Abstract
This paper presents simple weighted and fully augmented weighted estimators for the additive hazards model with missing covariates when they are missing at random. The additive hazards model estimates the difference in hazards and has an intuitive biological interpretation. The proposed weighted estimators for the additive hazards model use incomplete data nonparametrically and have close-form expressions. We show that they are consistent and asymptotically normal, and are more efficient than the simple weighted estimator which only uses the complete data. We illustrate their finite-sample performance through simulation studies and an application to study the progression from mild cognitive impairment to dementia using data from the Alzheimer’s Disease Neuroimaging Initiative as well as an application to the mouse leukemia study.
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Acknowledgements
The authors wish to thank the editor, associate editor, and the referee for their constructive comments and suggestions which have greatly improved the paper. The authors thank the ADNI study and database for allowing us to use their data (the full acknowledgements section of the ADNI study are in the online supplementary material). Many thanks to Dr. Danielle Harvey for her valuable help with obtaining the ADNI data set and suggestions regarding to the data analysis. Thanks to Wei Ran, Yueheng An, Yiming Hu, and Nan Bi for their help. Dr. Yichuan Zhao was partially supported by the NSF Grant DMS-1406163 and NSA Grant H98230-12-1-0209. Dr. Yanqing Sun was partially supported by the National Science Foundation Grant DMS-1208978, DMS-1513072, and the National Institute of Health NIAID Grant R37 AI054165.
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For the Alzheimer’s Disease Neuroimaging Initiative—Data used in preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this report. A complete listing of ADNI investigators can be found at: http://adni.loni.usc.edu/wp-content/uploads/how_to_apply/ADNI_Acknowledgement_List.pdf.
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Appendix
Appendix
The following regularity conditions are needed in the proofs of Theorems 1–4.
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(a1)
\(\Lambda _{0}(\tau ) < \infty \).
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(a2)
\(P\{Y(\tau )=1\} > 0\).
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(a3)
Z is time-independent and bounded.
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(a4)
The matrix \(\Sigma _A = E[\int _{0}^{\tau }\{Z-e(t)\}^{\otimes 2}\hbox {d}N(t)]\) is positive definite.
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(a5)
W has bounded support \(\mathcal{W}\). There exists a constant \(\pi _0>0\) such that \(\pi (w)>\pi _0\) for \(w\in \mathcal{W}\).
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(a6)
The selection probability \(\pi (w)\) has r continuous and bounded partial derivatives with respect to the continuous components of W a.e.
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(a7)
The probability density/mass function f(w) of w and the conditional probability density/mass function \(f_{W {\mid } V}(w)\) of \({W} {\mid } V\) have r continuous and bounded partial derivatives with respect to the continuous components of W a.s.
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(a8)
Conditional distributions \(f_{{W}{\mid } V=0}(w)\) and \(f_{{W} {\mid } V=1}(w)\) have the same support, and \(c(w) = f_{{ W} {\mid } V=0}(w)/f_{{W} {\mid } V=1}(w)\) is bounded over the support.
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(a9)
The conditional expectations \(E({Z}^{k} {\mid } {W}= {w})\), \(E\{(Z^{k})^{\otimes {2}}|{W}= {w}\}\), \(k = 0, 1\), have r continuous and bounded partial derivatives with respect to the continuous components of W a.e.
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(a10)
\(nh^{2d} \rightarrow \infty \) and \(nh^{2r} \rightarrow 0\), as \( n \rightarrow \infty \).
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Qi, L., Zhang, X., Sun, Y. et al. Weighted estimating equations for additive hazards models with missing covariates. Ann Inst Stat Math 71, 365–387 (2019). https://doi.org/10.1007/s10463-018-0648-y
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DOI: https://doi.org/10.1007/s10463-018-0648-y