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Mediation Analysis with Random Distribution as Mediator with an Application to iCOMPARE Trial

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Abstract

Physical activity has long been shown to be associated with biological and physiological performance and risk of diseases. It is of great interest to assess whether the effect of an exposure or intervention on an outcome is mediated through physical activity measured by modern wearable devices such as actigraphy. However, existing methods for mediation analysis focus almost exclusively on mediation variable that is in the Euclidean space, which cannot be applied directly to physical activity measured by wearable devices. Such data is best summarized in the form of a random distribution. In this paper, we develop a structural equation model to the setting where a random distribution is treated as the mediator. We provide sufficient conditions for identifying the average causal effects of a distribution mediator and present methods for estimating the direct and mediating effects of a random distribution mediator on the outcome. We apply our method to analyze the iCOMPARE data set that compares flexible duty-hour policies and standard duty-hour policies on interns’ sleep related outcomes and to investigate the mediation effect of physical activity on the causal path between flexible duty-hour policies and sleep related outcomes.

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Funding

The authors gratefully acknowledge supports from the NIH grants GM129781 and GM123056. The iCompare trial was supported by grants U01HL125388 and U01HL126088 from the National Heart, Lung, and Blood Institute and grants from the American Council for Graduate Medical Education.

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Authors and Affiliations

  1. University of Pennsylvania Perelman School of Medicine, Philadelphia, PA, 19104, USA

    **gru Zhang, Mathias Basner, Christopher W. Jones, David F. Dinges, Haochang Shou & Hongzhe Li

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  1. **gru Zhang
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  2. Mathias Basner
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  3. Christopher W. Jones
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  4. David F. Dinges
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  5. Haochang Shou
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  6. Hongzhe Li
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Corresponding author

Correspondence to Hongzhe Li.

Appendix

Appendix

1.1 Sensitivity Analysis in Sect. 3.3

Given \(\rho (t)\), we can estimate \(\beta _\rho ^{(s)}(t)\) as follows.

Step 1. Estimate \(\delta ^{(s)}(t,X_i)\), \(\alpha ^{(s)}(t,X_i)\), and \(\epsilon _i(t)\) in (5) by using the method in Sect. 3.1. Then \(\Sigma (u,v)=E(\epsilon _i(u)\epsilon _i(v))\) could be estimated.

Step 2. Given \(\beta _\rho ^{(s)}(t)\), let

$$\begin{aligned} {\tilde{Y}}_i\{Z_i,\varvec{Q}_i(Z_i)\} = Y_i(Z_i,\varvec{Q}_i(Z_i)) - \int \beta _\rho ^{(s)}(t)\left\{ \delta ^{(s)}(t,X_i)+\alpha ^{(s)}(t,X_i)Z_i\right\} dt. \end{aligned}$$

Regress \({\tilde{Y}}_i\{Z_i,\varvec{Q}_i(Z_i)\}\) on \((X_i^T,Z_iX_i^T)\) and obtain the estimate of \(U_i={\tilde{Y}}_i\{Z_i,\varvec{Q}_i(Z_i)\}-X_i^T\xi ^{(s)}-Z_iX_i^T\gamma ^{(s)}\).

Step 3. Obtain \(\eta _i=U_i-\int \beta _\rho ^{(s)}(t)\epsilon _i(t)dt\) and \(\sigma _1^2=var(\eta _i)\).

Step 4. Estimate \(\beta _\rho ^{(s)}(t)\) by plugging \(cov\{U_i,\epsilon _i(t)\}\), \(\Sigma (u,v)\), \(\sigma _1\), and \(\rho (t)\) into (12).

Step 5. Repeat Steps 2–4 until convergence.

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Zhang, J., Basner, M., Jones, C.W. et al. Mediation Analysis with Random Distribution as Mediator with an Application to iCOMPARE Trial. Stat Biosci 16, 107–128 (2024). https://doi.org/10.1007/s12561-023-09383-9

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