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\onecolumn  
\title{Addressing Treatment Switching Bias with
G-methods: Exploring the Impact of Model
Specification}
\author[1,2]{Amani Al Tawil\thanks{Correspondence: altawil@ibe.med.uni-muenchen.de}}
\author[3]{Sean McGrath}
\author[4]{Robin Ristl\thanks{Equally contributed}}
\author[1,2]{Ulrich Mansmann\textsuperscript{†}}
\affil[1]{Institute for Medical Information Processing, Biometry, and Epidemiology (IBE), Faculty of Medicine, Ludwig-Maximilians-Universität München}
\affil[2]{Pettenkofer School of Public Health, Faculty of Medicine, Ludwig-Maximilians-Universität München}
\affil[3]{Department of Biostatistics, Harvard T.H. Chan School of Public Health}
\affil[4]{Center for Medical Data Science, Medical University of Vienna}

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\section*{\centering \LARGE Electronic Supplementary Material 2}
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%\large \textbf{Hazard}
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\large \textbf{Hazard}
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\large \textbf{Hazard}
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\justify The following will specify the concepts of hazard ratio (HR), discrete hazard ratio (dHR) and cumulative hazard ratio (cHR) as used in this paper.
\vspace{20pt}

\noindent The hazard at time $t$ of an event time $T$ given covariates $x$ is defined by Equation \eqref{eq:hazard}.
%%% Eqn 1
\begin{equation}\label{eq:hazard}
\lim_{{\Delta\to0}} \frac{{P(T\in[t,t+\Delta]|T>t,x)}}{{\Delta}} = \lambda_x(t)\tag{1}
\end{equation}
\vspace{20pt}

\justify In case of a Proportional Hazards (PH) model it holds $\lambda_x(t) = \lambda_0(t) \cdot \text{HR}(x)$. Here, $\lambda_0(t)$ is the baseline hazards function and $\text{HR}(x)$ is a specific factor changing the hazard to a group of subjects with specific covariates $x$ (e.g., indicator of assignment to experimental versus control group). In the PH model, it is assumed that $\lambda_0(t)$  holds for a well-defined baseline group (where all components of $x$ are set to 0) and $\text{HR}(x)$ quantifies the change in hazard between a group of subjects defined by $x$ and the baseline group.
\vspace{20pt}

\justify Several of the approaches we apply model the expression $P(T \in [t, t+\Delta] \mid T>t, x)$ for small but fixed $\Delta$ by a logistic regression as shown in Equation \eqref{eq:prob}.
%%% Eqn 2
\begin{equation}\label{eq:prob}
P(T \in [t, t+\Delta] \mid T>t, x) = \frac{e^{LP(t, x)}}{1 + e^{LP(t, x)}}\tag{2}
\end{equation}
\vspace{20pt}

\justify Here the linear predictor $LP(t,x)$ depends on the time $t$ and the covariates $x$ (Equation \eqref{eq:linear}).
%%% Eqn 3
\begin{equation}\label{eq:linear}
LP(t, x) = a_0 + a_1 \cdot t + a_2 \cdot t^2 + \sum_{i=1}^{k} b_i \cdot x_i \tag{3}
\end{equation}
\vspace{20pt}

\justify Given the small time interval $[t, t+\Delta]$ the probability that the event may happen is assumed to be quite small (Equation \eqref{eq:probsmall}), 
%%% Eqn 4
\begin{equation}\label{eq:probsmall}
P(T \in [t, t+\Delta] \mid T>t, x) \sim e^{LP(t, x)} = f(t) \cdot g(x)\tag{4}
\end{equation}
\vspace{20pt}

\justify The term $\text{g}(x)$ is called discrete hazard ratio (dHR) which is the product of odds ratios defined by the vectors $b$ and $x$. It approximates the HR in case the proportional hazards assumption holds.
\vspace{20pt}

\justify The cumulative hazard ratio (cHR) between two groups (defined by $x$ and $y$) at time t is defined by Equation \eqref{eq:cHR}:
%%% Eqn 5
\begin{equation}\label{eq:cHR}
cHR_{t,x,y} = \frac{\log(P(T>t \mid x))}{\log(P(T>t \mid y))}\tag{5}
\end{equation}
\vspace{20pt}

\justify The expression is motivated by the structure of the PH model which implies (Equation \eqref{eq:prop})
%%% Eqn 6
\begin{equation}\label{eq:prop}
P(T>t \mid x) = \exp\left(-HR(x) \int_0^t \lambda_0(s) \, ds\right)\tag{6}
\end{equation}
\vspace{20pt}

\justify In case of a PH model the $cHR_{t,x,y} = HR(x - y)$ . It also holds that $dHR_{t,x,y} = g(x - y)$ which approximates the HR in case the PH assumption is fulfilled.


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