b$ has efficient influence function (EIF), relative to the
      nonparametric model $\M$:
      \begin{equation}\label{eif_ate}
        D_b(P_0)(o) = \left(\frac{I(a = 1)}{g(1 \mid w)} -
        \frac{I(a = 0)}{g(0 \mid w)}\right) \cdot \left[y_b - Q_0^b(a, w)\right]
        + \left(Q_0^b(1, w) - Q_0^b(0, w) - \Psi_b(P_0)(o)\right).
      \end{equation}
  \item A moderated test statistic \cite{smyth2004linear,hejazi2018+supervised}
      may be constructed for use with asymptotically linear estimators:
      \begin{equation}\label{mod_eif}
        \widetilde{t}_b = \frac{\sqrt{n}(\Psi_b(P_n) -
          \psi_{\text{null}})}{\widetilde{S}_{b,n}^2}
        \quad \text{where} \quad
        \widetilde{S}_{b,n}^2 = \frac{d_0S_0^2 + d_bS_b^2(D_{b,n})}{d_0 + d_b},
      \end{equation}
    $\{S_b^2, d_b\}$: var.~EIF and df for $b^{th}$ biomarker;
    $\{S_0^2, d_0\}$: var.~EIF and df for other $(B-1)$ biomarkers.
\end{itemize}

}

\end{poster}
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