Skip to main content
SpringerLink
Log in

Bounds on average controlled direct effects with an unobserved response variable

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

This paper considers the problem of estimating the bounds on the average controlled direct effects (ACDEs) of a treatment variable on an unobserved response variable in the presence of unobserved confounders between an intermediate variable and the response variable. When the response variable is observed, Cai, et al.(2008) derived the formulas for the sharp bounds on the ACDEs. When the response variable is unobserved, the authors propose a graphical criterion for selecting variables affected by the response variable to derive the formulas for the bounds on the ACDEs, which is an extension of the result of Kuroki(2005) to ACDEs. The results enable us not only to judge from the graph structure whether the bounds on the ACDEs can be expressed through observed variables when the response variable is unobserved, but also to provide their formulas when the answer is affirmative.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. L. Prentice, Surrogate endpoints in clinical trials: Definition and operational criteria, Statistics in Medicine, 1989, 8: 431–440.

    Article  Google Scholar 

  2. M. Buyse and G. Molenberghs, Criteria for the validation of surrogate endpoints in randomized experiments, Biometrics, 1998, 54: 1014–1029.

    Article  MATH  Google Scholar 

  3. W. Mu, S. Xiong, and X. Xu, Generalized confidence regions of fixed effects in the two-way ANOVA, Journal of Systems Science & Complexity, 2008, 21(2): 276–282.

    Article  MATH  MathSciNet  Google Scholar 

  4. H. Xie, G. Zou, and S. Wang, Price range volatility: A new indicator for risk measurement, Journal of Systems Science and Mathematical Sciences, 2009, 29: 1459–1465.

    MathSciNet  Google Scholar 

  5. J. Pearl, Causal diagrams for empirical research (with discussion), Biometrika, 1995, 83: 669–710.

    Article  MathSciNet  Google Scholar 

  6. J. Pearl, Causality: Models, Reasoning, and Inference, Cambridge University Press, New York, 2000.

    MATH  Google Scholar 

  7. M. Kuroki and M. Miyakawa, Covariate selection for estimating the causal effect of control plans by using causal diagrams, Journal of the Royal Statistical Society Series B, 2003, 65: 209–222.

    Article  MATH  MathSciNet  Google Scholar 

  8. S. Kaufman, J. S. Kaufman, R. F. Maclenose, et al., Improved estimation of controlled direct effects in the presence of unmeasured confounding of intermediate variables, Statistics in Medicine, 2005, 24: 1683–1702.

    Article  MathSciNet  Google Scholar 

  9. Z. Cai, M. Kuroki, J. Pearl, and J. Tian, Bounds on direct effects in the presence of confounded intermediate variables, Biometrics, 2008, 64: 695–701.

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Kuroki, Bounds on average causal effects in studies with a latent response variable, Metrika, 2005, 61: 63–71.

    Article  MATH  MathSciNet  Google Scholar 

  11. M. Kuroki, Graphical identifiability criteria for causal effects in studies with an unobserved treatment/ response variable, Biometrika, 2007, 94: 37–47.

    Article  MATH  MathSciNet  Google Scholar 

  12. J. Pearl, Probabilistic Reasoning in Intelligent Systems, Morgan Kaufmann, San Mateo, CA, 1988.

    Google Scholar 

  13. P. Spirtes, C. Glymour, and R. Scheines, Causation, Prediction, and Search, 2nd edition, MIT Press, 2000.

  14. The Lipid Research Clinics Coronary Primary Prevention Trial Results, I. Reduction in incidence of coronary heart disease, Journal of the American Medical Association, 1984, 251: 351–364.

    Article  Google Scholar 

  15. L. S. Freedman, B. I. Graubard, and A. Schatzkin, Statistical validation of intermediate endpoints for chronic diseases, Statistics in Medicine, 1992, 11: 167–178.

    Article  Google Scholar 

  16. A. Balke and J. Pearl, Bounds on treatment effects from studies with imperfect compliance, Journal of the American Statistical Association, 1997, 92: 1171–1176.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Key Laboratory for Applied Statistics of Ministry of Education, School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, China

    Na Shan & Jianhua Guo

  2. School of Basic Science, Changchun University of Technology, Changchun, 130012, China

    Na Shan

Authors
  1. Na Shan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jianhua Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Na Shan.

Additional information

This research was partially supported by the National Natural Science Foundation of China under Grant Nos. 10871038, 10926186, and 11025102, the National 973 Key Project of China under Grant No. 2007CB311002, and the Jilin Project (20100401).

This paper was recommended for publication by Editor Guohua ZOU.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shan, N., Guo, J. Bounds on average controlled direct effects with an unobserved response variable. J Syst Sci Complex 24, 1154–1164 (2011). https://doi.org/10.1007/s11424-011-9072-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-011-9072-6

Key words

Search

Navigation