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The Authors Respond

VanderWeele, Tyler J.; van der Laan, Mark J.; Kessler, Ronald C.

Author Information
doi: 10.1097/EDE.0000000000001190

To the Editor:

We thank Penning de Vries et al1 for the illustration of optimal subgroup selection for treatment in the case of ties. In our article,2 we had addressed this setting of ties by referring the reader to the formal treatment given in Luedke and van der Laan3–5 in our eAppendix; ( The discussion in our main article was intended to focus on the simpler cases to keep the exposition as accessible as possible, but it is indeed good to have these matters more explicitly discussed in the epidemiologic literature.

Tyler J. VanderWeele
Departments of Epidemiology and Biostatistics, Harvard T.H. Chan School of Public Health, Boston, MA,
Mark J. van der Laan
Department of Biostatistics, University of California, Berkeley, CA
Ronald C. Kessler
Department of Health Care Policy, Harvard Medical School, Boston, MA


1. Penning de Vries BBLP, Groenwold RHH, Luedtke A. Re: selecting optimal subgroups for treatment using many covariates. Epidemiology. 2020;31:e33–e34.
2. VanderWeele TJ, Luedtke AR, van der Laan MJ, Kessler RC. Selecting optimal subgroups for treatment using many covariates. Epidemiology. 2019;30:334–341.
3. Luedtke AR, van der Laan MJ. Targeted learning of the mean outcome under an optimal dynamic treatment rule. J Causal Inference. 2015;3:61–95.
4. Luedtke AR, van der Laan MJ. Statistical inference for the mean outcome under a possibly non-unique optimal treatment strategy. Ann Stat. 2016;44:713–742.
5. Luedtke AR, van der Laan MJ. Optimal individualized treatments in resource-limited settings. Int J Biostat. 2016;12:283–303.

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