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  \title[Rank-based estimation in the $\ell_1$-regularized partly linear model for censored data]{Supplementary material to Rank-based estimation in the $\ell_1$-regularized partly linear model for censored outcomes 
	with application to integrated analyses of clinical predictors and gene expression data}
  \author[B. A.~Johnson]{Brent A. Johnson}
  \address{Department of Biostatistics, 
Emory University, Atlanta, GA  30322, USA}
\email{bajohn3@emory.edu}

\begin{document}

%\begin{center}
%{\bf Supplementary material to} 
%Rank-based estimation in the $\ell_1$-regularized partly linear model for censored outcomes 
%with application to integrated analyses of clinical predictors and gene expression data \\
%\vspace*{10pt}
%Brent A. Johnson \\
%Department of Biostatistics, 
%Rollins School of Public Health, Emory University, Atlanta, GA  30322, U. S. A.
%\end{center}


\section{Simulation Studies}\label{sec:Sim}

Multiple simulation studies were conducted to assess the effect of 
model misspecification in rank-based variable selection for censored data.  
Specifically, we consider the effect of fitting the regularized Gehan estimator (Johnson, 2008; 
Cai and others, 2009) when the true underlying model is the partial linear model,
\[
\log T_i = \phi(Z_i) + \bfX_i'\bfbeta + \varepsilon_i.
\]
Our simulation study details are adapted from those given by Chen and others (2005) and  
all calculations were conducted in {\tt R} using the {\tt quantreg} package.
We assume that the true regression coefficients are 
$\bfbeta=(3, 3/2, 0, 0, 2, 0, 0, 0)'$, $\varepsilon_i\sim N(0,\sigma^2)$ and
mutually independent of $(Z_i,\bfX_i)$.  The predictors $\bfX_i$ followed a 
standard normal with the correlation
between the $j$th and $k$th components of $\bfX$ equal to $0.5^{|j-k|}$.  The random 
variable $Z_i$ was correlated with $\bfX_i$ through the relation 
$Z_i=\gamma_1X_{1i}+\gamma_2X_{2i}+\gamma_3X_{3i}+U_i$, where $(\gamma_1,\gamma_2,\gamma_3)=(1/4,1/4,1/2)$ and $U_i$ is 
$\mbox{Un}(-5,5)$ and completely independent of all other random variables.  As in 
Chen and others (2005), we considered quadratic and linear effects, $\phi(Z_i)=Z_i^2$ and 
$\phi(Z_i)=2Z_i$, respectively.  Finally, censoring random variables were simulated 
according to the rule, $C_i=\phi(Z_i) + \bfX_i'\bfbeta + U^{\ast}_i,$
where $U_i^{\ast}$ follows the uniform distribution $\mbox{Un}(0,\tau)$ with $\tau=8$.
Our simulation results are displayed in Table~\ref{tab:sim}.

Because the stratified estimator does not attempt to estimate the nonlinear effect $\phi$, 
standard definitions of model error do not apply.  Instead, we define the partial model 
error,  
$\mbox{PME}\equiv (\widehat\bfbeta_{\rm S(1)} - \bfbeta)'\left[E(\bfX\bfX') \right] (\widehat\bfbeta_{\rm S(1)} - \bfbeta),$
which has the heuristic interpretation as additional variation in the model resulting 
from $\widehat\bfbeta_{\rm S(1)}$ after adjusting for $Z_i$.  
The median of PME over Monte Carlo data set is reported in Table~\ref{tab:sim}.  
The relative median PME (RPME) is 
defined  as the lasso median PME divided by the unpenalized median PME and 
summarizes the average improvement 
in PME due to lasso regularization and variable selection.  In addition, we monitor the average 
number of correct (C) and incorrect (I) zeros over Monte Carlo data set.  For 
comparison purposes, we also computed the Gehan and Gehan lasso estimates assuming a linear 
model in $Z_i$ and $\bfX_i$.  In each instance of lasso, five-fold cross-validation was used 
to tune the regularization parameter $\lambda$.


\begin{table}
\caption{\label{tab:sim}Simulation results comparing regularized Gehan and stratified estimator}
%\centering
\begin{tabular}{lccccccccccc} \\ \hline\hline
& \multicolumn{5}{c}{Quadratic} & & \multicolumn{5}{c}{Linear} \\ \cline{2-6}\cline{8-12}
& \multicolumn{3}{c}{PME} & & & & \multicolumn{3}{c}{PME} & & \\ \cline{2-4}\cline{8-10}
Method & Unpen & lasso & RPME & C & I & & Unpen & lasso & RPME & C & I \\ \hline
\multicolumn{10}{l}{$n=75, \sigma=1.5$}\\
Gehan & 7.49 & 4.73 & 63.15 & 1.39 & 0.20 & & 0.30 & 0.24 & 80.00 & 1.32 & 0 \\ 
Strat. ($K_n=2$) & 6.98 & 4.18 & 59.89 & 2.40 & 0.32 & & 2.34 & 1.39 & 59.40 & 1.79 & 0.01 \\ 
Strat. ($K_n=4$) & 1.81 & 1.46 & 80.66 & 3.08 & 0.08 & & 0.83 & 0.51 & 61.45 & 2.51 & 0 \\ 
Strat.$^{\ast}$ ($K_n=8$) & 1.03 & 0.70 & 67.96 & 3.85 & 0.02 & & 0.51 & 0.24 & 47.05 & 3.58 & 0 \\[+2ex] 
\multicolumn{10}{l}{$n=75, \sigma=3$}\\
Gehan & 10.05 & 6.13 & 61.00 & 1.45 & 0.25 & & 1.17 & 0.83 & 70.94 & 1.33 & 0 \\ 
Strat. ($K_n=2$) & 9.66 & 5.23 & 54.14 & 2.48 & 0.38 & & 3.03 & 1.75 & 57.76 & 2.25 & 0.01 \\ 
Strat. ($K_n=4$) & 3.88 & 2.52 & 64.95 & 3.06 & 0.23 & & 1.58 & 1.08 & 68.35 & 2.85 & 0.01 \\ 
Strat.$^{\ast}$ ($K_n=8$) & 2.60 & 1.78 & 68.46 & 3.54 & 0.16 & & 1.66 & 1.05 & 63.25 & 3.55 & 0.04 \\[+2ex] 
\multicolumn{10}{l}{$n=100, \sigma=1.5$}\\
Gehan & 4.91 & 3.05 & 62.12 & 1.74 & 0.12 & & 0.20 & 0.15 & 75.00 & 1.60 & 0 \\ 
Strat. ($K_n=2$) & 4.67 & 2.74 & 58.67 & 2.56 & 0.17 & & 1.66 & 0.94 & 56.63 & 2.23 & 0 \\ 
Strat. ($K_n=4$) & 1.26 & 0.92 & 73.02 & 2.86 & 0.02 & & 0.58 & 0.32 & 55.17 & 2.91 & 0 \\ 
Strat.$^{\ast}$ ($K_n=8$) & 0.67 & 0.56 & 83.58 & 3.97 & 0.02 & & 0.35 & 0.22 & 62.86 & 3.52 & 0 \\[+2ex] 
\multicolumn{10}{l}{$n=100, \sigma=3$}\\
Gehan & 7.20 & 4.45 & 61.81 & 1.73 & 0.20 & & 0.87 & 0.61 & 70.11 & 1.44 & 0 \\ 
Strat. ($K_n=2$) & 7.04 & 3.72 & 52.84 & 2.72 & 0.28 & & 2.43 & 1.36 & 55.97 & 2.34 & 0 \\ 
Strat. ($K_n=4$)& 2.53 & 1.92 & 75.89 & 3.10 & 0.08 & & 1.35 & 0.78 & 57.78 & 2.65 & 0 \\ 
Strat.$^{\ast}$ ($K_n=8$) & 1.69 & 1.08 & 63.91 & 3.62 & 0.02 & & 1.03 & 0.65 & 63.11 & 3.38 & 0.01 \\[+1ex] \hline
\multicolumn{11}{l}{\footnotesize{$^{\ast}$ Tune $\lambda$ using generalized cross-validation in 
Johnson (2008)}} \\
\end{tabular}
\end{table}

First, when the effect $\phi$ is linear, the Gehan lasso beats the stratified estimator 
which confirms our intuition.  At the same time, it is interesting to note that stratified 
estimator gradually achieves similar operating characteristics as the unstratified estimator 
as the sample size increases and number of strata $K_n$ increases.  Nevertheless, the stratified 
estimator is far too cumbersome if the unknown function $\phi$ is indeed linear in $Z_i$.
Now, the big improvements in the stratified estimator are seen when the unknown 
function $\phi$ is nonlinear.   For example, when the sample size $n=75$ and $\sigma=1.5$, 
the partial model error is 7.49 and 4.73 for the unpenalized and regularized Gehan, 
respectively.  We compare this to the PME of the unpenalized and regularized stratified 
estimator, 1.03 and 0.70, respectively.  Hence, 
there is an average seven-fold increase in PME if we fit the Gehan lasso when the true 
underlying model is a nonlinear (i.e. quadratic) function of $Z_i$. 
Finally, it comes as no surprise that the stratified estimator performs better as the number 
of strata increase; such finding agrees with Chen and others (2005) and with stratified 
estimators, in general.  Of course, the small sample improvement comes with added computational 
burden.
 
\begin{thebibliography}{}

\bibitem{Cai09}
\textsc{Cai, T., Huang, J.} and \textsc{Tian, L.} (2009)
Regularized estimation for the accelerated failure time model.
{\it Biometrics} (In press).

\bibitem{CSY05}
\textsc{Chen, K., Shen, J.} and \textsc{Ying, Z.} (2005)
Rank estimation in partial linear model with censored data.
{\it Statistica Sinica} {\bf 15}  767--779.

\bibitem{baj2008}
\textsc{Johnson, B. A.} (2008)
Variable selection in semiparametric linear regression with censored data.
{\it J. R. Statist. Soc. Ser. B} {\bf 70} 351--370.

\end{thebibliography}

\end{document}