The Annals of Statistics

Statistical inference for the mean outcome under a possibly non-unique optimal treatment strategy

Alexander R. Luedtke and Mark J. van der Laan

Full-text: Open access

Abstract

We consider challenges that arise in the estimation of the mean outcome under an optimal individualized treatment strategy defined as the treatment rule that maximizes the population mean outcome, where the candidate treatment rules are restricted to depend on baseline covariates. We prove a necessary and sufficient condition for the pathwise differentiability of the optimal value, a key condition needed to develop a regular and asymptotically linear (RAL) estimator of the optimal value. The stated condition is slightly more general than the previous condition implied in the literature. We then describe an approach to obtain root-$n$ rate confidence intervals for the optimal value even when the parameter is not pathwise differentiable. We provide conditions under which our estimator is RAL and asymptotically efficient when the mean outcome is pathwise differentiable. We also outline an extension of our approach to a multiple time point problem. All of our results are supported by simulations.

Article information

Source
Ann. Statist., Volume 44, Number 2 (2016), 713-742.

Dates
Received: December 2014
Revised: September 2015
First available in Project Euclid: 17 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.aos/1458245733

Digital Object Identifier
doi:10.1214/15-AOS1384

Mathematical Reviews number (MathSciNet)
MR3476615

Zentralblatt MATH identifier
1338.62089

Subjects
Primary: 62G05: Estimation
Secondary: 62N99: None of the above, but in this section

Keywords
Efficient estimator non-regular inference online estimation optimal treatment pathwise differentiability semi parametric model optimal value

Citation

Luedtke, Alexander R.; van der Laan, Mark J. Statistical inference for the mean outcome under a possibly non-unique optimal treatment strategy. Ann. Statist. 44 (2016), no. 2, 713--742. doi:10.1214/15-AOS1384. https://projecteuclid.org/euclid.aos/1458245733


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Supplemental materials

  • Supplementary appendices: Proofs and extension to multiple time point case. Supplementary Appendix A contains all the proofs of all of the results in the main text. Supplementary Appendix B contains an outline of the extension to the multiple time point case. Supplementary Appendix C contains additional figures referenced in the main text.