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\begin{document}

\setcounter{footnote}{2}

%\date{{\it Received October} 2004. {\it Revised February} 2005.\newline 
%{\it Accepted March} 2005.}
%
%\pagerange{\pageref{firstpage}--\pageref{lastpage}} \pubyear{2006}
%
%\volume{59}
%\artmonth{December}
%\doi{10.1111/j.1541-0420.2005.00454.x}

\title{Supplementary Materials for ``TEAMS: \underline{T}oxicity- and \underline{E}fficacy-based
Dose Insertion Design with \underline{A}daptive \underline{M}odel
\underline{S}election for Phase I/II Dose-Escalation
Trials in Oncology"
}
%\subtitle{Do you have a subtitle?\\ If so, write it here}

\titlerunning{Supplementary materials for TEAMS design}        % if too long for running head

\author{Wentian Guo         \and
        Yang Ni   \and
        Yuan Ji %etc.
}

%\authorrunning{Short form of author list} % if too long for running head

\institute{Y. Ji \at
              Center for Biomedical Research Informatics, NorthShore University HealthSystem, Evanston, IL, USA\\
              Department of Health Studies, The University of Chicago, Chicago, USA\\
              TEL: 224.364.7312\\
              \email{jiyuan@uchicago.edu}           %  \\
%             \emph{Present address:} of F. Author  %  if needed
           \and
           W. Guo \at
              Department of Biostatistics, Fudan University, Shanghai, China
             \and
            Y. Ni \at
           Department of Statistics, Rice University, Houston, USA
}

\date{Received: date / Accepted: date}
\label{firstpage}

\maketitle
%
%\footnotetext{{\it Address for correspondence:} Yuan Ji, Center for Biomedical Research Informatics,
%NorthShore University HealthSystem,
%Evanston, IL 60201,
%USA.\\ Email: yji@health.bsd.uchicago.edu}
\renewcommand\thetable{\thesection.\arabic{table}}  
\renewcommand\theequation{\thesection.\arabic{equation}}  
%\appendix

\renewcommand\thefigure{\thesection.\arabic{figure}}    
\begin{appendices}
  \section{Sensitivity analysis for calibrating $\tau$} 
%%\appendixname{Appendix}
%\appendixname{Web Appendix}
%  \section{A: Sensitivity analysis for 
%%\yy 
%calibrating 
%%\jj 
%$\tau$}
\label{A}



%\section{Appendix A: Sensitivity analysis for choosing $\tau$}
%\label{}
%\subsection{A}
\indent To 
%\yy 
determine 
%\jj % search for
an appropriate $\tau$ value in the iMOM prior, %\cut % which can place equal preference on both true null and true alternative hypotheses,
 we perform a  sensitivity analysis % based on simulation studies, to
 and test how $\tau$ values impact 
%\yy 
the selection of %\jj % on selecting 
 the true dose-efficacy model. 
%\yy 
We use the first eight scenarios described in
 Section 4 and also add three more scenarios, in which the toxicity
 and efficacy outcomes follow the same logit models in equation (1) %\eqref{eq:tox}
 and equation (1) in Section 2.1. %\eqref{eq:eff}.
 The true parameters are given by
 \begin{itemize}
 \item[] *scenario 9:
  $(\beta_0,\beta_1,\beta_2)=(-3,4.8,-3),dose=(0.05,0.4,0.6,0.8,1)$;
  \item[] *scenario 10:
  $(\beta_0,\beta_1,\beta_2)=(-3,5.6,-4),dose=(0.05,0.3,0.5,0.7,1)$;
  \item[] *scenario 11:
  $(\beta_0,\beta_1,\beta_2)=(-3,6.3,-5),dose=(0.05,0.4,0.6,0.8,1)$.
\end{itemize}
 For each scenario we consider 11 different values of
 $\tau$ and for each combination of scenario and $\tau$, 1,000
 datasets are generated. For each dataset, we generate efficacy and
 toxicity responses for 10 patients on each dose. With five doses,
 each dataset then has responses from 50 patients. %\jj % After generating the efficacy responses according to equation (2), 
 Model selection using MCMC posterior samples is performed for each dataset, 
%\yy 
after which a linear or
 quadratic logit model is selected. The tested $\tau$ values are:
 0.5, 1, 5, 10, 20, 30, 50, 70, 100, 150, 200. Posterior inference
 follows the same path as in the main paper. % Regarding MCMC, we adopt a random walk Metropolis-Hasting algorithm with normal proposals, and for each chain 10,000 samples are drawn with a burn-in size of 5,000 iterations. All the other prior distributions are the same to Section 2.4.
%\yy 
Figure \ref{circo-app} presents the results 
%\jj 
of the sensitivity analysis.% \citep{Krzywinski2009}. % See Figure 8.
%\setcounter{figure}{0}    
\begin{figure}
\begin{center}
\includegraphics[width=1\textwidth]{sensitive2.ps}
\caption{%\yy 
A 
%\jj 
Circos plot of stacked barplots for 11 scenarios, showing the results of sensitive analysis for $\tau$. In each stacked barplot, the blue, yellow and green bar indicate frequency of selecting quadratic logistic models, selecting linear logistic models, and no preference between the two models. The color of the outer arc in the first eight scenarios corresponds to the true dose-efficacy model, i.e blue means quadratic and yellow means linear.} \label{circo-app}
\end{center}
\end{figure}

\indent From Figure \ref{circo-app}, we can see that all $\tau$ values have similar performance in the Scenarios 1-6 with quadratic logistic models, while larger $\tau$ values have a larger proportion in choosing the true null hypothesis in Scenarios 7 and 8 with true linear logistic models. However, as can be seen from Figure \ref{fig:imom}, if $\tau$ is larger, the area near 0 in which the iMOM prior places almost 0 density is wider. As a result, we could not choose too large $\tau$ in consideration of covering small $|\beta_2|'s$. To decide an appropriate $\tau$, we need to decide the range of $\beta_2$ to be taken into consideration.
\begin{figure}
\begin{center}
\includegraphics[scale=.8]{imom_density.ps}
\caption{Density functions for the iMOM prior, with varying $\tau$'s and fixed $k,\nu=1$ }
\label{fig:imom}
\end{center}
\end{figure}

From equation (2) in Section 2.1, %\eqref{eq:eff}, 
it is deduced that $q_E(x)$ is symmetric about $x_0=-\beta_1/(2\beta_2)$, where $q_E(x)$ peaks to the summit $\logit^{-1}(-\beta_0-\frac{\beta_1^2}{4\beta_2})$. If $x_0$ is close to 1, both quadratic and linear logistic models would generate similar results on curve fitting as the true curve is almost monotonically increasing, in which case it is not worth considering when deciding the appropriate $\tau$ value. To search for the appropriate $\tau$ to cover small  $|\beta_2|'s$, we set a boundary on the parameters of dose-efficacy models:
\begin{eqnarray}
x_0=-\frac{\beta_1}{2\beta_2}<0.8.\label{eq:cons1}
\end{eqnarray}
Also, because a new drug should have certain potential efficacy on patients before it is studied in the clinical trial, we assume a second boundary on the parameters:
\begin{eqnarray}
\logit^{-1}(-\beta_0-\frac{\beta_1^2}{4\beta_2})>0.25.\label{eq:cons2}
\end{eqnarray}
Substituting \eqref{eq:cons1} into \eqref{eq:cons2} yields (the quasi-equation is derived from $\beta_0\sim N(-3,0.5^2)$)
\begin{eqnarray*}
-\beta_2>1.5625\times(\beta_0-1.099)\approx2.97,
\end{eqnarray*}
which is the range of $\beta_2$ that we want to explore to decide the appropriate $\tau$.
The densities at $\beta_2=2.97$ for $\tau=(0.5, 1, 5, 10, 20, 30, 50, 70, 100, 150, 200)$ are $(0.043, 0.057, 0.081, 0.065, 0.030, 0.012, 0.002, 2e^{-4},
7.6e^{-6}, 3.2e^{-8}, 1.3e^{-10})$. we can see that $\tau's$ larger than 50 have less than 0.001 density
on 2.97, which is quite small.
%\setcounter{table}{0}    
%{\footnotesize 

% \begin{table}
% \caption{\label{tablensity} Densities at 2.97 for all tested
%   $\tau$'s}

 % \begin{tabular}{c}
 % abc
 % \end{tabular}
% \end{table}

%\clearpage
%\newpage
%%\begin{table}
%
% % \endfirsthead
% % \multicolumn{8}{c}{\tablename\ \thetable\ (continued)}
% % \endhead
%%\begin{center}
%%\begin{tabular}{cccccccccccc} \hline
%
%\begin{table}
%\caption{\label{table:density} Densities at 2.97 for all tested $\tau$'s}
%%\centering
%
%\begin{tabular}{cccccccccccc} \hline
%
%$\tau$&0.5&1& 5& 10& 20& 30& 50& 70& 100& 150& 200\\
%\hline
%$density$&0.043& 0.057& 0.081& 0.065& 0.030& 0.012& 0.002& $2e^{-4}$&
%$7.6e^{-6}$& $3.2e^{-8}$& $1.3e^{-10}$ \\ \hline%\end{tabular}
%%%\end{center}
%\end{tabular}
%%\end{center}
%\end{table}
%
%\clearpage

%}
%\end{longtable}

%we can see that $\tau's$ larger than 50 have less than 0.001 density
%on 2.97, which is quite small. %, close to zero.
To explore more cases with relatively
small $\beta_2$'s, % detailedly the performance of different $\tau$ values on small $\beta_2$'s,
we perform sensitivity analysis on Scenarios 9-11. See the results for Scenarios 9-11 in Figure \ref{circo-app}. Notice that the
ability of detecting the true alternative hypothesis decreases as
$\tau$ increases when $\tau>50$, since larger $\tau$'s place less
density on small values. In summary, we make a final decision to choose $\tau=50$ in our design.


\newpage


%\section{Detailed results of the simulation}


%\begin{table}[h]
%\caption{Simulations with model selection 1 }


\section{BMA-based TEAMS design} 
%\section{Detailed results of the simulation}
\label{B}
%\setcounter{table}{0}   
%\renewcommand\thetable{B.\arabic{table}}     
\doublespacing
The BMA-based design is summarized in Box B.1. The additional rules and the default dose escalation design used in TEAMS are also applied in the BMA-based design. All the equations mentioned below can be found in the manuscript.
%\clearpage
%\newpage
%\onecolumn
\singlespacing
\begin{center}
%\begin{minipage}{150mm}
\fbox{
\parbox{\hsize}
{
\textbf{Box B.1: The BMA-based design for phase I/II dose finding trials}
%\center{\includegraphics[width=8cm]{diflow.eps}}
\begin{itemize}
\item \textbf{Stage I} Apply a default dose escalation design when
  $\sum_{i=1}^{D}n_i< N/2$.
\item \textbf{Stage II} % Apply the TEAMS design 
  When $\sum_{i=1}^{D}n_i\geq N/2$.
\begin{itemize}
\item \textbf{II1} Treat a new cohort of patients at the current dose, and collect data on both efficacy and toxicity.

\item \textbf{II2} Using the collected data, generate MCMC posterior
  samples of the parameters under $M_1$ and $M_2$ respectively, i.e. $\{\boldsymbol{\theta}_{1}^{(r)}, r=1,\cdots,R\}$ and $\{\boldsymbol{\theta}_{2}^{(r)}, r=1,\cdots,R\}$.
Obtain a posterior sample of $x_{opt}$ under $M_{j}$,
  i.e. $\{\hat{x}_{opt,j}^{(r)}, r=1, \cdots, R\}$.% from (6).

\item \textbf{II3} For each dose $x_i$, test if $\frac{1-C\%}{2}<Pr(x_{opt}<x_i|Data)<\frac{1+C\%}{2}$ using (9) and consequently obtain the decision indicator $A$ given by (6).

\begin{itemize}
\item[] (a) If $A=1$, the new dose $\hat{x}_{opt}=\sum_{j=1}^2 Pr(M_j|Data) \sum_{r=1}^R x_{opt,j}^{(r)}/R$ is inserted in the trial and assigned to the next cohort.
\item[] (b) If $A=0$, compute the posterior mean of the penalty for
  each existing dose, i.e. $\bar{P}(x_i,
  \boldsymbol{\theta})=\sum_{j=1}^2 Pr(M_j|Data) \sum_{r=1}^R P(x,\bb{\theta}_j^{(r)})/R$ for $i=1, \cdots, D$. Treat
  the next cohort on the dose $\tilde{i}$ that has the lowest mean
  penalty, where $\tilde{i}=\argmin_i\{\bar{P}(x_i,
  \boldsymbol{\theta})\}$. %|i=1, \cdots, D\}$.
\end{itemize}
\item \textbf{II4} Repeat steps II1-II3 until the maximum sample size $N$ is achieved. 
\item \textbf{II5} Select as OBD the dose that has the lowest mean penalty among all tested doses, including prespecified doses and new inserted doses.
\end{itemize}
\end{itemize}
}}
%\end{minipage}
\end{center}
\doublespacing
\newpage
\section{Zoom-in illustration for scenario 1 in Figure 5}
\label{C}

\setcounter{figure}{0}    
\begin{figure}[ht]
\begin{center}
\includegraphics[width=1\textwidth]{append-C.ps}
\caption{A zoom-in illustration for the  circulator sector of scenario 1 in the circos plot.
}
\label{fig:example}
\end{center}
\end{figure}
\newpage
\section{TEAMS without model selection}

\begin{center}

% \fontsize{9}{11}\selectfont
\LTcapwidth=\textwidth
%\begin{tabular}{cccccccccc}
\begin{longtable}{cccccccc}

\caption{Simulation results for scenario 2 and 8 without model selection }\\ \hline
\endfirsthead
\multicolumn{8}{c}{\tablename\ \thetable\ (continued)}
\endhead

\hline
\multicolumn{8}{c}{Scenario 2 with linear model}\\
\multicolumn{8}{c}{Quadratic; Efficacy peaks
in the left side of MTD; Bracket; Insertion is
needed.\small}\\
\hline
\hline

\multicolumn{2}{c}{Tox Parameter
Estimation}&&\multicolumn{5}{c}{Eff Parameter
Estimation}\\
\multicolumn{2}{c}{(True: -3,
2.8)\small}&&\multicolumn{5}{c}{(True: -3, 17,
-18)}\\
\hline

$\hat{\alpha}_0$&$\hat{\alpha}_1$&&&&$\hat{\beta}_0^L$&$\hat{\beta}_1^L$\\
-2.95(0.21)&2.79(1.01)&&&&-1.73(0.50)&2.17(2.19)\\
\hline
\\
\multicolumn{6}{c}{Pre-specified Dose}&Dose
inserted&Dose selected\\
\multicolumn{6}{c}{(OBD=0.445)}&\%ins=0.6\%\\\hline
Dose
level&0.05&0.3&0.55&0.75&1.0&0.294(0.185)&0.536(0.300)\\
%\hline
%penalty&0.813&0.157&0.118&0.435&2.964&0.163&0.112\\
\%
pts&0.101&0.315&0.266&0.149&0.168&0.000&0.228\\%\hline
\% sel&0.208 &0.053& 0.361& 0.253& 0.119&
0.006
\\
\\
\\


\hline
\multicolumn{8}{c}{Scenario 2 with quadratic model}\\
\multicolumn{8}{c}{Quadratic; Efficacy peaks
in the left side of MTD; Bracket; Insertion is
needed.\small}\\
\hline
\hline

\multicolumn{2}{c}{Tox Parameter
Estimation}&&\multicolumn{5}{c}{Eff Parameter
Estimation}\\
\multicolumn{2}{c}{(True: -3,
2.8)\small}&&\multicolumn{5}{c}{(True: -3, 17,
-18)}\\
\hline

$\hat{\alpha}_0$&$\hat{\alpha}_1$&&&$\hat{\beta}_0^Q$&$\hat{\beta}_1^Q$&$\hat{\beta}_2^Q$\\
-2.96(0.22)&2.73(1.08)&&&-2.81(0.20)&15.50(2.68)&-16.33(4.02)\\
\hline
\\
\multicolumn{6}{c}{Pre-specified Dose}&Dose
inserted&Dose selected\\
\multicolumn{6}{c}{(OBD=0.445)}&\%ins=93.8\%\\\hline
Dose
level&0.05&0.3&0.55&0.75&1.0&0.434(0.033)&0.443(0.051)\\
%\hline
%penalty&0.813&0.157&0.118&0.435&2.964&0.094&0.094\\
\%
pts&0.060&0.134&0.132&0.125&0.171&0.378&0.384\\%\hline
\% sel&0.001 &0.019& 0.092& 0.000& 0.000&
0.888
\\
\\
\\
\hline
\multicolumn{8}{c}{Scenario 8 with linear model}\\
\multicolumn{8}{c}{Linear; All doses are safe
but too inefficient; Insertion is needed.}\\
\hline
\hline

\multicolumn{2}{c}{Tox Parameter
Estimation}&&\multicolumn{5}{c}{Eff Parameter
Estimation}\\
\multicolumn{2}{c}{(True: -3,
1.5)\small}&&\multicolumn{5}{c}{(True: -3,
3)}\\
\hline

$\hat{\alpha}_0$&$\hat{\alpha}_1$&&&&$\hat{\beta}_0^L$&$\hat{\beta}_1^L$\\
-2.96(0.17)&1.46(0.61)&&&&-3.02(0.18)&3.05(0.50)\\
\hline
\\
\multicolumn{6}{c}{Pre-specified Dose}&Dose
inserted&Dose selected\\
\multicolumn{6}{c}{(OBD=1.435)}&\%ins=61.1\%\\\hline
Dose
level&0.1&0.3&0.55&0.75&1.0&1.713(0.125)&1.200(0.361)\\
%\hline
%penalty&0.881&0.799&0.641&0.479&0.283&2.011&0.179\\
\%
pts&0.058&0.063&0.103&0.217&0.511&0.049&0.339\\%\hline
\% sel&0.003 &0.001& 0.010& 0.057& 0.613&
0.316
%%%%%%%%%%%with 30.8\% trials selecting dose higher than MTD as the OBD.

\\
\\
\\
\hline
\multicolumn{8}{c}{Scenario 8 with quadratic model}\\
\multicolumn{8}{c}{Linear; All doses are safe
but too inefficient; Insertion is needed.}\\
\hline
\hline

\multicolumn{2}{c}{Tox Parameter
Estimation}&&\multicolumn{5}{c}{Eff Parameter
Estimation}\\
\multicolumn{2}{c}{(True: -3,
1.5)\small}&&\multicolumn{5}{c}{(True: -3,
3)}\\
\hline

$\hat{\alpha}_0$&$\hat{\alpha}_1$&&&$\hat{\beta}_0^Q$&$\hat{\beta}_1^Q$&$\hat{\beta}_2^Q$\\
-2.93(0.16)&1.25(0.54)&&&-3.30(0.15)&7.56(0.79)&-5.05(0.91)\\
\hline
\\
\multicolumn{6}{c}{Pre-specified Dose}&Dose
inserted&Dose selected\\
\multicolumn{6}{c}{(OBD=1.435)}&\%ins=1.6\%\\\hline
Dose
level&0.1&0.3&0.55&0.75&1.0&0.567(0.104)&0.734(0.082)\\
%\hline
%penalty&0.881&0.799&0.641&0.479&0.283&0.628&0.493\\
\%
pts&0.056&0.066&0.169&0.439&0.268&0.049&0.003\\%\hline
\% sel&0.001 &0.006& 0.084& 0.879& 0.023&
0.007
\label{table:results}
\end{longtable}
\end{center}


\end{appendices}
\end{document}