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AEBilgrau committed Jul 18, 2015
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Expand Up @@ -335,7 +335,7 @@ A $(1-\alpha)100\%$ confidence interval can be computed as
\begin{equation*}
( \hat{\delta}_{(\alpha/2)} , \hat{\delta}_{(1-\alpha/2)} )
\end{equation*}
where e.g.\ $\hat{\delta}_{(\alpha/2)}$ denotes the $\alpha/2$-percentile of $\hat{\delta}_1, \ldots \hat{\delta}_B$.
where e.g.\ $\hat{\delta}_{(\alpha/2)}$ denotes the $\alpha/2$-percentile of $\hat{\delta}_1, \ldots, \hat{\delta}_B$.
The $p$-value for the null hypothesis of $\delta = 0$ is computed by
\begin{equation*}
2\min(\pi, 1 - \pi)
Expand Down Expand Up @@ -482,6 +482,29 @@ The mean of the bootstrap distribution seems consistently larger that the other

We see that the large number of dilution steps, as recommended, ensures a low impact of the AE induced standard error on the inference of the $\ddcq$.

% CIC fig
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{fig1}
\end{center}
\caption{
Overview of CIC experiment data.
A: Raw $C_q$-values for different cell lines (samples) for each gene type and sample type.
The point type and colour differentiates the different gene types.
B: Dilution data for
reference genes (\textit{ACTB}, \textit{GAPDH}) and
target genes (\textit{MGST1}, \textit{MMSET}).
}
\label{fig:cqCIC}
\end{figure}

% CIC table
\input{../output/Table1.tex}





\subsection{DLBCL study}

The $C_q$-values and dilution curves for the DLBCL study are depicted in Fig.~\ref{fig:cqTestis}, panels A--B, respectively.
Expand All @@ -501,6 +524,30 @@ The bootstrap method provides a standard deviation similar to the delta method a
Regarding the biological interest, we conclude there is evidence for a difference in \textit{miR-127} expression between testicular and nodal DLBCL whilst the data is not compatible with difference in \textit{miR-143} expression.
While the AE estimate had no influence in these cases a change in significance is easily imagined in other cases.

% DLBCL fig
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{fig2}
\end{center}
\caption{
Overview of DLBCL testis experiment data.
A: Raw $C_q$-values for different patient samples for each gene type and sample type.
The point type and colour differentiates the different gene types.
B: Dilution data for
reference genes (\textit{RNU-24}, \textit{RNU-6B}) and
target genes (\textit{miR-127}, \textit{miR-143}).
}
\label{fig:cqTestis}
\end{figure}

% DLBCL table
\input{../output/Table2.tex}






\subsection{Arabidopsis thaliana data}

The $C_q$-values and dilution data for the arabidopsis thaliana data are shown in Fig.~\ref{fig:cqYuan}.
Expand All @@ -517,6 +564,27 @@ Secondly, as dilution curves are used for each group the four group-specific AE
While this example was selected as a worst-case scenario, it should illustrate that although the standard curves are seemingly well determined, it is hard to intuitively predetermine the combined effect on the standard error of $\ddcq$.



% Arabidopsis thaliana fig
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{fig3}
\end{center}
\caption{
Overview of \citet{Yuan2008} experiment data.
$C_q$-values against the dilution step for case and control samples.
Dilution data are present for both the target (\textit{MT7}) and reference genes (Tublin, \textit{UBQ}).
The technical duplicates have been averaged out in the analysis.
}
\label{fig:cqYuan}
\end{figure}

% Arabidopsis thaliana table
\input{../output/Table3.tex}




\subsection{Simulation study}

First, we present results of a simulation study for a two-sided test for the null hypothesis of a vanishing $\ddcq$ at a $5\%$ significance level.
Expand Down Expand Up @@ -545,6 +613,22 @@ Overall, we see that the EC\&VA adjusted estimate is the only procedure consiste
Likewise, for many dilutions, the difference between the EC and EC\&VA procedures diminish as the uncertainty of the AE is relatively low.
Finally as expected a decrease in FPR corresponds to a decrease in TPR.

% Simulation tab
\input{../output/Table4.tex}

% Simulation fig
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{fig4}
\end{center}
\caption{
Plot of the false positive rates (FPR, black) and true positive rates (TPR, grey) and their 95 \% confidence intervals achieved simulation experiments for each method at various $p$-value cut-offs (0.05, 0.01, 0.1) shown by solid red horizontal lines.
The FPR and TPR are computed completely analogous to Table~\ref{tab:simexample}.
The rates are plotted for each combination of 4 or 8 samples with 4 or 8 fold dilution curves.
}
\label{fig:simstudy}
\end{figure}


\section{Discussion and conclusion}
% Main message
Expand Down Expand Up @@ -573,7 +657,7 @@ We recommend the conservative approach of always accounting for the uncertainty

\phantomsection
\addcontentsline{toc}{section}{Supplementary Material and Software}
\section*{Software and supplementary Material}
\section*{Supplementary Material and Software}

All statistical analysis were done with R \citep{Rproj} version \Sexpr{paste0(version$major, ".", version$minor)} and the contributed \texttt{lme4}-package \citep{Pinheiro2000a} was applied for random effects modelling.
All data, R code, LaTeX, and instructions to reproduce the present paper and results within are freely available at \url{http://www.github.org/AEBilgrau/effadj/} or upon personal request.
Expand Down Expand Up @@ -603,83 +687,5 @@ The Danish Agency for Science, Technology and Innovation, as well as Karen Elise
\bibliographystyle{biorefs}
\bibliography{references}

\newpage

% CIC

\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{fig1}
\end{center}
\caption{
Overview of CIC experiment data.
A: Raw $C_q$-values for different cell lines (samples) for each gene type and sample type.
The point type and colour differentiates the different gene types.
B: Dilution data for
reference genes (\textit{ACTB}, \textit{GAPDH}) and
target genes (\textit{MGST1}, \textit{MMSET}).
}
\label{fig:cqCIC}
\end{figure}

\input{../output/Table1.tex}


% DLBCL

\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{fig2}
\end{center}
\caption{
Overview of DLBCL testis experiment data.
A: Raw $C_q$-values for different patient samples for each gene type and sample type.
The point type and colour differentiates the different gene types.
B: Dilution data for
reference genes (\textit{RNU-24}, \textit{RNU-6B}) and
target genes (\textit{miR-127}, \textit{miR-143}).
}
\label{fig:cqTestis}
\end{figure}

\input{../output/Table2.tex}


% Arabidopsis thaliana

\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{fig3}
\end{center}
\caption{
Overview of \citet{Yuan2008} experiment data.
$C_q$-values against the dilution step for case and control samples.
Dilution data are present for both the target (\textit{MT7}) and reference genes (Tublin, \textit{UBQ}).
The technical duplicates have been averaged out in the analysis.
}
\label{fig:cqYuan}
\end{figure}

\input{../output/Table3.tex}


% Simulation

\input{../output/Table4.tex}

\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{fig4}
\end{center}
\caption{
Plot of the false positive rates (FPR, black) and true positive rates (TPR, grey) and their 95 \% confidence intervals achieved simulation experiments for each method at various $p$-value cut-offs (0.05, 0.01, 0.1) shown by solid red horizontal lines.
The FPR and TPR are computed completely analogous to Table~\ref{tab:simexample}.
The rates are plotted for each combination of 4 or 8 samples with 4 or 8 fold dilution curves.
}
\label{fig:simstudy}
\end{figure}



\end{document}

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