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\begin{document}
\begin{center}

{\scshape\Large{Supplementary Material for `Using a continuous biomarker to allocate patients in a response-adaptive clinical trial'}\par}
\vspace{0.4cm}
\end{center}

\section{Additional Simulation Results}

Below, we show the results of our simulations as described in the main paper for $N=40, 80$, and $120$.

\subsection{Ethical Measure}

\begin{figure}[H]
\vspace{-0.6cm}
	\begin{center}
		\subfloat[$N=40$]{\includegraphics[width = 2.12in]{Figure_SM1a.eps}} 
		\subfloat[$N=80$]{\includegraphics[width = 2.12in]{Figure_SM1b.eps}} 
		\subfloat[$N=120$]{\includegraphics[width = 2.12in]{Figure_SM1c.eps}} 
	\end{center}
		\caption{Simulated proportion of patients on their best treatment, when the trial size is $N=40$ (a), $N=80$ (b) and $N=120$ (c) for 6 scenarios. 
		\label{Fig: treat}}
	\vspace{-0.6cm}
\end{figure}

\subsection{Type I error and Power}

\subsubsection{One-sided:}

\begin{figure}[H]
\vspace{-0.6cm}
	\begin{center}
		\subfloat[$N=40$]{\includegraphics[width = 2.12in]{Figure_SM2a.eps}} 
		\subfloat[$N=80$]{\includegraphics[width = 2.12in]{Figure_SM2b.eps}} 
		\subfloat[$N=120$]{\includegraphics[width = 2.12in]{Figure_SM2c.eps}} 
	\end{center}
		\caption{Simulated one-sided type I error and overall power, when the trial size is $N=40$ (a), $N=80$ (b) and $N=120$ (c) for 6 scenarios. 
		\label{Fig: treat}}
	\vspace{-0.6cm}
\end{figure}

\begin{figure}[H]
\vspace{-0.6cm}
	\begin{center}
		\subfloat[$N=40$]{\includegraphics[width = 2.12in]{Figure_SM3a.eps}} 
		\subfloat[$N=80$]{\includegraphics[width = 2.12in]{Figure_SM3b.eps}} 
		\subfloat[$N=120$]{\includegraphics[width = 2.12in]{Figure_SM3c.eps}} 
	\end{center}
		\caption{Simulated one-sided type I error and power for biomarkers $x_{n}\geq 0$, when the trial size is $N=40$ (a), $N=80$ (b) and $N=120$ (c) for 6 scenarios. 
		\label{Fig: treat}}
	\vspace{-0.6cm}
\end{figure}

\begin{figure}[H]
\vspace{-0.6cm}
	\begin{center}
		\subfloat[$N=40$]{\includegraphics[width = 2.12in]{Figure_SM4a.eps}} 
		\subfloat[$N=80$]{\includegraphics[width = 2.12in]{Figure_SM4b.eps}} 
		\subfloat[$N=120$]{\includegraphics[width = 2.12in]{Figure_SM4c.eps}} 
	\end{center}
		\caption{Simulated one-sided power for biomarkers $x_{n}\geq X$, when the trial size is $N=40$ (a), $N=80$ (b) and $N=120$ (c) for 6 scenarios. 
		\label{Fig: treat}}
	\vspace{-0.6cm}
\end{figure}

\begin{figure}[H]
\vspace{-0.6cm}
	\begin{center}
		\subfloat[$N=40$]{\includegraphics[width = 2.12in]{Figure_SM5a.eps}} 
		\subfloat[$N=80$]{\includegraphics[width = 2.12in]{Figure_SM5b.eps}} 
		\subfloat[$N=120$]{\includegraphics[width = 2.12in]{Figure_SM5c.eps}} 
	\end{center}
		\caption{Simulated one-sided type I error and power for biomarkers $x_{n}< 0$, when the trial size is $N=40$ (a), $N=80$ (b) and $N=120$ (c) for 6 scenarios. 
		\label{Fig: treat}}
	\vspace{-0.6cm}
\end{figure}

\begin{figure}[H]
\vspace{-0.6cm}
	\begin{center}
		\subfloat[$N=40$]{\includegraphics[width = 2.12in]{Figure_SM6a.eps}} 
		\subfloat[$N=80$]{\includegraphics[width = 2.12in]{Figure_SM6b.eps}} 
		\subfloat[$N=120$]{\includegraphics[width = 2.12in]{Figure_SM6c.eps}} 
	\end{center}
		\caption{Simulated one-sided power for biomarkers $x_{n}< X$, when the trial size is $N=40$ (a), $N=80$ (b) and $N=120$ (c) for 6 scenarios. 
		\label{Fig: treat}}
	\vspace{-0.6cm}
\end{figure}

\subsubsection{Two-sided:}

\begin{figure}[H]
\vspace{-0.6cm}
	\begin{center}
		\subfloat[$N=40$]{\includegraphics[width = 2.12in]{Figure_SM7a.eps}} 
		\subfloat[$N=80$]{\includegraphics[width = 2.12in]{Figure_SM7b.eps}}
		\subfloat[$N=120$]{\includegraphics[width = 2.12in]{Figure_SM7c.eps}} 
	\end{center}
		\caption{Simulated two-sided type I error and overall power, when the trial size is $N=40$ (a), $N=80$ (b) and $N=120$ (c) for 6 scenarios. 
		\label{Fig: treat}}
	\vspace{-0.6cm}
\end{figure}

\begin{figure}[H]
\vspace{-0.6cm}
	\begin{center}
		\subfloat[$N=40$]{\includegraphics[width = 2.12in]{Figure_SM8a.eps}} 
		\subfloat[$N=80$]{\includegraphics[width = 2.12in]{Figure_SM8b.eps}}
		\subfloat[$N=120$]{\includegraphics[width = 2.12in]{Figure_SM8c.eps}} 
	\end{center}
		\caption{Simulated two-sided type I error and power for biomarkers $x_{n}\geq 0$, when the trial size is $N=40$ (a), $N=80$ (b) and $N=120$ (c) for 6 scenarios. 
		\label{Fig: treat}}
	\vspace{-0.6cm}
\end{figure}

\begin{figure}[H]
\vspace{-0.6cm}
	\begin{center}
		\subfloat[$N=40$]{\includegraphics[width = 2.12in]{Figure_SM9a.eps}} 
		\subfloat[$N=80$]{\includegraphics[width = 2.12in]{Figure_SM9b.eps}} 
		\subfloat[$N=120$]{\includegraphics[width = 2.12in]{Figure_SM9c.eps}} 
	\end{center}
		\caption{Simulated two-sided power for biomarkers $x_{n}\geq X$, when the trial size is $N=40$ (a), $N=80$ (b) and $N=120$ (c) for 6 scenarios. 
		\label{Fig: treat}}
	\vspace{-0.6cm}
\end{figure}

\begin{figure}[H]
\vspace{-0.6cm}
	\begin{center}
		\subfloat[$N=40$]{\includegraphics[width = 2.12in]{Figure_SM10a.eps}}  
		\subfloat[$N=80$]{\includegraphics[width = 2.12in]{Figure_SM10b.eps}}  
		\subfloat[$N=120$]{\includegraphics[width = 2.12in]{Figure_SM10c.eps}} 
	\end{center}
		\caption{Simulated two-sided type I error and power for biomarkers $x_{n}< 0$, when the trial size is $N=40$ (a), $N=80$ (b) and $N=120$ (c) for 6 scenarios. 
		\label{Fig: treat}}
	\vspace{-0.6cm}
\end{figure}

\begin{figure}[H]
\vspace{-0.6cm}
	\begin{center}
		\subfloat[$N=40$]{\includegraphics[width = 2.12in]{Figure_SM11a.eps}} 
		\subfloat[$N=80$]{\includegraphics[width = 2.12in]{Figure_SM11b.eps}} 
		\subfloat[$N=120$]{\includegraphics[width = 2.12in]{Figure_SM11c.eps}} 
	\end{center}
		\caption{Simulated two-sided power for biomarkers $x_{n}< X$, when the trial size is $N=40$ (a), $N=80$ (b) and $N=120$ (c) for 6 scenarios. 
		\label{Fig: treat}}
	\vspace{-0.6cm}
\end{figure}

\end{document}