Statistical Science

An Apparent Paradox Explained

Wen Wei Loh, Thomas S. Richardson, and James M. Robins

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Article information

Statist. Sci., Volume 32, Number 3 (2017), 356-361.

First available in Project Euclid: 1 September 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)


Loh, Wen Wei; Richardson, Thomas S.; Robins, James M. An Apparent Paradox Explained. Statist. Sci. 32 (2017), no. 3, 356--361. doi:10.1214/17-STS610.

Export citation


  • Aronow, P. M., Green, D. P. and Lee, D. K. K. (2014). Sharp bounds on the variance in randomized experiments. Ann. Statist. 42 850–871.
  • Bayarri, M. J. and Berger, J. O. (2000). $P$ values for composite null models. J. Amer. Statist. Assoc. 95 1127–1142, 1157–1170. With comments and a rejoinder by the authors.
  • Chiba, Y. (2015). Exact tests for the weak causal null hypothesis on a binary outcome in randomized trials. J. Biom. Biostat. 6 244.
  • Chung, E. and Romano, J. P. (2013). Exact and asymptotically robust permutation tests. Ann. Statist. 41 484–507.
  • Ding, P. (2016). Personal communication.
  • Ding, P. (2017). A paradox from randomization-based causal inference. Statist. Sci. 32 331–345.
  • Ding, P. and Dasgupta, T. (2016). A potential tale of two-by-two tables from completely randomized experiments. J. Amer. Statist. Assoc. 111 157–168.
  • Lang, J. B. (2015). A closer look at testing the “no-treatment-effect” hypothesis in a comparative experiment. Statist. Sci. 30 352–371.
  • Loh, W. W. and Richardson, T. S. (2017). Likelihood analysis for the finite population Neyman–Rubin binary causal model. In preparation.
  • Præstgaard, J. T. (1995). Permutation and bootstrap Kolmogorov–Smirnov tests for the equality of two distributions. Scand. J. Stat. 22 305–322.
  • Rigdon, J. and Hudgens, M. G. (2015). Randomization inference for treatment effects on a binary outcome. Stat. Med. 34 924–935.
  • Robins, J. M. (1988). Confidence intervals for causal parameters. Stat. Med. 7 773–785.
  • Robins, J. M., van der Vaart, A. and Ventura, V. (2000). Asymptotic distribution of $p$ values in composite null models. J. Amer. Statist. Assoc. 95 1143–1167, 1171–1172.

See also

  • Main article: A Paradox from Randomization-Based Causal Inference.