Supplementary materials: Comparing the performance of two-stage residual inclusion methods when using physician’s prescribing preference as an instrumental variable: unmeasured confounding and noncollapsibility

<b>These are peer-reviewed supplementary materials for the article '</b><b>Comparing the performance of two-stage </b><b>residual inclusion methods when using </b><b>physician’s prescribing preference as an </b><b>instrumental variable: unmeasured </b><b>confounding and noncollapsibility</b><b>' published in the</b><b> </b><b><i>Journal of Comparative Effectiveness Research</i></b><b>.</b><b>Figure S1. Results from prior 1 prescription as IV</b><b>Figure S2. Results from prior 2 prescriptions as IV</b><b>Figure S3. Results from prior 3 prescriptions as IV</b><b>Figure S4. Results from prior 4 prescriptions as IV</b><b>R Code</b><b>Aim: </b>The first objective is to compare the performance of two-stage residual inclusion (2SRI), two-stage least square (2SLS) with the multivariable generalized linear model (GLM) in terms of the reducing unmeasured confounding bias. The second objective is to demonstrate the ability of 2SRI and 2SPS in alleviating unmeasured confounding when noncollapsibility exists. <b>Materials &amp; methods:</b> This study comprises a simulation study and an empirical example from a real-world UK population health dataset (Clinical Practice Research Datalink). The instrumental variable (IV) used is based on physicians’ prescribing preferences (defined by prescribing history). <b>Results:</b> The percent bias of 2SRI in terms of treatment effect estimates to be lower than GLM and 2SPS and was less than 15% in most scenarios. Further, 2SRI was found to be robust to mild noncollapsibility with the percent bias less than 50%. As the level of unmeasured confounding increased, the ability to alleviate the noncollapsibility decreased. Strong IVs tended to be more robust to noncollapsibility than weak IVs. <b>Conclusion:</b> 2SRI tends to be less biased than GLM and 2SPS in terms of estimating treatment effect. It can be robust to noncollapsibility in the case of the mild unmeasured confounding effect.

本数据集为发表于<i>Journal of Comparative Effectiveness Research</i>（《比较效果研究杂志》）的论文《以医师处方偏好作为工具变量（instrumental variable, IV）时两阶段残差包含法（two-stage residual inclusion, 2SRI）的性能比较：未测混杂与不可折叠性》的同行评议补充材料。
<b>图S1：以既往1次处方作为工具变量的分析结果</b>
<b>图S2：以既往2次处方作为工具变量的分析结果</b>
<b>图S3：以既往3次处方作为工具变量的分析结果</b>
<b>图S4：以既往4次处方作为工具变量的分析结果</b>
<b>R代码</b>

<b>研究目标：</b>本研究的首要目标为对比两阶段残差包含法（2SRI）、两阶段最小二乘法（two-stage least square, 2SLS）与多变量广义线性模型（multivariable generalized linear model, GLM）在降低未测混杂偏倚方面的性能；次要目标为验证当存在不可折叠性时，2SRI与2SPS能够缓解未测混杂的能力。
<b>材料与方法：</b>本研究包含一项模拟研究与一项基于英国真实人群健康数据集——临床实践研究数据链（Clinical Practice Research Datalink）的实证分析。本研究使用的工具变量（IV）基于医师的处方偏好（由处方历史定义）。
<b>结果：</b>在治疗效应估计方面，2SRI的百分比偏倚低于GLM与2SPS，且在大多数场景下其偏倚小于15%。进一步分析显示，2SRI对轻度不可折叠性具有鲁棒性，其百分比偏倚低于50%。随着未测混杂程度升高，缓解不可折叠性的能力随之下降。相较于弱工具变量，强工具变量对不可折叠性的鲁棒性更强。
<b>结论：</b>在估计治疗效应时，2SRI的偏倚通常低于GLM与2SPS。在存在轻度未测混杂效应的场景下，2SRI可对不可折叠性保持鲁棒性。