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Semiparametric temporal process regression of survival-out-of-hospital

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Abstract

The recurrent/terminal event data structure has undergone considerable methodological development in the last 10–15 years. An example of the data structure that has arisen with increasing frequency involves the recurrent event being hospitalization and the terminal event being death. We consider the response Survival-Out-of-Hospital, defined as a temporal process (indicator function) taking the value 1 when the subject is currently alive and not hospitalized, and 0 otherwise. Survival-Out-of-Hospital is a useful alternative strategy for the analysis of hospitalization/survival in the chronic disease setting, with the response variate representing a refinement to survival time through the incorporation of an objective quality-of-life component. The semiparametric model we consider assumes multiplicative covariate effects and leaves unspecified the baseline probability of being alive-and-out-of-hospital. Using zero-mean estimating equations, the proposed regression parameter estimator can be computed without estimating the unspecified baseline probability process, although baseline probabilities can subsequently be estimated for any time point within the support of the censoring distribution. We demonstrate that the regression parameter estimator is asymptotically normal, and that the baseline probability function estimator converges to a Gaussian process. Simulation studies are performed to show that our estimating procedures have satisfactory finite sample performances. The proposed methods are applied to the Dialysis Outcomes and Practice Patterns Study (DOPPS), an international end-stage renal disease study.

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Acknowledgements

This work was supported in part by National Institutes of Health Grant R01 DK070869 and by an M-Cubed Grant from the University of Michigan. The DOPPS is administered by Arbor Research Collaborative for Health and supported by scientific research grants from Amgen (since 1996), Kyowa Hakko Kirin (since 1999, in Japan), Sanofi Renal (since 2009), Abbott (since 2009), Baxter (since 2011), and Vifor Fresenius Renal Pharma (since 2011), without restrictions on publications. The authors thank Arbor Research Collaborative for Health for providing access to the DOPPS data and, in particular, Keith McCullough for creating the analysis files.

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Authors and Affiliations

  1. Department of Biostatistics, University of Michigan, 1415 Washington Hts., Ann Arbor, MI, 48109-2029, USA

    Tianyu Zhan & Douglas E. Schaubel

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  1. Tianyu Zhan
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  2. Douglas E. Schaubel
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Correspondence to Douglas E. Schaubel.

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Appendix

Appendix

We first list notation that appeared in Sect. 3.1:

$$\begin{aligned} \widehat{{\varvec{\varOmega }}}({\varvec{\beta }})= & {} n^{-1}\sum _{i=1}^n\int _0^\tau \{\varvec{S}^{(2)}(t;{\varvec{\beta }})/{S}^{(0)}(t;{\varvec{\beta }})-{\bar{\varvec{Z}}}(t;{\varvec{\beta }})^{\otimes 2} \}A^0_i(t)I(C_i \ge t)dt.\\ {\varvec{\varOmega }}({\varvec{\beta }})= & {} E\left[ \int _0^\tau \{\varvec{s}^{(2)}(t;{\varvec{\beta }})/{s}^{(0)}(t;{\varvec{\beta }})-\bar{{\varvec{z}}}(t;{\varvec{\beta }})^{\otimes 2} \}A^0_1(t)I(C_1 \ge t)dt\right] .\\ {{\varvec{u}}}_i({\varvec{\beta }})= & {} \int _0^\tau \{{\varvec{Z}}_i(t)-\bar{{\varvec{z}}}(t;{\varvec{\beta }}) \}dM_i(t;\varvec{\beta }) .\\ dM_i(t;\varvec{\beta })= & {} A^0_i(t)I(C_i \ge t)dt-\text {exp}[\varvec{\beta }^T\varvec{Z}_i(t)]I(C_i \ge t)\pi _0(t)dt.\\ f_i^{\pi _1}(t; \varvec{\beta })= & {} A^0_i(t)I(C_i \ge t)-\text {exp}[\varvec{\beta }^T\varvec{Z}_i(t)]I(C_i \ge t)\pi _0(t).\\ f_i^{\pi _2}(t; \varvec{\beta })= & {} \varvec{s}^{(1)}(t; \varvec{\beta })^T\pi _0(t) {\varvec{f}}_i^{\varvec{\beta }}(\varvec{\beta }) .\\ {\varvec{f}}_i^{\varvec{\beta }}(\varvec{\beta })= & {} {\varvec{\varOmega }}(\varvec{\beta })^{-1} {\varvec{u}}_i(\varvec{\beta }). \end{aligned}$$

Formulas pertaining to Sect. 3.2 are as follows:

$$\begin{aligned} \widehat{{\varvec{\varOmega }}}^{\langle m \rangle }({\varvec{\beta }},{\varvec{\theta }})= & {} n^{-1}\sum _{i=1}^n\int _0^\tau \{\varvec{S}^{(2){\langle m \rangle }}(t;{\varvec{\beta }},{\varvec{\theta }})/{S}^{(0){\langle m \rangle }}(t;{\varvec{\beta }},{\varvec{\theta }})\\&-{\bar{\varvec{Z}}}^{\langle m \rangle }(t;{\varvec{\beta }},{\varvec{\theta }})^{\otimes 2} \}A^0_i(t)I(C_i^{\langle m \rangle } \ge t;{\varvec{\theta }})dt.\\ {\varvec{\varOmega }}^{\langle 1 \rangle }({\varvec{\beta }},{\varvec{\theta }})= & {} E\left[ \int _0^\tau \{\varvec{s}^{(2){\langle 1 \rangle }}(t;{\varvec{\beta }},{\varvec{\theta }})/{s}^{(0){\langle 1 \rangle }}(t;{\varvec{\beta }},{\varvec{\theta }})\right. \\&\left. -\bar{{\varvec{z}}}^{\langle 1 \rangle }(t;{\varvec{\beta }},{\varvec{\theta }})^{\otimes 2} \}A^0_1(t)I(C_1^{\langle 1 \rangle } \ge t;{\varvec{\theta }})dt\right] .\\ {{\varvec{u}}}^{\langle m \rangle }_i({\varvec{\beta }},{\varvec{\theta }})= & {} \int _0^\tau \{{\varvec{Z}}_i(t)-\bar{{\varvec{z}}}^{\langle 1 \rangle }(t;{\varvec{\beta }},{\varvec{\theta }}) \}dM^{\langle m \rangle }_i(t;\varvec{\beta }, {\varvec{\theta }}).\\ dM^{\langle m \rangle }_i(t;\varvec{\beta }, {\varvec{\theta }})= & {} A^0_i(t)I(C^{\langle m \rangle }_i \ge t ; {\varvec{\theta }})dt-\text {exp}[\varvec{\beta }^T\varvec{Z}_i(t)]I(C^{\langle m \rangle }_i\ge t; {\varvec{\theta }})\pi _0(t)dt.\\ f_i^{\pi _1}(t; \varvec{\beta }, \varvec{\theta }, M)= & {} M^{-1}\sum _{m=1}^MI(C^{\langle m \rangle }_i \ge t; \varvec{\theta })A^0_i(t)\\&-M^{-1}\sum _{m=1}^MI(C^{\langle m \rangle }_i \ge t; \varvec{\theta })\text {exp}[\varvec{\beta }^T\varvec{Z}_i(t)]\pi _0(t).\\ {f}^{\pi _2}_i(t; \varvec{\beta }, {\varvec{\theta }}, M)= & {} \varvec{s}^{(1)\langle 1 \rangle }(t; \varvec{\beta }, \varvec{\theta })^T\pi _0(t) {\varvec{f}}_i^{\varvec{\beta }}(\varvec{\beta }, \varvec{\theta }, M) .\\ {{\varvec{f}}}_i^{\varvec{\beta }}(\varvec{\beta },{\varvec{\theta }}, M)= & {} [{\varvec{\varOmega }}({\varvec{\beta }})]^{-1}M^{-1}\sum _{m=1}^M{{\varvec{u}}}^{\langle m \rangle }_i({\varvec{\beta }},{\varvec{\theta }}). \end{aligned}$$

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Zhan, T., Schaubel, D.E. Semiparametric temporal process regression of survival-out-of-hospital. Lifetime Data Anal 25, 322–340 (2019). https://doi.org/10.1007/s10985-018-9433-8

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