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%\section*{Epidemiology and Infection}
% \subsection*{Initially fewer bloodstream infections for allogeneic versus autologous stem-cell transplants in neutropenic patients}
% \subsubsection*{S. HIEKE, H. BERTZ, M. DETTENKOFER, M. SCHUMACHER AND J. BEYERSMANN}
\section*{Supplementary Material}
\subsection*{Computational details}

In order to investigate whether there is a significant evidence that allo-HCTs initially lead to a smaller number of patients with BSI compared to auto-HCTs, we studied simultaneous confidence bands for the difference of the cumulative infection probabilities. That is, we used time-simultaneous inference about the cumulative infection probabilities rather than considering the estimated probabilities at a fixed, albeit arbitrary, point in time. 
Following Lin, \textit{et al.} [18], we considered a martingale-based resampling technique where cause-specific martingales are approximated by a conditional normal random sample. Lin$\textquoteright$s approach based on a martingale representation of
\begin{equation}
\sqrt{n}(\hat{P}(T \leq t, \varepsilon =1) - P(T \leq t, \varepsilon =1)),
\end{equation}
where $T$ is the waiting time a patient spends in the initial state, $\varepsilon$ the type of first event ($\varepsilon=1$ occurrence of BSI) and $n$ the number of patients in the study. $P(T \leq t, \varepsilon =1)$ and $\hat{P}(T \leq t, \varepsilon =1)$ are the cumulative infection probability and the estimated cumulative infection probability via the Aalen-Johansen estimator. Using the resampling technique, the cause-specific martingales in the martingale representation of the expression above (1) are replaced with $N_{0\varepsilon,i}(t)G_{0\varepsilon,i}$, where $G_{0\varepsilon,i}$ denote independent standard normal variables and $N_{0\varepsilon,i}(t)$ denotes a counting process which counts the number of observed type $\varepsilon$ events in $[0,t]$, $i= 1,..n$ and $\varepsilon = 1,2$. Note that the competing endpoints 'end of neutropenia without BSI and death of neutropenia without BSI' are combined into one single endpoint $\varepsilon = 2$. This yields the rewritten martingale representation

\begin{equation}
\begin{split}
 & \sqrt{n}\sum_{i=1}^{n}\left[\int_{0}^{t}\frac{(1-\hat{P}(T \leq u,\varepsilon=2))G_{1i}\,dN_{1i}(u)}{Y(u)} 
+\int_{0}^{t}\frac{\hat{P}(T \leq u, \varepsilon =1)G_{2i}\,dN_{2i}(u)}{Y(u)}\right.\\
 & \qquad\qquad\left.-\hat{P}(T \leq t, \varepsilon =1)\int_{0}^{t}\frac{G_{1i}\,dN_{1i}(u)+G_{2i}\,dN_{2i}(u)}{Y(u)}\right]+o_{p}(1),
\end{split}
 \end{equation}
where $Y(t)$ denotes the at risks process which counts the number at risk at $t-$.\\
In order to approximate the distribution of the rewritten martingale representation (2), the data are kept fixed and a large number of replicates of $G_{0\varepsilon,i}$ are generated. The basis of the resampling technique is that the unconditional distribution of the martingale representation of (1)
asymptotically coincides with the conditional distribution of the rewritten martingale representation (2) keeping the data fixed.\\
The resampling technique can also be applied to the difference of the cumulative infection probabilities of the two transplant groups which enables to construct simultaneous confidence bands for their difference.\\
Practical implementation of the present resampling technique is easy. A typical data file comprises one line per patient. The data entries or columns are a patient's study entry time, the (potentially censored) event time and an event indicator $\varepsilon$. If an event has been observed, the event indicator $\varepsilon$ equals 1 or 2. The event indicator equals 0, if the patient has been censored. In order to generate one resampling replicate, two columns are added to the existing data file. These columns hold replicates of the individual standard normal variables $G_{0\varepsilon,i} \ (i= 1,..n \ \text{and}\ \varepsilon = 1,2)$, i.e. $G_{01,i} \ \text{and} \ G_{02,i}$ in the first and second new column, respectively. Based on the present data file, one replicate of the rewritten martingale representation (2) is computed. Such a replicate is a step function with jumps at the observed event times. Altogether a large number, e.g. 1000, replicates are computed by generating 1000 replicates of $G_{01,i} \ \text{and} \ G_{02,i}$.\\
Simultaneous confidence bands for the difference of the cumulative infection probabilities of the two transplant groups can be constructed by the means of the resampling technique, provided that the observations from the two transplant groups are independent. The key step to construct confidence bands for
\begin{equation*}
(P(T \leq t, \varepsilon =1)^{(1)} - P(T \leq t, \varepsilon =1)^{(2)}),
\end{equation*}
where $P(T \leq t, \varepsilon =1)^{(1)}$ is the CIF for auto-HCT and $P(T \leq t, \varepsilon =1)^{(2)}$ the CIF for allo-HCT, is to find quantiles $q_{\alpha}$ of the limit distribution of
\begin{equation*}
\begin{split}
&\sup_{t_{1}\leq t\leq t_{2}}|k(t)\sqrt{n}\left[( \hat{P}(T \leq t, \varepsilon =1)^{(1)}-\hat{P}(T \leq t, \varepsilon =1)^{(2)})\right.\\
 &\qquad \qquad  \qquad\left.-(P(T \leq t, \varepsilon =1)^{(1)}-P(T \leq t, \varepsilon =1)^{(2)})\right]|,
\end{split}
\end{equation*}
where $k(t)=\frac{Y_{1}(t)Y_{2}(t)}{Y_{1}(t)+Y_{2}(2)}$ is the weight function as in the log-rank test with $Y_{1}(t)$ and $Y_{2}(t)$ the risk sets in the auto-HCT and allo-HCT group, respectively. Given $\alpha$, the following probability 
\begin{equation*}
\begin{split}
&P\left(\sup_{t_{1}\leq t\leq t_{2}}|k(t)\sqrt{n}\left[( \hat{P}(T \leq t, \varepsilon =1)^{(1)}-\hat{P}(T \leq t, \varepsilon =1)^{(2)})\right.\right.\\
 &\qquad \qquad  \qquad \left.\left.-(P(T \leq t, \varepsilon =1)^{(1)}-P(T \leq t, \varepsilon =1)^{(2)})\right]|>q_{\alpha}\right) =\alpha
\end{split}
\end{equation*}
can be evaluated through Lin's resampling technique to obtain the boundary value $q_{\alpha}$. This is achieved by considering 1000 replicates of the rewritten martingale representation (2) for the auto-HCT and allo-HCT group, respectively, on the time interval $[t_1,t_2]$. The maximum of the absolute values of the difference of the rewritten martingale representations (2) for the auto-HCT and allo-HCT group in each replicate is computed. The quantile of the limited distribution is estimated by considering the empirical quantile of the resulting 1000 maximum values.\\
Thus, a simultaneous confidence band of the difference of the cumulative infection probabilities of the two transplant groups with weight function $k(t)$ is given by
\begin{equation*}
(\hat{P}(T \leq t, \varepsilon =1)^{(1)} - \hat{P}(T \leq t, \varepsilon =1)^{(2)})\pm\frac{q_{\alpha}(Y_{1}(t)+Y_{2}(t))}{\sqrt{n}Y_{1}(t)Y_{2}(t)}\quad(t_{1}\leq t\leq t_{2})
\end{equation*}


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