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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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\begin{document}
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\vspace{-5em} % Increase the space between title and author names
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\begin{center}
\section*{\centering \LARGE Electronic Supplementary Material 3}
\vspace{1.5em} % Increase the space between "Electronic Supplementary Material 1" and "Data Dictionary"
\large \textbf{Parameter Estimate Values and Odds Ratios for the Coefficients of IPCW Method}
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%\vspace{30pt}
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%\onecolumn









% Full model_LOFU
%\onecolumn
\begin{table}[H]
\centering
\begin{adjustwidth}{-2.5cm}{-2.5cm} % Adjust the values of X as needed
\centering
\begin{threeparttable}
\caption{Pooled logistic regression analysis on the probability of being censored defined as lost to follow-up/administrative censoring, conditional on baseline and TVCs, to calculate the denominator of the weights - Full model}
\label{table: OR LOFU full}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{>{\raggedright\arraybackslash}p{3cm}p{3.5cm}p{2.5cm}p{2.7cm}p{2.5cm}p{2.5cm}}
\hline
\textbf{Specification 1}\tnote{\ding{72}}  &   &  & \textbf{Loss to follow-up}  & & \\
 \hline
Variable & & \multicolumn{2}{c}{\textbf{Control arm}} &  \multicolumn{2}{c}{\textbf{Experimental arm}}\\
\cmidrule(lr){3-4}
\cmidrule(lr){5-6}
&  & \textbf{Estimate (SE)} & \textbf{OR (95$\%$ CI)}\tnote{\ding{61}} &\textbf{Estimate (SE)} & \textbf{OR (95$\%$ CI)}\tnote{\ding{61}}\\
\hline
{Intercept} &  &  1.505 (2.887) &  &  -7.957 (2.951) &  \\
\hline
{Age in yrs (continuous)} &   &  -0.301 (0.112) & 0.74 (0.595,0.921) &   0.136 (0.107) & 1.145 (0.928,1.414)\\
{Age in yrs (quadratic)} &   &  0.003 (0.001) & 1.003 (1.001,1.005)  &  -0.001 (0.001) & 0.999 (0.997,1.001) \\
\hline
{Female Sex} &   & NA & NA &   -0.426 (0.317) & 0.653 (0.351,1.216) \\
\hline
{Race} & Non-Asian   &NA  & NA&  Reference & Reference\\
& Asian   &  NA  &  NA &   0.297 (0.351) & 1.346 (0.676,2.679)  \\
\hline
{Initial diagnosis stage} 
& IV   &  Reference & Reference&  Reference & Reference\\
& IIIB  &   -0.045 (0.521) & 0.956 (0.344,2.653)  &   0.78 (0.54) & 2.181 (0.757,6.283)  \\
& IIIA  &   1.377 (0.675) & 3.961 (1.055,14.873)  &   -0.485 (0.851) & 0.615 (0.116,3.262) \\
& IA,IB,IIA  &   0.524 (0.551) & 1.688 (0.573,4.971) &  0.297 (0.69) & 1.346 (0.348,5.201)  \\
\hline
{Measurable intracranial CNS disease} & &  0.132 (0.828) & 1.141 (0.225,5.782) &   0.134 (0.559) & 1.143 (0.382,3.418) \\
\hline
{Lung involvement at study entry} 
& Both  & Reference  & Reference & Reference  & Reference\\
& Right  &    -0.062 (0.365) & 0.939 (0.46,1.92)&  -0.205 (0.352) & 0.815 (0.409,1.623)
  \\
& Left  &    -0.16 (0.382) & 0.852 (0.403,1.801)  & -0.604 (0.423) & 0.547 (0.239,1.252)
\\
& Lung not involved   &   -0.546 (0.714) & 0.579 (0.143,2.345)  &   -0.353 (0.58) & 0.703 (0.226,2.189) \\
\hline
{Baseline ECOG } 
& 0   & NA & NA &  Reference & Reference\\
& 1 or 2  &   NA &  NA &   -0.527 (0.395) & 0.59 (0.272,1.282)  \\
\hline
{Strata at randomization} 
& No iCNS Metastasis at baseline/No prior chemotherapy  & Reference & Reference & Reference  & Reference\\
& iCNS Metastasis at baseline/No prior chemotherapy &   -0.184 (0.777) & 0.832 (0.182,3.813) &   -0.428 (0.518) & 0.651 (0.236,1.8)\\
& No iCNS Metastasis at baseline/Prior chemotherapy &  -0.913 (0.535) & 0.401 (0.14,1.146)  &   -0.224 (0.418) & 0.799 (0.353,1.813) \\
& iCNS Metastasis at baseline/Prior chemotherapy  & -1.002 (0.789) & 0.367 (0.078,1.723)  &   -0.44 (0.711) & 0.644 (0.16,2.592) \\
\hline
{History of tobacco use} 
& Never smoked & Reference  & Reference &  Reference & Reference \\
& Ever  Smoked  &  0.223 (0.279) & 1.25 (0.724,2.159)  &   -0.837 (0.345) & 0.433 (0.22,0.851) \\
\hline
{Previous radiation therapy} &   & NA  & NA &   0.378 (0.667) & 1.46 (0.395,5.399)  \\
\hline
{Time in month (continuous)} &    &    -0.21 (0.046) & 0.81 (0.741,0.886)  & -0.354 (0.046) & 0.702 (0.641,0.769)
  \\
{Time in month (quadratic)} &    &   0.007 (0.001) & 1.007 (1.006,1.009)  &   0.01 (0.001) & 1.01 (1.008,1.012)  \\
\hline
{ECOG performance-status score at current visit} 
& 0   & Reference & Reference &  Reference & Reference\\
& 1    &    0.284 (0.301) & 1.329 (0.736,2.398) &   1.232 (0.37) & 3.427 (1.659,7.078)  \\
& 2+   &   -0.509 (0.963) & 0.601 (0.091,3.964) &   -0.235 (1.186) & 0.79 (0.077,8.076) \\
\hline
{Tumour lesion size at current visit}  &    &  0.012 (0.005) & 1.012 (1.001,1.023) & 0.004 (0.006) & 1.004 (0.993,1.016) \\
\hline
{Intracranial progression at current visit}  &    & NA  &  NA &  -0.794 (0.443) & 0.452 (0.19,1.076)
\\
\hline
{Disease progression at current visit}  &    &  0.455 (0.451) & 1.576 (0.651,3.818)   &  1.398 (0.339) & 4.046 (2.081,7.863) \\
\hline
{Treatment at current visit}  &    &  0.032 (0.414) & 1.032 (0.459,2.323)   & NA & NA\\
\hline
% Add more rows if needed
\end{tabular}
\begin{tablenotes}
\item[\ding{61}] Confidence Intervals (CIs) estimated using "sandwich" variance estimator
\item[\ding{72}] Specification 3: Same as specification 1, but replace the linear terms of time,
target lesion size and age with 5
knots splines\\ Specification 4: Same as specification 1, but replace the step function (3 categories) for time-varying ECOG
with 2 categories.
\end{tablenotes}
\end{threeparttable}
\end{adjustwidth}
\end{table}








% Restricted model_LOFU
\onecolumn
\begin{table}[H]
\centering
\begin{adjustwidth}{-2.5cm}{-2.5cm} % Adjust the values of X as needed
\centering
\begin{threeparttable}
\caption{Pooled logistic regression analysis on the probability of being censored defined as lost to follow-up/administrative censoring, conditional on baseline and TVCs, to calculate the denominator of the weights - Restricted model}
\label{table: OR LOFU restricted}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{>{\raggedright\arraybackslash}p{3cm}p{3.5cm}p{2.5cm}p{2.7cm}p{2.5cm}p{2.5cm}}
\hline
\textbf{Specification 2}\tnote{\ding{72}}  &   &  & \textbf{Loss to follow-up}  & & \\
 \hline
Variable & & \multicolumn{2}{c}{\textbf{Control arm}} &  \multicolumn{2}{c}{\textbf{Experimental arm}}\\
\cmidrule(lr){3-4}
\cmidrule(lr){5-6}
&  & \textbf{Estimate (SE)} & \textbf{OR (95$\%$ CI)}\tnote{\ding{61}} &\textbf{Estimate (SE)} & \textbf{OR (95$\%$ CI)}\tnote{\ding{61}}\\
\hline
{Intercept} &   &    1.47 (2.901) &   &   -8.286 (2.526) &  \\
\hline
{Age in yrs (continuous)} &    &    -0.29 (0.107) & 0.748 (0.606,0.924)  &   0.152 (0.092) & 1.165 (0.973,1.394)  \\
{Age in yrs (quadratic)} &    &    0.003 (0.001) & 1.003 (1.001,1.005)  &   -0.001 (0.001) & 0.999 (0.997,1) \\
\hline
{Female Sex} &   & NA   & NA  & -0.434 (0.293) & 0.648 (0.365,1.15)  \\
\hline
{Initial diagnosis stage} 
& IV   &  Reference & Reference & Reference & Reference\\
& IIIB   &   0.065 (0.455) & 1.067 (0.438,2.603)  &   0.788 (0.433) & 2.199 (0.941,5.137) \\
& IIIA   &   1.252 (0.617) & 3.498 (1.044,11.715) &  -0.093 (0.493) & 0.911 (0.346,2.396)\\
& IA,IB,IIA  &   0.352 (0.492) & 1.422 (0.542,3.725)  &   0.285 (0.619) & 1.33 (0.395,4.479) \\
\hline
%{Lung involvement at study entry} 
%& Both  & NA  &NA  &  Reference & Reference\\
%& Right  &  NA  & NA &   &  \\
%& Left  &   NA & NA &   &  \\
%& Lung not involved   &  NA  &NA  &   & \\
%\hline
{Baseline ECOG } 
& 0   &  NA& NA & Reference  & Reference\\
& 1 or 2  &  NA  & NA &   -0.482 (0.328) & 0.618 (0.325,1.175)  \\
\hline
{Strata at randomization} 
& No iCNS Metastasis at baseline/No prior chemotherapy  & Reference & Reference&  NA & NA\\
& iCNS Metastasis at baseline/No prior chemotherapy &  -0.247 (0.427) & 0.781 (0.338,1.802)
 &  NA & NA\\
& No iCNS Metastasis at baseline/Prior chemotherapy  & -1.008 (0.491) & 0.365 (0.139,0.956) &  NA & NA\\
& iCNS Metastasis at baseline/Prior chemotherapy   &  -0.826 (0.679) & 0.438 (0.116,1.659)
 &  NA & NA \\
\hline
{History of tobacco use} 
& Never smoked &  NA    &  NA & Reference  & Reference\\
& Ever  Smoked   & NA & NA &   -0.897 (0.323) & 0.408 (0.216,0.769) \\
\hline
{Time in month (continuous)} &   & -0.212 (0.045) & 0.809 (0.741,0.884)  &   -0.36 (0.046) & 0.698 (0.638,0.763)  \\
{Time in month (quadratic)} &    & 0.007 (0.001) & 1.007 (1.006,1.009) &   0.01 (0.001) & 1.01 (1.008,1.012) \\
\hline
{ECOG performance-status score at current visit} 
& 0   &  NA & NA &  Reference & Reference\\
& 1   &  NA & NA &   1.187 (0.339) & 3.278 (1.685,6.376) \\
& 2+  &  NA & NA &   -0.353 (1.004) & 0.703 (0.098,5.03) \\
\hline
{Tumour lesion size at current visit} &   &  0.012 (0.005) & 1.012 (1.003,1.022) & NA  &NA \\
\hline
{Intracranial progression at current visit} &   & NA & NA &  -0.859 (0.394) & 0.423 (0.195,0.917) \\
\hline
{Disease progression at current visit} &  &  0.52 (0.33) & 1.682 (0.881,3.211) &   1.394 (0.329) & 4.03 (2.115,7.677)\\
\hline
% Add more rows if needed
\end{tabular}
\begin{tablenotes}
\item[\ding{61}] Confidence Intervals (CIs) estimated using "sandwich" variance estimator
\item[\ding{72}] Specification 5: Same as specification 2, but replace the linear terms of time,
target lesion size and age with 5
knots splines.\\ Specification 6: Same as specification 2, but remove baseline covariates: sex, baseline ECOG and initial diagnosis stage in calculating the denominator of the experimental arm
\end{tablenotes}
\end{threeparttable}
\end{adjustwidth}
\end{table}



% Full model_XO
%\onecolumn
\begin{table}[H]
\centering
\begin{adjustwidth}{-2.5cm}{-2.5cm} % Adjust the values of X as needed
\centering
\begin{threeparttable}
\caption{Pooled logistic regression analysis on the probability of being censored defined as switching, conditional on baseline and time-varying covariates, to calculate the denominator of the weights - Full model}
\label{table: OR XO full}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{>{\raggedright\arraybackslash}p{3cm}p{7cm}p{3cm}p{3cm}}
\hline
\textbf{Specification 1}\tnote{\ding{72}} &   & \raggedleft\textbf{Treatment switching}& \\
 \hline
Variable & & \multicolumn{2}{c}{\centering\textbf{Control arm at and after progression}}\\
\cmidrule(lr){3-4}
&  & \textbf{Estimate (SE)} & \textbf{OR (95$\%$ CI)}\tnote{\ding{61}}\\
\hline
{Intercept} &  & -1.355 (1.116) & \\
\hline
{Age in yrs (continuous)} &   & -0.027 (0.014) & 0.973 (0.946,1.001) \\
\hline
{Initial diagnosis stage} 
& IV   &  Reference & Reference\\
& IIIB &   -0.878 (0.574) & 0.416 (0.135,1.280) \\
& IIIA  &  -0.373 (0.771) & 0.688 (0.152,3.122)  \\
& IA,IB,IIA  & -0.43 (0.796) & 0.65 (0.137,3.094) \\
\hline
{Measurable intracranial CNS disease} & & -0.312 (0.555) & 0.732 (0.247,2.171) \\
\hline
{Lung involvement at study entry} 
& Both  & Reference  & Reference\\
& Right  & 0.195 (0.486) & 1.216 (0.469,3.149)\\
& Left  &  0.418 (0.446) & 1.519 (0.634,3.643) \\
& Lung not involved & 0.088 (0.847) & 1.092 (0.208,5.748) \\
\hline
{Strata at randomization} 
& No iCNS Metastasis at baseline/No prior chemotherapy  & Reference & Reference\\
& iCNS Metastasis at baseline/No prior chemotherapy & 0.009 (0.563) & 1.009 (0.334,3.043)\\
& No iCNS Metastasis at baseline/Prior chemotherapy &  1.23 (0.498) & 3.42 (1.29,9.071) \\
& iCNS Metastasis at baseline/Prior chemotherapy & 0.436 (0.621) & 1.546 (0.458,5.22) \\
\hline
{History of tobacco use} 
& Never smoked & Reference  & Reference\\
& Ever  Smoked  & -0.069 (0.333) & 0.933 (0.486,1.791) \\
\hline
{Time in month (continuous)} & & -0.277 (0.039)  &  0.758 (0.702,0.819) \\
\hline
{ECOG performance-status score at current visit} 
& 0   & Reference & Reference \\
& 1  & -0.156 (0.33) & 0.855 (0.448,1.632) \\
& 2+  & -1.23 (1.01) & 0.292 (0.04,2.117)\\
\hline
{Tumour lesion size at current visit} &  & 0.008 (0.006) & 1.008 (0.997,1.02) \\
\hline
{Time to disease progression (continuous)} &  & 0.472 (0.089) & 1.604 (1.348,1.908) \\
{Time to disease progression (quadratic)} &  & -0.006 (0.003) & 0.994 (0.989,1)\\
\hline
% Add more rows if needed
\end{tabular}
\begin{tablenotes}
\item[\ding{61}] Confidence Intervals (CIs) estimated using "sandwich" variance estimator
\item[\ding{72}] Specification 3: Same as specification 1, but with 5 knots splines for time,
time to disease progression, target lesion size and baseline age\\ Specification 4: Same as specification 1, but replace the step function (3 categories) for time-varying ECOG with 2 categories\\ Specification 7: Same as specification 1, but without linear and quadratic terms for time to disease progression.
\end{tablenotes}
\end{threeparttable}
\end{adjustwidth}
\end{table}
%\end{landscape}








% Restricted model_XO
\onecolumn
\begin{table}[H]
\centering
\begin{adjustwidth}{-2.5cm}{-2.5cm} % Adjust the values of X as needed
\centering
\begin{threeparttable}
\caption{Pooled logistic regression analysis on the probability of being censored defined as switching, conditional on baseline and TVCs, to calculate the denominator of the weights - Restricted model}
\label{table: OR XO restricted}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{>{\raggedright\arraybackslash}p{3cm}p{7cm}p{3cm}p{3cm}}
\hline
\textbf{Specification 2}\tnote{\ding{72}} &   &\textbf{Treatment switching}& \\
 \hline
Variable & & \multicolumn{2}{c}{\textbf{Control arm at and after progression}}\\
\cmidrule(lr){3-4}
&  & \textbf{Estimate (SE)} & \textbf{OR (95$\%$ CI)}\tnote{\ding{61}}\\
\hline
{Intercept} &  &  -1.465 (0.928) & \\
\hline
{Age in yrs (continuous)} &  &  -0.025 (0.013) & 0.976 (0.952,1)
 \\
\hline
{Initial diagnosis stage} 
& IV   &  Reference & Reference\\
& IIIB  &  -0.902 (0.593) & 0.406 (0.127,1.297)  \\
& IIIA  &  -0.543 (0.703) & 0.581 (0.146,2.305)  \\
& IA,IB,IIA  &  -0.411 (0.704) & 0.663 (0.167,2.632)  \\
\hline
{Strata at randomization} 
& No iCNS Metastasis at baseline/No prior chemotherapy  & Reference & Reference\\
& iCNS Metastasis at baseline/No prior chemotherapy &  -0.139 (0.436) & 0.87 (0.37,2.048) \\
& No iCNS Metastasis at baseline/Prior chemotherapy & 1.165 (0.444) & 3.205 (1.342,7.653)\\
& iCNS Metastasis at baseline/Prior chemotherapy  &  0.218 (0.591) & 1.243 (0.39,3.959) \\
\hline
{Time in month (continuous)} &   &  -0.284 (0.036) & 0.752 (0.701,0.808) \\
\hline
{Tumour lesion size at current visit} &  &  0.008 (0.005) & 1.008 (0.999,1.018)\\
\hline
{Time to disease progression (continuous)} & & 0.486 (0.09) & 1.625 (1.362,1.94)
\\
{Time to disease progression (quadratic)} & & -0.006 (0.003) & 0.995 (0.989,1) \\
\hline
\end{tabular}
\begin{tablenotes}
\item[\ding{61}] Confidence Intervals (CIs) estimated using "sandwich" variance estimator
\item[\ding{72}] Specification 5: Same as specification 2, but with 5 knots splines for time,
time to disease progression, target lesion size and baseline age\\ Specification 6: Same as specification 2, but without step function (4 categories) for initial diagnosis stage\\ Specification 8: Same as specification 2, but
without linear and quadratic
terms for time to disease progression
\end{tablenotes}
\end{threeparttable}
\end{adjustwidth}
\end{table}







\end{document}
 